Search for non-Gaussianity in pixel, harmonic and wavelet space compared and combined

合集下载

MEMS振镜扫描共聚焦图像畸变机理分析及校正

2021年2月Feb. 2021第50卷第2期Vol.50 No.2红外与激光工程Infrared and Laser EngineeringMEMS 振镜扫描共聚焦图像畸变机理分析及校正缪新叫李航锋1,张运海心，王发民込施辛彳(1.中国科学技术大学，安徽合肥230026；2.中国科学院苏州生物医学工程技术研究所江苏省医用光学重点实验室，江苏苏州215163;3.苏州大学附属第二医院，江苏苏州215000)摘要：在皮肤反射式共聚焦显微成像过程中，针对MEMS 振镜二维扫描引起的共聚焦图像畸变，开展了光束偏转理论分析，得出了投影面扫描图像的具体形状表征,理论畸变图像与真实畸变图像一致，明确了畸变机理，提出一种有效的畸变校正算法，实现对图像二维畸变的校正。

首先记录原始光栅畸变图像，然后基于Hessian 矩阵提取光栅中心线，拾取特征点并设置基准参考线，通过基于最小二乘法的7次多项式插值法标定二维方向像素畸变校正量，采用加权平均法填补间隙像素灰度值，最终实现图像畸变校正。

利用网格畸变测试靶实验得出7次多项式插值后的校正决定系数最高、均方根误差值最低，整幅512行图像在7次多项式插值后最优行数占379行，比例为74%,通过残差分析，二维方向上残差最大为4个像素，最小为0个像素，平均为1.15个像素，校正结果较为精确。

皮肤在体实时成像实验显示，图像畸变校正后组织结构特征更加真实准确，表明这种校正算法有效可行，有助于皮肤疾病的准确诊断。

关键词：图像二维畸变；机理分析；Hessian 矩阵；光栅；多项式插值中图分类号：TH742.9文献标志码：A DOI ： 10.3788/IRLA20200206Analysis and correction of image distortion in MEMSgalvanometer scanning confocal systemMiao Xin 12, Li Hangfeng 1, Zhang Yunhai 1,2*, Wang Famin 1,2, Shi Xin 3(1. University of Science and Technology of China, Hefei 230026, China;2. Jiangsu Key Laboratory of Medical Optics, Suzhou Institute of Biomedical Engineering and Technology,Chinese Academy of Sciences, Suzhou 215163, China;3. The Second Affiliated Hospital of Soochow University, Suzhou 215000, China)Abstract: Aiming at the distorted confocal images caused by the two-dimensional scanning of MEMS galvanometer during skin imaging by reflectance confocal microscopy, the theoretical analysis of beam deflectionwas carried out, and the specific shape representation of projection plane scanning image was obtained. It was concluded that the theoretical distortion image was consistent with the real distortion image. The distortionmechanism was clarified and a distortion correction method was proposed. First, the original distorted grating image was recorded, then the center lines of grating were obtained based on the Hessian matrix, after that feature points were picked and datum reference lines were set. Finally, the correction to the distorted confocal images wasrealized by calibrating the corrections of the two-dimensional pixel distortions using polynomial interpolation收稿日期：2020-10-12；修订日期:2020-11-15基金项目:国家重点研发计划(2017YFC0110305)；山东省自然科学基金(ZR2019BF012)；济南市“高校20条”资助项目(2018GXRC018)；苏州市民生科技项目(SS201643)红外与激光工程第50卷第2期based on the least square method and filling the gray value of gap pixels by weighted average method.By the experiment of measuring target with grid distortion,the correction coefficient was the highest and the root mean square error was the lowest after polynomial interpolation of degree7.Also,the optimal number of512rows was 379,accounting for74%.The residual distortions were accurately evaluated,in two dimensional,the maximum value is4pixels,the minimum value was0pixel and the average value was L15pixels,so the results were accurate.The experiment of in vivo real-time skin imaging shows that the organizational structure features are more real and accurate after corrections.So this method is effective and feasible,which is helpful for accurate diagnosis of skin diseases.Key words:two-dimensional distortions of images;polynomial interpolation0引言作为一款新型影像学临床诊断设备，皮肤反射式共聚焦显振镜利用皮肤中血红蛋白、黑色素和角蛋白等不同组织成分的折射率差异进行成像，为皮肤组织的实时观测提供有效的技术手段，在恶性皮肤肿瘤早期诊断、治疗后随访等方面发挥了愈发重要的作用"役为进一步实现皮肤病检查时的便捷性,需要发展手持式皮肤共聚焦显振镜，由于受到系统体积和重量限制，系统中的核心部件扫描振镜要采用单镜面式MEMS振镜实现二维扫描成像,该扫描方式使得采集到的图像存在较为严重的二维畸变，扭曲了皮肤组织真实的结构形态.如不对这种图像畸变进行校正，将不利于医生观察皮损组织真实形态、边界轮廓、结构特征等信息，直接影响临床诊断结果因此，需要在分析产生图像畸变机理的基础之上，实现畸变校正，将真实图像信息准确呈现，为皮肤疾病诊断奠定基础。

