An Anti-Hasse Principle for Prime Twists
lec5_twinning_孪晶
Carmelo Giacovazzo: “ 孪晶是由组成相同的不同晶体在某些特定的方向结合组 成的规则的聚集体。”
图来源于MIT开放式课程
两个二维晶体由垂直的镜面相关联。这一镜面是晶胞的公制对 称的一部分，不是晶胞的组成部分。 使两个晶体融合就得到了孪晶。垂直镜面操作就是孪晶法则：1 0 （如果X向下移，y向右移） 0 -1
对于这种特殊情况下的Fc方程 式，x是Flack参数。
非缺面对称孪晶
孪晶法则不依赖晶系或公制对称或其他，它只是两个相似 的晶体（不一定同尺寸）变为一个。第二个通常相对于第一个 旋转了180°。衍射图中的一些反射不受影响，一些反射是两 个反射的完美重叠，一些是部分重叠（分裂反射）。
8
非缺面对称孪晶
孪晶法则不依赖于晶系或公制对称或其他，它只是两个相似 的晶体（不一定同尺寸）变为一个。第二个通常相对于第一个旋 转了180°。衍射图中的一些反射不受影响，一些反射是两个反 射的完美重叠，一些是部分重叠（分裂反射）。 在结合过程中，要确定(cell_now)和考虑(SAINT)两个或更多 个独立方位矩阵。每个重叠（或部分重叠）反射在带有两套不同 整数的hkl文件中出现两次（或更多次数目的孪晶组成 ），这需 要不同格式的hkl文件： HKLF 5.举例：书147页。 非缺面对称孪晶数据不可以合并。R int值对于非缺面对称孪 晶是不确定的。 在ins文件中：HKLF 5 替代 HKLF 4 ， n-1 BASF 参数替代 n用于孪晶处理 。
孪晶警告标识
书中所列重要的警告标识中： 公制对称比劳厄对称高。 高对称性劳厄群Rint比低对称性劳厄群稍高一些。 |E2-1|数据比期望的低。 三方和六方空间群。 系统消光不吻合任何空间群。 无解，未能解决能被发现或者精修后R值偏高。 非缺面对称孪晶特点： 一些反射裂分，一些不裂分。 发现和精修晶胞困难。 Fo>>Fc。 残余电子密度很大。
Combinatorial Nullstellensatz
h i gi .
s∈Si (xi
In the special case m = n, where each gi is a univariate polynomial of the form stronger conclusion holds, as follows.
− s), a
Theorem 1.1 Let F be an arbitrary field, and let f = f (x1 , . . . , xn ) be a polynomial in F [x1 , . . . , xn ]. Let S1 , . . . , Sn be nonempty subsets of F and define gi2 , . . . , sn ∈ Sn so that
In this paper we prove these two theorems, which may be called Combinatorial Nullstellensatz, and describe several combinatorial applications of them. After presenting the (simple) proofs of the above theorems in Section 2, we show, in Section 3 that the classical theorem of Chevalley and Warning on roots of systems of polynomials as well as the basic theorem of Cauchy and Davenport on the addition of residue classes follow as simple consequences. We proceed to describe additional applications in Additive Number Theory and in Graph Theory and Combinatorics in Sections 4,5,6,7 and 8. Many of these applications are known results, proved here in a unified way, and some are new. There are several known results that assert that a combinatorial structure satisfies certain combinatorial property if and only if an appropriate polynomial associated with it lies in a properly defined ideal. In Section 9 we apply our technique and obtain several new results of this form. The final Section 10 contains some concluding remarks and open problems.
goodstein定理
goodstein定理（原创实用版）目录1.Goodstein 定理的概述2.Goodstein 定理的证明方法3.Goodstein 定理的应用领域4.Goodstein 定理的意义和影响正文1.Goodstein 定理的概述Goodstein 定理，又称 Goodstein 引理，是由英国数学家 Ronald Goodstein 于 1944 年提出的一个数论定理。
该定理主要研究的是素数序列中的一些性质。
在数论研究中，Goodstein 定理起到了很大的作用，为许多问题的解决提供了关键性的思路。
2.Goodstein 定理的证明方法Goodstein 定理的证明方法相对简单。
首先，我们需要了解一些基本的数论概念，如素数、孪生质数对、Mersenne 数等。
然后，通过构造一个特殊的数列，利用数论中的迭代法证明该定理。
具体来说，Goodstein 定理的证明过程分为两个部分：第一部分是证明存在一个特殊的数列，第二部分是证明这个数列具有某种性质。
通过这两部分，我们可以得出Goodstein 定理的结论。
3.Goodstein 定理的应用领域Goodstein 定理在数论领域的许多问题中都有应用，如孪生质数猜想、素数定理、Mersenne 数等。
其中，最著名的应用是与孪生质数猜想的联系。
孪生质数猜想认为，存在无穷多对相差为 2 的素数。
虽然这个猜想至今尚未得到证明，但 Goodstein 定理为解决这个问题提供了一个重要的思路。
此外，Goodstein 定理还在计算机科学、密码学等领域有一定的应用。
4.Goodstein 定理的意义和影响Goodstein 定理在数论研究中具有重要的意义。
一方面，它为许多问题的解决提供了关键性的思路，推动了数论研究的发展；另一方面，它本身也是一个有趣的数学结论，丰富了数论领域的研究内容。
同时，Goodstein 定理的影响力也扩展到了其他领域，如计算机科学、密码学等，体现了数学在实际应用中的重要性。
【大学英语专业教材】高级英语（张汉熙主编）第四册课文及习题详解
u the thesis ----“ All colonial empires are founded upon this fact”
u the author's central idea
uColonializa on is based on the fact that people are very poor.
un l it is needed.
u= store
ustowst –ow away = hide
u 土豆要在阴凉避光处储藏。 u Potatoes must be stowed in a cool dark place.
u stow away:
u My jewellery is safely stowed away in the bank. u The hungry boy stowed away all the food on the table.
upome --- apple
ugranate ---- uhaving many seeds
chant ---words repeated in a monotonous tone of voice
u 1. a word or group of words that is repeated over and over again, usually by more
ufrenTzhieedre – isa daj . warp in her nature. ufull of uncontrolled excitement
u那狗狂吠著跳起扑向闯进来的人. uThe dog jumped at the intruder with frenzied barks. u--- make frenzied efforts
二语习得(L2)
Theories of habit formation were theories of learning in general. They could be and were applied to language learning.(L1 acquisition →imitating utterances, SLA could process in a similar way). Imitation and reinforcement were the means by which the learner identified the stimulus-response associations that constituted the habits of the L2. Habit-formation theory dominated discussion of both first and second language acquisition up to the 1960’s for it provides a theoretical account of how the learner’s L1 intruded into the process of SLA.
Introduction
Some popular believes Second language acquisition (SLA) is strongly influenced by the learner’s first language(L1). The role of L1 in SLA is a negative one. That is, the L1 gets in the way or interferes with the learning of the L2, such that features of the L1 are transferred into the L2.
