Scaling properties of three-dimensional magnetohydrodynamic turbulence

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Junctions of three quantum wires

2 tion on the nature of the ﬁxed points for a range of the parameter g , and that diﬀerent methods that we describe in the paper are needed to shed light onto the nature of the strong coupling ﬁxed points. In Sec. VI we establish the mapping of the Y-junction model onto the DHM and deduce its consequences for the phase diagram of the Yjunction model. In Sec. VII we introduce the study of the system by looking at the boundary conditions satisﬁed by the bosonic ﬁelds at the junction. In Sec. VIII we discuss the twisted structure of the Hilbert space that is manifest in the compactiﬁcation of the bosons so as to take proper care of the fermionic statistics of the electrons in the wires. In Sec. IX we introduce what we refer to as the method of delayed evaluation of boundary conditions (DEBC), which is less reliant on the apparatus of BCFT. In Sec. X we apply all the results from the previous sections to analyze the RG ﬁxed points in the junction problem. We also show in this section that asymmetries in the couplings between the three wires are irrelevant at low energies and temperatures, so the simple Z3 symmetric model of the system may indeed be a good description of realistic junctions. In Sec. XI we use simple arguments based on energy conservation to obtain constraints on the conductance tensor. We show that all the ﬁxed points that we are able to understand in this paper correspond to physical situations where no energy is transfered to neutral (non-charge carrying) modes in the quantum wires. In Sec. XII we discuss how the conductance tensors are modiﬁed if the Luttinger liquid wires are connected to Fermi liquid leads far from the junction. Depending on details of the potential experiment, this may be a better model. In Sec. XIII we present and discuss open problems. In Appendix A we give results on the non-interacting version of our model. In Appendix B we review boundary conformal ﬁeld theory. In Appendix C we review the boundary conformal ﬁeld theory of standard free bosons. In Appendix D we review the boundary conformal ﬁeld theory approach to calculating the conductance. In Appendix E we discuss the related problem of a Y-junction of quantum Hall edge states.

Three-dimensional Tutte embedding

this paper.
Theorem A (Steinitz, see Grunbaum 3]) A graph G is 3-polytopal if and only
if G is planar and 3-connected.
Theorem B (Tutte 7]) If G is 3-connected planar graph, then the facial cycles in
Conditions are given for a graph to have a convex representation in three dimensions. This extends Tutte's barycentric embedding. Introduction
Tutte 7] proved that a 3-connected planar graph has a nice embedding in the plane in the sense that it can be drawn so that the boundary of every face is a convex polygon and every edge is a straight line segment. This is called a convex embedding or convex representation. More details on Tutte embedding are given in the next section. The main objective of this work is to visualize nice drawings of nonplanar graphs in three-dimensional space and to introduce some ideas for future research. Now we describe some basic notation to be used throughout this paper. Let G = (V; E ) represent a nite graph with vertex set V and edge set E . A cycle C in G is a collection of distinct vertices v1; v2; : : :; vn such that vi is adjacent to vi+1 for 1 i n, where the indices are considered modulo n. If there are no edges in G between any two nonconsecutive edges of C , then C is chordless. If G can be drawn on the plane so that the edges are Jordan arcs whose end points are vertices and no two edges meet except at the vertices, then G is called a planar graph. Given a plane drawing of a planar graph G, the graph G itself (the vertices and the edges together) can be viewed as a subset of <2. It is convenient to call this subset G. Then <2 ? G is a union of simply connected regions called faces of G, and exactly one of these regions is in nite. The boundary of any face of a 2-connected plane graph corresponds to a cycle of the graph. If the cycle C = v1; v2; : : :; vn corresponds to a face, then we call C a facial cycle . The facial cycle that corresponds to the in nite face is called the outer cycle. Suppose G is a graph and H is a subgraph of G, then we say H is nonseparating if G ? V (H ) (also written G ? H ) is connected. Let Q be a convex polygon in the plane with vertices x1; x2; : : :; xn. We say that the vertices of Q are listed in cyclic order if xixi+1 is a side of Q for 1 i n, where the indices are considered modulo n. An n-wheel is a graph Wn+1 consisting of a cycle v1; v2; : : :; vn and a vertex v adjacent to every vertex of the cycle. This paper presents an approach for the display of certain nonplanar graphs in 3-space. In Section 2, we have a detailed discussion of barycentric embeddings of which Tutte's embedding is a special case. In Section 3, we have drawings of several graphs using NETPAD ( a computer program developed at Bellcore). In Section 4 we consider two di erent barycentric embeddings of G K2 where G is 3-connected and prove that these embeddings are nice in a certain well-de ned sense. In Section 5, we give a negative answer to a question posed by Tutte 7]. The reader is referred to any standard text book on Graph Theory for the unde ned terms. Several of the terms may also be found in Thomassen's article 5].

Compositeness, Triviality and Bounds on Critical Exponents for Fermions and Magnets