磁荷对Hayward-Anti-deSitter黑洞的全息互信息的影响

第41卷第6期2023年12月沈阳师范大学学报(自然科学版)J o u r n a l o f S h e n y a n g N o r m a lU n i v e r s i t y(N a t u r a l S c i e n c eE d i t i o n)V o l.41N o.6D e c.2023文章编号:16735862(2023)06056804磁荷对H a y w a r d-A n t i-d eS i t t e r黑洞的全息互信息的影响李慧玲,张宁,张宝琪,李瑶(沈阳师范大学物理科学与技术学院,沈阳110034)摘要:H a y w a r d黑洞是爱因斯坦引力非线性耦合一个携带磁荷的电磁场的解析解,是非线性磁单极子引力场的简并结构㊂一般情况下,黑洞的内部会存在奇点,而非奇异黑洞是一种内部没有奇点的黑洞㊂H a y w a r d黑洞属于非奇异黑洞,此规则黑洞的对称性由磁势决定,带磁荷和不带磁荷的黑洞具有不同的微观结构㊂利用纠缠熵讨论磁荷对非奇异H a y w a r d-A n t i-d eS i t t e r黑洞中全息互信息的影响㊂结果表明,随着条带宽度(子区域)的增加,2个渐进子系统纠缠增大,且全息互信息随磁荷的增加而降低㊂除此之外,存在临界磁荷使得全息互信息为零,此时对偶的子区域之间不存在纠缠,磁荷取不同值时,全息互信息消失的条带宽度临界值是不同的㊂关键词:全息互信息;纠缠熵;磁荷;A d S黑洞中图分类号:P145.8文献标志码:Ad o i:10.3969/j.i s s n.16735862.2023.06.014E f f e c to fm a g n e t i cc h a r g e so nh o l o g r a p h i cm u t u a l i n f o r m a t i o no fH a y w a r d-A n t i-d e S i t t e r b l a c kh o l e sL IH u i l i n g,Z HA N GN i n g,Z HA N GB a o q i,L IY a o(C o l l e g e o f P h y s i c a l S c i e n c e a n dT e c h n o l o g y,S h e n y a n g N o r m a lU n i v e r s i t y,S h e n y a n g110034,C h i n a)A b s t r a c t:H a y w a r d b l a c k h o l ei sa n a n a l y t i c a ls o l u t i o n o f E i n s t e i n s g r a v i t a t i o n a ln o n l i n e a rc o u p l i n g o f a n e l e c t r o m a g n e t i c f i e ld c a r r y i n g m a g ne t i c c h a r g e s.I n g e n e r a l,t h e r e w i l l b es i n g u l a r i t i e s i n s i d eb l a c kh o l e s,a n d"n o n-s i n g u l a r"b l a c kh o l e sa r eb l a c kh o l e s w i t h o u t i n t e r n a l s i n g u l a r i t i e s,a n d H a y w a r d b l a c k h o l e sa r en o n-s i n g u l a r.H a y w a r d b l a c k h o l ei sad e g e n e r a t e s t r u c t u r eo f t h e n o n l i n e a rm a g n e t i cm o n o p o l e g r a v i t a t i o n a l f i e l d.T h e s y mm e t r y o f t h e r e g u l a r b l a c kh o l e i s d e t e r m i n e db y t h em a g n e t i c p o t e n t i a l,a n d t h em a g n e t i c c h a r g e a n d t h e n o n-m a g n e t i c c h a r g eh a v e d i f f e r e n tm i c r o s t r u c t u r e s.W e d i s c u s s t h e i n f l u e n c e o fm a g n e t i c c h a r g e o nh o l o g r a p h i cm u t u a li n f o r m a t i o n i nn o n s i n g u l a rH a y w a r d-A n t i-d eS i t t e r b l a c kh o l eb y u s i n g e n t a n g l e m e n t e n t r o p y.T h er e s u l t s s h o wt h a t t h e e n t a n g l e m e n t o f t h e t w o a s y m p t o t i c s u b s y s t e m s i n c r e a s e sw i t h t h e i n c r e a s e o f t h e s t r i p w i d t h(s u b r e g i o n),a n d t h e h o l o g r a p h i cm u t u a l i n f o r m a t i o nd e c r e a s e sw i t h t h e i n c r e a s e o f m a g n e t i c c h a r g e,I n a d d i t i o n,t h e r e i s a c r i t i c a lm a g n e t i c c h a r g e t h a tm a k e s t h e h o l o g r a p h i cm u t u a li n f o r m a t i o n z e r o,a n d t h e r e i s n o e n t a n g l e m e n t b e t w e e n t h e s u b r e g i o n s o f t h e d u a l i t y,a n dw h e n t h em a g n e t i c c h a r g et a k e sd i f f e r e n tv a l u e s,t h ec r i t i c a lv a l u eo ft h es t r i p e w i d t ho ft h eh o l o g r a p h i c m u t u a l i n f o r m a t i o nd i s a p p e a r i n g i s d i f f e r e n t.K e y w o r d s:h o l o g r a p h i cm u t u a l i n f o r m a t i o n;e n t a n g l e d e n t r o p y;m a g n e t i c c h a r g e;A n t i-d e S i t t e r(A d s)b l a c kh o l e收稿日期:20230713基金项目:辽宁省教育厅科学研究经费项目(L J KM20221474)㊂作者简介:李慧玲(1977 ),女,辽宁沈阳人,沈阳师范大学教授,博士㊂黑洞是广义相对论中最具有深远意义的预言之一,多年来,人们一直在研究宇宙中这个神秘天体㊂对于非奇异H a y w a r d -A d S 黑洞,带磁荷和不带磁荷的黑洞具有不同的微观结构㊂全息互信息测量了量子信息理论中2个子系统之间的相关性[1],可以通过计算连接一个永恒A d S 黑洞两侧的虫洞长度[2]获得全息互信息㊂热场二重态(t h e r m a l f i e l dd o u b l e s t a t e ,T F D )可以描述黑洞两侧纠缠态[3],即Ψ>ʉðie -β2E ii >L 췍i >R其中:β是温度的倒数;i >L 和i >R 是两侧的A d S 黑洞上相同的量子态㊂假设黑洞的每一侧存在2个完全相同的类空子区域A 和B ,则A 和B 之间的全息互信息I (A ,B )可以表示为[4]I (A ,B )ʉS (A )+S (B )-S (A ɣB )其中:S (A ),S (B )是最小表面A 和B 上类空区域的纠缠熵;S (A ɣB )是穿过事件视界连接A 和B 的区域的纠缠熵㊂1 非奇异H a y w a r d -A d S 黑洞的全息互信息非奇异H a y w a r d -A d S 黑洞[56]是爱因斯坦引力与携带磁荷的电磁场非线性耦合的解析解㊂在四维A d S 背景下的H a y w a r d -A d S 黑洞解[78]为d s 2=-f (r )d t 2+d r 2f (r)+r 2dΩ2(1)其中,d Ω2=d θ2+s i n 2θd φ2,度规函数为f (r )=1-2M r 2g 3+r3-Λ3r 2(2)式(2)中得到的参数g 与黑洞总磁荷Q m 有关,即Q m =g22α(3)α为自由积分常数㊂永恒黑洞有2个渐近的A d S 区域,其可以用2个相同的㊁无相互作用的共形场论[4]的T F D 来进行全息描述㊂为了方便计算,令左渐近边界上的子区域A 和右渐近边界上的B 完全相同,即A =B ㊂在四维背景下,将A d S 黑洞边界参数化为(x ,y )的二维空间㊂将子区域A 或B 看作一条带,其宽度为x ɪ(0,x 0),且沿y 方向延伸,长度为Y ,Y ʉ1㊂因此,子区A 的纠缠熵S (A )=R e g i o n A /4,其中R e g i o n A 是最小表面的面积,即R e g i o n A =ʏx 00d x r ᶄ21-2M r 2g 3+r3+r 2+r 2(4)其中,r ᶄ=d r /d x ㊂如果将等式(4)中的被积函数看作拉格朗日函数 L ,定义一个与x 方向平移相关的守恒量为r3r 2+r ᶄ21-2M r 2g 3+r3+r 2=r 2m i n(5)r m i n 为r ᶄ=0时的转折点㊂根据其表面对称性,转折点位于x =x 0/2㊂根据守恒方程(5),x 0为x 0=ʏx 0d x =2ʏɕr m i n d r r 1-2M r 2g 3+r3r 21(r /r m i n )4-1(6)式(6)中的最小面积为R e g i o n A =2ʏɕr m i n d r 1-2M r 2g 3+r3+r 21(r /r m i n )4-1(7) 由于B 与A 相同,所以R e g i o n B 与R e g i o n A 也是相同的㊂通过黑洞视界连接区域A (左)和B (右)的最小表面面积为R e g i o n A ɣB ,对应的纠缠熵为S (A ɣB )965第6期李慧玲,等:磁荷对H a y w a r d -A n t i -d eS i t t e r 黑洞的全息互信息的影响=R e g i o n A ɣB /4㊂两侧的总面积R e g i o n A ɣB 可表示为R e g i o n A ɣB =4ʏɕr hd r 11-2M r 2g 3+r3+r 2r (8) 根据全息互信息表达式I (A ,B )ʉS (A )+S (B )-S (A ɣB ),结合式(7)和式(8),得到I (g )=ʏɕr m i n d r r 1-2M r 2g 3+r3+r 211-(r m i n /r )4-ʏɕr hd rr1-2M r 2g 3+r3+r 2(9)此即H a y w a r d -A d S 黑洞的全息互信息㊂2 磁荷对全息互信息的影响要讨论静态A d S 背景下磁荷对全息互信息的影响,首先要研究非奇异H a yw a r d 黑洞的全息互信息与条带宽度的关系㊂将式(6)代入到式(8)中,得到条带的宽度x 0与全息互信息I (x 0,g )的关系I (x 0,g )=12x 0r 2m i n +ʏɕr m i nd rr 1-2M r 2g 3+r3+r 21-(r m i n/r )4-ʏɕr h d rr 1-2M r 2g 3+r3+r 2㊂(10) 当I (x 0,g )=0时全息互信息会消失㊂因此,可得到全息互信息消失时条带宽度的临界值x 0c 为x 0c =2r 2m i n ʏɕr h d r r1-2M r 2g 3+r3+r 2-ʏɕr m i nd rr 1-2M r 2g 3+r3+r 21-(r m i n/r )éëêêêùûúúú4(11)图1 条带的宽度x 0和转折点位置r m i n 之间的关系(令r h =1,g =0.4)F i g .1 T h e r e l a t i o n s h i p be t w e e n t h ew i d t hof t h e s t r i p a n d t h e l o c a t i o no f t h e t u r n i n gpo i n t 图2 全息互信息I (x 0,g )和转折点位置r m i n 之间的关系(令r h =1,g =0.4)F i g .2 T h e r e l a t i o n s h i p b e t w e e n h o l o g r a ph i cm u t u a l i n f o r m a t i o na n d t u r n i n gpo i n t l o c a t i o n 首先研究条带的宽度x 0对转折点位置r m i n 的影响㊂根据式(6)中条带宽度x 0与转折点位置r m i n 之间的关系,绘制如图1所示的图像㊂如图1所示,当r m i n ңr h 时,条带宽度x 0的积分发散,式(6)中所表现出的趋势与图1中的图像是一致的㊂r m i n 越大对应的条带宽度越小㊂由式(10)和式(11)可知,当r m i n ңr h 时2个边界上条带的宽度几乎是发散的,所以全息互信息也是发散的,这与在图2中所绘制的趋势也是一致的㊂发散条带的全息互信息也会是发散的㊂如图2所示,当r m i n ʈ1.269时,全息互信息消失㊂也就说明全息互信息消失存在一个临界值㊂结合图1和图2,发现全息互信息I (x 0,g )和条带宽度x 0之间也存在密切关系,从而作出图3进行进一步研究㊂在图3中,可以清晰地看出,当磁荷g 取不同值时,全息互信息I (x 0,g )消失的条带宽度临界值x 0c 是不同的㊂其数值结果为g =0.4时,x 0c ʈ1.77;g =0.6时,x 0c ʈ1.89;g =0.8时,x 0c ʈ2.15㊂磁荷g 越大,其临界宽度的值x 0c 也越大㊂且当全息互信息I (x 0,g )为某一值时,对应的磁荷g 不同㊂意味着磁荷g 也会对全息互信息I (x 0,g )产生重要影响㊂从图3中还发现全息互信息I (x 0,g )总是随着条带宽度x 0的增加而增加㊂可见2个渐近边界075沈阳师范大学学报(自然科学版) 第41卷上的子系统更大,纠缠也更大㊂(从上到下磁荷g 分别为g =0.4,g =0.6,g =0.8)图3 全息互信息I (x 0,g )和条带宽度x 0之间的关系F i g .3 T h e r e l a t i o n s h i p b e t w e e n h o l o g r a ph i cm u t u a l i n f o r m a t i o na n d s t r i pew i d t h (从上到下的曲线对应于r m i n 从1.18增加到1.21,步长为0.01)图4 全息互信息I (x 0,g )和磁荷g 之间的关系F i g .4 T h e r e l a t i o n s h i p b e t w e e n h o l o g r a ph i cm u t u a l i n f o r m a t i o na n dm a g n e t i c c h a r ge 接下来,研究非奇异H a y w a r d 黑洞的磁荷g 对全息互信息I (x 0,g )的影响㊂由图4可见,对于每条曲线,全息互信息随着磁荷的增加而减小㊂存在一个临界磁荷g c 使得全息互信息为零,此时对偶的子区域之间不存在纠缠㊂图4中的4条曲线从上到下对应r m i n 从1.18增加到1.21,步长为0.01㊂由此发现,当磁荷g 为固定值时,r m i n 越大,全息互信息I (x 0,g )越小㊂且由图1可知,r m i n 与条带宽度x 0有关,r m i n 越大,边界上的条带宽度x 0越小,图4中的结论与图2中的结果相一致㊂当条带宽度固定时,随着温度的升高,2条条带的全息互信息也会增加㊂条带的临界宽度x 0c 是使得互信息消失的宽度,即I (x 0c ,g )=0㊂在图4中,随着磁荷的增大,全息互信息单调递减㊂3 结论对非奇异H a y w a r d -A d S 黑洞的全息互信息的研究表明:在静态情况下,当r m i n ңr h 时全息互信息是发散的,且随着条带宽度的增加而增加,说明2个渐进子系统更大则纠缠也更大;磁荷对全息互信息有直接影响,全息互信息会随着磁荷的增加而减小,当磁荷增加到临近值g c 时,I (x 0c ,g )=0,即全息互信息消失,可见磁荷对H a y w a r d 黑洞全息互信息产生重要影响㊂参考文献:[1]N I E L S E N M A ,C HU A N GI .Q u a n t u m c o m p u t a t i o na n d q u a n t u m i n f o r m a t i o n [J ].A m JP h y s ,2002,70(5):558559.[2]C A IR G ,Z E N G X X ,Z HA N G H Q.I n f l u e n c eo f i n h o m o g e n e i t i e so nh o l o g r a p h i cm u t u a l i n f o r m a t i o na n db u t t e r f l y e f f e c t [J ].JH i g hE n e r g y P h y s ,2017,2017(7):120.[3]MA L D A C E N AJ ,S U S S K I N DL .C o o l h o r i z o n s f o r e n t a n g l e db l a c kh o l e s [J ].F o r t s c h rP h y s ,2013,61(9):781811.[4]S H E N K E RS H ,S T A N F O R D D.B l a c kh o l e sa n dt h eb u t t e r f l y e f f e c t [J ].J H i g h E n e r g y P h y s ,2013,2014(3):125.[5]T O R R E S R.N o n s i n g u l a rb l a c kh o l e s ,t h ec o s m o l o g i c a l c o n s t a n t ,a n da s y m p t o t i cs a f e t y [J ].P h y sR e vD ,2017,95(12):124004.[6]Z E N G XX ,L I U X M ,L I U W B .H o l o g r a p h i c t h e r m a l i z a t i o nw i t h a c h e m i c a l p o t e n t i a l i nG a u s s -B o n n e t g r a v i t y [J ].J H i g hE n e r g y P h y s ,2014,2014(3):124.[7]Z E N G XX ,L I U W B .H o l o g r a p h i c t h e r m a l i z a t i o n i nG a u s s -B o n n e t g r a v i t y [J ].P h y sL e t t B ,2013,726(6):481487.[8]P A R KC .H o l o g r a p h i c r e n o r m a l i z a t i o n i nd e n s em e d i u m [J ].A d vH i g hE n e r g y P h ys ,2014,2014(9):565219.175第6期李慧玲,等:磁荷对H a y w a r d -A n t i -d eS i t t e r 黑洞的全息互信息的影响。