分球悖论
Francis Edward Su
(submitted in fulfillment of the “Minor Thesis” requirement for the Ph.D. program)
Harvard University 18 December 1990
[0, 2) = [0, 1) ∪ [1, 2), the union of two sets each of which is “equal” to the original by a “stretching” bijection. But we are not very shocked that we can stretch [0, 1) to get [0, 2). However, when we restrict the number of pieces to be finite and the allowable transformations to be isometries of the ambient space, any paradox that persists is hightly counterintuitive. Notice that the previous paradoxes depended on the set of allowable transformations. Hence we shall demand that our definition of paradoxical be dependent on a group whose action on the set produces the transformations. Definition 1 Let G be a group acting on a set X and suppose E ⊆ X . E is said to be G-paradoxical if for some m,n there exist g1 , ..., gm and h1 , ..., hn ∈ G and pairwise disjoint A1 , ..., Am and B1 , ..., Bn ⊆ E such that E = ∪gi (Ai ) = ∪hj (Bj ). When the isometry group is omitted, we understand it to be the isometry group of X . Note that in our definition, {Ai } ∪ {Bi } may not cover all of E , and the {gi (Ai )} (or {hi (Bj )}) may not be pairwise disjoint. This distinction is artificial, however, because we may take the Ai , Bj to be smaller to ensure the pairwise disjointness, and we will prove later that we can use all of E in the decomposition. Theorem 1 (The Banach-Tarski Paradox) Any ball in R3 is paradoxical. Paradoxes first emerged in the study of measures. In fact, they were constructed to show the non-existence of certain kinds of measures, such as in the following example. Theorem 2 S 1 is countably SO2 -paradoxical (i.e., paradoxical with a countable number of pieces). 4
弱π缠绕模的Maschke型定理
缠 绕模 是一 类更 广义 的模 ，作 为缠 绕 模 的推 广 ，王栓 宏 Ｈ在 Ｈ ｐ 一 ｏｆｒ余代 数 中 引入 了 ７缠 绕 模 ． ７ ｒ 一 类 似地 ，文献 ［ ］ 入 了弱 Ｈ ｐ 一 代数 中的弱 ７缠 绕模 ． ２引 ｏｆｒ ７余 ｒ 一
在 经典 表示 论 中 ，Ｍ ｓｈ ｅ 理是 一个 重要 的结 论 ，即有 限群 上 的群 代 数 为 半单 的充要 条 件 是 域 ａｃ ｋ 定
第５ ０卷
第 ４期
吉 林 大 学 学 报 （理 学 版 ）
Ｊｕｎｌ ｆ ｉｎＵ ｉ ｒ ｔ Ｓｉ ｃ ｄｔ ｎ ｏ ｒａ ｏ Ｊ ｉ ｎｖ ｓｙ（ ｃ ｎｅＥ ｉｏ ） ｌ ｅｉ ｅ ｉ
Ｖ ０． ０ Ｎｏ． １５ ４
２ｔ ０ ２年 ７月
Ｊ ｌ ２ １ ｕｙ ０ ２
弱 缠 绕 模 的 Ｍａｃ ｋ ｓｈ ｅ型 定 理
（ｅｐ ｐｍ ｒｈｓ ｓｌｓａ ｎＡ 一 ｏ ｕｅｍ ｒｈｓ ． ｒｓ ．ｅ ｉ ｏｐ ｉ ｍ） ｐｉ ｓ ｍ ｄ ｌ ｏｐ ｉ ｔ ａ ｍ
ｓｌｓｉ ｐｈ ｍｏｏ ｏｐｉ ｅ ｍ
Ｋｅ ｒ ｓ：ｗ ａ ｐ 『 ｃ ａｇ ｂ ａ ；ｗｅ ｋ ７ 一 ｎ ｗ ｎ ｄ ｍｏ ｕ ｅ ；Ｍａ ｃ ｋ — ｐ ｈ ｏ ｅ ｙ ｗｏ ｄ ｅ ｋ Ｈｏ ｆ７一 ｏ ｌ ｅ ｒ ｓ ａ 『ｅ ｔ ｉ ｅ ｄ ｌ ｓ ｓ ｈ ｅ ｔ ｅ ｔ ｅ ｒ ｍ ｙ
１ ， １） ）（ ｍ ， 是 一 数 ， 代
，｛ ｝ △ 邮 ， 满 足下 列条 件 ： ）
２ ）余 乘法 满 足 ： （）对任 意 的 Ｏ， ∈７，（ ⑧ ｉ ） ｉ ／ ， ｒ △ ｄ 。△
收稿 日期 ： ０ １ ９０ ， ２ １ － －５ ０
用连分数定义莫尔条纹“准周期”
第４０卷第９期大 学 物 理Ｖｏｌ．４０Ｎｏ．９２０２１年９月ＣＯＬＬＥＧＥ ＰＨＹＳＩＣＳＳｅｐ．２０２１ 收稿日期：２０２１－０２－２２；修回日期：２０２１－０４－１１ 基金项目：２０２０年国家级大学生创新创业训练计划项目（２０２０１０２６９０７０Ｇ）资助 作者简介：叶政君（２０００—），男，重庆人，华东师范大学２０１８级本科生． 通信作者：夏成杰，Ｅ－ｍａｉｌ：ｃｊｘｉａ＠ｐｈｙ．ｅｃｎｕ．ｅｄｕ．ｃｎ用连分数定义莫尔条纹“准周期”叶政君，祝怡然，黄泽江，夏成杰（华东师范大学物理与电子科学学院，上海 ２００２４１）摘要：由两组平行周期条纹叠加而成的莫尔条纹并非严格的周期结构，其近似的周期性在数学上等价于对任意实数的最佳有理逼近．通过将两条纹周期之比近似为其连分数的各阶渐进分数，可系统性地严格定义莫尔条纹各阶“准周期”并计算其长度；实际观察到的莫尔条纹的周期，是满足非周期程度低于经验阈值的最低阶准周期．基于莫尔条纹与连分数展开之间的对应关系，可以找到一类具有严格周期性的莫尔条纹，以及一类“周期性最差”的黄金比例莫尔条纹．本文建立了莫尔条纹与实数基本性质的联系，对莫尔条纹现象的本质提供了新的理解，对所有周期叠加问题都具有普适意义．关键词：莫尔条纹；连分数；周期叠加中图分类号：Ｏ５９ 文献标识码：Ａ 文章编号：１０００ ０７１２（２０２１）０９ ００５２ ０６【ＤＯＩ】１０．１６８５４／ｊ．ｃｎｋｉ．１０００ ０７１２．２１００８５莫尔条纹是两组周期相近的条纹叠加形成的“周期”结构［１］．如图１（ａ）所示，两组黑白条纹平行、重叠放置，可观察到具有更长周期的莫尔条纹．这一放大周期的特性使莫尔条纹现象具有广泛的应用，如对角度、位移、材料应变的精密测量，放大晶格缺陷实现纳米探伤，纸币防伪等［２ ５］．同时，近年在凝聚态物理领域发现了与莫尔条纹现象直接相关的石墨烯“魔角”特性———通过材料的层间晶格差异或者转角形成莫尔超晶格结构，引入额外的大周期势场，产生新型能带调制，在某些情况下与高温超导现象类似［６，７］．莫尔条纹周期长度的计算是理解并应用这一现象的基础．在大多数相关文献中，常用遮光原理求解莫尔条纹的周期［８］．我们指出，这一方法虽然在大多数情况下都能给出正确的结论，但会与某些特殊条件下的观察结果不符：例如，当两组条纹的各自的周期长度之比，接近一对相差２的互质正整数之比的情况，如５／７等；并且，其推导过程先验地暗含了“莫尔条纹的周期性是严格的”这一假设，而实际上更应该被视为一项近似，在逻辑和数学上需要更为严谨的讨论．针对这些问题，本文通过对两组条纹的周期之比进行连分数展开，严格推导了它们所形成的莫尔条纹的周期的表达式，为莫尔条纹的周期性给出了严谨的数学定义和新的理解．据我们所知，这是第一次建立起莫尔条纹现象与连分数这一基础数学形式之间的联系．１ 莫尔条纹周期计算方法的谬误图１中的莫尔条纹由周期分别为ＴＡ和ＴＢ的两组平行条纹Ａ、Ｂ叠加而成，不失一般性，设ＴＡ＞ＴＢ，且它们的第０根线条重合．由于两组条纹周期不同，重合线条右侧的线条逐渐错开，遮光（黑色）区域的面积逐渐增加；继续向右，条纹Ａ的第ｎ根线条距离条纹Ｂ的第ｎ＋１根线条的间距，开始比两组条纹各自第ｎ根线条的间距更近，即遮光面积又逐渐减少；直至条纹Ａ的第ｎＡ根线条与条纹Ｂ的第ｎＢ＝ｎＡ＋１根线条几乎重合．这就形成了一套“亮－暗－亮”交错的图案，其中两组条纹的线条几乎重合之处为莫尔条纹中的亮纹，两组条纹的线条最不重合之处为暗纹．通常，人们基于上述遮光原理推导莫尔条纹的周期长度ＴＭ＝ｎＡＴＡ＝ｎＢＴＢ（１）其中ｎＢ＝ｎＡ＋１（２）这表示两组条纹各自经过ｎＡ和ｎＡ＋１个周期后再次重合．联立式（１）、（２）可解得ＴＭ＝ＴＡＴＢＴＡ－ＴＢ（３）对于绝大多数的周期ＴＡ、ＴＢ组合，式（３）都可第９期叶政君，等：用连分数定义莫尔条纹“准周期”５３ 给出正确的结果，但也存在例外．例如，我们在图１（ｃ）中展示利用ＡｄｏｂｅＩｌｌｕｓｔｒａｔｏｒ（Ａｉ）软件绘制的两组周期之比为ＴＢ／ＴＡ＝０．７１４的重叠平行条纹，直接观察图１（ｃ）可知，这组莫尔条纹的周期约为ＴＭ＝５ＴＡ，而根据式（３）计算所得的周期是ＴＭ＝２．４９６５ＴＡ；约为计算结果的１／２．可见，对于某些ＴＡ和ＴＢ的组合，基于式（３）计算所得的莫尔条纹周期与实际观察结果不符．这一错误本质上是由于上述推导过程中存在一系列不严谨的近似和假设，一些明显的数学漏洞被忽视了．首先，从数学角度而言，当且仅当α＝ＴＢ／ＴＡ为有理数时，式（１）存在整数解ｎＡ＝Ａ，ｎＢ＝Ｂ，其中正整数Ａ、Ｂ是表示成既约分数（即最简分数）的α＝Ａ／Ｂ的分子和分母．所以，对于任意两个周期ＴＡ，ＴＢ，关于ｎＡ，ｎＢ的方程式（１）不一定存在正整数解；甚至，由于有理数在实数轴上测度为０，严格来说，两组条纹周期之比α为有理数的概率为０．更进一步，即便在某些式（１）存在整数解的特殊情况下，直接由式（１）计算出的莫尔条纹周期也可能不正确．譬如，当ＴＢ／ＴＡ＝０．９０１＝９０１／１０００时，式（１）的正整数解所对应的莫尔条纹周期为９０１ＴＡ＝１０００ＴＢ，显然不符合观察结果（图１（ｂ））．所以，式（１）只能是近似成立．但是，这一近似的程度，以及做此近似的合理性均未得到充分说明．第二，式（２）也隐含着人为的假设：两组条纹刚好“错开”一条时，对应于莫尔条纹的一个周期．这一假设一方面限制了能使式（１）成立的α必须具有Ａ／Ａ＋１p y 的形式，同时其本身的合理性与必要性也尚待讨论．２ 莫尔条纹“准周期”的定义上述分析表明，周期分别为ＴＡ和ＴＢ的两组条纹各自经过ｎＡ和ｎＢ个周期后通常只能是近似重合．所以，莫尔条纹并非是严格的周期结构，将其周期称为“准周期”更为合理．由莫尔条纹遮光原理的图像可知，计算莫尔条纹的周期长度，本质上是在寻找一组不太大的正整数ｎＡ和ｎＢ，使ｎＡＴＡ－ｎＢＴＢ＝ＴＡｎＡ－αｎＢ（４）足够小．其中，ｎＡＴＡ－ｎＢＴＢ为两组条纹各自经过ｎＡ和ｎＢ个周期后的间距．在本节，我们为“不太大”和“足够小”这两个近似建立严格的数学定义．其他计算莫尔条纹周期的方法本质上都未加说明地采取了同样的近似［９］．所以，如何在数学上明确这一近似的含义，并找到一组符合实际观察结果的ｎＡ和ｎＢ，是一个普遍存在的问题．ＴＢ／ＴＡ＝９／１０＝［０，１，９］ＴＢ／ＴＡ＝９０１／１０００＝［０，１，９，９，１，９］ＴＢ／ＴＡ＝０．７１４＝［０，１，２，２，７１］ＴＢ／ＴＡ＝０．７１５＝［０，１，２，１，１，２８］图１ 两组周期分别为ＴＡ和ＴＢ的平行条纹形成的莫尔条纹２．１ 有理数逼近由式（４）可知，计算莫尔条纹的周期本质上是在寻找α＝ＴＢ／ＴＡ的有理数逼近，即用一个有理数ｎＡ／ｎＢ来近似α，从而使ｎＡ－αｎＢ足够接近于零．这种对数字精度的取舍存在于所有实际问题中．最为通用的逼近方法是“四舍五入”，例如将图１（ｂ）中的０．９０１＝９０１／１０００近似为０．９＝９／１０后便可得到符合观察结果的莫尔条纹周期ＴＭ＝９ＴＡ．但是，对于莫尔条纹，“四舍五入”有时并非最佳的有理逼近方式．例如，若ＴＡ＝１，ＴＢ＝０．９３１，小数α＝ＴＢ／ＴＡ＝０．９３１四舍五入近似为０．９＝９／１０，得到ｎＡ＝９，ｎＢ＝１０．代入式（４）可知两组条纹各经过ｎＡ和ｎＢ个周期后的距离相差ｎＡＴＡ－ｎＢＴＢ＝０．３１．对比之下，如果用ｎＡ／ｎＢ＝１３／１４≈０．９２９来近似α，则得到ｎＡＴＡ－ｎＢＴＢ≈０．０３４，近似的精度提高了一个量级．数学上，通过对实数α＝ＴＢ／ＴＡ进行连分数展开，可以系统性地给出所有关于α的有理数逼近；对于莫尔条纹，将恰好给出所有使式（４）近似为零的ｎＡ和ｎＢ．２．２ 连分数展开任何一个正实数α都可以被表示成如下（简单）连分数展开的形式５４ 大 学 物 理 第４０卷α＝ａ０＋１ａ１＋１ａ２＋１ （５）可简写为ａ０，ａ１，…，ａｎ，…C o ，其中展开系数ａｎ为正整数．一个有理数的连分数展开为有限阶，无理数的连分数展开为无穷阶．连分数展开是收敛的，其形式是唯一的，（对于一个有理数存在两种等价的表示形式：ａ０，ａ１，…，ａｎC o 与ａ０，ａ１，…，ａｎ－１，１C o ），它是对于实数的一种“纯粹”的表示方式———其各阶展开系数与进制无关．本文主要利用连分数展开的以下性质，相应的证明可参见引文［１０，１１］．首先，将连分数ａ０，ａ１，…，ａｎ，…C o 在第ｋ阶截断，得到ａ０，ａ１，…，ａｋC o ，等于分数ｐｋ／ｑｋ，被称为连分数的第ｋ阶渐进分数，其中正整数ｐｋ和ｑｋ满足递推关系ｐ０＝ａ０，ｐ１＝ａ０ａ１＋１ｑ０＝１，ｑ１＝ａ１ｐｋ＝ａｋｐｋ－１＋ｐｋ－２ｋ≥２p y ｑｋ＝ａｋｑｋ－１＋ｑｋ－２ ｋ≥２p y（６）可以证明，由式（６）定义的渐近分数ｐｋ／ｑｋ为既约分数；并且，随着阶数ｋ增大，各阶渐近分数与α之差的绝对值严格递减且收敛于０．