a rXiv:h ep-th/93127v11D ec1993ILL-(TH)-93-21COMPOSITENESS,TRIVIALITY AND BOUNDS ON CRITICAL EXPONENTS FOR FERMIONS AND MAGNETS Aleksandar KOCI ´C and John KOGUT Loomis Laboratory of Physics,University of Illinois,Urbana,Il 61801Abstract We argue that theories with fundamental fermions which undergo chiral symmetry breaking have several universal features which are qualitatively diﬀerent than those of theories with fundamental scalars.Several bounds on the critical indices δand ηfollow.We observe that in four dimensions the logarithmic scaling violations enter into the Equation of State of scalar theories,such as λφ4,and fermionic models,such as Nambu-Jona-Lasinio,in qualitatively diﬀerent ways.These observations lead to useful approaches for analyzing lattice simulations of a wide class of model ﬁeld theories.Our results imply that λφ4cannot be a good guide to understanding the possible triviality of spinor QED .1.IntroductionThere are two classes of theories in the literature that are used to model the Higgs sector of the Standard Model[1,2].One is based on the self-interactingφ4scalar theory in which the Higgs is elementary.The other is based on strongly interacting constituent fermiﬁelds in which the Higg’s particle is a fermion-antifermion bound state.A recent proposal for the second realization uses Nambu-Jona-Lasinio(NJL)models in which composite scalars emerge as a consequence of spontaneous chiral symmetry breaking[2].In four dimensions both types of models have a trivial continuum limit and are meaningful only as eﬀective theories with aﬁnite cutoﬀ.This restriction places constraints on the low-energy parameters e.g.bounds on the masses.We wish to point out in this letter that triviality in the two models is realized in diﬀerent ways.The diﬀerences between theories of composite and elementary mesons can be expressed in terms of the critical indicesδandη,and several inequalities and bounds on these indices will follow.These results should prove useful in theoretical,phenomenological and lattice simulation studies of a wide class of modelﬁeld theories.We begin with a few comments about the physics in each model.In a NJL model[3],as a result of spontaneous chiral symmetry breaking,the pion-fermion coupling is given by the Goldberger-Treiman relation gπN=M N/fπ,where M N is the fermion mass and fπis the pion decay constant.Being the wavefunction of the pion,fπdetermines its radius as well:rπ∼1/fπ.The Goldberger-Treiman can then be written in the suggestive form gπN∼M N rπ.Thus,the coupling between pions and fermions vanishes as the size of the pion shrinks to zero.The origin of triviality of the Nambu-Jona-Lasinio model is precisely the loss of compositeness of the mesons[3].The force between the fermions is so strong that the constituents collapse onto one another producing pointlike mesons and a noninteracting continuum theory.In a self-interacting scalar theory,likeφ4,the mesons are elementary and the reason for triviality is diﬀerent[4,5].At short distances the interaction is repulsive,so there is no collapse.The structure of the scalars,needed for the interaction to survive the continuum limit,should be built by weakly interacting bosons.In four dimensions, the short rangedλφ4interaction fails to provide a physical size for the mesons.It cannot be felt by the particles because of the short-distance repulsion–they cannot meet where collective behavior can set in and produce macroscopicﬂuctuations.In this way the cutoﬀremains the only scale and the continuum limit is trivial.Consider two simple,soluble examples:the large-N limits of the O(N)σ-and the four-fermi model [6,7].They exhibit a phase transition atﬁnite coupling for2<d<4.Their critical exponents are given in Table1.As is apparent from the Table,the two sets of critical indices evolve diﬀerently when d is reduced below4.Atﬁrst glance this might be surprising since both models break the same symmetry spontaneously and one expects that they describe the same low energy physics.The purpose of this paper is to show that this diﬀerence between the critical exponents is generally valid,irrespective of the approximations employed.As a consequence of this it will be possible to establish a bound on the exponentδwhich for scalar theories isδ≥3,and for fermionic theories isδ≤3.Although in four dimensions the two sets of exponents coincide,they are accompanied by logarithmic corrections due to scaling violations.Itconsequently the scaling violations have opposite signs in the two classes of theories.These bounds are a consequence of diﬀerent realizations of symmetry breaking,the essential diﬀerence being the fact that for scalar theories mesons are elementary,while in the case of the chiral transition in fermionic theories,they are composite.The bounds onδare just another way of expressing this diﬀerence in terms of universal quantities. Finally,we will discuss the implications of these results on triviality in both models in four dimensions.Table1Leading order critical exponents for the spherical and four-fermi modelexponentσ-model four-fermiβ1d−2ν1d−2δd+21d−2η04−d2.Mass ratios and bounds onδTo approach the problem,it is convenient to adopt a particular view of the phase transition[8].Instead of the order parameter we will use mass ratios to distinguish the two phases.While the order parameter is a useful quantity to parametrize the phase diagram,the spectrum carries direct information about the response of the system in it’s diﬀerent phases and its form does not change in the presence of an external symmetry-breakingﬁeld.In what follows we will switch from magnetic to chiral notation without notice.The correspondence is:magneticﬁeld(h)⇔bare mass(m);magnetization(M)⇔chiral condensate(<¯ψψ>); longitudinal and transverse modes⇔(σ,π);h→0⇔chiral limit.Theories that treat scalars as elementary will be referred to as’magnets’and those that give rise to composite mesons as a consequence of spontaneous chiral symmetry breaking will be refered to as’fermions’.Consider the eﬀect of spontaneous symmetry breaking on the spectrum from a physical point of view. In the symmetric phase,there is no preferred direction and symmetry requires the degeneracy between longitudinal and transverse modes(chiral partners).Therefore,in the zero-ﬁeld(chiral)limit the ratio R=M2T/M2L=M2π/M2σ→1.As the magneticﬁeld(bare mass)increases,the ratio decreases(because of level ordering,σis always heavier thenπ).In the broken phase,however,the ratio vanishes in the chiral limit because the pion is a Goldstone boson.This time,the ratio clearly increases away from the chiral limit.dynamics.The value of the ratio at large h is less sensitive to variations in the coupling.The qualitative behavior of the mass ratio is sketched in Fig.1.The important property of the mass ratio,in this context, is that its properties follow completely from the properties of the order parameter[8].This,after all,comes as no surprise since both quantities,M and R,contain the same physics and merely reﬂect two aspects of one phenomenon.The essential ingredients are the Equation of State(EOS)and the Ward identity which follows from it.h a=M a Mδ−1f t/M1/β ,χ−1T=h/M,χ−1L=∂hR =χ−1LOne way to determine the sign of the scaling violations in four dimensions is to proceed in the spirit of the ǫ-expansion i.e.to approach four dimensions from below [9].The transcription to the language of scaling violations is established by the replacement ǫ→1/log Λin the limit ǫ→0.Thus,the extension of the arguments made before for d <4can be made by simply taking the limit d →4.In this way we anticipate that the two inequalities prevail and suggest that the scaling violations have diﬀerent signs in the two theories.The diﬀerence in the sign of the scaling violations in fermions and magnets has a simple explanation and lies at the root of the diﬀerence between the patterns of symmetry breaking in the two systems.Imagine that we ﬁx the temperature to its critical value T =T c and approach the critical point (T =T c ,h =0)in the (T,h )plane from the large-h region.The possible similarity of the two models is related to their symmetry.This is apparent in the chiral (zero ﬁeld)limit where this symmetry is manifest.By going away from this limit chiral symmetry is violated and the two models diﬀer.Consider the behavior of the mass ratio for magnets in a strong magnetic ﬁeld,away from the scaling region.In this regime,the temperature factor can be neglected and the hamiltonian describes free spins in an external ﬁeld (H →h i S i ).The energy of longitudinal excitations is proportional to the ﬁeld-squared,χ−1L ∼h 2,while the transverse mass remains ﬁxed by the symmetry,namely χ−1T =h/M for any value of h .The eﬀect of the external ﬁeld is to introduce a preferred direction and its increase results in ampliﬁcation of the diﬀerence between the longitudinal and transverse dirrections.For large h the ratio scales as R ∼1/h .Therefore,an increase in magnetic ﬁeld reduces the ratio towards zero.The critical isotherm in this case bends down (Fig.3).For fermions,the mesons are fermion-antifermion composites.Close to the chiral limit,they are collec-tive.However,as the constituent mass increases,they turn into atomic states and the main contribution to the meson mass comes from the rest energy of its constituents.In the limit of inﬁnite bare mass,interactions are negligible and M →2m regardless of the channel.Thus,outside of the scaling region,an increase in m drives the ratio to 1(Fig.3).Thus,the scaling violations for magnetis and fermionis have opposite signs .They contain knowledge of the physics away from the chiral limit where the two models are quite diﬀerent and these diﬀerences remain as small corrections close to the chiral limit.To establish the connection between scaling violations and triviality,we introduce the renormalized cou-pling.It is a dimensionless low-energy quantity that contains information about the non-gaussian character of the theory.It is conventionally deﬁned as [10]g R =−χ(nl )∂h 3= 123<φ(0)φ(1)φ(2)φ(3)>c (4)The normalization fators,χ= x <φ(0)φ(x )>c and ξd ,in eq.(3)take care of the four ﬁelds and the three integrations.In a gaussian theory all higher-point functions factorize,so g R ing the hyperscalingg R∼ξ(2∆−γ−dν)/ν(5)where∆=β+γis the gap exponent.Being dimensionless,g R should be independent ofξifξis the only scale.Thus,the validity of hyperscaling requires that the exponent must vanish.It implies the relation, 2∆−γ−dν=0,between the critical indices.In general,it is known that the following inequality[11]holds2∆≤γ+dν(6)The exponent in the expression for g R is always non-positive,so that violations of hyperscaling imply that the resulting theory is non-interacting.Above four dimensions,the exponents are gaussian(γ=1,∆=3/2,ν=1/2).In this case,it is easy to verify the above inequality:3≤1+d/2,which amounts to d≥4.In four dimensions mostﬁeld theoretical models have mean-ﬁeld critical exponents,but with logarithmic corrections that drive g R to zero.Scaling violations in any thermodynamic quantity propagate into the renormalized coupling and,according to eq.(6), these violations lead to triviality.Instead of using the ideas of theǫ-expansion where scaling is always respected and where equalities between exponents hold,we willﬁx d=4and compute the logarithmic corrections to the critical exponents. In order to focus on the problem in question,we analyze two simple models:φ4and(¯ψψ)2theories both in the large-N limit.The results that will be discussed are completely general and the two models are chosen just to make the argument simple.The eﬀective actions for the two models are[12,13]V(M)=−14M42t<¯ψψ>2+<¯ψψ>4log(1/<¯ψψ>)(7b) This is the leading log contribution only.In theﬁrst example,it is clear how log-corrections lead to triviality. The logarithm can simply be thought of as coming from the running coupling–quantum corrections lead to the replacementλ→λR.The vanishing of the renormalized coupling is then manifest from eq.(7a).In the case of fermions,eq.(7b),the details are completely diﬀerent–the analogous reasoning would lead to an erroneous conclusion that the renormalized coupling increases in the infrared.In eq.(7b)the explicit coupling is absent from theﬂuctuating term–it is already absorbed in the curvature.Once the curvature isﬁxed,the eﬀective coupling is independent of the bare one.The vanishing of the renormalized coupling here follows from the wave funciton renormalization constant Z∼1/ln(1/<¯ψψ>)[13].In both cases the renormalized coupling is obtained through the nonlinear susceptibility.For simplicity, we work in the symmetric phase where the odd-point functions vanish.The correlation length is related to the susceptibility byξ2=χ/Z.For magnets the folowing relations holdχ(nl)∼χ4λln(1/M)∼1For fermions,on the other hand,we have1χ(nl)∼χ4ln(1/<¯ψψ>),Z∼(9b)ln(1/<¯ψψ>)In this context the following point should be made.The nonlinear susceptibility is a connected four-point function for the composite¯ψψﬁeld.The free fermionic theory is not gaussian in¯ψψ,so even in freeﬁeld theory g R does not vanish.The fact that g R→0near the critical point indicates that the resulting theory is indeed gaussian in the compositeﬁeld which results in a free bosonic theory in the continuum.The Equation of State(EOS)is obtained from the eﬀective potential by simple diﬀerentiation.To make the connection withδ,we take t=0.The critical EOS for the magnets is[12]M3h∼(11)log(1/M)Thus,on the ratio plot,Fig.2a,the critical isotherm is no longerﬂat,but goes down,as the h-ﬁeld(order parameter)increases.The result of eq.(11)is well known in the literature and has been obtained in the past using theǫ-expansion:δ=3+ǫ[12],where the correspondence with eq.(10)is made after the replacement 1/ǫ→log.Such corrections toδ,as eq.(11),can never occur in the case of the chiral transition.For the four-fermi model the critical EOS[14]reads,m∼<¯ψψ>3log(1/<¯ψψ>)(12) Unlike scalar theories,the log’s appear in the numerator–the right hand side in eq.(12)vanishes slower then the pure power and the”eﬀective”δis smaller then the(pure)mean-ﬁeld value.˜δ=3−1We have seen that the two diﬀerent realizations of spontaneous symmetry breaking can be expressed simply in terms of universal quantities.In this way,many apparently complicated dynamical questions become transparent.In addition,some of our observations lead to practical applications–they can be used for extracting the properties of the continuum limit of theories with newﬁxed points,especially when clear theoretical ideas about the low-energy physics of the theory are missing.Of special importance is the knowledge of the position of the logarithms when triviality is studied on the lattice.It is extremely diﬃcult to establish the presence of the logarithms for aﬁnite system and to disentangle them fromﬁnite size eﬀects. The bounds obtained in this paper establish some criteria in this direction as far as chiral transitions are concerned.Recently,they were proven to be decisive in studies of the chiral transition of QED[15]and in establishing triviality of the NJL model in four dimensions by computer simulations[16].The literature on fermionic QED abounds with loose statements such as”QED is ultimately trivial and reduces toλφ4”.If QED suﬀers from complete screening(the Moscow zero),then we expect the NJL model to describe its triviality.One result of this paper is thatλφ4cannot be a good guide to fermionic QED under any circumstances!There are several physical implications that the two bounds onδimply and we discuss some of them brieﬂy below.The wavefunction renormalization constant respects the Lehmann bound:0≤Z≤1.Roughly speaking, Z is the probability that the scalarﬁeld creates a single particle from the vacuum.The limit Z=1 corresponds to a noninteracting theory whereas the compositeness condition,Z=0,sets an upper bound on the eﬀective coupling[17].The anomalous dimensionηdetermines the scaling of Z in the critical region Z∼ξ−η.It describes the scaling of the correlation function in the massless limit:D(x)∼1/|x|d−2+η.Small ηis associated with weak coupling andη=O(1)with the strong coupling limit of the theory.It is related to the exponentδthrough the hyperscaling relationd+2−ηδ=especially visible from the behavior of the correlation functions in the two theories.In the scalar theory, where anomalous dimensions are small,the scaling of the correlation functions is weakly aﬀected at short distances.They behave almost like free particle propagators,D(x)∼1/|x|d−2.For fermions,however, there are nontrivial changes in scaling due to strongly interacting dynamics at short distances.In four-fermi theory,for example,at large-Nη=4−d and D(x)∼1/|x|2irrespective of d.Another consequence of the diﬀerence in the bounds onδconcerns the physics in the broken phase. As an eﬀect of spontaneous symmetry breaking a trilinear coupling is generated and the decayσ→ππis a dominant decay mode in the Goldstone phase.The pole in theσpropagator is buried in the continuum states and the pion state saturates all the correlation functions.This is especially visible in lower dimensions and persists even atﬁnite h.For fermions,however,this is not necessarily the case for the simple reason that theπ−σmass-squared ratio in the broken phase is bounded by1/δ>1/3,away from the chiral limit and for theσ→ππdecay to occur the masses must satisfy M2π/M2σ≤1/4.Thus,for appropriately chosen couplings and bare fermion masses,the decay becomes kinematically forbidden even in the broken phase.Regarding the usage of perturbation theory,the rules are diﬀerent for magnets and fermions.The applicability of perturbation theory to magnets was noted long ago[18].Its success near two and four dimensions is not surprising.Below four dimensions,φ4posesses an infraredﬁxed point at coupling g c∼O(ǫ). This coupling is an upper bound on the renormalized coupling i.e.g R∼ǫ.So,theǫ-expansion is in eﬀect renormalized perturbation theory.The critical exponents receive corrections of the typeδ=3+O(g R),β= 1/2+O(g R),etc..Thus,inφ4the success of perturbation theory is a consequence of the fact that the infraredﬁxed point moves to the origin as d→4.In the non-linearσ-model the critical coupling is an ultravioletﬁxed point that moves to the origin as d→2.The weak coupling phase is at low-temperatures. Due to the presence of Goldstone bosons,all the correlation functions are saturated with massless states and the entire low-temperature phase is massless.Every point is a critical point in the limit of vanishing magneticﬁeld.Thus,the low-temperature expansion is an expansion in powers of T.Terms of the form exp(−M/T)are absent and there is no danger that they will be omitted by using perturbation theory.In this way,in principle,the critical region can be accessed through perturbation theory[18].Clearly,such reasoning can not be applied to fermions simply because the weak coupling phase is symmetric.Thus,no matter how small the coupling is,perturbation theory omits the Goldstone physics as a matter of principle. It can not produce bound states that accompany the chiral transition and its applicability is questionable in general.Especially,it is diﬃcult to imagine how perturbation theory could give a mass ratio that is constant, independent of the bare parameters,once the bare coupling is tuned to the critical value.Even if this were possible,the renormalized coupling would be sensitive to the variation of the bare mass leading to conﬂicting renormalization group trajectories as a consequence.Finally,we comment on one possible use of theδ<3bound for controllingﬁnite size eﬀects in lattice studies of chiral transitions.As was argued in[8],it is convenient to introduce a plotχ−1πversus<¯ψψ>2. The usefulness of this plot becomes clear if we write the critical EOS.From m∼<¯ψψ>δandχ−1π=m/<¯ψψ>,it follows thatχ−1π∼(<¯ψψ>2)(δ−1)/2(15)concave downwards.On a small lattice the order parameter is smaller and pion mass is bigger then in the thermodynamic limit.Thus,small volume distortions are always in the direction of opposite concavity of the plot and the wrong concavity of this plot is a clear sign of the presence ofﬁnite size eﬀects[8].Acknowledgement We wish to acknowledge the discussions with E.Fradkin,A.Patrascioiou and E.Seiler. This work is supported by NSF-PHY92-00148.References[1]See for example The Standard Model Higgs Boson,Edited by M.Einhorn,(North-Holland,Amsterdam, 1991).[2]Y.Nambu,in New Trends in Physics,proceedings of the XI International Symposium on Elementary Particle Physics,Kazimierz,Poland,1988,edited by Z.Ajduk S.Pokorski and A.Trautman(World Scientiﬁc, Singapore,1989);V.Miransky,M.Tanabashi and K.Yamawaki,Mod.Phys.Lett.A4,1043(1989);W. Bardeen,C.Hill and M.Lindner,Phys.Rev.D41,1647(1990).[3]Y.Nambu and G.Jona-Lasinio,Phys.Rev.122,345(1961).[4]M.Aizenman,Comm.Math.Phys.86,1(1982);C.Arag˜a o de Carvalho,S.Caracciolo and J.Fr¨o hlich, Nuc.Phys.B215[FS7],209(1983).[5]M.L¨u scher and P.Weisz,Nuc.Phys.B290[FS20],25(1987)[6]S.K.Ma,in Phase Transitions and Critical Phenomena Vol.6,eds.C.Domb and M.Green(Academic Press,London,1976).[7]S.Hands,A.Koci´c and J.B.Kogut,Phys.Lett.B273(1991)111.[8]A.Koci´c,J.B.Kogut and M.-P.Lombardo,Nuc.Phys.B398,376(1993).[9]K.Wilson and J.Kogut,Phys.Rep.12C,75(1974).[10]See,for example,C.Itzykson and J.-M.Drouﬀe,Statistical Field Theory(Cambridge University Press, 1989);V.Privman,P.C.Hohenberg and A.Aharony,in Phase Transitions and Critical Phenomena Vol.14, eds.C.Domb and J.L.Lebowitz(Academic Press,London,1991).[11]B.Freedman and G.A.Baker Jr,J.Phys.A15(1982)L715;R.Schrader,Phys.Rev.B14(1976)172;B.D.Josephson,Proc.Phys.Soc.92(1967)269,276.[12]E.Brezin,J.-C.Le Guillou and J.Zinn-Justin,in Phase Transitions and Critical Phenomena Vol.6,eds.C.Domb and M.Green(Academic Press,London,1976).[13]T.Eguchi,Phys.Rev.D17,611(1978).[14]S.Hands,A.Koci´c and J.B.Kogut,Ann.Phys.224,29(1993).[15]A.Koci´c,J.B.Kogut and K.C.Wang,Nucl.Phys.B398(1993)405.[16]S.Kim,A.Koci´c and J.Kogut(unpublished)[17]S.Weinberg,Phys.Rev.130,776(1963).[18]E.Brezin and J.Zinn-Justin,Phys.Rev.B14,3110(1976).Figure captions1.Susceptibility ratio as a function of magneticﬁeld(bare mass)forﬁxed values of the temperature (coupling).2.The behavior of the critical mass ratio,R(t=0,h)=1/δ,for diﬀerent values of d,in a)magnets and b) in the case of chiral transition.3.Critical mass ratio,R(t=0,h),for fermions and magnets in four dimensions over extended range of magneticﬁeld(mass).This figure "fig1-1.png" is available in "png" format from: /ps/hep-th/9312007v1This figure "fig1-2.png" is available in "png" format from: /ps/hep-th/9312007v1This figure "fig1-3.png" is available in "png" format from: /ps/hep-th/9312007v1。