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。

基于亮度先验的星图杂散光噪声去除方法

文章编号：1002-2082 (2021) 03-0454-08基于亮度先验的星图杂散光噪声去除方法张　新1，林　彬1，杨　夏1，王鲲鹏2，张小虎1（1. 中山大学航空航天学院，广州 510006；2. 北京跟踪与通信技术研究所，北京 100094）摘要：针对地基可见光观测图像中存在的杂散光干扰问题，提出了一种基于星图亮度先验的杂散光噪声去除方法。

首先，通过分析杂散光形成的原因及其在星图中的空间分布特征，建立星图在杂散光干扰下的退化模型；然后利用星图的亮度先验，估计大气的深度信息并去除分布不均的杂散光噪声；最后，在地基光学望远镜拍摄的实际星图上进行验证。

与现有的算法相比，对于受不同程度杂散光干扰的目标，该方法在背景抑制和目标信杂比提升上均获得了更好的实验效果。

其中，针对序列星图中信杂比为2.05以上的空间目标，处理后能够获得7.39以上的信杂比增益和1.92以上的背景抑制因子。

关键词：地基光电探测系统；杂散光噪声；亮度先验；星图；背景抑制因子中图分类号：TN201；V556.5 文献标志码：A DOI ：10.5768/JAO202142.0302002Stray light noise removal method of star maps based on intensity priorZHANG Xin 1，LIN Bin 1，YANG Xia 1，WANG Kunpeng 2，ZHANG Xiaohu 1（1. School of Aeronautics and Astronautics, Sun Yat-Sen University, Guangzhou 510006, China ；2. Beijing Institute ofTracking and Telecommunications Technology, Beijing 100094, China ）Abstract ：Aiming at the problem of stray light interference in ground-based visible light observation images, a stray light noise removal method of star maps based on intensity prior was proposed. First, by analyzing the causes of stray light and its spatial distribution characteristics in star maps, a degradation model of the star maps under stray light interference was established; then, the intensity prior of star maps was used to estimate the depth information of the atmosphere and remove the unevenly distributed stray light noise; finally, the method was verified on the actual star maps taken by the ground-based optical telescopes. Compared with the existing algorithms, the proposed method obtained better experimental results in terms of background suppression and target signal-to-clutter ratio (SCR) improvement for targets interfered by different degrees of stray light. Among them, for the space target with a SCR of 2.05 or more in the sequence star maps, the proposed method can obtain the SCR gain of 7.39 or more and the background suppression factor (BSF) of 1.92 or more after processing.Key words ：ground-based photoelectric detection system ；stray light noise ；intensity prior ；star maps ；background suppression factor引言近年来，地球轨道上分布的废弃卫星、火箭残骸、航天器碎片等人造物体大量增加，严重影响了航天器在轨运行的安全。