第二，首先定义：对正实数α以及既约分数ｐ／ｑ，如果所有不大于ｑ的正整数ｑ′以及任意ｐ′，都满足ｐ′－αｑ′≥ｐ－αｑ（７）那么ｐ／ｑ是实数α的第二类最佳逼近．可以证明如下定理：α的连分数的各阶渐进分数ｐｋ／ｑｋ都是其第二类最佳逼近；并且，α的所有第二类最佳逼近都是其连分数展开的渐进分数．例如，２．１节中的１３／１４是０．９３１的第二阶渐进分数，可以验证，它是０．９３１的一个第二类最佳逼近．由上述定义和定理并对比式（４）、（７）可知，对于莫尔条纹，α＝ＴＢ／ＴＡ的连分数展开的第ｋ阶渐进分数ｐｋ／ｑｋ将正好给出使式（４）取其极小值的解：ｎＡ＝ｐｋ，ｎＢ＝ｑｋ；此处“极小值”的含义是指：第ｑｋ根Ｂ条纹与第ｐｋ根Ａ条纹的间距小于第ｑｋ根Ｂ条纹之前任意第ｑ′（０＜ｑ′＜ｑｋ）根Ｂ条纹与任意一根Ａ条纹之间的距离．从遮光原理的角度来说，第ｐｋ根Ａ条纹与第ｑｋ根Ｂ条纹对应于莫尔条纹中的一个亮纹，且比它之前除第０根条纹之外的所有其余位置“更亮”．所以，从第０根条纹到第ｐｋ根Ａ条纹（或第ｑｋ根Ｂ条纹）可定义为莫尔条纹的一个准周期．式（４）、（７）在形式上的巧合，以及连分数的渐进分数与第二类最佳逼近的充要性，使得连分数展开自然而然地成为定义及计算莫尔条纹周期的最合适的数学方法．此外，我们指出，利用看似更为直接的第一类最佳逼近，无法给出符合实际观察结果的莫尔条纹周期．（第一类最佳逼近指的是：如果对于一个既约分数ｐ／ｑ，对任意ｑ′≤ｑ以及ｐ′都有ｐ′／ｑ′－α≥ｐ／ｑ－α，则ｐ／ｑ是α的最佳逼近．可以证明，α的连分数的各阶渐进分数都是α的第一类最佳逼近；但并非所有α的第一类最佳逼近都是其渐进分数．）２．３ 各阶准周期上述讨论表明，α＝ＴＢ／ＴＡ的连分数的各阶渐进分数都可以对应一个莫尔条纹周期，且阶数ｋ越小，莫尔条纹周期（ｐｋＴＡ或ｑｋＴＢ）越短，但ｐｋ－αｑｋ越大，即近似程度越低．各阶准周期的表达式如下．为简化起见，我们令ＴＡ＝１，且只讨论１／２＜ＴＢ＜ＴＡ的情形，此时α＝ＴＢ／ＴＡ的连分数展开系数中有：ａ０＝０，ａ１＝１．若１／３＜ＴＢ≤１／２，可将相邻两条Ｂ条纹视作一条，从而将其周期扩大一倍，则回到上述情形；其余情况也可类似转化．α的第１阶渐进分数为ｐ１／ｑ１＝１／１，从而得到ｎＡ＝ｎＢ＝１，即ＴＭ≈ＴＡ≈ＴＢ，是一个平凡的、无意义的解．α的第２阶渐进分数为ｐ２／ｑ２＝ａ２／ａ２＋１p y ，从而得到ｎＡ＝ａ２，ｎＢ＝ａ２＋１，以及莫尔条纹的周期ＴＭ＝ａ２ＴＡ，或等价的ＴＭ＝ａ２＋１p y ＴＢ．ａ２的数值为?α／１－αp y 」，其中?ｘ」表示ｘ向下取整．如代入小数α／１－αp y ＝ＴＢ／ＴＡ－ＴＢp y 的值以近似计算ａ２，可得ＴＭ＝ＴＢ／ＴＡ－ＴＢp y 」ＴＡ．此式与式（３）几乎一致，最多相差一个ＴＡ．如果考虑更高阶的渐进分数，比如对于第３阶渐进分数有：α≈ａ２ａ３＋１p y ／ａ２ａ３＋ａ３＋１p y ，得到莫尔条纹第３阶准周期ＴＭ＝ａ２ａ３＋１p y ＴＡ．以此类推，可以定义莫尔条纹的第ｋ阶准周期ＴＭ＝ｐｋＴＡ（或等价的ＴＭ＝ｑｋＴＢ）．在各阶准周期中，第２阶准周期最为特殊：只有在第２阶渐进分数α≈ａ２／ａ２＋１p y 的形式下，Ａ条纹刚好比Ｂ条纹少经过一个周期而第一次达到近似重合的状态；并且，Ａ条纹与其最近的Ｂ条纹的距离先单调增加后单调减小，这样的单次“亮－暗－亮”的变化最容易被人眼分辨．对于更高阶的准周期，其长度通常远大于２阶准周期长度，且每个周期内部还存在多次“亮－暗”的转变，对应于多个仅有微小区别的低阶准周期图案．所以人眼往往不易辨别出高阶准周期的存在，而通常能观察到的莫尔条纹的周第９期叶政君，等：用连分数定义莫尔条纹“准周期”５５ 期就对应于连分数的２阶渐进分数，也就是式（３）所给出的最为通用的表达式．但是我们还是指出，存在一些较为特殊的α，观察到的莫尔条纹周期确实对应于更高阶的渐进分数．２．４ 非周期程度我们定义莫尔条纹各阶非周期程度ｆｋ（α）＝ｐｋ－αｑｋ（ｋ≥２），它表示Ａ、Ｂ两组条纹经过一个第ｋ阶莫尔条纹准周期后的间距与ＴＡ之比，反映了它们近似重合的程度．由连分数渐进分数的性质可知：低阶准周期非周期程度大，但周期短，所以易观察；高阶准周期的非周期程度小，即近似的精度高，但周期太长，不易观察．可见，实际观察到的莫尔条纹周期，在非周期程度（即精度）与准周期长度之间达到某种平衡．基于以上考虑，我们给出莫尔条纹周期的严格定义ＴＭ＝ｐｋＴＡ，ｋ＝ｍｉｎｋ′r i ，ｓ．ｔ．ｆｋ′（α）≤Ｅ（８）表示：ｋ为满足ｆｋ（α）≤Ｅ的最小值，其中ｐｋ由式（６）的递推关系给出，而Ｅ是可观察到莫尔条纹周期的经验阈值．考虑到人类通常能够较为准确地分辨出一段给定长度的１／１０，我们认为Ｅ比较合理的取值应在０．１左右．图２ （ａ）２阶非周期程度ｆ２（α）；（ｂ）３阶非周期程度ｆ３（α）．虚线表示α＝ ≈０．６１８以及α＝ ２≈０．７２４ｆ２（α）和ｆ３（α）的图像如图２所示．对于某一阶ｆｋ（α），其函数的图像形式为无数个直角三角形，每个直角三角形又刚好覆盖了高一阶的ｆｋ＋１（α）的图像中一组面积依次改变的直角三角形．全体ｆｋ（α）函数表现出一种分形的自相似特征．由图２可知，当两组条纹的周期非常接近时（如α≥０．９），ｆ２（α）＜０．１始终成立，所以莫尔条纹周期为第２阶准周期，基本上与式（３）等价．对于某些较小的α，经过一个２阶准周期后Ａ条纹和Ｂ条纹重合程度不高，即ｆ２（α）较大，而只有经过更高阶的准周期后，才满足ｆｋ（α）≤Ｅ，从而对应于人眼观察到的莫尔条纹．对于图１中所列举的各条纹周期之比，可验证，取Ｅ＝０．１时，由式（８）定义的莫尔条纹周期与实际观察结果均相符：如图１（ａ），α＝０．９，ｆ２（０．９）＝０，表示两条纹严格重合，所以有ＴＭ＝ｐ２ＴＡ＝９ＴＡ；图１（ｂ），α＝０．９０１，ｆ２（０．９０１）＝０．０１＜Ｅ，所以只需展开到二阶，同样有ＴＭ＝ｐ２ＴＡ＝９ＴＡ；图１（ｃ），α＝０．７１４，ｆ２（０．７１４）＝０．１４２＞Ｅ而ｆ３（０．７１４）＝０．００２＜Ｅ，所以需要展开到第三阶，即ＴＭ＝ｐ３ＴＡ＝５ＴＡ；图１（ｄ），α＝０．７１５，有ｆ２（０．７１５）＝０．１４５＞Ｅ，ｆ３（０．７１５）＝０．１４＞Ｅ，而ｆ４（０．７１５）＝０．００５＜Ｅ，所以ＴＭ＝ｐ４ＴＡ＝５ＴＡ．图１（ｃ）、（ｄ）展示的两种莫尔条纹正是由通用的式（３）无法给出符合实际观察结果的情况．３ 由连分数展开得到的推论这一节，我们基于上述结果讨论几类特殊的条纹周期之比α及其莫尔条纹的周期性质．为验证理论推论，我们首先利用菲林打印机在透明塑料薄片上打印出由Ａｉ软件绘制出的不同周期的条纹，随后将各组周期长度不同的条纹平行、重叠放置于一块平板ＬＥＤ灯前，用单反相机拍摄它们所构成的莫尔条纹（如图３、４）．３．１ 周期性二阶莫尔条纹由上述莫尔条纹周期的连分数计算方法可知，若α＝ＴＢ／ＴＡ的连分数为０，１，ａ２C o ，即所有ｎ＞２的系数ａｎ均为０，那么其二阶渐进分数就等于α，此时由式（３）计算所得的周期是严格的，所形成的莫尔条纹的周期性也是严格的．满足这一条件的α具有ａ２／（ａ２＋１）的形式（见图２（ａ）），即第１节中提到的使式（１）存在正整数解、且满足ｎＢ＝ｎＡ＋１的情况．此类周期性二阶莫尔条纹如图３（ａ）、（ｂ）所示．可以看到，即便当α较小（如２／３）时也可以观察到明显的莫尔条纹，并且每个莫尔条纹周期都表现为简单的“亮－暗－亮”图样．当α在这些特殊比值附近时，显然也可以观察到较为明显的莫尔条纹现象．所以，通常认为只有当两组条纹的周期长度非常接近时才能观察到莫尔条纹现象的想法并不严谨．３．２ 周期性高阶莫尔条纹若α＝ＴＢ／ＴＡ的连分数为０，１，ａ２，２C o ，此时其二阶渐进分数为ａ２／（ａ２＋１），当ａ２较小时ｆ２（α）较大，所以不应直接由式（３）计算莫尔条纹的周期；而其三阶渐进分数（２ａ２＋１）／（２ａ２＋３）就等于α（见图２５６ 大 学 物 理 第４０卷第９期 叶政君，等：用连分数定义莫尔条纹“准周期”５７４ 总结与展望本文探讨了莫尔条纹的“准周期”结构与基于连分数展开的有理数逼近之间的深刻联系．研究发现，对于两组周期不等的平行条纹，它们叠加所形成的莫尔条纹具有各阶准周期，并恰好对应于两者周期之比的连分数的各阶渐近分数；其中，满足非周期程度低于经验阈值、即条纹重合精度足够高的最低阶渐进分数，是人眼所观察到的莫尔条纹．以上对莫尔条纹周期的严格定义，可避免过去通用的周期计算表达式在某些特殊情况下的错误结果．此外，基于连分数的有理逼近性质，可以建立起莫尔条纹非周期程度与实数基本特性之间的映射，从而可以系统性地分析出两类特殊的莫尔条纹：具有严格周期性的莫尔条纹，与各阶非周期程度都较大的黄金比例莫尔条纹．最后，我们指出，本文连分数展开的分析方法对所有周期性时空结构叠加的问题（如拍现象、历法置润等）都具有普适意义．本文主要讨论了符合人眼观察结果的较为低阶的莫尔条纹，而连分数的各阶渐进分数可以系统性地预言周期叠加的全部性质，从而能帮助探索更高阶的准周期结构在凝聚态物理、信号处理等方面的潜在应用．本文只讨论了两组一维平行条纹的莫尔条纹现象．两组二维周期性网格能够叠加产生更为复杂的莫尔条纹，它们所形成的图样也具有类似的近似周期性，所以连分数展开的分析方法也可以推广到更高维度的莫尔条纹中．并且，本文给出了一组非周期程度最大的黄金比例莫尔条纹，黄金比例直接对应于五重旋转对称性的特征，（７２°＝３６０°／５，ｃｏｓ７２°p y＝ ／２），而五重旋转对称性又是典型的准晶结构特征．这些巧合暗示着实数的连分数表示、莫尔条纹、晶体或准晶结构、超晶格，这些看似无关的数学、物理概念背后存在着非常深刻的联系，并且可能借由最新的转角电子学（Ｔｗｉｓｔｒｏｎｉｃｓ）的相关研究揭示出来［１３，１４］．参考文献：［１］ ＡｍｉｄｒｏｒＩ．ＴｈｅＴｈｅｏｒｙｏｆｔｈｅＭｏｉｒéＰｈｅｎｏｍｅｎｏｎ［Ｍ］．ＳｐｒｉｎｇｅｒＮｅｔｈｅｒｌａｎｄｓ，２００７：１ ３．［２］ 张继友，范天泉，曹学东．光电自准直仪研究现状与展望［Ｊ］．计量技术，２００４，（７）：２７ ２９．［３］ 徐海英，缪长宗，陈慧琴．莫尔条纹应用的初探［Ｊ］．南京工程学院学报，２００６，（１）：６０ ６５．［４］ 朱超逸，陆炜鑫，高红，等．基于横向莫尔条纹的自准直测角方法［Ｊ］．大学物理，２０２０，３９（５）：７４ ７７．［５］ 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ｊｉｅ（ＳｃｈｏｏｌｏｆＰｈｙｓｉｃｓａｎｄＥｌｅｃｔｒｏｎｉｃＳｃｉｅｎｃｅ，ＥａｓｔＣｈｉｎａＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ，Ｓｈａｎｇｈａｉ２００２４１，Ｃｈｉｎａ）Ａｂｓｔｒａｃｔ：ＡＭｏｉｒｅｆｒｉｎｇｅｆｏｒｍｅｄｂｙｔｈｅｓｕｐｅｒｐｏｓｉｔｉｏｎｏｆｔｗｏｐａｒａｌｌｅｌｐｅｒｉｏｄｉｃａｒｒａｙｓｏｆｌｉｎｅｓｉｓｎｏｔｓｔｒｉｃｔｌｙｐｅｒｉ ｏｄｉｃ，ｗｈｏｓｅａｐｐｒｏｘｉｍａｔｅｐｅｒｉｏｄｉｃｉｔｙｃｏｒｒｅｓｐｏｎｄｓｔｏｂｅｓｔａｐｐｒｏｘｉｍａｔｉｏｎｓｔｏａｒｅａｌｎｕｍｂｅｒ．Ｔｈｅｑｕａｓｉ－ｐｅｒｉｏｄｉｃｉｔｙｏｆａＭｏｉｒｅｆｒｉｎｇｅｃａｎｂｅｒｉｇｏｒｏｕｓｌｙｄｅｆｉｎｅｄｂｙｅｘｐｒｅｓｓｉｎｇｔｈｅｒａｔｉｏｏｆｔｈｅｒｅｓｐｅｃｔｉｖｅｐｅｒｉｏｄｓｏｆｔｈｅｔｗｏａｒｒａｙｓｉｎｔｈｅｆｏｒｍｏｆａｃｏｎｔｉｎｕｅｄｆｒａｃｔｉｏｎｅｘｐａｎｓｉｏｎ，ａｎｄｉｔｓｑｕａｓｉ－ｐｅｒｉｏｄｓｃａｎｂｅｄｅｒｉｖｅｄｂｙａｐｐｒｏｘｉｍａｔｉｎｇｔｈｅｒａｔｉｏｔｏｃｏｎｖｅｒｇｅｎｔｓｏｆｄｉｆｆｅｒｅｎｔｏｒｄｅｒｓ．Ｍｅａｎｗｈｉｌｅ，ｔｈｅｏｂｓｅｒｖｅｄｐｅｒｉｏｄｉｓｔｈｅｌｏｗｅｓｔｑｕａｓｉ－ｐｅｒｉｏｄｗｉｔｈａｄｅｇｒｅｅｏｆａｐｅｒｉｏｄｉｃｉｔｙｓｍａｌｌｅｒｔｈａｎａｎｅｍｐｉｒｉｃａｌｃｏｎｓｔａｎｔ．ＢａｓｅｄｏｎｔｈｉｓｄｉｒｅｃｔｃｏｒｒｅｓｐｏｎｄｅｎｃｅｂｅｔｗｅｅｎａＭｏｉｒｅｆｒｉｎｇｅａｎｄｃｏｎｔｉｎｕｅｄｆｒａｃｔｉｏｎｏｆａ（下转６２页）６２大 学 物 理 第４０卷１８ ２０，４９ ５５，２０４ ２０８．［４］ Ｐ．Ｊ．Ｃｏｏｎｅｙ，Ｅ．Ｐ．Ｋａｎｔｅｒ，Ｚ．Ｖａｇｅｒ．Ｃｏｎｖｅｎｉｅｎｔｎｕ ｍｅｒｉｃａｌｔｅｃｈｎｉｑｕｅｆｏｒｓｏｌｖｉｎｇｔｈｅｏｎｅ ｄｉｍｅｎｓｉｏｎａｌＳｃｈｒ ｄｉｎｇｅｒｅｑｕａｔｉｏｎｆｏｒｂｏｕｎｄｓｔａｔｅｓ［Ｊ］．ＡｍｅｒｉｃａｎＪｏｕｒ ｎａｌｏｆＰｈｙｓｉｃｓ，１９８１，４９（１）：７６ ７７．［５］ ＪｕａｎＦ．ＶａｎｄｅｒＭａｅｌｅｎＵｒíａ，ＳａｎｔｉａｇｏＧａｒｃíａ Ｇｒａｎｄａ，ＡｍａｄｏｒＭｅｎｅｎｄｅｚ Ｖｅｌａｚｑｕｅｚ．Ｓｏｌｖｉｎｇｏｎｅ ｄｉｍｅｎｓｉｏｎａｌＳｃｈｒ ｄｉｎｇｅｒ ｌｉｋｅｅｑｕａｔｉｏｎｓｕｓｉｎｇａｎｕｍｅｒｉｃａｌｍａｔｒｉｘｍｅｔｈｏｄ［Ｊ］．ＡｍｅｒｉｃａｎＪｏｕｒｎａｌｏｆＰｈｙｓｉｃｓ，１９９６，６４（３）：３２７ ３３２．［６］ 陈皓，高明，汪青杰．用有限差分法解薛定谔方程［Ｊ］．沈阳航空工业学院学报，２００５，２２（１）：８７ ８８．［７］ 刘建军，翟利学．有限差分法解能量本征方程［Ｊ］．北京工业大学学报，２００８，３４（３）：３２５ ３３１．［８］ 宫建平．有限差分法求解薛定谔方程［Ｊ］．晋中学院学报，２０１４，３１（３）：１ ６．［９］ 陈杰，张琴，朱占武．外电场作用下“Ｗ”型势阱的能级［Ｊ］．大学物理，２０１５，３４（１１）：１８ ２１．［１０］ 李培咸，郝跃，范隆，等．基于量子微扰的ＡｌＧａＮ／ＧａＮ异质结波函数半解析求解［Ｊ］．物理学报，２００３，５２（１２）：２９８５ ２９８８．［１１］ 哈斯花，班士良．电子－空穴气屏蔽影响下有限深量子阱中电子与空穴的本征态［Ｊ］．内蒙古大学学报（自然科学版），２００７，３８（３）：２７２ ２７７．［１２］ 宋晏蓉，华玲玲，张鹏，等．ＩｎＧａＡｓ／ＧａＡｓ应变量子阱能带结构的计算［Ｊ］．北京工业大学学报，２０１１，３７（４）：５６５ ５６９．［１３］ 林洽武．求解定态薛定谔方程的有限差分法［Ｊ］．广东第二师范学院学报，２０１３，３３（３）：４５ ４８．