ABSTRACT Progressive Simplicial Complexes

Progressive Simplicial Complexes Jovan Popovi´c Hugues HoppeCarnegie Mellon University Microsoft ResearchABSTRACTIn this paper,we introduce the progressive simplicial complex(PSC) representation,a new format for storing and transmitting triangu-lated geometric models.Like the earlier progressive mesh(PM) representation,it captures a given model as a coarse base model together with a sequence of reﬁnement transformations that pro-gressively recover detail.The PSC representation makes use of a more general reﬁnement transformation,allowing the given model to be an arbitrary triangulation(e.g.any dimension,non-orientable, non-manifold,non-regular),and the base model to always consist of a single vertex.Indeed,the sequence of reﬁnement transforma-tions encodes both the geometry and the topology of the model in a uniﬁed multiresolution framework.The PSC representation retains the advantages of PM’s.It deﬁnes a continuous sequence of approx-imating models for runtime level-of-detail control,allows smooth transitions between any pair of models in the sequence,supports progressive transmission,and offers a space-efﬁcient representa-tion.Moreover,by allowing changes to topology,the PSC sequence of approximations achieves betterﬁdelity than the corresponding PM sequence.We develop an optimization algorithm for constructing PSC representations for graphics surface models,and demonstrate the framework on models that are both geometrically and topologically complex.CR Categories:I.3.5[Computer Graphics]:Computational Geometry and Object Modeling-surfaces and object representations.Additional Keywords:model simpliﬁcation,level-of-detail representa-tions,multiresolution,progressive transmission,geometry compression.1INTRODUCTIONModeling and3D scanning systems commonly give rise to triangle meshes of high complexity.Such meshes are notoriously difﬁcult to render,store,and transmit.One approach to speed up rendering is to replace a complex mesh by a set of level-of-detail(LOD) approximations;a detailed mesh is used when the object is close to the viewer,and coarser approximations are substituted as the object recedes[6,8].These LOD approximations can be precomputed Work performed while at Microsoft Research.Email:jovan@,hhoppe@Web:/jovan/Web:/hoppe/automatically using mesh simpliﬁcation methods(e.g.[2,10,14,20,21,22,24,27]).For efﬁcient storage and transmission,meshcompression schemes[7,26]have also been developed.The recently introduced progressive mesh(PM)representa-tion[13]provides a uniﬁed solution to these problems.In PM form,an arbitrary mesh M is stored as a coarse base mesh M0together witha sequence of n detail records that indicate how to incrementally re-ﬁne M0into M n=M(see Figure7).Each detail record encodes theinformation associated with a vertex split,an elementary transfor-mation that adds one vertex to the mesh.In addition to deﬁninga continuous sequence of approximations M0M n,the PM rep-resentation supports smooth visual transitions(geomorphs),allowsprogressive transmission,and makes an effective mesh compressionscheme.The PM representation has two restrictions,however.First,it canonly represent meshes:triangulations that correspond to orientable12-dimensional manifolds.Triangulated2models that cannot be rep-resented include1-d manifolds(open and closed curves),higherdimensional polyhedra(e.g.triangulated volumes),non-orientablesurfaces(e.g.M¨o bius strips),non-manifolds(e.g.two cubes joinedalong an edge),and non-regular models(i.e.models of mixed di-mensionality).Second,the expressiveness of the PM vertex splittransformations constrains all meshes M0M n to have the same topological type.Therefore,when M is topologically complex,the simpliﬁed base mesh M0may still have numerous triangles(Fig-ure7).In contrast,a number of existing simpliﬁcation methods allowtopological changes as the model is simpliﬁed(Section6).Ourwork is inspired by vertex uniﬁcation schemes[21,22],whichmerge vertices of the model based on geometric proximity,therebyallowing genus modiﬁcation and component merging.In this paper,we introduce the progressive simplicial complex(PSC)representation,a generalization of the PM representation thatpermits topological changes.The key element of our approach isthe introduction of a more general reﬁnement transformation,thegeneralized vertex split,that encodes changes to both the geometryand topology of the model.The PSC representation expresses anarbitrary triangulated model M(e.g.any dimension,non-orientable,non-manifold,non-regular)as the result of successive reﬁnementsapplied to a base model M1that always consists of a single vertex (Figure8).Thus both geometric and topological complexity are recovered progressively.Moreover,the PSC representation retains the advantages of PM’s,including continuous LOD,geomorphs, progressive transmission,and model compression.In addition,we develop an optimization algorithm for construct-ing a PSC representation from a given model,as described in Sec-tion4.1The particular parametrization of vertex splits in[13]assumes that mesh triangles are consistently oriented.2Throughout this paper,we use the words“triangulated”and“triangula-tion”in the general dimension-independent sense.Figure 1:Illustration of a simplicial complex K and some of its subsets.2BACKGROUND2.1Concepts from algebraic topologyTo precisely deﬁne both triangulated models and their PSC repre-sentations,we ﬁnd it useful to introduce some elegant abstractions from algebraic topology (e.g.[15,25]).The geometry of a triangulated model is denoted as a tuple (K V )where the abstract simplicial complex K is a combinatorial structure specifying the adjacency of vertices,edges,triangles,etc.,and V is a set of vertex positions specifying the shape of the model in 3.More precisely,an abstract simplicial complex K consists of a set of vertices 1m together with a set of non-empty subsets of the vertices,called the simplices of K ,such that any set consisting of exactly one vertex is a simplex in K ,and every non-empty subset of a simplex in K is also a simplex in K .A simplex containing exactly d +1vertices has dimension d and is called a d -simplex.As illustrated pictorially in Figure 1,the faces of a simplex s ,denoted s ,is the set of non-empty subsets of s .The star of s ,denoted star(s ),is the set of simplices of which s is a face.The children of a d -simplex s are the (d 1)-simplices of s ,and its parents are the (d +1)-simplices of star(s ).A simplex with exactly one parent is said to be a boundary simplex ,and one with no parents a principal simplex .The dimension of K is the maximum dimension of its simplices;K is said to be regular if all its principal simplices have the same dimension.To form a triangulation from K ,identify its vertices 1m with the standard basis vectors 1m ofm.For each simplex s ,let the open simplex smdenote the interior of the convex hull of its vertices:s =m:jmj =1j=1jjsThe topological realization K is deﬁned as K =K =s K s .The geometric realization of K is the image V (K )where V :m 3is the linear map that sends the j -th standard basis vector jm to j 3.Only a restricted set of vertex positions V =1m lead to an embedding of V (K )3,that is,prevent self-intersections.The geometric realization V (K )is often called a simplicial complex or polyhedron ;it is formed by an arbitrary union of points,segments,triangles,tetrahedra,etc.Note that there generally exist many triangulations (K V )for a given polyhedron.(Some of the vertices V may lie in the polyhedron’s interior.)Two sets are said to be homeomorphic (denoted =)if there ex-ists a continuous one-to-one mapping between them.Equivalently,they are said to have the same topological type .The topological realization K is a d-dimensional manifold without boundary if for each vertex j ,star(j )=d .It is a d-dimensional manifold if each star(v )is homeomorphic to either d or d +,where d +=d:10.Two simplices s 1and s 2are d-adjacent if they have a common d -dimensional face.Two d -adjacent (d +1)-simplices s 1and s 2are manifold-adjacent if star(s 1s 2)=d +1.Figure 2:Illustration of the edge collapse transformation and its inverse,the vertex split.Transitive closure of 0-adjacency partitions K into connected com-ponents .Similarly,transitive closure of manifold-adjacency parti-tions K into manifold components .2.2Review of progressive meshesIn the PM representation [13],a mesh with appearance attributes is represented as a tuple M =(K V D S ),where the abstract simpli-cial complex K is restricted to deﬁne an orientable 2-dimensional manifold,the vertex positions V =1m determine its ge-ometric realization V (K )in3,D is the set of discrete material attributes d f associated with 2-simplices f K ,and S is the set of scalar attributes s (v f )(e.g.normals,texture coordinates)associated with corners (vertex-face tuples)of K .An initial mesh M =M n is simpliﬁed into a coarser base mesh M 0by applying a sequence of n successive edge collapse transforma-tions:(M =M n )ecol n 1ecol 1M 1ecol 0M 0As shown in Figure 2,each ecol uniﬁes the two vertices of an edgea b ,thereby removing one or two triangles.The position of the resulting uniﬁed vertex can be arbitrary.Because the edge collapse transformation has an inverse,called the vertex split transformation (Figure 2),the process can be reversed,so that an arbitrary mesh M may be represented as a simple mesh M 0together with a sequence of n vsplit records:M 0vsplit 0M 1vsplit 1vsplit n 1(M n =M )The tuple (M 0vsplit 0vsplit n 1)forms a progressive mesh (PM)representation of M .The PM representation thus captures a continuous sequence of approximations M 0M n that can be quickly traversed for interac-tive level-of-detail control.Moreover,there exists a correspondence between the vertices of any two meshes M c and M f (0c f n )within this sequence,allowing for the construction of smooth vi-sual transitions (geomorphs)between them.A sequence of such geomorphs can be precomputed for smooth runtime LOD.In addi-tion,PM’s support progressive transmission,since the base mesh M 0can be quickly transmitted ﬁrst,followed the vsplit sequence.Finally,the vsplit records can be encoded concisely,making the PM representation an effective scheme for mesh compression.Topological constraints Because the deﬁnitions of ecol and vsplit are such that they preserve the topological type of the mesh (i.e.all K i are homeomorphic),there is a constraint on the min-imum complexity that K 0may achieve.For instance,it is known that the minimal number of vertices for a closed genus g mesh (ori-entable 2-manifold)is (7+(48g +1)12)2if g =2(10if g =2)[16].Also,the presence of boundary components may further constrain the complexity of K 0.Most importantly,K may consist of a number of components,and each is required to appear in the base mesh.For example,the meshes in Figure 7each have 117components.As evident from the ﬁgure,the geometry of PM meshes may deteriorate severely as they approach topological lower bound.M 1;100;(1)M 10;511;(7)M 50;4656;(12)M 200;1552277;(28)M 500;3968690;(58)M 2000;14253219;(108)M 5000;029010;(176)M n =34794;0068776;(207)Figure 3:Example of a PSC representation.The image captions indicate the number of principal 012-simplices respectively and the number of connected components (in parenthesis).3PSC REPRESENTATION 3.1Triangulated modelsThe ﬁrst step towards generalizing PM’s is to let the PSC repre-sentation encode more general triangulated models,instead of just meshes.We denote a triangulated model as a tuple M =(K V D A ).The abstract simplicial complex K is not restricted to 2-manifolds,but may in fact be arbitrary.To represent K in memory,we encode the incidence graph of the simplices using the following linked structures (in C++notation):struct Simplex int dim;//0=vertex,1=edge,2=triangle,...int id;Simplex*children[MAXDIM+1];//[0..dim]List<Simplex*>parents;;To render the model,we draw only the principal simplices ofK ,denoted (K )(i.e.vertices not adjacent to edges,edges not adjacent to triangles,etc.).The discrete attributes D associate amaterial identiﬁer d s with each simplex s(K ).For the sake of simplicity,we avoid explicitly storing surface normals at “corners”(using a set S )as done in [13].Instead we let the material identiﬁer d s contain a smoothing group ﬁeld [28],and let a normal discontinuity (crease )form between any pair of adjacent triangles with different smoothing groups.Previous vertex uniﬁcation schemes [21,22]render principal simplices of dimension 0and 1(denoted 01(K ))as points and lines respectively with ﬁxed,device-dependent screen widths.To better approximate the model,we instead deﬁne a set A that associates an area a s A with each simplex s 01(K ).We think of a 0-simplex s 00(K )as approximating a sphere with area a s 0,and a 1-simplex s 1=j k 1(K )as approximating a cylinder (with axis (j k ))of area a s 1.To render a simplex s 01(K ),we determine the radius r model of the corresponding sphere or cylinder in modeling space,and project the length r model to obtain the radius r screen in screen pixels.Depending on r screen ,we render the simplex as a polygonal sphere or cylinder with radius r model ,a 2D point or line with thickness 2r screen ,or do not render it at all.This choice based on r screen can be adjusted to mitigate the overhead of introducing polygonal representations of spheres and cylinders.As an example,Figure 3shows an initial model M of 68,776triangles.One of its approximations M 500is a triangulated model with 3968690principal 012-simplices respectively.3.2Level-of-detail sequenceAs in progressive meshes,from a given triangulated model M =M n ,we deﬁne a sequence of approximations M i :M 1op 1M 2op 2M n1op n 1M nHere each model M i has exactly i vertices.The simpliﬁcation op-erator M ivunify iM i +1is the vertex uniﬁcation transformation,whichmerges two vertices (Section 3.3),and its inverse M igvspl iM i +1is the generalized vertex split transformation (Section 3.4).Thetuple (M 1gvspl 1gvspl n 1)forms a progressive simplicial complex (PSC)representation of M .To construct a PSC representation,we ﬁrst determine a sequence of vunify transformations simplifying M down to a single vertex,as described in Section 4.After reversing these transformations,we renumber the simplices in the order that they are created,so thateach gvspl i (a i)splits the vertex a i K i into two vertices a i i +1K i +1.As vertices may have different positions in the different models,we denote the position of j in M i as i j .To better approximate a surface model M at lower complexity levels,we initially associate with each (principal)2-simplex s an area a s equal to its triangle area in M .Then,as the model is simpliﬁed,wekeep constant the sum of areas a s associated with principal simplices within each manifold component.When2-simplices are eventually reduced to principal1-simplices and0-simplices,their associated areas will provide good estimates of the original component areas.3.3Vertex uniﬁcation transformationThe transformation vunify(a i b i midp i):M i M i+1takes an arbitrary pair of vertices a i b i K i+1(simplex a i b i need not be present in K i+1)and merges them into a single vertex a i K i. Model M i is created from M i+1by updating each member of the tuple(K V D A)as follows:K:References to b i in all simplices of K are replaced by refer-ences to a i.More precisely,each simplex s in star(b i)K i+1is replaced by simplex(s b i)a i,which we call the ancestor simplex of s.If this ancestor simplex already exists,s is deleted.V:Vertex b is deleted.For simplicity,the position of the re-maining(uniﬁed)vertex is set to either the midpoint or is left unchanged.That is,i a=(i+1a+i+1b)2if the boolean parameter midp i is true,or i a=i+1a otherwise.D:Materials are carried through as expected.So,if after the vertex uniﬁcation an ancestor simplex(s b i)a i K i is a new principal simplex,it receives its material from s K i+1if s is a principal simplex,or else from the single parent s a i K i+1 of s.A:To maintain the initial areas of manifold components,the areasa s of deleted principal simplices are redistributed to manifold-adjacent neighbors.More concretely,the area of each princi-pal d-simplex s deleted during the K update is distributed toa manifold-adjacent d-simplex not in star(a ib i).If no suchneighbor exists and the ancestor of s is a principal simplex,the area a s is distributed to that ancestor simplex.Otherwise,the manifold component(star(a i b i))of s is being squashed be-tween two other manifold components,and a s is discarded. 