稳定的高功率激光系统在高级引力波探测器中的应用

Stabilized high-power laser system forthe gravitational wave detector advancedLIGOP.Kwee,1,∗C.Bogan,2K.Danzmann,1,2M.Frede,4H.Kim,1P.King,5J.P¨o ld,1O.Puncken,3R.L.Savage,5F.Seifert,5P.Wessels,3L.Winkelmann,3and B.Willke21Max-Planck-Institut f¨u r Gravitationsphysik(Albert-Einstein-Institut),Hannover,Germany2Leibniz Universit¨a t Hannover,Hannover,Germany3Laser Zentrum Hannover e.V.,Hannover,Germany4neoLASE GmbH,Hannover,Germany5LIGO Laboratory,California Institute of Technology,Pasadena,California,USA*patrick.kwee@aei.mpg.deAbstract:An ultra-stable,high-power cw Nd:Y AG laser system,devel-oped for the ground-based gravitational wave detector Advanced LIGO(Laser Interferometer Gravitational-Wave Observatory),was comprehen-sively ser power,frequency,beam pointing and beamquality were simultaneously stabilized using different active and passiveschemes.The output beam,the performance of the stabilization,and thecross-coupling between different stabilization feedback control loops werecharacterized and found to fulﬁll most design requirements.The employedstabilization schemes and the achieved performance are of relevance tomany high-precision optical experiments.©2012Optical Society of AmericaOCIS codes:(140.3425)Laser stabilization;(120.3180)Interferometry.References and links1.S.Rowan and J.Hough,“Gravitational wave detection by interferometry(ground and space),”Living Rev.Rel-ativity3,1–3(2000).2.P.R.Saulson,Fundamentals of Interferometric Gravitational Wave Detectors(World Scientiﬁc,1994).3.G.M.Harry,“Advanced LIGO:the next generation of gravitational wave detectors,”Class.Quantum Grav.27,084006(2010).4. B.Willke,“Stabilized lasers for advanced gravitational wave detectors,”Laser Photon.Rev.4,780–794(2010).5.P.Kwee,“Laser characterization and stabilization for precision interferometry,”Ph.D.thesis,Universit¨a t Han-nover(2010).6.K.Somiya,Y.Chen,S.Kawamura,and N.Mio,“Frequency noise and intensity noise of next-generationgravitational-wave detectors with RF/DC readout schemes,”Phys.Rev.D73,122005(2006).7. B.Willke,P.King,R.Savage,and P.Fritschel,“Pre-stabilized laser design requirements,”internal technicalreport T050036-v4,LIGO Scientiﬁc Collaboration(2009).8.L.Winkelmann,O.Puncken,R.Kluzik,C.Veltkamp,P.Kwee,J.Poeld,C.Bogan,B.Willke,M.Frede,J.Neu-mann,P.Wessels,and D.Kracht,“Injection-locked single-frequency laser with an output power of220W,”Appl.Phys.B102,529–538(2011).9.T.J.Kane and R.L.Byer,“Monolithic,unidirectional single-mode Nd:Y AG ring laser,”Opt.Lett.10,65–67(1985).10.I.Freitag,A.T¨u nnermann,and H.Welling,“Power scaling of diode-pumped monolithic Nd:Y AG lasers to outputpowers of several watts,”mun.115,511–515(1995).11.M.Frede,B.Schulz,R.Wilhelm,P.Kwee,F.Seifert,B.Willke,and D.Kracht,“Fundamental mode,single-frequency laser ampliﬁer for gravitational wave detectors,”Opt.Express15,459–465(2007).#161737 - $15.00 USD Received 18 Jan 2012; revised 27 Feb 2012; accepted 4 Mar 2012; published 24 Apr 2012 (C) 2012 OSA7 May 2012 / Vol. 20, No. 10 / OPTICS EXPRESS 1061712. A.D.Farinas,E.K.Gustafson,and R.L.Byer,“Frequency and intensity noise in an injection-locked,solid-statelaser,”J.Opt.Soc.Am.B12,328–334(1995).13.R.Bork,M.Aronsson,D.Barker,J.Batch,J.Heefner,A.Ivanov,R.McCarthy,V.Sandberg,and K.Thorne,“New control and data acquisition system in the Advanced LIGO project,”Proc.of Industrial Control And Large Experimental Physics Control System(ICALEPSC)conference(2011).14.“Experimental physics and industrial control system,”/epics/.15.P.Kwee and B.Willke,“Automatic laser beam characterization of monolithic Nd:Y AG nonplanar ring lasers,”Appl.Opt.47,6022–6032(2008).16.P.Kwee,F.Seifert,B.Willke,and K.Danzmann,“Laser beam quality and pointing measurement with an opticalresonator,”Rev.Sci.Instrum.78,073103(2007).17. A.R¨u diger,R.Schilling,L.Schnupp,W.Winkler,H.Billing,and K.Maischberger,“A mode selector to suppressﬂuctuations in laser beam geometry,”Opt.Acta28,641–658(1981).18. B.Willke,N.Uehara,E.K.Gustafson,R.L.Byer,P.J.King,S.U.Seel,and R.L.Savage,“Spatial and temporalﬁltering of a10-W Nd:Y AG laser with a Fabry-Perot ring-cavity premode cleaner,”Opt.Lett.23,1704–1706 (1998).19.J.H.P¨o ld,“Stabilization of the Advanced LIGO200W laser,”Diploma thesis,Leibniz Universit¨a t Hannover(2009).20. E.D.Black,“An introduction to Pound-Drever-Hall laser frequency stabilization,”Am.J.Phys.69,79–87(2001).21.R.W.P.Drever,J.L.Hall,F.V.Kowalski,J.Hough,G.M.Ford,A.J.Munley,and H.Ward,“Laser phase andfrequency stabilization using an optical resonator,”Appl.Phys.B31,97–105(1983).22. A.Bullington,ntz,M.Fejer,and R.Byer,“Modal frequency degeneracy in thermally loaded optical res-onators,”Appl.Opt.47,2840–2851(2008).23.G.Mueller,“Beam jitter coupling in Advanced LIGO,”Opt.Express13,7118–7132(2005).24.V.Delaubert,N.Treps,ssen,C.C.Harb,C.Fabre,m,and H.-A.Bachor,“TEM10homodynedetection as an optimal small-displacement and tilt-measurement scheme,”Phys.Rev.A74,053823(2006). 25.P.Kwee,B.Willke,and K.Danzmann,“Laser power noise detection at the quantum-noise limit of32A pho-tocurrent,”Opt.Lett.36,3563–3565(2011).26. A.Araya,N.Mio,K.Tsubono,K.Suehiro,S.Telada,M.Ohashi,and M.Fujimoto,“Optical mode cleaner withsuspended mirrors,”Appl.Opt.36,1446–1453(1997).27.P.Kwee,B.Willke,and K.Danzmann,“Shot-noise-limited laser power stabilization with a high-power photodi-ode array,”Opt.Lett.34,2912–2914(2009).28. ntz,P.Fritschel,H.Rong,E.Daw,and G.Gonz´a lez,“Quantum-limited optical phase detection at the10−10rad level,”J.Opt.Soc.Am.A19,91–100(2002).1.IntroductionInterferometric gravitational wave detectors[1,2]perform one of the most precise differential length measurements ever.Their goal is to directly detect the faint signals of gravitational waves emitted by astrophysical sources.The Advanced LIGO(Laser Interferometer Gravitational-Wave Observatory)[3]project is currently installing three second-generation,ground-based detectors at two observatory sites in the USA.The4kilometer-long baseline Michelson inter-ferometers have an anticipated tenfold better sensitivity than theirﬁrst-generation counterparts (Inital LIGO)and will presumably reach a strain sensitivity between10−24and10−23Hz−1/2.One key technology necessary to reach this extreme sensitivity are ultra-stable high-power laser systems[4,5].A high laser output power is required to reach a high signal-to-quantum-noise ratio,since the effect of quantum noise at high frequencies in the gravitational wave readout is reduced with increasing circulating laser power in the interferometer.In addition to quantum noise,technical laser noise coupling to the gravitational wave channel is a major noise source[6].Thus it is important to reduce the coupling of laser noise,e.g.by optical design or by exploiting symmetries,and to reduce laser noise itself by various active and passive stabilization schemes.In this article,we report on the pre-stabilized laser(PSL)of the Advanced LIGO detector. The PSL is based on a high-power solid-state laser that is comprehensively stabilized.One laser system was set up at the Albert-Einstein-Institute(AEI)in Hannover,Germany,the so called PSL reference system.Another identical PSL has already been installed at one Advanced LIGO site,the one near Livingston,LA,USA,and two more PSLs will be installed at the second #161737 - $15.00 USD Received 18 Jan 2012; revised 27 Feb 2012; accepted 4 Mar 2012; published 24 Apr 2012 (C) 2012 OSA7 May 2012 / Vol. 20, No. 10 / OPTICS EXPRESS 10618site at Hanford,WA,USA.We have characterized the reference PSL and theﬁrst observatory PSL.For this we measured various beam parameters and noise levels of the output beam in the gravitational wave detection frequency band from about10Hz to10kHz,measured the performance of the active and passive stabilization schemes,and determined upper bounds for the cross coupling between different control loops.At the time of writing the PSL reference system has been operated continuously for more than18months,and continues to operate reliably.The reference system delivered a continuous-wave,single-frequency laser beam at1064nm wavelength with a maximum power of150W with99.5%in the TEM00mode.The active and passive stabilization schemes efﬁciently re-duced the technical laser noise by several orders of magnitude such that most design require-ments[5,7]were fulﬁlled.In the gravitational wave detection frequency band the relative power noise was as low as2×10−8Hz−1/2,relative beam pointingﬂuctuations were as low as1×10−7Hz−1/2,and an in-loop measurement of the frequency noise was consistent with the maximum acceptable frequency noise of about0.1HzHz−1/2.The cross couplings between the control loops were,in general,rather small or at the expected levels.Thus we were able to optimize each loop individually and observed no instabilities due to cross couplings.This stabilized laser system is an indispensable part of Advanced LIGO and fulﬁlled nearly all design goals concerning the maximum acceptable noise levels of the different beam pa-rameters right after installation.Furthermore all or a subset of the implemented stabilization schemes might be of interest for many other high-precision optical experiments that are limited by laser noise.Besides gravitational wave detectors,stabilized laser systems are used e.g.in the ﬁeld of optical frequency standards,macroscopic quantum objects,precision spectroscopy and optical traps.In the following section the laser system,the stabilization scheme and the characterization methods are described(Section2).Then,the results of the characterization(Section3)and the conclusions(Section4)are presented.ser system and stabilizationThe PSL consists of the laser,developed and fabricated by Laser Zentrum Hannover e.V.(LZH) and neoLASE,and the stabilization,developed and integrated by AEI.The optical components of the PSL are on a commercial optical table,occupying a space of about1.5×3.5m2,in a clean,dust-free environment.At the observatory sites the optical table is located in an acoustically isolated cleanroom.Most of the required electronics,the laser diodes for pumping the laser,and water chillers for cooling components on the optical table are placed outside of this cleanroom.The laser itself consists of three stages(Fig.1).An almostﬁnal version of the laser,the so-called engineering prototype,is described in detail in[8].The primary focus of this article is the stabilization and characterization of the PSL.Thus only a rough overview of the laser and the minor modiﬁcations implemented between engineering prototype and reference system are given in the following.Theﬁrst stage,the master laser,is a commercial non-planar ring-oscillator[9,10](NPRO) manufactured by InnoLight GmbH in Hannover,Germany.This solid-state laser uses a Nd:Y AG crystal as the laser medium and resonator at the same time.The NPRO is pumped by laser diodes at808nm and delivers an output power of2W.An internal power stabilization,called the noise eater,suppresses the relaxation oscillation at around1MHz.Due to its monolithic res-onator,the laser has exceptional intrinsic frequency stability.The two subsequent laser stages, used for power scaling,inherit the frequency stability of the master laser.The second stage(medium-power ampliﬁer)is a single-pass ampliﬁer[11]with an output power of35W.The seed laser beam from the NPRO stage passes through four Nd:YVO4crys-#161737 - $15.00 USD Received 18 Jan 2012; revised 27 Feb 2012; accepted 4 Mar 2012; published 24 Apr 2012 (C) 2012 OSA7 May 2012 / Vol. 20, No. 10 / OPTICS EXPRESS 10619power stabilizationFig.1.Pre-stabilized laser system of Advanced LIGO.The three-staged laser(NPRO,medium power ampliﬁer,high power oscillator)and the stabilization scheme(pre-mode-cleaner,power and frequency stabilization)are shown.The input-mode-cleaner is not partof the PSL but closely related.NPRO,non-planar ring oscillator;EOM,electro-optic mod-ulator;FI,Faraday isolator;AOM,acousto-optic modulator.tals which are longitudinally pumped byﬁber-coupled laser diodes at808nm.The third stage is an injection-locked ring oscillator[8]with an output power of about220W, called the high-power oscillator(HPO).Four Nd:Y AG crystals are used as the active media. Each is longitudinally pumped by sevenﬁber-coupled laser diodes at808nm.The oscillator is injection-locked[12]to the previous laser stage using a feedback control loop.A broadband EOM(electro-optic modulator)placed between the NPRO and the medium-power ampliﬁer is used to generate the required phase modulation sidebands at35.5MHz.Thus the high output power and good beam quality of this last stage is combined with the good frequency stability of the previous stages.The reference system features some minor modiﬁcations compared to the engineering proto-type[8]concerning the optics:The external halo aperture was integrated into the laser system permanently improving the beam quality.Additionally,a few minor designﬂaws related to the mechanical structure and the optical layout were engineered out.This did not degrade the output performance,nor the characteristics of the locked laser.In general the PSL is designed to be operated in two different power modes.In high-power mode all three laser stages are engaged with a power of about160W at the PSL output.In low-power mode the high-power oscillator is turned off and a shutter inside the laser resonator is closed.The beam of the medium-power stage is reﬂected at the output coupler of the high power stage leaving a residual power of about13W at the PSL output.This low-power mode will be used in the early commissioning phase and in the low-frequency-optimized operation mode of Advanced LIGO and is not discussed further in this article.The stabilization has three sections(Fig.1:PMC,PD2,reference cavity):A passive resonator, the so called pre-mode-cleaner(PMC),is used toﬁlter the laser beam spatially and temporally (see subsection2.1).Two pick-off beams at the PMC are used for the active power stabilization (see subsection2.2)and the active frequency pre-stabilization,respectively(see subsection2.3).In general most stabilization feedback control loops of the PSL are implemented using analog electronics.A real-time computer system(Control and Data Acquisition Systems,CDS,[13]) which is common to many other subsystems of Advanced LIGO,is utilized to control and mon-itor important parameters of the analog electronics.The lock acquisition of various loops,a few #161737 - $15.00 USD Received 18 Jan 2012; revised 27 Feb 2012; accepted 4 Mar 2012; published 24 Apr 2012 (C) 2012 OSA7 May 2012 / Vol. 20, No. 10 / OPTICS EXPRESS 10620slow digital control loops,and the data acquisition are implemented using this computer sys-tem.Many signals are recorded at different sampling rates ranging from16Hz to33kHz for diagnostics,monitoring and vetoing of gravitational wave signals.In total four real-time pro-cesses are used to control different aspects of the laser system.The Experimental Physics and Industrial Control System(EPICS)[14]and its associated user tools are used to communicate with the real-time software modules.The PSL contains a permanent,dedicated diagnostic instrument,the so called diagnostic breadboard(DBB,not shown in