［１４］ 朱萧霄，崔艳波，丁鑫，等．有限差分法解薛定谔方程及其应用［Ｊ］．常州工学院学报，２０１４，２７（４）：３７ ４１．［１５］ 黄晓亚，杜亚冰．磁场和量子点尺寸对方形杂质量子点中电子性质的影响［Ｊ］．量子光学学报，２０１６，２２（１）：９８ １０２．［１６］ Ｚｈｉ ＨａｉＺｈａｎｇ，ＧｕｏｃｅＺｈｕａｎｇ，Ｋａｎｇ ＸｉａｎＧｕｏ，ｅｔａｌ．Ｄｏｎｏｒ ｉｍｐｕｒｉｔｙ ｒｅｌａｔｅｄｏｐｔｉｃａｌａｂｓｏｒｐｔｉｏｎａｎｄｒｅｆｒａｃｔｉｖｅｉｎｄｅｘｃｈａｎｇｅｓｉｎＧａＡｓ／ＡｌＧａＡｓｃｏｒｅ／ｓｈｅｌｌｓｐｈｅｒｉｃａｌｑｕａｎｔｕｍｄｏｔｓ［Ｊ］．ＳｕｐｅｒｌａｔｔｉｃｅｓａｎｄＭｉｃｒｏｓｔｒｕｃｔｕｒｅｓ，２０１６，１００：４４０ ４４７．［１７］ 陈真梅，陈妮，莫运海，等．球形ＧａＡｓ量子点中心类氢杂质基态束缚能的研究［Ｊ］．现代物理，２０１７，７（４）：１５５ １６２．［１８］ 陈鄂生，李明明．量子力学基础教程［Ｍ］．６版．济南：山东大学出版社，２０１４：４４ ５４．［１９］ 曾谨言．量子力学教程［Ｍ］．３版．北京：科学出版社，２０１４：３１ ４９．Ｓｏｌｖｉｎｇｔｉｍｅ－ｉｎｄｅｐｅｎｄｅｎｔＳｃｈｒ ｄｉｎｇｅｒｅｑｕａｔｉｏｎｂｙｔｈｅｆｏｕｒｔｈ－ｏｒｄｅｒａｃｃｕｒａｔｅｄｉｆｆｅｒｅｎｃｅｍｅｔｈｏｄＬＩＵＺｈａｎ ｙｕａｎ，ＧＵＡＮＣｈｅｎｇ ｂｏ，ＬＵＹｉｎｇ ｂｏ，ＺＨＡＮＧＰｅｎｇ，ＣＯＮＧＷｅｉ ｙａｎ（ＳｃｈｏｏｌｏｆＳｐａｃｅＳｃｉｅｎｃｅａｎｄＰｈｙｓｉｃｓ，ＳｈａｎｄｏｎｇＵｎｉｖｅｒｓｉｔｙ，Ｗｅｉｈａｉ，Ｓｈａｎｄｏｎｇ２６４２０９，Ｃｈｉｎａ）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｅｆｉｎｉｔｅｄｉｆｆｅｒｅｎｃｅｃａｌｃｕｌａｔｉｏｎｓｏｆｔｈｅｔｉｍｅ－ｉｎｄｅｐｅｎｄｅｎｔＳｃｈｒ ｄｉｎｇｅｒｅｑｕａｔｉｏｎ，ｔｈｅｍｏｓｔｌｙｕｓｅｄｄｉｆｆｅｒｅｎｃｅｆｏｒｍｕｌａｉｓｔｈｅｃｅｎｔｒａｌｄｉｆｆｅｒｅｎｃｅｆｏｒｍｕｌａ，ｗｈｉｃｈｉｓａｃｃｏｍｐａｎｉｅｄｗｉｔｈａｔｒｕｎｃａｔｉｏｎｅｒｒｏｒｏｎｔｈｅｓｅｃｏｎｄ－ｏｒ ｄｅｒｏｆｓｔｅｐ－ｓｉｚｅ．Ｉｎｔｈｉｓｐａｐｅｒ，ｔｈｅｆｏｕｒｔｈ－ｏｒｄｅｒａｃｃｕｒａｔｅｄｉｆｆｅｒｅｎｃｅｆｏｒｍｕｌａｓｏｆｔｈｅｄｅｒｉｖａｔｉｖｅｓａｒｅｄｅｒｉｖｅｄｂｙｔｈｅｆｉｖｅ－ｐｏｉｎｔｐｏｌｙｎｏｍｉａｌｉｎｔｅｒｐｏｌａｔｉｏｎ，ａｎｄｕｓｅｄｔｏｓｏｌｖｅｔｉｍｅ－ｉｎｄｅｐｅｎｄｅｎｔＳｃｈｒ ｄｉｎｇｅｒｅｑｕａｔｉｏｎｉｎｓｅｖｅｒａｌｃｏｍｍｏｎｐｏｔｅｎｔｉａｌｗｅｌｌｓ．Ｔｈｅｎｕｍｅｒｉｃａｌｒｅｓｕｌｔｓｓｈｏｗｔｈａｔ，ｔｈｅｆｏｕｒｔｈ－ｏｒｄｅｒａｃｃｕｒａｔｅｄｉｆｆｅｒｅｎｃｅｆｏｒｍｕｌａｈａｓｂｅｔｔｅｒｃｏｎｖｅｒｇｅｎｃｅａｎｄｈｉｇｈｅｒｐｒｅｃｉｓｉｏｎｔｈａｎｔｈｅｃｏｍｍｏｎｃｅｎｔｒａｌｄｉｆｆｅｒｅｎｃｅｆｏｒｍｕｌａ．Ｋｅｙｗｏｒｄｓ：ｈｉｇｈｐｒｅｃｉｓｉｏｎ；ｄｉｆｆｅｒｅｎｃｅｍｅｔｈｏｄ；ｉｎｔｅｒｐｏｌａｔｉｏｎ；ｐｏｔｅｎｔｉａｌｗｅｌｌ；ｇｒｏｕｎｄ－ｓｔａｔｅｅｎｅｒｇｙ（上接５７页）ｒｅａｌｎｕｍｂｅｒ，ａｓｅｔｏｆｒｉｇｏｒｏｕｓｌｙ－ｐｅｒｉｏｄｉｃＭｏｉｒｅｆｒｉｎｇｅｓａｎｄａｎｏｔｈｅｒｓｅｔｏｆ“ｗｏｒｓｔｐｅｒｉｏｄｉｃ”ｇｏｌｄｅｎ－ｒａｔｉｏｆｒｉｎｇｅｓｃａｎｂｅｉｄｅｎｔｉｆｉｅｄ．ＴｈｅｐｒｅｓｅｎｔｗｏｒｋｃｏｎｎｅｃｔｓＭｏｉｒｅｆｒｉｎｇｅｓｗｉｔｈｔｈｅｂａｓｉｃｐｒｏｐｅｒｔｉｅｓｏｆｒｅａｌｎｕｍｂｅｒｓ，ａｎｄｔｈｅｒｅｆｏｒｅｐｒｏ ｖｉｄｅｓｎｅｗｕｎｄｅｒｓｔａｎｄｉｎｇｓｆｏｒｔｈｅＭｏｉｒｅｆｒｉｎｇｅｐｈｅｎｏｍｅｎｏｎ，ａｎｄｉｓｏｆｇｅｎｅｒａｌｓｉｇｎｉｆｉｃａｎｃｅｔｏａｌｌｓｏｒｔｓｏｆｐｅｒｉｏｄｓｓｕｐｅｒｐｏｓｉｔｉｏｎｐｒｏｂｌｅｍｓ．Ｋｅｙｗｏｒｄｓ：Ｍｏｉｒｅｆｒｉｎｇｅ；ｃｏｎｔｉｎｕｅｄｆｒａｃｔｉｏｎ；ｐｅｒｉｏｄｓｓｕｐｅｒｐｏｓｉｔｉｏｎ。
Atomic Decomposition by Basis pursuit
SIAM R EVIEWc2001Society for Industrial and Applied Mathematics Vol.43,No.1,pp.129–159Atomic Decomposition by BasisPursuit ∗Scott Shaobing Chen †David L.Donoho ‡Michael A.Saunders §Abstract.The time-frequency and time-scale communities have recently developed a large number ofovercomplete waveform dictionaries—stationary wavelets,wavelet packets,cosine packets,chirplets,and warplets,to name a few.Decomposition into overcomplete systems is not unique,and several methods for decomposition have been proposed,including the method of frames (MOF),matching pursuit (MP),and,for special dictionaries,the best orthogonal basis (BOB).Basis pursuit (BP)is a principle for decomposing a signal into an “optimal”superpo-sition of dictionary elements,where optimal means having the smallest l 1norm of coef-ficients among all such decompositions.We give examples exhibiting several advantages over MOF,MP,and BOB,including better sparsity and superresolution.BP has interest-ing relations to ideas in areas as diverse as ill-posed problems,abstract harmonic analysis,total variation denoising,and multiscale edge denoising.BP in highly overcomplete dictionaries leads to large-scale optimization problems.With signals of length 8192and a wavelet packet dictionary,one gets an equivalent linear program of size 8192by 212,992.Such problems can be attacked successfully only because of recent advances in linear and quadratic programming by interior-point methods.We obtain reasonable success with a primal-dual logarithmic barrier method and conjugate-gradient solver.Key words.overcomplete signal representation,denoising,time-frequency analysis,time-scale anal-ysis, 1norm optimization,matching pursuit,wavelets,wavelet packets,cosine pack-ets,interior-point methods for linear programming,total variation denoising,multiscale edges,MATLAB code AMS subject classifications.94A12,65K05,65D15,41A45PII.S003614450037906X1.Introduction.Over the last several years,there has been an explosion of in-terest in alternatives to traditional signal representations.Instead of just represent-ing signals as superpositions of sinusoids (the traditional Fourier representation)we now have available alternate dictionaries—collections of parameterized waveforms—of which the wavelets dictionary is only the best known.Wavelets,steerable wavelets,segmented wavelets,Gabor dictionaries,multiscale Gabor dictionaries,wavelet pack-∗Publishedelectronically February 2,2001.This paper originally appeared in SIAM Journal onScientific Computing ,Volume 20,Number 1,1998,pages 33–61.This research was partially sup-ported by NSF grants DMS-92-09130,DMI-92-04208,and ECS-9707111,by the NASA Astrophysical Data Program,by ONR grant N00014-90-J1242,and by other sponsors./journals/sirev/43-1/37906.html†Renaissance Technologies,600Route 25A,East Setauket,NY 11733(schen@).‡Department of Statistics,Stanford University,Stanford,CA 94305(donoho@).§Department of Management Science and Engineering,Stanford University,Stanford,CA 94305(saunders@).129D o w n l o a d e d 08/09/14 t o 58.19.126.38. R e d i s t r i b u t i o n s u b j e c t t o S I A M l i c e n s e o r c o p y r i g h t ; s e e h t t p ://w w w .s i a m .o r g /j o u r n a l s /o j s a .p h p130S.S.CHEN,D.L.DONOHO,AND M.A.SAUNDERSets,cosine packets,chirplets,warplets,and a wide range of other dictionaries are now available.Each such dictionary D is a collection of waveforms (φγ)γ∈Γ,with γa parameter,and we envision a decomposition of a signal s ass =γ∈Γαγφγ,(1.1)or an approximate decomposition s =m i =1αγi φγi +R (m ),(1.2)where R (m )is a residual.Depending on the dictionary,such a representation de-composes the signal into pure tones (Fourier dictionary),bumps (wavelet dictionary),chirps (chirplet dictionary),etc.Most of the new dictionaries are overcomplete ,either because they start out that way or because we merge complete dictionaries,obtaining a new megadictionary con-sisting of several types of waveforms (e.g.,Fourier and wavelets dictionaries).The decomposition (1.1)is then nonunique,because some elements in the dictionary have representations in terms of other elements.1.1.Goals of Adaptive Representation.Nonuniqueness gives us the possibility of adaptation,i.e.,of choosing from among many representations one that is most suited to our purposes.We are motivated by the aim of achieving simultaneously the following goals .