3.4Generalized vertex split transformation Constructing the PSC representation involves recording the infor-mation necessary to perform the inverse of each vunify i.This inverse is the generalized vertex split gvspl i,which splits a0-simplex a i to introduce an additional0-simplex b i.(As mentioned previously, renumbering of simplices implies b i i+1,so index b i need not be stored explicitly.)Each gvspl i record has the formgvspl i(a i C K i midp i()i C D i C A i)and constructs model M i+1from M i by updating the tuple (K V D A)as follows:K:As illustrated in Figure4,any simplex adjacent to a i in K i can be the vunify result of one of four conﬁgurations in K i+1.To construct K i+1,we therefore replace each ancestor simplex s star(a i)in K i by either(1)s,(2)(s a i)i+1,(3)s and(s a i)i+1,or(4)s,(s a i)i+1and s i+1.The choice is determined by a split code associated with s.Thesesplit codes are stored as a code string C Ki ,in which the simplicesstar(a i)are sortedﬁrst in order of increasing dimension,and then in order of increasing simplex id,as shown in Figure5. V:The new vertex is assigned position i+1i+1=i ai+()i.Theother vertex is given position i+1ai =i ai()i if the boolean pa-rameter midp i is true;otherwise its position remains unchanged.D:The string C Di is used to assign materials d s for each newprincipal simplex.Simplices in C Di ,as well as in C Aibelow,are sorted by simplex dimension and simplex id as in C Ki. A:During reconstruction,we are only interested in the areas a s fors01(K).The string C Ai tracks changes in these areas.Figure4:Effects of split codes on simplices of various dimensions.code string:41422312{}Figure5:Example of split code encoding.3.5PropertiesLevels of detail A graphics application can efﬁciently transitionbetween models M1M n at runtime by performing a sequence ofvunify or gvspl transformations.Our current research prototype wasnot designed for efﬁciency;it attains simpliﬁcation rates of about6000vunify/sec and reﬁnement rates of about5000gvspl/sec.Weexpect that a careful redesign using more efﬁcient data structureswould signiﬁcantly improve these rates.Geomorphs As in the PM representation,there exists a corre-spondence between the vertices of the models M1M n.Given acoarser model M c and aﬁner model M f,1c f n,each vertexj K f corresponds to a unique ancestor vertex f c(j)K cfound by recursively traversing the ancestor simplex relations:f c(j)=j j cf c(a j1)j cThis correspondence allows the creation of a smooth visual transi-tion(geomorph)M G()such that M G(1)equals M f and M G(0)looksidentical to M c.The geomorph is deﬁned as the modelM G()=(K f V G()D f A G())in which each vertex position is interpolated between its originalposition in V f and the position of its ancestor in V c:Gj()=()fj+(1)c f c(j)However,we must account for the special rendering of principalsimplices of dimension0and1(Section3.1).For each simplexs01(K f),we interpolate its area usinga G s()=()a f s+(1)a c swhere a c s=0if s01(K c).In addition,we render each simplexs01(K c)01(K f)using area a G s()=(1)a c s.The resultinggeomorph is visually smooth even as principal simplices are intro-duced,removed,or change dimension.The accompanying video demonstrates a sequence of such geomorphs.Progressive transmission As with PM’s,the PSC representa-tion can be progressively transmitted by ﬁrst sending M 1,followed by the gvspl records.Unlike the base mesh of the PM,M 1always consists of a single vertex,and can therefore be sent in a ﬁxed-size record.The rendering of lower-dimensional simplices as spheres and cylinders helps to quickly convey the overall shape of the model in the early stages of transmission.Model compression Although PSC gvspl are more general than PM vsplit transformations,they offer a surprisingly concise representation of M .Table 1lists the average number of bits re-quired to encode each ﬁeld of the gvspl records.Using arithmetic coding [30],the vertex id ﬁeld a i requires log 2i bits,and the boolean parameter midp i requires 0.6–0.9bits for our models.The ()i delta vector is quantized to 16bitsper coordinate (48bits per),and stored as a variable-length ﬁeld [7,13],requiring about 31bits on average.At ﬁrst glance,each split code in the code string C K i seems to have 4possible outcomes (except for the split code for 0-simplex a i which has only 2possible outcomes).However,there exist constraints between these split codes.For example,in Figure 5,the code 1for 1-simplex id 1implies that 2-simplex id 1also has code 1.This in turn implies that 1-simplex id 2cannot have code 2.Similarly,code 2for 1-simplex id 3implies a code 2for 2-simplex id 2,which in turn implies that 1-simplex id 4cannot have code 1.These constraints,illustrated in the “scoreboard”of Figure 6,can be summarized using the following two rules:(1)If a simplex has split code c12,all of its parents havesplit code c .(2)If a simplex has split code 3,none of its parents have splitcode 4.As we encode split codes in C K i left to right,we apply these two rules (and their contrapositives)transitively to constrain the possible outcomes for split codes yet to be ing arithmetic coding with uniform outcome probabilities,these constraints reduce the code string length in Figure 6from 15bits to 102bits.In our models,the constraints reduce the code string from 30bits to 14bits on average.The code string is further reduced using a non-uniform probability model.We create an array T [0dim ][015]of encoding tables,indexed by simplex dimension (0..dim)and by the set of possible (constrained)split codes (a 4-bit mask).For each simplex s ,we encode its split code c using the probability distribution found in T [s dim ][s codes mask ].For 2-dimensional models,only 10of the 48tables are non-trivial,and each table contains at most 4probabilities,so the total size of the probability model is small.These encoding tables reduce the code strings to approximately 8bits as shown in Table 1.By comparison,the PM representation requires approximately 5bits for the same information,but of course it disallows topological changes.To provide more intuition for the efﬁciency of the PSC repre-sentation,we note that capturing the connectivity of an average 2-manifold simplicial complex (n vertices,3n edges,and 2n trian-gles)requires ni =1(log 2i +8)n (log 2n +7)bits with PSC encoding,versus n (12log 2n +95)bits with a traditional one-way incidence graph representation.For improved compression,it would be best to use a hybrid PM +PSC representation,in which the more concise PM vertex split encoding is used when the local neighborhood is an orientableFigure 6:Constraints on the split codes for the simplices in the example of Figure 5.Table 1:Compression results and construction times.Object#verts Space required (bits/n )Trad.Con.n K V D Arepr.time a i C K i midp i (v )i C D i C Ai bits/n hrs.drumset 34,79412.28.20.928.1 4.10.453.9146.1 4.3destroyer 83,79913.38.30.723.1 2.10.347.8154.114.1chandelier 36,62712.47.60.828.6 3.40.853.6143.6 3.6schooner 119,73413.48.60.727.2 2.5 1.353.7148.722.2sandal 4,6289.28.00.733.4 1.50.052.8123.20.4castle 15,08211.0 1.20.630.70.0-43.5-0.5cessna 6,7959.67.60.632.2 2.50.152.6132.10.5harley 28,84711.97.90.930.5 1.40.453.0135.7 3.52-dimensional manifold (this occurs on average 93%of the time in our examples).To compress C D i ,we predict the material for each new principalsimplex sstar(a i )star(b i )K i +1by constructing an ordered set D s of materials found in star(a i )K i .To improve the coding model,the ﬁrst materials in D s are those of principal simplices in star(s )K i where s is the ancestor of s ;the remainingmaterials in star(a i )K i are appended to D s .The entry in C D i associated with s is the index of its material in D s ,encoded arithmetically.If the material of s is not present in D s ,it is speciﬁed explicitly as a global index in D .We encode C A i by specifying the area a s for each new principalsimplex s 01(star(a i )star(b i ))K i +1.To account for this redistribution of area,we identify the principal simplex from which s receives its area by specifying its index in 01(star(a i ))K i .The column labeled in Table 1sums the bits of each ﬁeld of the gvspl records.Multiplying by the number n of vertices in M gives the total number of bits for the PSC representation of the model (e.g.500KB for the destroyer).By way of compari-son,the next column shows the number of bits per vertex required in a traditional “IndexedFaceSet”representation,with quantization of 16bits per coordinate and arithmetic coding of face materials (3n 16+2n 3log 2n +materials).4PSC CONSTRUCTIONIn this section,we describe a scheme for iteratively choosing pairs of vertices to unify,in order to construct a PSC representation.Our algorithm,a generalization of [13],is time-intensive,seeking high quality approximations.It should be emphasized that many quality metrics are possible.For instance,the quadric error metric recently introduced by Garland and Heckbert [9]provides a different trade-off of execution speed and visual quality.As in [13,20],we ﬁrst compute a cost E for each candidate vunify transformation,and enter the candidates into a priority queueordered by ascending cost.Then,in each iteration i =n 11,we perform the vunify at the front of the queue and update the costs of affected candidates.4.1Forming set of candidate vertex pairs In principle,we could enter all possible pairs of vertices from M into the priority queue,but this would be prohibitively expensive since simpliﬁcation would then require at least O(n2log n)time.Instead, we would like to consider only a smaller set of candidate vertex pairs.Naturally,should include the1-simplices of K.Additional pairs should also be included in to allow distinct connected com-ponents of M to merge and to facilitate topological changes.We considered several schemes for forming these additional pairs,in-cluding binning,octrees,and k-closest neighbor graphs,but opted for the Delaunay triangulation because of its adaptability on models containing components at different scales.We compute the Delaunay triangulation of the vertices of M, represented as a3-dimensional simplicial complex K DT.We deﬁne the initial set to contain both the1-simplices of K and the subset of1-simplices of K DT that connect vertices in different connected components of K.During the simpliﬁcation process,we apply each vertex uniﬁcation performed on M to as well in order to keep consistent the set of candidate pairs.For models in3,star(a i)has constant size in the average case,and the overall simpliﬁcation algorithm requires O(n log n) time.(In the worst case,it could require O(n2log n)time.)4.2Selecting vertex uniﬁcations fromFor each candidate vertex pair(a b),the associated vunify(a b):M i M i+1is assigned the costE=E dist+E disc+E area+E foldAs in[13],theﬁrst term is E dist=E dist(M i)E dist(M i+1),where E dist(M)measures the geometric accuracy of the approximate model M.Conceptually,E dist(M)approximates the continuous integralMd2(M)where d(M)is the Euclidean distance of the point to the closest point on M.We discretize this integral by deﬁning E dist(M)as the sum of squared distances to M from a dense set of points X sampled from the original model M.We sample X from the set of principal simplices in K—a strategy that generalizes to arbitrary triangulated models.In[13],E disc(M)measures the geometric accuracy of disconti-nuity curves formed by a set of sharp edges in the mesh.For the PSC representation,we generalize the concept of sharp edges to that of sharp simplices in K—a simplex is sharp either if it is a boundary simplex or if two of its parents are principal simplices with different material identiﬁers.The energy E disc is deﬁned as the sum of squared distances from a set X disc of points sampled from sharp simplices to the discontinuity components from which they were sampled.Minimization of E disc therefore preserves the geom-etry of material boundaries,normal discontinuities(creases),and triangulation boundaries(including boundary curves of a surface and endpoints of a curve).We have found it useful to introduce a term E area that penalizes surface stretching(a more sophisticated version of the regularizing E spring term of[13]).Let A i+1N be the sum of triangle areas in the neighborhood star(a i)star(b i)K i+1,and A i N the sum of triangle areas in star(a i)K i.The mean squared displacement over the neighborhood N due to the change in area can be approx-imated as disp2=12(A i+1NA iN)2.We let E area=X N disp2,where X N is the number of points X projecting in the neighborhood. To prevent model self-intersections,the last term E fold penalizes surface folding.We compute the rotation of each oriented triangle in the neighborhood due to the vertex uniﬁcation(as in[10,20]).If any rotation exceeds a threshold angle value,we set E fold to a large constant.Unlike[13],we do not optimize over the vertex position i a, but simply evaluate E for i a i+1a i+1b(i+1a+i+1b)2and choose the best one.This speeds up the optimization,improves model compression,and allows us to introduce non-quadratic energy terms like E area.5RESULTSTable1gives quantitative results for the examples in theﬁgures and in the video.Simpliﬁcation times for our prototype are measured on an SGI Indigo2Extreme(150MHz R4400).Although these times may appear prohibitive,PSC construction is an off-line task that only needs to be performed once per model.Figure9highlights some of the beneﬁts of the PSC representa-tion.The pearls in the chandelier model are initially disconnected tetrahedra;these tetrahedra merge and collapse into1-d curves in lower-complexity approximations.Similarly,the numerous polyg-onal ropes in the schooner model are simpliﬁed into curves which can be rendered as line segments.The straps of the sandal model initially have some thickness;the top and bottom sides of these straps merge in the simpliﬁcation.Also note the disappearance of the holes on the sandal straps.The castle example demonstrates that the original model need not be a mesh;here M is a1-dimensional non-manifold obtained by extracting edges from an image.6RELATED WORKThere are numerous schemes for representing and simplifying tri-angulations in computer graphics.A common special case is that of subdivided2-manifolds(meshes).Garland and Heckbert[12] provide a recent survey of mesh simpliﬁcation techniques.Several methods simplify a given model through a sequence of edge col-lapse transformations[10,13,14,20].With the exception of[20], these methods constrain edge collapses to preserve the topological type of the model(e.g.disallow the collapse of a tetrahedron into a triangle).Our work is closely related to several schemes that generalize the notion of edge collapse to that of vertex uniﬁcation,whereby separate connected components of the model are allowed to merge and triangles may be collapsed into lower dimensional simplices. Rossignac and Borrel[21]overlay a uniform cubical lattice on the object,and merge together vertices that lie in the same cubes. Schauﬂer and St¨u rzlinger[22]develop a similar scheme in which vertices are merged using a hierarchical clustering algorithm.Lue-bke[18]introduces a scheme for locally adapting the complexity of a scene at runtime using a clustering octree.In these schemes, the approximating models correspond to simplicial complexes that would result from a set of vunify transformations(Section3.3).Our approach differs in that we order the vunify in a carefully optimized sequence.More importantly,we deﬁne not only a simpliﬁcation process,but also a new representation for the model using an en-coding of gvspl=vunify1transformations.Recent,independent work by Schmalstieg and Schauﬂer[23]de-velops a similar strategy of encoding a model using a sequence of vertex split transformations.Their scheme differs in that it tracks only triangles,and therefore requires regular,2-dimensional trian-gulations.Hence,it does not allow lower-dimensional simplices in the model approximations,and does not generalize to higher dimensions.Some simpliﬁcation schemes make use of an intermediate vol-umetric representation to allow topological changes to the model. He et al.[11]convert a mesh into a binary inside/outside function discretized on a three-dimensional grid,low-passﬁlter this function,。