Fig.1)[15].This instrument is used to analyze two different beams,pick-off beams of the medium power stage and of the HPO.Two shutters are used to multiplex these to the DBB.We are able to measureﬂuctuations in power,frequency and beam pointing in an automated way with this instrument.In addition the beam quality quantiﬁed by the higher order mode content of the beam was measured using a modescan technique[16].The DBB is controlled by one real-time process of the CDS.In contrast to most of the other control loops in the PSL,all DBB control loops were implemented digitally.We used this instrument during the characterization of the laser system to measure the mentioned laser beam parameters of the HPO.In addition we temporarily placed an identical copy of the DBB downstream of the PMC to characterize the output beam of the PSL reference system.2.1.Pre-mode-cleanerA key component of the stabilization scheme is the passive ring resonator,called the pre-mode-cleaner(PMC)[17,18].It functions to suppress higher-order transverse modes,to improve the beam quality and the pointing stability of the laser beam,and toﬁlter powerﬂuctuations at radio frequencies.The beam transmitted through this resonator is the output beam of the PSL, and it is delivered to the subsequent subsystems of the gravitational wave detector.We developed and used a computer program[19]to model theﬁlter effects of the PMC as a function of various resonator parameters in order to aid its design.This led to a resonator with a bow-tie conﬁguration consisting of four low-loss mirrors glued to an aluminum spacer. The optical round-trip length is2m with a free spectral range(FSR)of150MHz.The inci-dence angle of the horizontally polarized laser beam is6◦.Theﬂat input and output coupling mirrors have a power transmission of2.4%and the two concave high reﬂectivity mirrors(3m radius of curvature)have a transmission of68ppm.The measured bandwidth was,as expected, 560kHz which corresponds to aﬁnesse of133and a power build-up factor of42.The Gaussian input/output beam had a waist radius of about568µm and the measured acquired round-trip Gouy phase was about1.7rad which is equivalent to0.27FSR.One TEM00resonance frequency of the PMC is stabilized to the laser frequency.The Pound-Drever-Hall(PDH)[20,21]sensing scheme is used to generate error signals,reusing the phase modulation sidebands at35.5MHz created between NPRO and medium power ampliﬁer for the injection locking.The signal of the photodetector PD1,placed in reﬂection of the PMC, is demodulated at35.5MHz.This photodetector consists of a1mm InGaAs photodiode and a transimpedance ampliﬁer.A piezo-electric element(PZT)between one of the curved mirrors and the spacer is used as a fast actuator to control the round-trip length and thereby the reso-nance frequencies of the PMC.With a maximum voltage of382V we were able to change the round-trip length by about2.4µm.An analog feedback control loop with a bandwidth of about 7kHz is used to stabilize the PMC resonance frequency to the laser frequency.In addition,the electronics is able to automatically bring the PMC into resonance with the laser(lock acquisition).For this process a125ms period ramp signal with an amplitude cor-responding to about one FSR is applied to the PZT of the PMC.The average power on pho-todetector PD1is monitored and as soon as the power drops below a given threshold the logic considers the PMC as resonant and closes the analog control loop.This lock acquisition proce-#161737 - $15.00 USD Received 18 Jan 2012; revised 27 Feb 2012; accepted 4 Mar 2012; published 24 Apr 2012 (C) 2012 OSA7 May 2012 / Vol. 20, No. 10 / OPTICS EXPRESS 10621dure took an average of about65ms and is automatically repeated as soon as the PMC goes off resonance.One real-time process of CDS is dedicated to control the PMC electronics.This includes parameters such as the proportional gain of the loop or lock acquisition parameters.In addition to the PZT actuator,two heating foils,delivering a maximum total heating power of14W,are attached to the aluminum spacer to control its temperature and thereby the roundtrip length on timescales longer than3s.We measured a heating and cooling1/e time constant of about2h with a range of4.5K which corresponds to about197FSR.During maintenance periods we heat the spacer with7W to reach a spacer temperature of about2.3K above room temperature in order to optimize the dynamic range of this actuator.A digital control loop uses this heater as an actuator to off-load the PZT actuator allowing compensation for slow room temperature and laser frequency drifts.The PMC is placed inside a pressure-tight tank at atmospheric pressure for acoustic shield-ing,to avoid contamination of the resonator mirrors and to minimize optical path length changes induced by atmospheric pressure variations.We used only low-outgassing materials and fabri-cated the PMC in a cleanroom in order to keep the initial mirror contamination to a minimum and to sustain a high long-term throughput.The PMCﬁlters the laser beam and improves the beam quality of the laser by suppress-ing higher order transverse modes[17].The acquired round-trip Gouy phase of the PMC was chosen in such a way that the resonance frequencies of higher order TEM modes are clearly separated from the TEM00resonance frequency.Thus these modes are not resonant and are mainly reﬂected by the PMC,whereas the TEM00mode is transmitted.However,during the design phase we underestimated the thermal effects in the PMC such that at nominal circu-lating power the round-trip Gouy-phase is close to0.25FSR and the resonance of the TEM40 mode is close to that of the TEM00mode.To characterize the mode-cleaning performance we measured the beam quality upstream and downstream of the PMC with the two independent DBBs.At150W in the transmitted beam,the circulating power in the PMC is about6.4kW and the intensity at the mirror surface can be as high as1.8×1010W m−2.At these power levels even small absorptions in the mirror coatings cause thermal effects which slightly change the mirror curvature[22].To estimate these thermal effects we analyzed the transmitted beam as a function of the circulating power using the DBB.In particular we measured the mode content of the LG10and TEM40mode.Changes of the PMC eigenmode waist size showed up as variations of the LG10mode content.A power dependence of the round-trip Gouy phase caused a variation of the power within the TEM40mode since its resonance frequency is close to a TEM00mode resonance and thus the suppression of this mode depends strongly on the Gouy phase.We adjusted the input power to the PMC such that the transmitted power ranged from100W to 150W corresponding to a circulating power between4.2kW and6.4kW.We used our PMC computer simulation to deduce the power dependence of the eigenmode waist size and the round-trip Gouy phase.The results are given in section3.1.At all circulating power levels,however,the TEM10and TEM01modes are strongly sup-pressed by the PMC and thus beam pointingﬂuctuations are reduced.Pointingﬂuctuations can be expressed toﬁrst order as powerﬂuctuations of the TEM10and TEM01modes[23,24].The PMC reduces theﬁeld amplitude of these modes and thus the pointingﬂuctuations by a factor of about61according to the measuredﬁnesse and round-trip Gouy phase.To keep beam point-ingﬂuctuations small is important since they couple to the gravitational wave channel by small differential misalignments of the interferometer optics.Thus stringent design requirements,at the10−6Hz−1/2level for relative pointing,were set.To verify the pointing suppression effect of the PMC we used DBBs to measure the beam pointingﬂuctuations upstream and downstream #161737 - $15.00 USD Received 18 Jan 2012; revised 27 Feb 2012; accepted 4 Mar 2012; published 24 Apr 2012 (C) 2012 OSA7 May 2012 / Vol. 20, No. 10 / OPTICS EXPRESS 10622Fig.2.Detailed schematic of the power noise sensor setup for theﬁrst power stabilizationloop.This setup corresponds to PD2in the overview in Fig.1.λ/2,waveplate;PBS,polar-izing beam splitter;BD,glassﬁlters used as beam dump;PD,single element photodetector;QPD,quadrant photodetector.of the PMC.The resonator design has an even number of nearly normal-incidence reﬂections.Thus the resonance frequencies of horizontal and vertical polarized light are almost identical and the PMC does not act as polarizer.Therefore we use a thin-ﬁlm polarizer upstream of the PMC to reach the required purity of larger than100:1in horizontal polarization.Finally the PMC reduces technical powerﬂuctuations at radio frequencies(RF).A good power stability between9MHz and100MHz is necessary as the phase modulated light in-jected into the interferometer is used to sense several degrees of freedom of the interferometer that need to be controlled.Power noise around these phase modulation sidebands would be a noise source for the respective stabilization loop.The PMC has a bandwidth(HWHM)of about 560kHz and acts toﬁrst order as a low-passﬁlter for powerﬂuctuations with a-3dB corner frequency at this frequency.To verify that the suppression of RF powerﬂuctuations is sufﬁ-cient to fulﬁll the design requirements,we measured the relative power noise up to100MHz downstream of the PMC with a dedicated experiment involving the optical ac coupling tech-nique[25].In addition the PMC serves the very important purpose of deﬁning the spatial laser mode for the downstream subsystem,namely the input optics(IO)subsystem.The IO subsystem is responsible,among other things,to further stabilize the laser beam with the suspended input mode cleaner[26]before the beam will be injected into the interferometer.Modiﬁcations of beam alignment or beam size of the laser system,which were and might be unavoidable,e.g., due to maintenance,do not propagate downstream of the PMC toﬁrst order due to its mode-cleaning effect.Furthermore we beneﬁt from a similar isolating effect for the active power and frequency stabilization by using the beams transmitted through the curved high-reﬂectivity mirrors of the PMC.2.2.Power stabilizationThe passiveﬁltering effect of the PMC reduces powerﬂuctuations signiﬁcantly only above the PMC bandwidth.In the detection band from about10Hz to10kHz good power stability is required sinceﬂuctuations couple via the radiation pressure imbalance and the dark-fringe offset to the gravitational wave channel.Thus two cascaded active control loops,theﬁrst and second power stabilization loop,are used to reduce powerﬂuctuations which are mainly caused by the HPO stage.Theﬁrst loop uses a low-noise photodetector(PD2,see Figs.1and2)at one pick-off port #161737 - $15.00 USD Received 18 Jan 2012; revised 27 Feb 2012; accepted 4 Mar 2012; published 24 Apr 2012 (C) 2012 OSA7 May 2012 / Vol. 20, No. 10 / OPTICS EXPRESS 10623of the PMC to measure the powerﬂuctuations downstream of the PMC.An analog electronics feedback control loop and an AOM(acousto-optic modulator)as actuator,located upstream of the PMC,are used to stabilize the power.Scattered light turned out to be a critical noise source for thisﬁrst loop.Thus we placed all required optical and opto-electronic components into a box to shield from scattered light(see Fig.2).The beam transmitted by the curved PMC mirror has a power of about360mW.This beam isﬁrst attenuated in the box using aλ/2waveplate and a thin-ﬁlm polarizer,such that we are able to adjust the power on the photodetectors to the optimal operation point.Afterwards the beam is split by a50:50beam splitter.The beams are directed to two identical photode-tectors,one for the control loop(PD2a,in-loop detector)and one for independent out-of-loop measurements to verify the achieved power stability(PD2b,out-of-loop detector).These pho-todetectors consist of a2mm InGaAs photodiode(PerkinElmer C30642GH),a transimpedance ampliﬁer and an integrated signal-conditioningﬁlter.At the chosen operation point a power of about4mW illuminates each photodetector generating a photocurrent of about3mA.Thus the shot noise is at a relative power noise of10−8Hz−1/2.The signal conditioningﬁlter has a gain of0.2at very low frequencies(<70mHz)and ampliﬁes the photodetector signal in the im-portant frequency range between3.3Hz and120Hz by about52dB.This signal conditioning ﬁlter reduces the electronics noise requirements on all subsequent stages,but has the drawback that the range between3.3Hz and120Hz is limited to maximum peak-to-peak relative power ﬂuctuations of5×10−3.Thus the signal-conditioned channel is in its designed operation range only when the power stabilization loop is closed and therefore it is not possible to measure the free running power noise using this channel due to saturation.The uncoated glass windows of the photodiodes were removed and the laser beam hits the photodiodes at an incidence angle of45◦.The residual reﬂection from the photodiode surface is dumped into a glassﬁlter(Schott BG39)at the Brewster angle.Beam positionﬂuctuations in combination with spatial inhomogeneities in the photodiode responsivity is another noise source for the power stabilization.We placed a silicon quadrant photodetector(QPD)in the box to measure the beam positionﬂuctuations of a low-power beam picked off the main beam in the box.The beam parameters,in particular the Gouy phase,at the QPD are the same as on the power sensing detectors.Thus the beam positionﬂuctuations measured with the QPD are the same as the ones on the power sensing photodetectors,assuming that the positionﬂuctuations are caused upstream of the QPD pick-off point.We used the QPD to measure beam positionﬂuctuations only for diagnostic and noise projection purposes.In a slightly modiﬁed experiment,we replaced one turning mirror in the path to the power sta-bilization box by a mirror attached to a tip/tilt PZT element.We measured the typical coupling between beam positionﬂuctuations generated by the PZT and the residual relative photocurrent ﬂuctuations measured with the out-of-the-loop photodetector.This coupling was between1m−1 and10m−1which is a typical value observed in different power stabilization experiments as well.We measured this coupling factor to be able to calculate the noise contribution in the out-of-the-loop photodetector signal due to beam positionﬂuctuations(see Subsection3.3).Since this tip/tilt actuator was only temporarily in the setup,we are not able to measure the coupling on a regular basis.Both power sensing photodetectors are connected to analog feedback control electronics.A low-pass(100mHz corner frequency)ﬁltered reference value is subtracted from one signal which is subsequently passed through several control loopﬁlter stages.With power stabilization activated,we are able to control the power on the photodetectors and thereby the PSL output power via the reference level on time scales longer than10s.The reference level and other important parameters of these electronics are controlled by one dedicated real-time process of the CDS.The actuation or control signal of the electronics is passed to an AOM driver #161737 - $15.00 USD Received 18 Jan 2012; revised 27 Feb 2012; accepted 4 Mar 2012; published 24 Apr 2012 (C) 2012 OSA7 May 2012 / Vol. 20, No. 10 / OPTICS EXPRESS 10624。