•Sparsity.We should obtain the sparsest possible representation of the object—the one with the fewest significant coefficients.•Superresolution.We should obtain a resolution of sparse objects that is much higher resolution than that possible with traditional nonadaptive approaches.An important constraint ,which is perhaps in conflict with both the goals,follows.•Speed.It should be possible to obtain a representation in order O (n )or O (n log(n ))time.1.2.Finding a Representation.Several methods have been proposed for obtain-ing signal representations in overcomplete dictionaries.These range from general approaches,like the method of frames (MOF)[9]and the method of matching pursuit (MP)[29],to clever schemes derived for specialized dictionaries,like the method of best orthogonal basis (BOB)[7].These methods are described briefly in section 2.3.In our view,these methods have both advantages and shortcomings.The principal emphasis of the proposers of these methods is on achieving sufficient computational speed.While the resulting methods are practical to apply to real data,we show below by computational examples that the methods,either quite generally or in important special cases,lack qualities of sparsity preservation and of stable superresolution.1.3.Basis Pursuit.Basis pursuit (BP)finds signal representations in overcom-plete dictionaries by convex optimization:it obtains the decomposition that minimizes the 1normof the coefficients occurring in the representation.Because of the nondif-ferentiability of the 1norm,this optimization principle leads to decompositions that can have very different properties fromthe MOF—in particular,they can be m uch sparser.Because it is based on global optimization,it can stably superresolve in ways that MP cannot.D o w n l o a d e d 08/09/14 t o 58.19.126.38. R e d i s t r i b u t i o n s u b j e c t t o S I A M l i c e n s e o r c o p y r i g h t ; s e e h t t p ://w w w .s i a m .o r g /j o u r n a l s /o j s a .p h pATOMIC DECOMPOSITION BY BASIS PURSUIT131BP can be used with noisy data by solving an optimization problem trading offa quadratic misfit measure with an 1normof coefficients.Examples show that it can stably suppress noise while preserving structure that is well expressed in the dictionary under consideration.BP is closely connected with linear programming.Recent advances in large-scale linear programming—associated with interior-point methods—can be applied to BP and can make it possible,with certain dictionaries,to nearly solve the BP optimization problem in nearly linear time.We have implemented primal-dual log barrier interior-point methods as part of a MATLAB [31]computing environment called Atomizer,which accepts a wide range of dictionaries.Instructions for Internet access to Atomizer are given in section 7.3.Experiments with standard time-frequency dictionaries indicate some of the potential benefits of BP.Experiments with some nonstandard dictionaries,like the stationary wavelet dictionary and the heaviside dictionary,indicate important connections between BP and methods like Mallat and Zhong’s [29]multiscale edge representation and Rudin,Osher,and Fatemi’s [35]total variation-based denoising methods.1.4.Contents.In section 2we establish vocabulary and notation for the rest of the article,describing a number of dictionaries and existing methods for overcomplete representation.In section 3we discuss the principle of BP and its relations to existing methods and to ideas in other fields.In section 4we discuss methodological issues associated with BP,in particular some of the interesting nonstandard ways it can be deployed.In section 5we describe BP denoising,a method for dealing with problem (1.2).In section 6we discuss recent advances in large-scale linear programming (LP)and resulting algorithms for BP.For reasons of space we refer the reader to [4]for a discussion of related work in statistics and analysis.2.Overcomplete Representations.Let s =(s t :0≤t <n )be a discrete-time signal of length n ;this may also be viewed as a vector in R n .We are interested in the reconstruction of this signal using superpositions of elementary waveforms.Traditional methods of analysis and reconstruction involve the use of orthogonal bases,such as the Fourier basis,various discrete cosine transformbases,and orthogonal wavelet bases.Such situations can be viewed as follows:given a list of n waveforms,one wishes to represent s as a linear combination of these waveforms.The waveforms in the list,viewed as vectors in R n ,are linearly independent,and so the representation is unique.2.1.Dictionaries and Atoms.A considerable focus of activity in the recent sig-nal processing literature has been the development of signal representations outside the basis setting.We use terminology introduced by Mallat and Zhang [29].A dic-tionary is a collection of parameterized waveforms D =(φγ:γ∈Γ).The waveforms φγare discrete-time signals of length n called atoms .Depending on the dictionary,the parameter γcan have the interpretation of indexing frequency,in which case the dictionary is a frequency or Fourier dictionary,of indexing time-scale jointly,in which case the dictionary is a time-scale dictionary,or of indexing time-frequency jointly,in which case the dictionary is a time-frequency ually dictionaries are complete or overcomplete,in which case they contain exactly n atoms or more than n atoms,but one could also have continuum dictionaries containing an infinity of atoms and undercomplete dictionaries for special purposes,containing fewer than n atoms.Dozens of interesting dictionaries have been proposed over the last few years;we focusD o w n l o a d e d 08/09/14 t o 58.19.126.38. R e d i s t r i b u t i o n s u b j e c t t o S I A M l i c e n s e o r c o p y r i g h t ; s e e h t t p ://w w w .s i a m .o r g /j o u r n a l s /o j s a .p h p132S.S.CHEN,D.L.DONOHO,AND M.A.SAUNDERSin this paper on a half dozen or so;much of what we do applies in other cases as well.2.1.1.T rivial Dictionaries.We begin with some overly simple examples.The Dirac dictionary is simply the collection of waveforms that are zero except in one point:γ∈{0,1,...,n −1}and φγ(t )=1{t =γ}.This is of course also an orthogonal basis of R n —the standard basis.The heaviside dictionary is the collection of waveforms that jump at one particular point:γ∈{0,1,...,n −1};φγ(t )=1{t ≥γ}.Atoms in this dictionary are not orthogonal,but every signal has a representation s =s 0φ0+n −1 γ=1(s γ−s γ−1)φγ.