菲涅耳非相干关联全息图(综述)

Fresnel incoherent correlation hologram-a reviewInvited PaperJoseph Rosen,Barak Katz1,and Gary Brooker2∗∗1Department of Electrical and Computer Engineering,Ben-Gurion University of the Negev,P.O.Box653,Beer-Sheva84105,Israel2Johns Hopkins University Microscopy Center,Montgomery County Campus,Advanced Technology Laboratory, Whiting School of Engineering,9605Medical Center Drive Suite240,Rockville,MD20850,USA∗E-mail:rosen@ee.bgu.ac.il;∗∗e-mail:gbrooker@Received July17,2009Holographic imaging oﬀers a reliable and fast method to capture the complete three-dimensional(3D) information of the scene from a single perspective.We review our recently proposed single-channel optical system for generating digital Fresnel holograms of3D real-existing objects illuminated by incoherent light.In this motionless holographic technique,light is reﬂected,or emitted from a3D object,propagates througha spatial light modulator(SLM),and is recorded by a digital camera.The SLM is used as a beam-splitter of the single-channel incoherent interferometer,such that each spherical beam originated from each object point is split into two spherical beams with two diﬀerent curve radii.Incoherent sum of the entire interferences between all the couples of spherical beams creates the Fresnel hologram of the observed3D object.When this hologram is reconstructed in the computer,the3D properties of the object are revealed.OCIS codes:100.6640,210.4770,180.1790.doi:10.3788/COL20090712.0000.1.IntroductionHolography is an attractive imaging technique as it oﬀers the ability to view a complete three-dimensional (3D)volume from one image.However,holography is not widely applied to the regime of white-light imaging, because white-light is incoherent and creating holograms requires a coherent interferometer system.In this review, we describe our recently invented method of acquiring incoherent digital holograms.The term incoherent digi-tal hologram means that incoherent light beams reﬂected or emitted from real-existing objects interfere with each other.The resulting interferogram is recorded by a dig-ital camera and digitally processed to yield a hologram. This hologram is reconstructed in the computer so that 3D images appear on the computer screen.The oldest methods of recording incoherent holograms have made use of the property that every incoherent ob-ject is composed of many source points,each of which is self-spatial coherent and can create an interference pattern with light coming from the point’s mirrored image.Under this general principle,there are vari-ous types of holograms[1−8],including Fourier[2,6]and Fresnel holograms[3,4,8].The process of beam interfering demands high levels of light intensity,extreme stability of the optical setup,and a relatively narrow bandwidth light source.More recently,three groups of researchers have proposed computing holograms of3D incoherently illuminated objects from a set of images taken from diﬀer-ent points of view[9−12].This method,although it shows promising prospects,is relatively slow since it is based on capturing tens of scene images from diﬀerent view angles. Another method is called scanning holography[13−15],in which a pattern of Fresnel zone plates(FZPs)scans the object such that at each and every scanning position, the light intensity is integrated by a point detector.The overall process yields a Fresnel hologram obtained as a correlation between the object and FZP patterns.How-ever,the scanning process is relatively slow and is done by mechanical movements.A similar correlation is ac-tually also discussed in this review,however,unlike the case of scanning holography,our proposed system carries out a correlation without movement.2.General properties of Fresnel hologramsThis review concentrates on the technique of incoher-ent digital holography based on single-channel incoher-ent interferometers,which we have been involved in their development recently[16−19].The type of hologram dis-cussed here is the digital Fresnel hologram,which means that a hologram of a single point has the form of the well-known FZP.The axial location of the object point is encoded by the Fresnel number of the FZP,which is the technical term for the number of the FZP rings along the given radius.To understand the operation principle of any general Fresnel hologram,let us look on the diﬀerence between regular imaging and holographic systems.In classical imaging,image formation of objects at diﬀerent distances from the lens results in a sharp image at the image plane for objects at only one position from the lens,as shown in Fig.1(a).The other objects at diﬀerent distances from the lens are out of focus.A Fresnel holographic system,on the other hand,as depicted in Fig.1(b),1671-7694/2009/120xxx-08c 2009Chinese Optics Lettersprojects a set of rings known as the FZP onto the plane of the image for each and every point at every plane of the object being viewed.The depth of the points is en-coded by the density of the rings such that points which are closer to the system project less dense rings than distant points.Because of this encoding method,the 3D information in the volume being imaged is recorded into the recording medium.Thus once the patterns are decoded,each plane in the image space reconstructed from a Fresnel hologram is in focus at a diﬀerent axial distance.The encoding is accomplished by the presence of a holographic system in the image path.At this point it should be noted that this graphical description of pro-jecting FZPs by every object point actually expresses the mathematical two-dimensional (2D)correlation (or convolution)between the object function and the FZP.In other words,the methods of creating Fresnel holo-grams are diﬀerent from each other by the way they spatially correlate the FZP with the scene.Another is-sue to note is that the correlation should be done with a FZP that is somehow “sensitive”to the axial locations of the object points.Otherwise,these locations are not encoded into the hologram.The system described in this review satisﬁes the condition that the FZP is depen-dent on the axial distance of each and every objectpoint.parison between the Fresnel holography principle and conventional imaging.(a)Conventional imaging system;(b)fresnel holographysystem.Fig.2.Schematic of FINCH recorder [16].BS:beam splitter;L is a spherical lens with focal length f =25cm;∆λindicates a chromatic ﬁlter with a bandwidth of ∆λ=60nm.This means that indeed points,which are closer to the system,project FZP with less cycles per radial length than distant points,and by this condition the holograms can actually image the 3D scene properly.The FZP is a sum of at least three main functions,i.e.,a constant bias,a quadratic phase function and its complex conjugate.The object function is actually corre-lated with all these three functions.However,the useful information,with which the holographic imaging is real-ized,is the correlation with just one of the two quadratic phase functions.The correlation with the other quadratic phase function induces the well-known twin image.This means that the detected signal in the holographic system contains three superposed correlation functions,whereas only one of them is the required correlation between the object and the quadratic phase function.Therefore,the digital processing of the detected image should contain the ability to eliminate the two unnecessary terms.To summarize,the deﬁnition of Fresnel hologram is any hologram that contains at least a correlation (or convolu-tion)between an object function and a quadratic phase function.Moreover,the quadratic phase function must be parameterized according to the axial distance of the object points from the detection plane.In other words,the number of cycles per radial distance of each quadratic phase function in the correlation is dependent on the z distance of each object point.In the case that the object is illuminated by a coherent wave,this correlation is the complex amplitude of the electromagnetic ﬁeld directly obtained,under the paraxial approximation [20],by a free propagation from the object to the detection plane.How-ever,we deal here with incoherent illumination,for which an alternative method to the free propagation should be applied.In fact,in this review we describe such method to get the desired correlation with the quadratic phase function,and this method indeed operates under inco-herent illumination.The discussed incoherent digital hologram is dubbed Fresnel incoherent correlation hologram (FINCH)[16−18].The FINCH is actually based on a single-channel on-axis incoherent interferometer.Like any Fresnel holography,in the FINCH the object is correlated with a FZP,but the correlation is carried out without any movement and without multiplexing the image of the scene.Section 3reviews the latest developments of the FINCH in the ﬁeld of color holography,microscopy,and imaging with a synthetic aperture.3.Fresnel incoherent correlation holographyIn this section we describe the FINCH –a method of recording digital Fresnel holograms under incoher-ent illumination.Various aspects of the FINCH have been described in Refs.[16-19],including FINCH of re-ﬂected white light [16],FINCH of ﬂuorescence objects [17],a FINCH-based holographic ﬂuorescence microscope [18],and a hologram recorder in a mode of a synthetic aperture [19].We brieﬂy review these works in the current section.Generally,in the FINCH system the reﬂected incoher-ent light from a 3D object propagates through a spatial light modulator (SLM)and is recorded by a digital cam-era.One of the FINCH systems [16]is shown in Fig.2.White-light source illuminates a 3D scene,and the reﬂected light from the objects is captured by a charge-coupled device (CCD)camera after passing through a lens L and the SLM.In general,we regard the system as an incoherent interferometer,where the grating displayed on the SLM is considered as a beam splitter.As is com-mon in such cases,we analyze the system by following its response to an input object of a single inﬁnitesimal point.Knowing the system’s point spread function (PSF)en-ables one to realize the system operation for any general object.Analysis of a beam originated from a narrow-band inﬁnitesimal point source is done by using Fresnel diﬀraction theory [20],since such a source is coherent by deﬁnition.A Fresnel hologram of a point object is obtained when the two interfering beams are two spherical beams with diﬀerent curvatures.Such a goal is achieved if the SLM’s reﬂection function is a sum of,for instance,constant and quadratic phase functions.When a plane wave hits the SLM,the constant term represents the reﬂected plane wave,and the quadratic phase term is responsible for the reﬂected spherical wave.A point source located at some distance from a spher-ical positive lens induces on the lens plane a diverging spherical wave.This wave is split by the SLM into two diﬀerent spherical waves which propagate toward the CCD at some distance from the SLM.Consequently,in the CCD plane,the intensity of the recorded hologram is a sum of three terms:two complex-conjugated quadratic phase functions and a constant term.This result is the PSF of the holographic recording system.For a general 3D object illuminated by a narrowband incoherent illumination,the intensity of the recorded hologram is an integral of the entire PSFs,over all object intensity points.Besides a constant term,thehologramFig.3.(a)Phase distribution of the reﬂection masks dis-played on the SLM,with θ=0◦,(b)θ=120◦,(c)θ=240◦.(d)Enlarged portion of (a)indicating that half (randomly chosen)of the SLM’s pixels modulate light with a constant phase.(e)Magnitude and (f)phase of the ﬁnal on-axis digi-tal hologram.(g)Reconstruction of the hologram of the three characters at the best focus distance of ‘O’.(h)Same recon-struction at the best focus distance of ‘S’,and (i)of ‘A’[16].expression contains two terms of correlation between an object and a quadratic phase,z -dependent,function.In order to remain with a single correlation term out of the three terms,we follow the usual procedure of on-axis digital holography [14,16−19].Three holograms of the same object are recorded with diﬀerent phase con-stants.The ﬁnal hologram is a superposition of the three holograms containing only the desired correlation between the object function and a single z -dependent quadratic phase.A 3D image of the object can be re-constructed from the hologram by calculating theFresnelFig.4.Schematics of the FINCH color recorder [17].L 1,L 2,L 3are spherical lenses and F 1,F 2are chromaticﬁlters.Fig.5.(a)Magnitude and (b)phase of the complex Fres-nel hologram of the dice.Digital reconstruction of the non-ﬂuorescence hologram:(c)at the face of the red dots on the die,and (d)at the face of the green dots on the die.(e)Magnitude and (f)phase of the complex Fresnel hologram of the red dots.Digital reconstruction of the red ﬂuorescence hologram:(g)at the face of the red dots on the die,and (h)at the face of the green dots on the die.(i)Magnitude and (j)phase of the complex Fresnel hologram of the green dots.Digital reconstruction of the green ﬂuorescence hologram:(k)at the face of the red dots on the die,and (l)at the face of the green dots on the position of (c),(g),(k)and that of (d),(h),(l)are depicted in (m)and (n),respectively [17].Fig.6.FINCHSCOPE schematic in uprightﬂuorescence microscope[18].propagation formula.The system shown in Fig.2has been used to record the three holograms[16].The SLM has been phase-only, and as so,the desired sum of two phase functions(which is no longer a pure phase)cannot be directly displayed on this SLM.To overcome this obstacle,the quadratic phase function has been displayed randomly on only half of the SLM pixels,and the constant phase has been displayed on the other half.The randomness in distributing the two phase functions has been required because organized non-random structure produces unnecessary diﬀraction orders,therefore,results in lower interference eﬃciency. The pixels are divided equally,half to each diﬀractive element,to create two wavefronts with equal energy.By this method,the SLM function becomes a good approx-imation to the sum of two phase functions.The phase distributions of the three reﬂection masks displayed on the SLM,with phase constants of0◦,120◦and240◦,are shown in Figs.3(a),(b)and(c),respectively.Three white-on-black characters i th the same size of 2×2(mm)were located at the vicinity of rear focal point of the lens.