基于插值和周期图法的高动态信号载波频偏粗估计

收稿日期：２０２１０７０３；修回日期：２０２１０９０１作者简介：魏苗苗（１９８７），女（通信作者），河南鹿邑人，讲师，博士，主要研究方向为时序信号处理（６５４２＠ｚｕｔ．ｅｄｕ．ｃｎ）；刘洲峰（１９６２），男，河南郑州人，教授，硕导，博士，主要研究方向为数字图像处理；李春雷（１９７９），男，教授，博士，主要研究方向为智能信息处理；孙俊（１９８２），男，河南洛阳人，副教授，博士研究生，主要研究方向为信道测量和噪声估计．基于插值和周期图法的高动态信号载波频偏粗估计魏苗苗１，２，刘洲峰１，李春雷１，孙　俊１，２（１．中原工学院电子信息学院，郑州４５０００７；２．郑州大学信息工程学院，郑州４５０００１）摘　要：针对卫星通信系统中接收信号载波动态范围大、信噪比低造成的信号载波同步困难的问题进行了研究。

基于联合插值和频域移位平均周期图法的载波频偏估计算法，通过对半符号周期频域移位平均周期图法中各并行支路输出的功率谱峰值波形进行双谱线插值，以进一步降低载波频偏变化率估计误差，进而改善原算法捕获概率。

仿真结果显示，当比特信噪比为２．５ｄＢ时，相比于半符号周期频域移位平均周期图法，该算法只增加了一次插值计算就可以实现将载波频偏变化率估计误差降低２７％。

在同等估计精度和参数设置下，相比于半符号周期频域移位平均周期图法和带补零频域移位评价周期图法，基于联合插值和周期图法的载波频偏粗估计算法可达到更高的捕获概率。

关键词：频偏估计；载波同步；频域移位；插值估计；高动态中图分类号：ＴＮ９２７＋．２３文献标志码：Ａ文章编号：１００１３６９５（２０２２）０２０３８０５４８０４ｄｏｉ：１０．１９７３４／ｊ．ｉｓｓｎ．１００１３６９５．２０２１．０７．０３０３ＣｏａｒｓｅｃａｒｒｉｅｒｏｆｆｓｅｔｅｓｔｉｍａｔｉｏｎｏｆｈｉｇｈｄｙｎａｍｉｃａｌｓｉｇｎａｌｂａｓｅｄｏｎｉｎｔｅｒｐｏｌａｔｉｏｎａｎｄｐｅｒｉｏｄｏｇｒａｍａｌｇｏｒｉｔｈｍＷｅｉＭｉａｏｍｉａｏ１，２，ＬｉｕＺｈｏｕｆｅｎｇ１，ＬｉＣｈｕｎｌｅｉ１，ＳｕｎＪｕｎ１，２（１．ＳｃｈｏｏｌｏｆＥｌｅｃｔｒｏｎｉｃｓ＆Ｉｎｆｏｒｍａｔｉｏｎ，ＺｈｏｎｇｙｕａｎＵｎｉｖｅｒｓｉｔｙｏｆＴｅｃｈｎｏｌｏｇｙ，Ｚｈｅｎｇｚｈｏｕ４５０００７，Ｃｈｉｎａ；２．ＳｃｈｏｏｌｏｆＩｎｆｏｒｍａｔｉｏｎＥｎｇｉｎｅｅｒｉｎｇ，ＺｈｅｎｇｚｈｏｕＵｎｉｖｅｒｓｉｔｙ，Ｚｈｅｎｇｚｈｏｕ４５０００１，Ｃｈｉｎａ）Ａｂｓｔｒａｃｔ：Ｉｎｓａｔｅｌｌｉｔｅｃｏｍｍｕｎｉｃａｔｉｏｎｓｙｓｔｅｍｓ，ｔｈｅｒｅｃｅｉｖｅｄｓｉｇｎａｌｕｓｕａｌｌｙｈａｓｔｈｅｃｈａｒａｃｔｅｒｉｓｔｉｃｓｏｆｈｉｇｈｄｙｎａｍｉｃｒａｎｇｅａｎｄｌｏｗｓｉｇｎａｌｔｏｎｏｉｓｅｒａｔｉｏ（ＳＮＲ），ｗｈｉｃｈｌｅａｄｓｔｏｄｉｆｆｉｃｕｌｔｙｏｆｃａｒｒｉｅｒｓｙｎｃｈｒｏｎｉｚａｔｉｏｎ．Ｔｈｅｃａｒｒｉｅｒｅｓｔｉｍａｔｉｏｎａｌｇｏｒｉｔｈｍｂａｓｅｄｏｎｉｎｔｅｒｐｏｌａｔｉｏｎａｎｄｆｒｅｑｕｅｎｃｙｄｏｍａｉｎｓｈｉｆｔｅｄａｖｅｒａｇｅｐｅｒｉｏｄｏｇｒａｍｍｅｔｈｏｄｃｏｕｌｄｒｅｄｕｃｅｔｈｅｅｓｔｉｍａｔｉｏｎｅｒｒｏｒｏｆｆｒｅｑｕｅｎｃｙｏｆｆｓｅｔｄｅｒｉｖａｔｉｖｅ，ａｎｄｉｎｃｒｅａｓｅｔｈｅａｃｑｕｉｓｉｔｉｏｎｐｒｏｂａｂｉｌｉｔｙｂｙｂｉｓｐｅｃｔｒｕｍｉｎｔｅｒｐｏｌａｔｉｏｎｏｎｔｈｅｐｅａｋｗａｖｅｆｏｒｍｏｆｐｏｗｅｒｓｐｅｃｔｒｕｍｏｕｔｐｕｔｂｙｐａｒａｌｌｅｌｂｒａｎｃｈｅｓｉｎｔｈｅｓｅｍｉｓｙｍｂｏｌｆｒｅｑｕｅｎｃｙｄｏｍａｉｎｓｈｉｆｔｅｄａｖｅｒａｇｅｐｅｒｉｏｄｏｇｒａｍｍｅｔｈｏｄ．ＴｈｅｓｉｍｕｌａｔｉｏｎｒｅｓｕｌｔｓｓｈｏｗｔｈａｔｗｈｅｎｂｉｔＳＮＲｉｓ２．５ｄＢ，ｃｏｍｐａｒｅｄｗｉｔｈｔｈｅｓｅｍｉｓｙｍｂｏｌｆｒｅｑｕｅｎｃｙｄｏｍａｉｎｓｈｉｆｔｅｄａｖｅｒａｇｅｐｅｒｉｏｄｏｇｒａｍｍｅｔｈｏｄ，ｔｈｅｅｓｔｉｍａｔｉｏｎｅｒｒｏｒｏｆｔｈｅｆｒｅｑｕｅｎｃｙｏｆｆｓｅｔｄｅｒｉｖａｔｉｖｅｃａｎｂｅｒｅｄｕｃｅｄｂｙ２７％ｗｉｔｈｏｎｌｙｏｎｅｉｎｔｅｒｐｏｌａｔｉｏｎｃａｌｃｕｌａｔｉｏｎａｄｄｅｄ．Ｗｉｔｈｔｈｅｓａｍｅａｃｃｕｒａｃｙｒｅｑｕｉｒｅｍｅｎｔａｎｄｐａｒａｍｅｔｅｒｓｅｔｔｉｎｇ，ｃｏｍｐａｒｅｄｗｉｔｈｓｅｍｉｓｙｍｂｏｌｆｒｅｑｕｅｎｃｙｄｏｍａｉｎｓｈｉｆｔｅｄａｖｅｒａｇｅｐｅｒｉｏｄｏｇｒａｍｍｅｔｈｏｄａｎｄｚｅｒｏｐａｄｄｉｎｇｆｒｅｑｕｅｎｃｙｄｏｍａｉｎｓｈｉｆｔｅｄａｖｅｒａｇｅｐｅｒｉｏｄｏｇｒａｍｍｅｔｈｏｄ，ｔｈｅｐｒｏｐｏｓｅｄａｌｇｏｒｉｔｈｍｒｅａｃｈｅｓａｈｉｇｈｅｒｐｒｏｂａｂｉｌｉｔｙｏｆａｃｑｕｉｓｉｔｉｏｎ．Ｋｅｙｗｏｒｄｓ：ｆｒｅｑｕｅｎｃｙｂｉａｓｅｓｔｉｍａｔｉｏｎ；ｃａｒｒｉｅｒｓｙｎｃｈｒｏｎｉｚａｔｉｏｎ；ｆｒｅｑｕｅｎｃｙｄｏｍａｉｎｓｈｉｆｔ；ｉｎｔｅｒｐｏｌａｔｉｏｎｅｓｔｉｍａｔｉｏｎ；ｈｉｇｈｄｙｎａｍｉｃｓ０　引言面对近年来日益增高的卫星应用需求，实现超远距离下的可靠通信是保证空间探测系统有效运行的关键，但是有效载荷通信信号普遍存在运动速度极高、信噪比极低的特点，以火星等深空探测活动为例，接收信号比特信噪比可低至２ｄＢ以下，并伴有复杂运动情况［１］，致使载波频偏参量不仅包含频率偏差，而且还有更高阶分量［２～４］。