(2.1)2.1.2.Frequency Dictionaries.A Fourier dictionary is a collection of sinusoidalwaveforms φγindexed by γ=(ω,ν),where ω∈[0,2π)is an angular frequency variable and ν∈{0,1}indicates phase type:sine or cosine.In detail,φ(ω,0)=cos(ωt ),φ(ω,1)=sin(ωt ).For the standard Fourier dictionary,we let γrun through the set of all cosines with Fourier frequencies ωk =2πk/n ,k =0,...,n/2,and all sines with Fourier frequencies ωk ,k =1,...,n/2−1.This dictionary consists of n waveforms;it is in fact a basis,and a very simple one:the atoms are all mutually orthogonal.An overcomplete Fourier dictionary is obtained by sampling the frequencies more finely.Let be a whole number >1and let Γ be the collection of all cosines with ωk =2πk/( n ),k =0,..., n/2,and all sines with frequencies ωk ,k =1,..., n/2−1.This is an -fold overcomplete system.We also use complete and overcomplete dictionaries based on discrete cosine transforms and sine transforms.2.1.3.Time-Scale Dictionaries.There are several types of wavelet dictionaries;to fix ideas,we consider the Haar dictionary with “father wavelet”ϕ=1[0,1]and “mother wavelet”ψ=1(1/2,1]−1[0,1/2].The dictionary is a collection of transla-tions and dilations of the basic mother wavelet,together with translations of a father wavelet.It is indexed by γ=(a,b,ν),where a ∈(0,∞)is a scale variable,b ∈[0,n ]indicates location,and ν∈{0,1}indicates gender.In detail,φ(a,b,1)=ψ(a (t −b ))·√a,φ(a,b,0)=ϕ(a (t −b ))·√a.For the standard Haar dictionary,we let γrun through the discrete collection ofmother wavelets with dyadic scales a j =2j /n ,j =j 0,...,log 2(n )−1,and locations that are integer multiples of the scale b j,k =k ·a j ,k =0,...,2j −1,and the collection of father wavelets at the coarse scale j 0.This dictionary consists of n waveforms;it is an orthonormal basis.An overcomplete wavelet dictionary is obtained by sampling the locations more finely:one location per sample point.This gives the so-called sta-tionary Haar dictionary,consisting of O (n log 2(n ))waveforms.It is called stationary since the whole dictionary is invariant under circulant shift.A variety of other wavelet bases are possible.The most important variations are smooth wavelet bases,using splines or using wavelets defined recursively fromtwo-scale filtering relations [10].Although the rules of construction are more complicated (boundary conditions [33],orthogonality versus biorthogonality [10],etc.),these have the same indexing structure as the standard Haar dictionary.In this paper,we use symmlet -8smooth wavelets,i.e.,Daubechies nearly symmetric wavelets with eight vanishing moments;see [10]for examples.D o w n l o a d e d 08/09/14 t o 58.19.126.38. R e d i s t r i b u t i o n s u b j e c t t o S I A M l i c e n s e o r c o p y r i g h t ; s e e h t t p ://w w w .s i a m .o r g /j o u r n a l s /o j s a .p h pATOMIC DECOMPOSITION BY BASIS PURSUIT133Time 00.5100.20.40.60.81(c) Time DomainFig.2.1Time-frequency phase plot of a wavelet packet atom.2.1.4.Time-Frequency Dictionaries.Much recent activity in the wavelet com-munities has focused on the study of time-frequency phenomena.The standard ex-ample,the Gabor dictionary,is due to Gabor [19];in our notation,we take γ=(ω,τ,θ,δt ),where ω∈[0,π)is a frequency,τis a location,θis a phase,and δt is the duration,and we consider atoms φγ(t )=exp {−(t −τ)2/(δt )2}·cos(ω(t −τ)+θ).Such atoms indeed consist of frequencies near ωand essentially vanish far away from τ.For fixed δt ,discrete dictionaries can be built fromtim e-frequency lattices,ωk =k ∆ωand τ = ∆τ,and θ∈{0,π/2};with ∆τand ∆ωchosen sufficiently fine these are complete.For further discussions see,e.g.,[9].Recently,Coifman and Meyer [6]developed the wavelet packet and cosine packet dictionaries especially to meet the computational demands of discrete-time signal pro-cessing.For one-dimensional discrete-time signals of length n ,these dictionaries each contain about n log 2(n )waveforms.A wavelet packet dictionary includes,as special cases,a standard orthogonal wavelets dictionary,the Dirac dictionary,and a collec-tion of oscillating waveforms spanning a range of frequencies and durations.A cosine packet dictionary contains,as special cases,the standard orthogonal Fourier dictio-nary and a variety of Gabor-like elements:sinusoids of various frequencies weighted by windows of various widths and locations.In this paper,we often use wavelet packet and cosine packet dictionaries as exam-ples of overcomplete systems,and we give a number of examples decomposing signals into these time-frequency dictionaries.A simple block diagram helps us visualize the atoms appearing in the decomposition.This diagram,adapted from Coifman and Wickerhauser [7],associates with each cosine packet or wavelet packet a rectangle in the time-frequency phase plane.The association is illustrated in Figure 2.1for a cer-tain wavelet packet.When a signal is a superposition of several such waveforms,we indicate which waveforms appear in the superposition by shading the corresponding rectangles in the time-frequency plane.D o w n l o a d e d 08/09/14 t o 58.19.126.38. R e d i s t r i b u t i o n s u b j e c t t o S I A M l i c e n s e o r c o p y r i g h t ; s e e h t t p ://w w w .s i a m .o r g /j o u r n a l s /o j s a .p h p134S.S.CHEN,D.L.DONOHO,AND M.A.SAUNDERS2.1.5.Further Dictionaries.We can always merge dictionaries to create mega-dictionaries;examples used below include mergers of wavelets with heavisides.2.2.Linear Algebra.Suppose we have a discrete dictionary of p waveforms and we collect all these