‘O’was at z=–24mm,‘S’was at z=–48 mm,and‘A’was at z=–72mm.These characters were illuminated by a mercury arc lamp.The three holo-grams,each for a diﬀerent phase constant of the SLM, were recorded by a CCD camera and processed by a computer.Theﬁnal hologram was calculated accord-ing to the superposition formula[14]and its magnitude and phase distributions are depicted in Figs.3(e)and (f),respectively.The hologram was reconstructed in the computer by calculating the Fresnel propagation toward various z propagation distances.Three diﬀerent recon-struction planes are shown in Figs.3(g),(h),and(i).In each plane,a diﬀerent character is in focus as is indeed expected from a holographic reconstruction of an object with a volume.In Ref.[17],the FINCH has been capable to record multicolor digital holograms from objects emittingﬂuo-rescent light.Theﬂuorescent light,speciﬁc to the emis-sion wavelength of variousﬂuorescent dyes after excita-tion of3D objects,was recorded on a digital monochrome camera after reﬂection from the SLM.For each wave-length ofﬂuorescent emission,the camera sequentially records three holograms reﬂected from the SLM,each with a diﬀerent phase factor of the SLM’s function.The three holograms are again superposed in the computer to create a complex-valued Fresnel hologram of eachﬂu-orescent emission without the twin image problem.The holograms for eachﬂuorescent color are further combined in a computer to produce a multicoloredﬂuorescence hologram and3D color image.An experiment showing the recording of a colorﬂuo-rescence hologram was carried out[17]on the system in Fig. 4.The phase constants of0◦,120◦,and240◦were introduced into the three quadratic phase functions.The magnitude and phase of theﬁnal complex hologram,su-perposed from theﬁrst three holograms,are shown in Figs.5(a)and(b),respectively.The reconstruction from theﬁnal hologram was calculated by using the Fresnel propagation formula[20].The results are shown at the plane of the front face of the front die(Fig.5(c))and the plane of the front face of the rear die(Fig.5(d)).Note that in each plane a diﬀerent die face is in focus as is indeed expected from a holographic reconstruction of an object with a volume.The second three holograms were recorded via a redﬁlter in the emissionﬁlter slider F2 which passed614–640nmﬂuorescent light wavelengths with a peak wavelength of626nm and a full-width at half-maximum,of11nm(FWHM).The magnitude and phase of theﬁnal complex hologram,superposed from the‘red’set,are shown in Figs.5(e)and(f),respectively. The reconstruction results from thisﬁnal hologram are shown in Figs.5(g)and(h)at the same planes as those in Figs.5(c)and(d),respectively.Finally,an additional set of three holograms was recorded with a greenﬁlter in emissionﬁlter slider F2,which passed500–532nmﬂuo-rescent light wavelengths with a peak wavelength of516 nm and a FWHM of16nm.The magnitude and phase of theﬁnal complex hologram,superposed from the‘green’set,are shown in Figs.5(i)and(j),respectively.The reconstruction results from thisﬁnal hologram are shown in Figs.5(k)and(l)at the same planes as those in Fig. 5(c)and(d),positions of Figs.5(c), (g),and(k)and Figs.5(d),(h),and(l)are depicted in Figs.5(m)and(n),respectively.Note that all the colors in Fig.5(colorful online)are pseudo-colors.These last results yield a complete color3D holographic image of the object including the red and greenﬂuorescence. While the optical arrangement in this demonstration has not been optimized for maximum resolution,it is im-portant to recognize that even with this simple optical arrangement,the resolution is good enough to image the ﬂuorescent emissions with goodﬁdelity and to obtain good reﬂected light images of the dice.Furthermore, in the reﬂected light images in Figs.5(c)and(m),the system has been able to detect a specular reﬂection of the illumination from the edge of the front dice. Another system to be reviewed here is theﬁrst demon-stration of a motionless microscopy system(FINCH-SCOPE)based upon the FINCH and its use in record-ing high-resolution3Dﬂuorescent images of biological specimens[18].By using high numerical aperture(NA) lenses,a SLM,a CCD camera,and some simpleﬁlters, FINCHSCOPE enables the acquisition of3D microscopic images without the need for scanning.A schematic diagram of the FINCHSCOPE for an upright microscope equipped with an arc lamp sourceFig.7.FINCHSCOPE holography of polychromatic beads.(a)Magnitude of the complex hologram 6-µm beads.Images reconstructed from the hologram at z distances of (b)34µm,(c)36µm,and (d)84µm.Line intensity proﬁles between the beads are shown at the bottom of panels (b)–(d).(e)Line intensity proﬁles along the z axis for the lower bead from reconstructed sections of a single hologram (line 1)and from a wideﬁeld stack of the same bead (28sections,line 2).Beads (6µm)excited at 640,555,and 488nm with holograms reconstructed (f)–(h)at plane (b)and (j)–(l)at plane (d).(i)and (m)are the combined RGB images for planes (b)and (d),respectively.(n)–(r)Beads (0.5µm)imaged with a 1.4-NA oil immersion objective:(n)holographic camera image;(o)magnitude of the complex hologram;(p)–(r)reconstructed image at planes 6,15,and 20µm.Scale bars indicate image size [18].Fig.8.FINCHSCOPE ﬂuorescence sections of pollen grains and Convallaria rhizom .The arrows point to the structures in the images that are in focus at various image planes.(b)–(e)Sections reconstructed from a hologram of mixed pollen grains.(g)–(j)Sections reconstructed from a hologram of Convallaria rhizom .(a),(f)Magnitudes of the complex holograms from which the respective image planes are reconstructed.Scale bars indicate image size [18].is shown in Fig. 6.The beam of light that emerges from an inﬁnity-corrected microscope objective trans-forms each point of the object being viewed into a plane wave,thus satisfying the ﬁrst requirement of FINCH [16].A SLM and a digital camera replace the tube lens,reﬂec-tive mirror,and other transfer optics normally present in microscopes.Because no tube lens is required,inﬁnity-corrected objectives from any manufacturer can be used.A ﬁlter wheel was used to select excitation wavelengths from a mercury arc lamp,and the dichroic mirror holder and the emission ﬁlter in the microscope were used to direct light to and from the specimen through an inﬁnity-corrected objective.The ability of the FINCHSCOPE to resolve multicolor ﬂuorescent samples was evaluated by ﬁrst imaging poly-chromatic ﬂuorescent beads.A ﬂuorescence bead slidewith the beads separated on two separate planes was con-structed.FocalCheck polychromatic beads(6µm)were used to coat one side of a glass microscope slide and a glass coverslip.These two surfaces were juxtaposed and held together at a distance from one another of∼50µm with optical cement.The beads were sequentially excited at488-,555-,and640-nm center wavelengths(10–30nm bandwidths)with emissions recorded at515–535,585–615,and660–720nm,respectively.Figures7(a)–(d) show reconstructed image planes from6µm beads ex-cited at640nm and imaged on the FINCHSCOPE with a Zeiss PlanApo20×,0.75NA objective.Figure7(a) shows the magnitude of the complex hologram,which contains all the information about the location and in-tensity of each bead at every plane in theﬁeld.The Fresnel reconstruction from this hologram was selected to yield49planes of the image,2-µm apart.Two beads are shown in Fig.7(b)with only the lower bead exactly in focus.Figure7(c)is2µm into theﬁeld in the z-direction,and the upper bead is now in focus,with the lower bead slightly out of focus.The focal diﬀerence is conﬁrmed by the line proﬁle drawn between the beads, showing an inversion of intensity for these two beads be-tween the planes.There is another bead between these two beads,but it does not appear in Figs.7(b)or(c) (or in the intensity proﬁle),because it is48µm from the upper bead;it instead appears in Fig.7(d)(and in the line proﬁle),which is24sections away from the section in Fig.7(c).Notice that the beads in Figs.7(b)and(c)are no longer visible in Fig.7(d).In the complex hologram in Fig.7(a),the small circles encode the close beads and the larger circles encode the distant central bead. Figure7(e)shows that the z-resolution of the lower bead in Fig.7(b),reconstructed from sections created from a single hologram(curve1),is at least comparable to data from a wideﬁeld stack of28sections(obtained by moving the microscope objective in the z-direction)of the same ﬁeld(curve2).The co-localization of theﬂuorescence emission was conﬁrmed at all excitation wavelengths and at extreme z limits,as shown in Figs.7(f)–(m)for the 6-µm beads at the planes shown in Figs.7(b)((f)–(i)) and(d)((j)–(m)).In Figs.7(n)–(r),0.5-µm beads imaged with a Zeiss PlanApo×631.4NA oil-immersion objective are shown.Figure7(n)presents one of the holo-grams captured by the camera and Fig.7(o)shows the magnitude of the complex hologram.Figures7(p)–(r) show diﬀerent planes(6,15,and20µm,respectively)in the bead specimen after reconstruction from the complex hologram of image slices in0.5-µm steps.Arrows show the diﬀerent beads visualized in diﬀerent z image planes. The computer reconstruction along the z-axis of a group ofﬂuorescently labeled pollen grains is shown in Figs. 8(b)–(e).As is expected from a holographic reconstruc-tion of a3D object with volume,any number of planes can be reconstructed.In this example,a diﬀerent pollen grain was in focus in each transverse plane reconstructed from the complex hologram whose magnitude is shown in Fig.8(a).In Figs.8(b)–(e),the values of z are8,13, 20,and24µm,respectively.A similar experiment was performed with the autoﬂuorescent Convallaria rhizom and the results are shown in Figs.8(g)–(j)at planes6, 8,11,and12µm.The most recent development in FINCH is a new lens-less incoherent holographic system operating in a syn-thetic aperture mode[19].Synthetic aperture is a well-known super-resolution technique which extends the res-olution capabilities of an imaging system beyond thetheoretical Rayleigh limit dictated by the system’s ac-tual ing this technique,several patternsacquired by an aperture-limited system,from variouslocations,are tiled together to one large pattern whichcould be captured only by a virtual system equippedwith a much wider synthetic aperture.The use of optical holography for synthetic apertureis usually restricted to coherent imaging[21−23].There-fore,the use of this technique is limited only to thoseapplications in which the observed targets can be illu-minated by a laser.Synthetic aperture carried out by acombination of several oﬀ-axis incoherent holograms inscanning holographic microscopy has been demonstratedby Indebetouw et al[24].However,this method is limitedto microscopy only,and although it is a technique ofrecording incoherent holograms,a specimen should alsobe illuminated by an interference pattern between twolaser beams.Our new scheme of holographic imaging of incoher-ently illuminated objects is dubbing a synthetic aperturewith Fresnel elements(SAFE).This holographic lens-less system contains only a SLM and a digital camera.SAFE has an extended synthetic aperture in order toimprove the transverse and axial resolutions beyond theclassic limitations.The term synthetic aperture,in thepresent context,means time(or space)multiplexing ofseveral Fresnel holographic elements captured from vari-ous viewpoints by a system with a limited real aperture.The synthetic aperture is implemented by shifting theSLM-camera set,located across theﬁeld of view,be-tween several viewpoints.At each viewpoint,a diﬀerentmask is displayed on the SLM,and a single element ofthe Fresnel hologram is recorded(Fig.9).The variouselements,each of which is recorded by the real aperturesystem during the capturing time,are tiled together sothat theﬁnal mosaic hologram is eﬀectively consideredas being captured from a single synthetic aperture,whichis much wider than the actual aperture.An example of such a system with the synthetic aper-ture three times wider than the actual aperture can beseen in Fig.9.For simplicity of the demonstration,the synthetic aperture was implemented only along thehorizontal axis.In principle,this concept can be gen-eralized for both axes and for any ratio of synthetic toactual apertures.Imaging with the synthetic apertureis necessary for the cases where the angular spectrumof the light emitted from the observed object is widerthan the NA of a given imaging system.In the SAFEshown in Fig.9,the SLM and the digital camera movein front of the object.The complete Fresnel hologramof the object,located at some distance from the SLM,isa mosaic of three holographic elements,each of which isrecorded from a diﬀerent position by the system with thereal aperture of the size A x×A y.The complete hologram tiled from the three holographic Fresnel elements has thesynthetic aperture of the size3(·A x×A y)which is three times larger than the real aperture at the horizontal axis.The method to eliminate the twin image and the biasterm is the same as that has been used before[14,16−18];。