现代信号处理第6章连续小波变换

分形
小波
小波分形技术原理与离散信号盒维数的计算
设离散信号是n维欧氏空间Rn上的闭集。将Rn划分成尽可能细的Δ网格，若是网格宽度N Δ为Δ的离散空间上集合X的网格计数。盒维数定义为 :
由于离散信号的最高分辩率为采样间隔Δ t，所以上式的极限是无法按其定义Δ→0求出。实际计算时一般采用近似方法，即将Δ网格视为最小网格，然后逐步放大为kΔ网格，k∈Z+，令
6.1.5 谐波小波应用
小波分形技术原理与离散信号盒维数的计算
分形的自相似仿射算子r与小波变换的伸缩因子a是作用相同，小波变换从低分辨到高分辨的过渡原则与分形过程的从总体向局部、从宏观向微观深化分析原则是一致的，小波和分形都具有自相似性，两者结合是可行的。小波分形技术原理是应用小波包变换将机械振动信号分解到正交的、独立的频带内，然后分别计算出每个频带信号的盒维数，用盒维数衡量小波包分解每个频带信号的复杂程度由于一维离散信号的盒维数是介于1和2之间的一个实数，信号越复杂维数越大
谐波小波滤波能够在低频频带和高频频带内都具有足够的数据点数。
6.1.4 谐波小波滤波
6.1.4 谐波小波滤波
谐波小波实际上是一个完全理想的带通滤波器，可以用下面的方法定义谐波小波
其中m, n决定了谐波小波变换的尺度（j），且n = 2m，当m = 0时，n = 1。
谐波小波的光滑性，“盒形”谱特性，零相移特性以及明显的数学表达式，使得我们可构造出不同尺度下各频段序列数据点数不变、采样频率不变的算法，最终成功应用于转子轴心轨迹分析
谐波小波的定义及正交性
谐波小波的定义及正交性
实偶函数we(t)和实奇函数wo(t) , 它们的傅里叶变换分别为
谐波小波的定义及正交性

非均匀阵列的相干信源DOA估计研究与FPGA实现

摘要近几十年来，波达方向估计（DOA）的研究取得了大量的成果，并在工程中得到了广泛的应用。

非均匀阵列在实际应用中具有更大的孔径和更灵活的阵元间距设置，逐渐成为研究的热点。

但已有的子空间类DOA估计算法不适用于相干信源入射的情况，且算法中包含很多的非线性运算，硬件实现复杂，很难满足实际工程的需求。

因此本文主要针对非均匀阵列接收相干信源时的DOA估计算法和现场可编程门阵列（FPGA）硬件实现方案进行研究，主要内容如下：首先给出了非均匀阵列信号接收的数学模型，介绍了四种非均匀阵列结构，分析了非均匀阵列中角度模糊的问题。

接着针对最小阵元间距受限的对称非均匀阵列进行研究，该阵列中存在多个相同结构的子阵，可以采用空间平滑的方法进行解相干处理，之后利用多重信号分类（MUSIC）算法实现DOA估计。

通过仿真实验验证了该算法的有效性及DOA估计性能。

然后针对非对称的非均匀阵列DOA估计问题，研究了基于Khatri-Rao（KR）积的阵列扩展算法。

对非相干信源的KR-MUSIC算法的DOA估计性能进行仿真验证。

针对入射信源中存在相干信源时，该算法失效的问题，提出一种转换参考阵元的矩阵重构解相干算法。

利用虚拟阵列平移的方法得到不同参考阵元下的阵列接收数据，然后利用矩阵重构的方法恢复协方差矩阵的秩，最后利用KR-MUSIC算法实现DOA估计。

通过仿真验证了该算法可以有效的实现非均匀阵列相干信源的DOA估计。

最后研究了对称非均匀线阵空间平滑MUSIC算法的FPGA实现。

根据算法步骤，设计了满秩协方差矩阵计算、特征值分解、噪声子空间估计和谱峰搜索四个模块。

满秩协方差矩阵模块实现了协方差矩阵的计算，空间平滑解相干处理及实数化处理。

特征值分解模块利用并行雅克比旋转的方式实现，采用3级清扫的模式来减少该模块的耗时。

噪声子空间模块利用阈值判断信源个数并确定噪声子空间。

谱峰搜索模块根据输入的信源频率得到导向矢量，构造伪谱，搜索谱峰，输出DOA估计角度，并通过划分搜索角度范围来减少耗时。

利用ZYGO干涉仪进行非球面检测技术研究_龙夫年

级衍射波聚焦在光轴上不同的位置. 在 - 1级衍
射成份聚焦点设置光阑滤波, 把不用的衍射成份滤掉, 只让 - 1级衍射光波通过.
因为计算全息板是一个衍射光学元件, 光波
照射时存在多级衍射的现象. 一般, 我们只用它
的 - 1级衍射波, 将其它衍射成份滤掉. 在这种自准直光路中, 滤波显得尤为重要 [ 5] . 根据计算全
利用计算机模拟出的计算全息板如图 4 所示, 它的最大空间频率为 fm ax = 1 061p /mm, 这很容易制作, 且能达到较高的制作精度 [ 6] .
图 4 计算全息的模拟图
3结论
1) 检测光路中采用 ZYGO 数字干涉仪, 能实时快速、高精度检测非球面.
2) 透镜补偿和计算全息结合共同再现理想非球面波前, 该波前由实际被检非球面返回, 返回波前直接反映了被检非球面的误差.
器, 来实现对大口径高次非球面的高精度检测.
# 232#
哈尔滨工业大学学报
第 37卷
1 基本原理
在 ZYGO数字干涉仪的基础上设计一套附加光路, 通过设计制备透镜补偿镜和计算全息, 可在 ZYGO数字干涉仪上检测大口径深度高次非球面, 整个检测光路如图 1所示.
的分布图, 这时恰好实现了各衍射级次互不重叠. 即当 - 1级衍射成份光斑大小为 s时, 要使得它与 - 2级衍射成份分开, 则 - 2级衍射成份光斑边缘与零级光斑距离至少为 4s.
第 2期
龙夫年, 等: 利用 ZYGO 干涉仪进行非球面检测技术研究
# 233#
设计透镜补偿部分, 然后在检测光路中加入计算全息. 计算全息的设计要使得 - 1 级衍射成份与补偿透镜结合产生与理想非球面面型相同的检测波前, 而其他衍射级次用空间滤波器滤掉. 整个检测光路设计时必须考虑滤波的要求. 检测光路光线追迹和设计用光学设计软件 codeⅤ完成, 经过反复的调整, 最终设计出整个检测系统. 检测系统中各元件的参数数据见表 1, 系统剩余波像差如图 3, 其 RM S值为 0101K, P- V 值为 0105K.