waveforms as columns of an n -by-p matrix Φ,say.The decompo-sition problem(1.1)can be written Φα=s ,(2.2)where α=(αγ)is the vector of coefficients in (1.1).When the dictionary furnishes a basis,then Φis an n -by-n nonsingular matrix and we have the unique representation α=Φ−1s .When the atoms are,in addition,mutually orthonormal,then Φ−1=ΦT and the decomposition formula is very simple.2.2.1.Analysis versus Synthesis.Given a dictionary of waveforms,one can dis-tinguish analysis from synthesis .Synthesis is the operation of building up a signal by superposing atoms;it involves a matrix that is n -by-p :s =Φα.Analysis involves the operation of associating with each signal a vector of coefficients attached to atoms;it involves a matrix that is p -by-n :˜α=ΦT s .Synthesis and analysis are very differ-ent linear operations,and we must take care to distinguish them.One should avoid assuming that the analysis operator ˜α=ΦT s gives us coefficients that can be used as is to synthesize s .In the overcomplete case we are interested in,p n and Φis not invertible.There are then many solutions to (2.2),and a given approach selects a particular solution.One does not uniquely and automatically solve the synthesis problemby applying a sim ple,linear analysis operator.We now illustrate the difference between synthesis (s =Φα)and analysis (˜α=ΦTs ).Figure 2.2a shows the signal Carbon .Figure 2.2b shows the time-frequency structure of a sparse synthesis of Carbon ,a vector αyielding s =Φα,using a wavelet packet dictionary.To visualize the decomposition,we present a phase-plane display with shaded rectangles,as described above.Figure 2.2c gives an analysis of Carbon ,with the coefficients ˜α=ΦT s ,again displayed in a phase plane.Once again,between analysis and synthesis there is a large difference in sparsity.In Figure 2.2d we compare the sorted coefficients of the overcomplete representation (synthesis)with the analysis coefficients.putational Complexity of Φand ΦT .Different dictionaries can im-pose drastically different computational burdens.In this paper we report compu-tational experiments on a variety of signals and dictionaries.We study primarily one-dimensional signals of length n ,where n is several thousand.Signals of this length occur naturally in the study of short segments of speech (a quarter-second to a half-second)and in the output of various scientific instruments (e.g.,FT-NMR spec-trometers).In our experiments,we study dictionaries overcomplete by substantial factors,say,10.Hence the typical matrix Φwe are interested in is of size “thousands”by “tens-of-thousands.”The nominal cost of storing and applying an arbitrary n -by-p matrix to a p -vector is a constant times np .Hence with an arbitrary dictionary of the sizes we are interested in,simply to verify whether (1.1)holds for given vectors αand s would require tens of millions of multiplications and tens of millions of words of memory.In contrast,most signal processing algorithms for signals of length 1000require only thousands of memory words and a few thousand multiplications.Fortunately,certain dictionaries have fast implicit algorithms .By this we mean that Φαand ΦT s can be computed,for arbitrary vectors αand s ,(a)without everD o w n l o a d e d 08/09/14 t o 58.19.126.38. R e d i s t r i b u t i o n s u b j e c t t o S I A M l i c e n s e o r c o p y r i g h t ; s e e h t t p ://w w w .s i a m .o r g /j o u r n a l s /o j s a .p h pATOMIC DECOMPOSITION BY BASIS PURSUIT135Time0.5100.20.40.60.81Time0.5100.20.40.60.81(d) Sorted CoefficientsSynthesis: SolidAnalysis: Dashed Fig.2.2Analysis versus synthesis of the signal Carbon .storing the matrices Φand ΦT ,and (b)using special properties of the matrices to accelerate computations.The most well-known example is the standard Fourier dictionary for which we have the fast Fourier transform algorithm.A typical implementation requires 2·n storage locations and 4·n ·J multiplications if n is dyadic:n =2J .Hence for very long signals we can apply Φand ΦT with much less storage and time than the matrices would nominally require.Simple adaptation of this idea leads to an algorithm for overcomplete Fourier dictionaries.Wavelets give a more recent example of a dictionary with a fast implicit algorithm;if the Haar or S8-symmlet is used,both Φand ΦT may be applied in O (n )time.For the stationary wavelet dictionary,O (n log(n ))time is required.Cosine packets and wavelet packets also have fast implicit algorithms.Here both Φand ΦT can be applied in order O (n log(n ))time and order O (n log(n ))space—much better than the nominal np =n 2log 2(n )one would expect fromnaive use of the m atrix definition.For the viewpoint of this paper,it only makes sense to consider dictionaries with fast implicit algorithms.Among dictionaries we have not discussed,such algorithms may or may not exist.2.3.Existing Decomposition Methods.There are several currently popular ap-proaches to obtaining solutions to (2.2).2.3.1.Frames.The MOF [9]picks out,among all solutions of (2.2),one whose coefficients have minimum l 2norm:min α 2subject toΦα=s .(2.3)The solution of this problemis unique;label it α†.Geometrically,the collection of all solutions to (2.2)is an affine subspace in R p ;MOF selects the element of this subspace closest to the origin.It is sometimes called a minimum-length solution.There is aD o w n l o a d e d 08/09/14 t o 58.19.126.38. R e d i s t r i b u t i o n s u b j e c t t o S I A M l i c e n s e o r c o p y r i g h t ; s e e h t t p ://w w w .s i a m .o r g /j o u r n a l s /o j s a .p h p136S.S.CHEN,D.L.DONOHO,AND M.A.SAUNDERSTime0.5100.20.40.60.81Time0.5100.20.40.60.81Fig.2.3MOF representation is not sparse.matrix Φ†,the generalized inverse of Φ,that calculates the minimum-length solution to a systemof linear equations:α†=Φ†s =ΦT (ΦΦT )−1s .