a flexible three-dimensional force sensor skin

The design and fabrication ofaﬂexible three-dimensional force sensor skin∗Jian Hua Shan∗†,Tao Mei∗‡,Lei Sun∗,De Yi Kong∗,Zheng Yong Zhang∗,Lin Ni∗,Max Meng∗,Jia Ru Chu†∗State Key Laboratories of Transducer TechnologyInstitute of Intelligent Machines,Chinese Academy of SciencesHefei,Anhui230031,China†Department of Precision Machinery and Precision InstrumentationUniversity of Science and Technology of ChinaHefei,Anhui230027,China‡Corresponding author:Tel:(86)551–5591100,Fax:(86)551–5592420;Email:tmei@Abstract—To obtain shear and normal stress information on non-planar surfaces has long been a signiﬁcant challenge.This paper described a new version ofﬂexible tactile sensor skin for three-dimensional force measurement.The sensor array has been fabricated by MEMS technology.The sensor skin which can be bended90◦includes4×4force sensor cells.Each cell is hybrid integrated toﬂexible printing circuit board which consists of a E-shape diaphragm(50µm thick and4000µm wide ).Each cell exhibits a sensitivity of228mv/N to normal force and34mv/N to shear force in the designed force range of2N. Design analysis,fabrication processes,and experimental results are presented in this paper.Index Terms—MEMS sensor,three-dimensional force,ﬂexible skinI.I NTRODUCTIONMeasurement and processing of contact three-dimensional stress information is critical for full grasp force,torque determination,and determination of object slip,as well as estimation of static coefﬁcient of friction in robotic sys-tem.However,only several three-dimensional tactile sensors have been developed by B.J.Kane[1],[2],Z.Chu[3], W.L.Jin[4]and L.Wang[5].They are all silicon-based sensors and rigid,which limited them to be used only on ﬂat surfaces.For most of objects,the surfaces are not co-planar,sometimes they are even highly cured.It is needed to developﬂexible sensor array.Some researchers focus on the development of aﬂexible MEMS technology to produce ﬂexible skins with integrated MEMS device that can be easily taped or glued on non-planar surfaces.Fukang Jiang[6],[7] developed aﬂexible polyimide-packaged shear-stress sensor skin that was used for measuring the wall shear stress exerted by viscousﬂow inﬂow measurementﬁeld for delta planform unmanned aerial vehicles.It is composed of many silicon islands that are interconnected by two layers of polyimide ﬁlm.It can not measure the contact force and is rather fragile when contacted.Jonathan Engel[8]designed a polyimide ﬂexible tactile sensor skin,with micromachined thin-ﬁlm metal strain gauges positioned on the edges of polyimide ∗The research is supported by National Nature Science Foundation of China(No.60275027)and National Hightech Program(863)of China(No. 2003AA404080)membranes,which can only sense the normal force withlimited precision.Tao Mei[9]developed an integrated three-dimensional4×8tactile sensor with large force range usingMEMS and standard COMS technology.Reasonable detectionaccuracy of three force direction and tactile image correspon-dence are achieved.But it is also rigid and can not apply tonon-planar surface.Our objective is to develop aﬂexible force sensor skinfor measuring both normal and shear of contact forces basedon Tao Mei’s work.In order to realizeﬂexibility,individualsensor cells are hybrid integrated into aﬂexible PCB withﬂip-chip method.II.S ENSOR STRUCTUREThe sensor skin is constructed with rubber protectionsurface,ﬂexible PCB,force concentrating columns,sensorarray,gap-deﬁneﬁlm,protection steel plates andﬂexible baseas illustrated in Fig.1.A2-mm-thick rubber is used forabsorbing shock and protecting the inner devices.It is made ofhighly elastic rubber used in fast-attacking type table tennispaddles.The rubber layer is glued on the top of the forceconcentrating columns byﬁlling703-type silicone rubberglue on the surface ofﬂexible PCB.The703-type siliconerubber is a gel at room temperature,and by heating it to 50◦C for4h,it becomes a soft rubber.This rubber glue has good electrical insulation property.The force concentratingcolumns are made by precision machining and positioned inthe center of the tactile sensor cells.Each sensor cell isﬂip-chip solder bonded to theﬂexible PCB.The columns arethicker thanﬂexible PCB,which contact with the rubberprotection surface through holes made in theﬂexible PCB.Since the columns are surrounded by silicone rubber,a softsilicone rubber layer is formed around the columns.Thissilicone rubber glue layer provides large glue surface for therubber surface and prevents moving friction on the sensorarray surface.Therefore,the sensor array may achieve morerobustness while concentrating most of the distributed tactileforces onto the center of the sensor cells in the tactile sensorarray.Fig.1.Illustration of the cross-sectional view for the individual cells of the tactile sensor skinThere are4×4sensor cells that can detect three-dimensional force in the sensor array.Each sensor cell has an E-shaped square membrane structure show in Fig.2 fabricated by Inductively Coupled Plasma(ICP).ICP can etch vertical sidewalls with high aspect ratio and high etch speed.Fig.3shows SEM graph of silicon etch results by Oxford Instruments Plasmalab System100ICP180.So ICP can make each cell more smaller than wet etch method and effectively improve the density of the array.Each cell does not contact with other cells.The gap between two cells makes the sensor skin bend freely.When a force is applied on the top surface of the sensor,a cross-talk signal is generated. So the piezoresistors should be placed at the points where the cross-talk is minimum.By strain and stressﬁnite-element analysis(FEA)of the E-shaped membrane,suitable positions for the piezoresistors are determined to obtain optimal circuit output for F x,F y and F z.The FEA mesh model for a single cell and its displacement contour due to2N F z are shown in Fig.4.The Young’s modulus and Poisson’s ratio of silicon at 100 direction are assumed to be127GPa and0.278, respectively.The displacement at the center of a50-µm-thick membrane is about20µm due to2N normal force.The strain contour which indicates an asymmetric strain distribution on the membrane due to F z and F x is shown in Fig.5.This asymmetric distribution is the key to using the membrane as the three-dimensional force sensor.In addition,based on the FEA analysis,the cross-talk is10%when vertical force or shear force are applied.Strain induced by the applied force on the membrane is sensed by three groups of integrated piezoresistors in the silicon membrane.Locations of the piezoresistors on the membrane are shown in Fig. 2.The electric bridges of piezoresistors in each sensor cell are shown in Fig.6.When F z is applied to the sensor cell,the stress and strain on the membrane where R x1,R x2,R y1and R y2arelocated Fig. 2.Mechanical structure and piezoresistors locations of a three-dimensional force sensingcellFig.3.SEM photo of Silicon etch result by ICP(a)meshmodel(b)displacement contourFig. 4.(a)Mesh of the sensor cell for ﬁnite-element analysis.(b)Displacement contour of the sensor cell with 2N force applied in the Z direction on the force concentration column.The diaphragm thickness is 50µm,the force concentration column is 1mm ×1mm.The sensor cell membrane is 4mm ×4mm.are equal theoretically,thus these piezoresistors have equal resistance change.Consequently,the output V x and V y of circuit C x and circuit C y have no response to F z .However,under the load of F z ,R z 1and R z 2are under compressive stress while R z 3and R z 4are under tensile stress.Thus,R z 1and R z 2decrease while R z 3and R z 4increase under F z ,and the output V z varies accordingly.When F x is applied on the sensor cells,R x 1is compressed while R x 2is elongated,and the output V x varies correspondingly.In addition,R y 1and R y 2have the same resistance change in this case,therefore,C y has no response to F x .The contributions from R z 1and R z 2are balanced in this case,since R z 1and R z 2are located on opposite but equal absolute value stress zones.For the same reason,the resistance change of R z 3and R z 4balanced also.Therefore,V z has no response to F x .Similarly,when F y is applied,V y varies,but V x and V z have no response.The gap-deﬁne ﬁlm is a 25µm thick DuPont kapton HN ﬁlm with holes made in the center under the E-shape di-aphragm.Under this ﬁlm,a steel plate is aligned to the center of the E-shape diaphragm and adheres to the ﬁlm.So the thickness of the gap-deﬁne ﬁlm deﬁnes the gap to allowfor(a)strain stress under 2N Fz(b)strain stress under 1N F xFig.5.(a)Strain contour of X direction of the sensor cell diaphragm under 2N force in the Z direction.(b)Strain contour of X direction of the sensor cell diaphragm under 1N force in the Xdirection.Fig.6.Electric bridge of piezoresistors in sensor cell.R is the dunny resistor located outside of the membrane.Fig.7.Major steps of sensor arrayfabricationFig.8.Photo of theﬂexible PCBthe membrane deformation.The steel plates can protect the membranes from excessive deformation.Theﬂexible base is a125µm thick DuPont kapton HNﬁlm.III.S ENSOR SKIN FABRICATION AND ASSEMBLE The sensor array is fabricated using microfabrication tech-nology.The major fabrication steps(Fig.7)are given followed by typical geometric and process parameters.The fabrication process begins with a double-side polished 3in.n-type(100)silicon wafer.There are twelve serpentine resistors(30µm wide,1000Ω)which are made via Boron ion implantation for30min at950◦C.A1µm thick thermally grown oxide layer is used as the implantation mask.A1500˚Athick aluminumﬁlm is deposited via thermal evaporation. The contact pads and leads are formed via a lift-off process.A backside photolithographic procedure is followed to open etching windows for the E-shaped ing ICP etch to obtain50µm thick membranes with a1µm aluminum layer as mask.Finally,the sensor array is cut to individual cells.To connect the sensor signals output,a specialﬂexible PCB,as shown in Fig.8,is designed.At the right end,there are4×4cells of solder-bonding pads,pads of each cell match the pads of each cell of sensor array.At the other end of theﬂexible PCB,there are two columns of large soldering pads which are compatible with the standard connector.TheFig.9.The skin assembling processsolder bonding between sensor cells andﬂexible PCB usesﬂip-chip method.The method starts with conductive epoxyapplied on the pads of theﬂexible PCB using screen print.Then,the sensor cell is aligned to theﬂexible PCB under astereo microscope.Theﬂexible PCB and sensor array are laidon the top of a hotplate and a uniform pressure is applied onthe area of pads.Finally,the hotplate is heated up to150◦C. After cooling down,the sensor array is bonded toﬂexiblePCB.The assembling process of the sensor skin is describedbelow.The procedure is illustrated in Fig.9.1)The sensor array is bonded toﬂexible PCB by theﬂip-chip method described as above.2)Silicone rubber isﬁlled in the gaps between thesensor cells,therefore a soft silicone rubber layer isformed around the sensor cells.This rubber layerprevents moving friction on the sensor array surface.Epoxy glue is coated on the gap-deﬁneﬁlm bythick-ﬁlm printing technique,then theﬁlm is alignedand pressed to the base of sensor array.A50g weightis applied on the top of theﬁlm under60◦C for4h to getﬁrm connection.3)Epoxy glue is printed on the surface ofﬂexible baseFig.10.The back side of the bending sensor without the protection base.The vertical sidewall of E-shape diaphragm is shown clearly.The distance of each cell is1cm.Fig.11.Skin scanning circuit.and pressed to the gap-deﬁne ﬁlm.Then,a 50g weight is applied on the top of the ﬁlm under 60◦C for 4h to get ﬁrm connection.4)Epoxy glue is printed in the hole of the ﬂexible PCB,the force concentrating column is aligned and pressed to the hole of the ﬂexible PCB.Then,a 50g weight is applied on the top of the column array under 60◦C for 4h to get ﬁrm connection.5)Finally,the rubber protection layer is adhered to the top surface by silicone glue.The sensor array is shown in Fig.10.Multiplexing electronics are used to scan the array of sensors(Fig.11).A control circuit sequentially connects each driving column to a drive voltage V DRIV E and ground while the rest of the lines are equipotential to V out through feedback lines.The signals of each sensor cell on the driving column is multiplexed to an operational ampliﬁer circuit,then to an 16-bit,high speed,analog to digital converter,whose output is feed into a computer for further data analysis.IV.E XPERIMENTAL RESULTSThe sensor cells are calibrated to determine their sensi-tivity before silicone rubber covers them.A standard two-dimensional force sensor is used as the shear force reference.The tactile sensor is ﬁxed on an X-Y table.The forces on the X and Y directions could be applied by moving the tactile sensor in the X and Y direction when the head of the standard force sensor is pressed on the surface.The Z-direction force is applied through a vertical force table on which standard mass block is applied.Representative results are presented in Fig.12.The Figure demonstrates that the sensor response to applied shear and normal stresses is linear with a correlation coefﬁcient,R,of 0.999in all three directions.The sensor with 50µm thick diaphragm exhibits a sensitivity of 228mv/N to applied normal force and 34mv/N to applied shear force.Testing is performed to verify the ﬂexibility of sensor array.After more than 100times of bending 90◦,the ﬂexible skin is still in good condition.This shows that our technology for making ﬂexible sensor skin is practicable.To determine the relative size of the cross-talk that occurs between the three measurement directions,a matric C can be constructed which maps the applied stresses to the voltage measures of those stresses.The relationship is shown in Eq.(1).Inspection of C indications that the off-diagonal terms are about 10%of the diagonal terms.u z u x u y =C f z f x f y (1)whereC =2.10.20.20.020.40.030.040.030.35(2)V.C ONCLUSIONA ﬂexible three-dimensional force sensor skin is success-fully developed with MEMS technology.There are 4×4three dimensional force sensors with protection plates,a ﬂexible PCB,force concentrating column array,and a rubber cover in the sensor skin.The silicon force sensor is etched by ICP technique to reduce the sensor size and obtain good technique compatibility with the on-chip IC.The sensitivity of the force sensor is 228mv/N to normal force and 34mv/N to shear force with 2N force range,which is suitable for robotic systems.By employing the ﬂexible PCB and ﬂip-chip solder bonding techniques,the sensor skin gains very good ﬂexibility(a)Force is applied in the z-directiononly(b)Force is applied in the x-directiononly(c)Force is applied in the y-direction onlyFig.12.Sensor outputs(before amplify)vary with applied forces linearly.(a)Z-direction (b)X-direction(c)Y-directionand can be bended to 90◦in X or Y direction.The design and fabrication of the ﬂexible three dimensional force sensor skin have been proven practicable.Future research will be performed to decrease the cell size to gain higher space resolution and increase the ﬂexibility in all directions.R EFERENCES[1] B.J.Kane,Mark R.Cutkosky,Gregory T. A.Kovacs,”CMOS-compatible traction stress sensor for use in high-resolution tactile imag-ing,”Sensors and Actuators,A54(1996),pp.511–516[2] B.J.Kane,Mark R.Cutkosky,Gregory T.A.Kovacs ”A Traction StressSensor Array for Use in High-Resolution Robotic Tactile Imaging,”Journal of Microelectromechanical Systems,vol.9,no.4,December 2000pp.424–434[3]Z.Chu,P.M.Sarro,S.Middelhoek,”Silicon three-axial tactile sensor,”Sensors and Actuators,A54(1996),pp.505–510[4]W.L.Jin,C.D.Mote,”Development and calibration of a sub-millimeterthree-component force sensor,”Sensors and Actuators A65(1998),pp.88–94[5]Lin Wang,David J.Beebe,”A silicon-based shear force sensor:devel-opment and characterization,”Sensors and Actuators,A84(2000),pp.33–44[6]Fukang Jiang,Gwo-Bin Lee,Yu-Chong Tai,Chih-Ming Ho,”A ﬂexiblemicromachine-based shear-stress sensor array and its application to separation-point detection,”Sensors and Actuators,A79(2000),pp.194–203[7]Yong Xu,Fukang Jiang,et al.”Flexible shear-stress sensor skin andits application to unmanned aerial vehicles,”Sensors and Actuators,A105(2003),pp.321–329[8]Jonathan Engel,Jack Chen,Chang Liu,”Development of polyimideﬂexible tactile sensor skin,”Journal of Micromechanics and microengi-neering,13(2003),pp.359–366[9]Tao Mei,Wen J.Li,et al.”An integrated MEMS three-dimensionaltactile sensor with large force range,”Sensors and Actuators A80(2000)pp.155–162。

专业外语数字图像基础Digital Image Fundamentals

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6. Dilation and Erosion
Dilation adds a layer of pixels to all objects in an image. Erosion removes one layer of pixels from all objects. When dilation is followed by erosion, gaps in broken boundaries identified through edge detection can be filled in. Conversely, when erosion is followed by dilation, spots of noise in an image are removed.
Successfully detecting the edges in an image is the first step towards confident identification of object boundaries and then objects recognition. From boundary information, shape characteristics like perimeter and area can be calculated, which can be used to classify an object.
由于屏幕上的点、线和面都是由像素组成的，显示器可显示的像素越多，画面就越精细，同样的屏幕区域内能显示的信息也越多，所以分辨率是个非常重要的性能指标之一。

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The Optimal Isodual Lattice Quantizer in Three Dimensions

V
x · x dx
(1)
lattice has the largest packing radius and the smallest covering radius. The m.c.c. lattice, denoted here by M3 , has Gram matrix √ −1 −1 1+ 2 √ √ 1 −1 1+ 2 1− 2 2 √ √ −1 1− 2 1+ 2 determinant 1, packing radius
(3)
where α, β , h are any real numbers satisfying 0 ≤ 2h ≤ α ≤ β , αβ − h2 = 1. In the indecomposable case the nonzero conorms are: p01 = αβ 2α(1 − β ) α(2 − β ) , p02 = , p03 = , γ γ βγ 2β (1 − α) 2(1 − α)(1 − β ) β (2 − α) , p13 = , p23 = , αγ γ γ (4)
The Optimal Isodual Lattice Quantizer in Three Dimensions
J. H. Conway Mathematics Department Princeton University Princeton, NJ 08544 N. J. A. Sloane AT&T Shannon Labs 180 Park Avenue Florham Park, NJ 07932-0971 Email: conway@, njas@ Jan 02 2006. Abstract The mean-centered cuboidal (or m.c.c.) lattice is known to be the optimal packing and covering among all isodual three-dimensional lattices. In this note we show that it is also the best quantizer. It thus joins the isodual lattices Z, A2 and (presumably) D4 , E8 and the Leech lattice in being simultaneously optimal with respect to all three criteria. Keywords: quantizing, self-dual lattice, isodual lattice, f.c.c. lattice, b.c.c. lattice, m.c.c. lattice AMS 2000 Classiﬁcation: 52C07 (11H55, 94A29)

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a r X i v :p h y s i c s /9906003v 1 [p h y s i c s .f l u -d y n ] 1 J u n 1999Scaling properties of three-dimensional magnetohydrodynamic turbulenceWolf–Christian M¨u ller and Dieter BiskampMax-Planck-Institut f¨u r Plasmaphysik,85748Garching,GermanyThe scaling properties of three-dimensional magnetohydrodynamic turbulence are obtained from direct numerical simulations of decaying turbulence using 5123modes.The results indicate that the turbulence does not follow the Iroshnikov-Kraichnan phenomenology.The spectrum is consistent with k −5/3.In the case of hyperresistivity the structure functions exhibit a clear scaling range yielding absolute values of the scaling exponents ζp ,in particular ζ3≃1,consistent with a recent analytical result.The scaling exponents agree with a modiﬁed She-Leveque model ζp =p/9+1−(1/3)p/3,corresponding to Kolmogorov scaling but sheet-like geometry of the dissipative structures.PACS:47.27Gs;47.65+a;47.27EqMagnetic turbulence is the natural state of a plasma in motion,especially in astrophysical systems.The con-venient framework to describe such turbulence is magne-tohydrodynamics (MHD).For high magnetic Reynolds number Rm =vl 0/η,where v is a typical turbulent ve-locity,l 0the integral scale and ηthe magnetic diﬀusivity,there is a broad range of scales l between l 0,and the dis-sipative scale length l d ,l 0≫l ≫l d ,called the inertial range which exhibits characteristic self-similarity or scal-ing properties.The concept of inertial-range scaling was introduced by Kolmogorov [1]for hydrodynamic turbulence,which is called the Kolmogorov (K41)phenomenology.As-suming homogeneity and isotropy of the turbulence as well as locality of the turbulent cascade process,he ob-tains ǫ∼(δv l )2/τl =(δv l )3/l yielding the scaling law δv l ∼ǫ1/3l 1/3.Here ǫis the energy dissipation rate and,to be speciﬁc,δv l =[v (x +l )−v (x )]·l /l is the longi-tudinal velocity increment.A direct consequence is the Kolmogorov energy spectrum E k ∼ǫ2/3k −5/3.For MHD turbulence the Iroshnikov-Kraichnan (IK)phenomenology [2],[3]takes into account the Alfv´e n ef-fect,the coupling of small-scale velocity and magnetic ﬂuctuations by the integral-scale ﬁeld B 0.Hence the natural variables are the Els¨a sser ﬁelds z ±=v ±B ,which describe Alfv´e n waves.In the IK phenomenol-ogy the spectral transfer is reduced by the factor τA /τl ,ǫ∼(τA /τl )(δz l )2/τl ,where τA =l/v A ,v A =Alfv´e n ve-locity in the ﬁeld B 0,δz l ∼δv l ∼δB l ∼(ǫv A )1/4l 1/4,and the IK energy spectrum becomes E k ∼(ǫv A )1/2k −3/2.This spectrum can also be written in the form E k ∼ǫ2/3k −5/3(kl 0)1/6with the integral scale l 0deﬁned byl 0=v 3A /ǫ,which illustrates the nonlocal character of the energy cascade in MHD turbulence.It is,however,well known that these qualitative scaling relations for δv l or δz l are not exactly valid in a statisti-cal sense because of intermittency,which implies that the distribution of turbulent scales is not strictly self-similar.A quantitative measure is provided by the scaling expo-nents ζp of the structure functions,the moments of theﬁeld increments.For hydrodynamic turbulence She and Leveque [4]proposed a model leading to the expression ζSL p =p/9+2[1−(2/3)p/3],which ﬁts the experimental results surprisingly well,reproducing in particular the exact result ζ3=1.This model has been modiﬁed for MHD incorporating the IK eﬀect [5],[6],which yields ζIK p =p/8+1−(1/2)p/4,in particular ζIK 4=1.The IK phenomenology has been supported by direct numerical simulations of 2D MHD turbulence at moder-ate Reynolds numbers [7].However,recent developments in MHD turbulence theory cast some doubt on the gen-eral validity of the IK scaling.2D simulations at con-siderably higher Reynolds numbers reveal an anomalous scaling behavior [8],[9],indicating that the results of Ref.[7]are not asymptotic.There have also been theoretical arguments in favor of a Kolmogorov scaling,e.g.,[10],[11].Even more convincingly,exact relations have been derived for moments of certain triple products of δz l [12],which are shown to be proportional to l ,i.e.,ζ3=1,anal-ogous to the well-known 4/5-relation in hydrodynamic turbulence,thus excluding the IK result ζ4=1.Scaling exponents for MHD turbulence have also been derived from observations in the solar wind [13].Here agreement with the IK exponents has been claimed [5],but in this comparison the observational results were normalized as-suming ζ4=1.Actually the error bars seem to be too large to reach a deﬁnite conclusion.To clarify the issue of scaling in 3D MHD turbulence direct numerical simulations are desirable with higher Reynolds numbers than studied previously,for instance in [14],[15],[16],[17],[18].In this Letter we present a nu-merical study of freely decaying turbulence with spatial resolution of 5123modes.The scaling properties are ana-lyzed by considering the time-averages of the normalized spectra and structure functions.We solve the incom-pressible MHD equations∂t B −∇×(v ×B )=ην(−1)ν−1∇2νB ,(1)1∂t w−∇×(v×w)−∇×(j×B)=µν(−1)ν−1∇2νw,(2)w=∇×v,j=∇×B,∇·v=∇·B=0,by applying a pseudo-spectral method with spherical mode truncation as conveniently used in3D turbulencesimulations(instead of full dealiasing by the2/3rule used in most2D simulations).The generalized magneticPrandtl numberην/µνhas been set equal to unity.Initialconditions areB k=a e−k2/k20−iαk,v k=b e−k2/k20−iβk,(3) which are characterized by random phasesαk,βk andsatisfy the conditions k·B k=k·v k=0,E=E V+E M= 1and E V/E M=1.Further restrictions on B k ariseby requiring a speciﬁc value of the magnetic helicityH= d3x A·B.We believe thatﬁnite magnetic helicity is more typical than H≃0,since MHD turbulence usu-ally occurs in rotating systems.The wavenumber k0,thelocation of the maximum of the initial energy spectrum, is chosen k0=4,which allows the inverse cascade of H k to develop freely during the simulation time of10time units(about7eddy turnover times,deﬁning the eddy turnover time as the time required to reach the maxi-mum dissipation from the smooth initial state).Though this choice implies a certain loss of inertial range,the sacriﬁce is unavoidable in the presence of inverse cascade dynamics,since assuming k0∼1would lead to magnetic condensation in the lowest-k state,which would also af-fect the dynamics of higher-k modes.Both normal dif-fusionν=1and hyperdiﬀusionν=2have been used,ν=1to discuss the spectral properties andν=2to de-termine the scaling of the structure functions.All runs presented in this Letter haveﬁnite H,H/H max≃0.6, and negligible alignment.Table I lists the important parameters of the simulation runs,where the magnetic Taylor Reynolds number is Rmλ=Rm1/2.Since Rmλis not stricly constant during turbulence decay but in-creases slowly Rmλ∼t1/8,we give the values taken at a speciﬁc time t=4.Weﬁrst discuss the spectral properties considering the angle-averaged energy spectrum E k.Figure1shows the scatter plot of the normalized spectrum(the normaliza-tion is discussed below),compensated by k3/2,taken from run3over the period t=4−10of fully developed turbulence.The spectrum exhibits a clear scaling range of almost one decade with a spectral law,which is deﬁ-nitely steeper than the IK spectrum k−3/2,close to(in fact slightly steeper than)k−5/3indicated by the dashed line.In order to form the time average in a system of decaying turbulence,the spectrum must be normalized to eliminate the time variation of the macroscopic quan-tities.In hydrodynamic turbulence the only such quan-tity isǫ,which leads to the universal form of the Kol-mogorov spectrum E( k)=E k/(ǫη5)1/4, k=kl K,where l K=(η3/ǫ)1/4is the Kolmogorov length.However,when normalized in this way the MHD energy spectrum is found to change during turbulence decay and even more strongly so when comparing runs of diﬀerent Rmλ.For ﬁnite magnetic helicity the spectrum may also depend on H,which introduces a second macroscopic length scale l1=H/v2A in addition to l0,i.e,the spectrum may con-tain some function of l0/l1.To determine this function we propose the following argument.Since the Alfv´e n eﬀect is clearly present in the simulations,kinetic and magnetic energy spectrum being nearly equal at small scales,while on the other hand the scaling E k∼k−5/3 is observed,we modify the nonlinear transfer in the IK ansatz by a factor(l/l1)δ,τAl1δδz2lquasi-constant parts determine the scaling coeﬃcients,the dashed horizontal lines indicating the most probablevalues ζ+3≃0.95,ζ+4≃1.15.These results are consistent with the spectrallaw derived from Fig.1and are close to the analytical prediction ζ3=1.It is true that the analytical theory refers to third-order moments diﬀerent from |δz |p discussed here,but the scaling coeﬃcients should not depend thereof (the scaling range,however,does).One might object that the use of hyperdiﬀusion aﬀects the inertial-range scaling,if the scaling range is not very broad.In fact,the energy spectrum law tends to be polluted by the bottleneck eﬀect,which is particu-larly pronounced for hyperdiﬀusion (see e.g.,[19]).Thus the energy spectrum in run 4(not shown)is eﬀectively ﬂatter than k −5/3expected from the value of ζ2.How-ever,there is,to our knowledge,no argument for a similar eﬀect in the structure functions.Assuming the exact result ζ3=1allows to obtain rather accurate values of ζp by using the property of ESS (extended self-similarity)[20]plotting S p as function of S 3.(It should be noted that ESS usually results in almost perfect scaling behavior,but the scaling coeﬃcients thus derived vary in time,hence time averaging is required.)The results are shown in Fig.4,which gives the ESSresults of ζ+p for run 4(diamonds)and the ESS values ξ+3p =ζ+p /ζ+3from [21]for 2D MHD turbulence (trian-gles).(In 2D MHD the absolute values of ζp are found in [9]to decrease with Rm ,while the relative values ξ3p appear to be independent of Rm .)The results indicate that in 3D MHD turbulence is less intermittent than 2D,but it is more intermittent than hydrodynamic turbu-lence,the continuous curve,which gives the She-Levequeresult ζSLp .As shown by Politano and Pouquet [6],the She-Leveque concept contains eﬀectively three parame-ters:g related to the scaling δz l ∼l 1/g ,x related to the energy transfer time at the smallest,the dissipative scales t l ∼l x ,and C ,the codimension of the dissipative structures,ζp =p[1]A.Kolmogorov,Dokl.Akad.Nauk SSSR 31,5381941.[2]P.S.Iroshnikov,Astron.Zh.40,742(1963),[Sov.As-tron.7,568(1964)].[3]R.H.Kraichnan,Phys.Fluids 8,1385(1965).[4]Zh.-S.She and E.Leveque,Phys.Rev.Lett.72,336(1994).[5]R.Grauer,J.Krug,and C.Marliani,Phys.Lett.A 195,335(1994).[6]H.Politano and A.Pouquet,Phys.Rev.E 52,636(1995).[7]D.Biskamp and H.Welter,Phys.Fluids B 1,1964(1989).[8]D.Biskamp,E.Schwarz,and A.Celani,Phys.Rev.Lett.81,4855(1998).[9]D.Biskamp and E.Schwarz,to be published.[10]M.Verma,M.L.Goldstein,S.Gosh,and W.T.Stribling,J.Geophys.Res.101,21619(1996).[11]P.Goldreich and S.Sridhar,Astrophys.J.485,680(1997).[12]H.Politano and A.Pouquet,Phys.Rev.E 57,R21(1998),and Geophys.Res.Lett.25,273(1998).[13]L.F.Burlaga,J.Geophys.Res.96,5847(1991).[14]A.Pouquet,M.Meneguzzi,and U.Frisch,Phys.Rev.A33,4266(1986).[15]S.Kida,S.Yanase,and J.Mizushima,Phys.Fluids A 3,457(1991).[16]M.Hossain,P. C.Gary, D.H.Pontius,and W.H.Matthaeus,Phys.Fluids 7,2886(1995).[17]H.Politano,A.Pouquet,and P.L.Sulem,Phys.Plasmas2,2931(1995).[18]A.Brandenburg,R.J.Jennings,A.Nordlund,M.Rieu-tord,R.F.Stein,and I.Tuominen,J.Fluid Mech.306,325(1996).[19]V.Borue and S.A.Orszag,Europhys.Lett.29,687(1995).[20]R.Benzi,S.Ciliberto,R.Tripiccione,C.Baudet,F.Mas-saioli,and S.Succi,Phys.Rev.E 48,R29(1993).[21]H.Politano, A.Pouquet,and V.Carbone,Europhys.Lett.43,516(1998).3TABLE I.Summary of the simulation runs.run No NνηνRmλ。