(2.4)For so-called tight frame dictionaries MOF is available in closed form.A nice example is the standard wavelet packet dictionary.One can compute that for all vectors v ,ΦT v 2=L n · v 2,L n =log 2(n ).In short Φ†=L −1n ΦT .Notice that ΦTis simply the analysis operator.There are two key problems with the MOF.First,MOF is not sparsity preserving .If the underlying object has a very sparse representation in terms of the dictionary,then the coefficients found by MOF are likely to be very much less sparse.Each atom in the dictionary that has nonzero inner product with the signal is,at least potentially and also usually,a member of the solution.Figure 2.3a shows the signal Hydrogen made of a single atom in a wavelet packet dictionary.The result of a frame decomposition in that dictionary is depicted in a phase-plane portrait;see Figure 2.3c.While the underlying signal can be synthesized from a single atom,the frame decomposition involves many atoms,and the phase-plane portrait exaggerates greatly the intrinsic complexity of the object.Second,MOF is intrinsically resolution limited .No object can be reconstructed with features sharper than those allowed by the underlying operator Φ†Φ.Suppose the underlying object is sharply localized:α=1{γ=γ0}.The reconstruction will not be α,but instead Φ†Φα,which,in the overcomplete case,will be spatially spread out.Figure 2.4presents a signal TwinSine consisting of the superposition of two sinusoids that are separated by less than the so-called Rayleigh distance 2π/n .We analyze these in a fourfold overcomplete discrete cosine dictionary.In this case,reconstruction by MOF (Figure 2.4b)is simply convolution with the Dirichlet kernel.The result is the synthesis fromcoefficients with a broad oscillatory appearance,consisting not of twoD o w n l o a d e d 08/09/14 t o 58.19.126.38. R e d i s t r i b u t i o n s u b j e c t t o S I A M l i c e n s e o r c o p y r i g h t ; s e e h t t p ://w w w .s i a m .o r g /j o u r n a l s /o j s a .p h pATOMIC DECOMPOSITION BY BASIS PURSUIT137Fig.2.4Analyzing TwinSine with a fourfold overcomplete discrete cosine dictionary.but of many frequencies and giving no visual clue that the object may be synthesized fromtwo frequencies alone.2.3.2.Matching Pursuit.Mallat and Zhang [29]discussed a general method for approximate decomposition (1.2)that addresses the sparsity issue directly.Starting froman initial approxim ation s (0)=0and residual R (0)=s ,it builds up a sequence of sparse approximations stepwise.At stage k ,it identifies the dictionary atomthat best correlates with the residual and then adds to the current approximation a scalar multiple of that atom,so that s (k )=s (k −1)+αk φγk ,where αk = R (k −1),φγk and R (k )=s −s (k ).After m steps,one has a representation of the form(1.2),with residual R =R (m ).Similar algorithms were proposed by Qian and Chen [39]for Gabor dictionaries and by Villemoes [48]for Walsh dictionaries.A similar algorithm was proposed for Gabor dictionaries by Qian and Chen [39].For an earlier instance of a related algorithm,see [5].An intrinsic feature of the algorithmis that when stopped after a few steps,it yields an approximation using only a few atoms.When the dictionary is orthogonal,the method works perfectly.If the object is made up of only m n atoms and the algorithmis run for m steps,it recovers the underlying sparse structure exactly.When the dictionary is not orthogonal,the situation is less clear.Because the algorithmis m yopic,one expects that,in certain cases,it m ight choose wrongly in the first few iterations and end up spending most of its time correcting for any mistakes made in the first few terms.In fact this does seem to happen.To see this,we consider an attempt at superresolution.Figure 2.4a portrays again the signal TwinSine consisting of sinusoids at two closely spaced frequencies.When MP is applied in this case (Figure 2.4c),using the fourfold overcomplete discrete cosine dictionary,the initial frequency selected is in between the two frequencies making up the signal.Because of this mistake,MP is forced to make a series of alternating corrections that suggest a highly complex and organized structure.MPD o w n l o a d e d 08/09/14 t o 58.19.126.38. R e d i s t r i b u t i o n s u b j e c t t o S I A M l i c e n s e o r c o p y r i g h t ; s e e h t t p ://w w w .s i a m .o r g /j o u r n a l s /o j s a .p h p138S.S.CHEN,D.L.DONOHO,AND M.A.SAUNDERSFig.2.5Counterexamples for MP.misses entirely the doublet structure.One can certainly say in this case that MP has failed to superresolve.Second,one can give examples of dictionaries and signals where MP is arbitrarily suboptimal in terms of sparsity.While these are somewhat artificial,they have a character not so different fromthe superresolution exam ple.DeVore and Temlyakov’s Example.Vladimir Temlyakov,in a talk at the IEEE Confer-ence on Information Theory and Statistics in October 1994,described an example in which the straightforward greedy algorithmis not sparsity preserving.In our adapta-tion of this example,based on Temlyakov’s joint work with DeVore [12],one constructs a dictionary having n +1atoms.The first n are the Dirac basis;the final atomin-volves a linear combination of the first n with decaying weights.The signal s has an exact decomposition in terms of A atoms,but the greedy algorithm goes on forever,with an error of size O (1/√m )after m steps.We illustrate this decay in Figure 2.5a.For this example we set A =10and choose the signal s t =10−1/2·1{1≤t ≤10}.The dictionary consists of Dirac elements φγ=δγfor 1≤γ≤n andφn +1(t )=c,1≤t ≤10,c/(t −10),10<t ≤n,with c chosen to normalize φn +1to unit norm.Shaobing Chen’s Example.The DeVore–Temlyakov example applies to the original MP algorithmas announced by Mallat and Zhang in 1992.A later refinem ent of the algorithm(see Pati,Rezaiifar,and Krishnaprasad [38]and Davis,Mallat,and Zhang [11])involves an extra step of orthogonalization.One takes all m terms that have entered at stage m and solves the least-squares problemmin (αi )s −m i =1αi φγi2D o w n l o a d e d 08/09/14 t o 58.19.126.38. R e d i s t r i b u t i o n s u b j e c t t o S I A M l i c e n s e o r c o p y r i g h t ; s e e h t t p ://w w w .s i a m .o r g /j o u r n a l s /o j s a .p h p。
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