拉曼与AFM联用 TERS

AFM-microRaman and nanoRaman TMIntroductionThe use of Raman microscopy has become animportant tool for the analysis of materials on themicron scale. The unique confocal and spatialresolution of the LabRAM series has enabled opticalfar field resolution to be pushed to its limits withoften sub-micron resolution achievable.The next step to material analysis on a smallerscale has been the combination of Ramanspectroscopic analysis with near field optics and anAtomic force microscope (AFM). The hybridRaman/AFM combination enables nanometrictopographical information to be coupled to chemical(spectroscopic) information. The unique designsdeveloped by HORIBA Jobin Yvon enable in-situRaman measurements to be made upon variousdifferent AFM units, and for the exploration of newand evolving techniques such as nanoRamanspectroscopy based on the TERS (tip enhancedRaman spectroscopy) effect.AFM image of nano-structures on a SiN sampleHORIBA Jobin Yvon offers both off-axis and on-axisAFM/Raman coupling to better match your sampleand analysis requirements.Off-axis and inverted on-axis configurations forAFM/Raman coupling showing the laser (blue) andRaman (pink) optical pathThe LabRAM-Nano Series is based on the provenLabRAM HR system providing unsurpassedperformance for classical Raman analysis. With theAFM coupling option, it becomes the platform ofchoice for AFM/Raman experiments. The off-axisgeometry offers large sample handling capabilitiesand is ideally suited for the analysis ofsemiconductor materials, wafers and more generallyopaque samples.For biological and life science applications, theLabRAM-Nano operates in inverted on-axisconfiguration with a confocal inverted Ramanmicroscope on top of which the AFM unit is directlymounted. This system is ideally suited for the studyof transparent biological samples such as singlecells, tissue samples and bio-polymers.In both systems, AFM and SNOM fluorescencemeasurements can be combined with Ramananalysis to provide a more completecharacterisation of sample chemistry andmorphology on the same area. Several AFMsystems from leading AFM manufacturers can beadapted on these two instruments. Please contactus to find out which one is best for you!AFM- microRaman dual analysisThe seamless integration of hardware and software of both systems onto the same platform enables fast and user-friendly operation of both systems at the same time. Furthermore, the AFM/Raman coupling does not compromise the individual capabilities of either system and the imaging modes of the AFM remain available (EFM, MFM, Tapping Mode, etc.)The operator has direct access to both the nanometric topography of a sample given by the AFM, and the chemical information from the micro-Raman measurement. An AFM image can berecorded as an initial survey map, in which regions of interest can be defined for further Raman analysis, using the same software.An example of such analysis is illustrated below by an AFM image of Carbon Nanotubes (CNTs) giving information on the CNTs’ length, diameters and aggregation state. A more detailed AFM image is then obtained in which Raman analysis can be performed.Carbon nanotubes AFM images with a gold-coated tip in contact mode. The diameter of the bundles of nanotubes is between 10 and 30 nm.NanoRaman for TERS experimentsSurface Enhance Raman Scattering (SERS) has long been used to enhance weak Raman signals by means of surface plasmon resonance using nanoparticle colloids or rough metallic substrates, allowing to detect chemical species at ppm levels.The TERS effect is based on the same principle, but uses a metal-coated AFM tip (instead of nanoparticles) as an antenna that enhances the Raman signal coming from the sample area which is in contact (near-field). Although not yet fully understood, the TERS effect has attracted a lot of interest, as it holds the promise of producing chemical images with nanometric resolution.The LabRAM-Nano offers an ideal platform,combining state-of-the-art AFMs with our Raman expertise to perform exploratory TERS experiments with confidence.Raman signal TERS enhancement on a Silicon sample with far field suppression thanks to adequate polarization configuration. Red : Far field + Near Field (tip in contact)– Blue : Far field only (tip withdrawn)Technical specificationsFlexure guided scanner is used to maintain zero background curvature below 2 nm out-of-planeFor non-TERS measurements, classical Raman measurements can be made on the same spot as AFM images by translating the sample with a high-accuracy positioning stage from the AFM setup to the Raman setup (and vice et versa). The AFM map can be used to define a region of interest for the Raman analysisusing a common software.LabRAM-Nano coupled with Veeco’s Dimension 3100 AFMThe on-axis coupling configuration enables both AFM-microRaman dual analysis and TERS measurementson transparent and biological samples. The AFM is directly coupled onto the inverted microscope and directlyinterfaced to the LabRAM HR microprobe. It can also be taken off the optical microscope to obtain AFMimages in a different location. Seamless software integration is realized to provide a common platform to bothsystems for both AFM and Raman analysis of the same area and TERS investigation.Bioscope II from VeecoLabRAM-Nano coupled with Park Systems(formerly PSIA) XE-120Off-axis coupling for AFM-microRaman and nanoRaman (TERS)For both dual AFM-microRaman dual analysis and TERS measurements, the off-axis coupling is ideally suited for opaque and large samples. For opaque samples, the inverted on-axis coupling is not possible as the sample will not transmit the laser beam. This can be solved by setting the microscope objective at some angle to avoid “shadowing” effects from the AFM cantilever. Here also, seamless software integration is realized to provide a common platform to both systems. The AFM can be controlled by the Raman software (LabSpec), and mapping areas can be defined on AFM images for further Raman analysis.France : HORIBA Jobin Yvon S.A.S., 231 rue de Lille, 59650 Villeneuve d’Ascq. Tel : +33 (0)3 20 59 18 00, Fax : +33 (0)3 20 59 18 08. Email : raman@jobinyvon.fr www.jobinyvon.frUSA : HORIBA Jobin Yvon Inc., 3880 Park Avenue, Edison, NJ 08820-3012. Tel : +1-732-494-8660, Fax : +1-732-549-2571. Email : raman@ Japan : HORIBA Ltd., JY Optical Sales Dept., 1-7-8 Higashi-kanda, Chiyoda-ku, Tokyo 101-0031. Tel: +81 (0)3 3861 8231, Fax: +81 (0)3 3861 8259. Email: raman@ LabRAM-Nano coupled with Park Systems (formerly PSIA) XE-100Combined polarized Raman and atomic force microscopy:In situ study of point defects and mechanical properties in individual ZnO nanobelts Marcel Lucas,1Zhong Lin Wang,2and Elisa Riedo1,a͒1School of Physics,Georgia Institute of Technology,Atlanta,Georgia30332-0430,USA2School of Materials Science and Engineering,Georgia Institute of Technology,Atlanta,Georgia30332-0245,USA͑Received8June2009;accepted23June2009;published online4August2009͒We present a method,polarized Raman͑PR͒spectroscopy combined with atomic force microscopy͑AFM͒,to characterize in situ and nondestructively the structure and the physical properties ofindividual nanostructures.PR-AFM applied to individual ZnO nanobelts reveals the interplaybetween growth direction,point defects,morphology,and mechanical properties of thesenanostructures.In particular,weﬁnd that the presence of point defects can decrease the elasticmodulus of the nanobelts by one order of magnitude.More generally,PR-AFM can be extended todifferent types of nanostructures,which can be in as-fabricated devices.©2009American Instituteof Physics.͓DOI:10.1063/1.3177065͔Nanostructured materials,such as nanotubes,nanobelts ͑NBs͒,and thinﬁlms,have potential applications as elec-tronic components,catalysts,sensors,biomarkers,and en-ergy harvesters.1–5The growth direction of single-crystal nanostructures affects their mechanical,6–8optoelectronic,9 transport,4catalytic,5and tribological properties.10Recently, ZnO nanostructures have attracted a considerable interest for their unique piezoelectric,optoelectronic,andﬁeld emission properties.1,2,11,12Numerous experimental and theoretical studies have been undertaken to understand the properties of ZnO nanowires and NBs,11,12but several questions remain open.For example,it is often assumed that oxygen vacancies are present in bulk ZnO,and that their presence reduces the mechanical performance of ZnO materials.13However,no direct observation has supported the idea that point defects affect the mechanical properties of individual nanostructures.Only a few combinations of experimental techniques en-able the investigation of the mechanical properties,morphol-ogy,crystallographic structure/orientation and presence of defects in the same individual nanostructure,and they are rarely implemented due to technical challenges.Transmis-sion electron microscopy͑TEM͒can determine the crystal-lographic structure and morphology of nanomaterials that are thin enough for electrons to transmit through,4,14–17but suf-fers from some limitations.For example,characterization of point defects is rather challenging.14–17Also,the in situ TEM characterization of the mechanical and electronic properties of nanostructures is very challenging or impossible.15–17 Alternatively,atomic force microscopy͑AFM͒is well suited for probing the morphology,mechanical,magnetic, and electronic properties of nanostructures from the micron scale down to the atomic scale.3,6,7,10In parallel, Raman spectroscopy is effective in the characterization of the structure,mechanical deformation,and thermal proper-ties of nanostructures,18,19as well as the identiﬁcation of impurities.20Furthermore,polarized Raman͑PR͒spectros-copy was recently used to characterize the crystal structure and growth direction of individual single-crystal nanowires.21Here,an AFM is combined to a Raman microscope through an inverted optical microscope.The morphology and the mechanical properties of individual ZnO NBs are deter-mined by AFM,while polarized Raman spectroscopy is used to characterize in situ and nondestructively the growth direc-tion and randomly distributed defects in the same individual NBs.Weﬁnd that the presence of point defects can decrease the elastic modulus of the NBs by almost one order of mag-nitude.The ZnO NBs were prepared by physical vapor deposi-tion͑PVD͒without catalysts14and deposited on a glass cover slip.For the PR studies,the cover slip was glued to the bottom of a Petri dish,in which a hole was drilled to allow the laser beam to go through it.The round Petri dish was then placed on a sample plate below the AFM scanner,where it can be rotated by an angle␸,or clamped͑see Fig.1͒.The morphology and mechanical properties of the ZnO NBs were characterized with an Agilent PicoPlus AFM.The AFM was placed on top of an Olympus IX71inverted optical micro-scope using a quickslide stage͑Agilent͒.A silicon AFM probe͑PointProbe NCHR from Nanoworld͒,with a normal cantilever spring constant of26N/m and a radius of about 60nm,was used to collect the AFM topography and modulated nanoindentation data.The elastic modulus of the NBs was measured using the modulated nanoindentation method22by applying normal displacement oscillations at the frequency of994.8Hz,at the amplitude of1.2Å,and by varying the normal load.PR spectra were recorded in the backscattering geometry using a laser spot small enough ͑diameter of1–2␮m͒to probe one single NB at a time.The incident polarization direction can be rotated continuouslywith a half-wave plate and the scattered light is analyzedalong one of two perpendicular directions by a polarizer atthe entrance of the spectrometer͑Fig.1͒.Series of PR spec-tra from the bulk ZnO crystals and the individual ZnO NBswere collected with varying sample orientation␸͑the NBs are parallel to the incident polarization at␸=0͒,in the co-͑parallel incident and scattered analyzed polarizations͒and cross-polarized͑perpendicular incident and scattered ana-lyzed polarizations͒conﬁgurations.For the ZnO NBs,addi-tional series of PR spectra were collected where the incidenta͒Electronic mail:elisa.riedo@.APPLIED PHYSICS LETTERS95,051904͑2009͒0003-6951/2009/95͑5͒/051904/3/$25.00©2009American Institute of Physics95,051904-1polarization is rotated and the ZnO NB axis remained paral-lel or perpendicular to the analyzed scattered polarization ͑see supplementary information 25͒.The exposure time for each Raman spectrum was 10s for the bulk crystals and 20min for NBs.After each rotation of the NBs,the laser spot is recentered on the same NB and at the same location along the NB.Prior to the PR characterization of ZnO NBs,PR data were collected on the c -plane and m -plane of bulk ZnO crystals ͓Fig.2͑a ͔͒.In ambient conditions,ZnO has a wurtzite structure ͑space group C 6v 4͒.Group theory predicts four Raman-active modes:one A 1,one E 1,and two E 2modes.11,20,23The polar A 1and E 1modes split into transverse ͑TO ͒and longitudinal optical branches.On the c -plane ͑0001͒-oriented sample,only the E 2modes,at 99͑not shown ͒and 438cm −1,are observed,and their intensity is independent of the sample orientation ␸͓Fig.2͑a ͔͒.On them -plane ͑101¯0͒-oriented sample,the E 2,E 1͑TO ͒,and A 1͑TO ͒modes are observed at 99,438,409,and 377cm −1,respectively ͓Fig.2͑a ͔͒,and their intensity depends on ␸.Peaks at 203and 331cm −1in both crystals are assigned to multiple phonon scattering processes.The intensity,center,and width of the peaks at 438,409,and 377cm −1were obtained by ﬁtting the experimental PR spectra with Lorent-zian lines ͑see supplementary information 25͒.The successful ﬁts of the angular dependencies by using the group theory and crystal symmetry 23indicate that PR data can be used to characterize the growth direction of ZnO NBs.It is noted that the ZnO NBs studied here have dimensions over 300nm,so the determination of the growth direction is not ex-pected to be affected by any enhancement of the polarized Raman signal due to their high aspect ratio.24AFM images and PR data of three individual ZnO NBs are presented in Figs.2͑b ͒–2͑d ͒.These NBs,labeled NB1,NB2,and NB3,have different dimensions and properties assummarized in Table I .A comparison of the PR spectra in Figs.2͑a ͒–2͑d ͒reveals differences between bulk ZnO and individual NBs.First,the glass cover slip gives rise to a weak broadband centered around 350cm −1on the Raman spectra of the NBs ͓see bottom of Fig.2͑d ͔͒.Second,there are additional Raman bands around 224and 275cm −1for NB2and NB3.These bands are observed in doped or ion-implanted ZnO crystals.11,20Their appearance is explained by the disorder in the crystal lattice due to randomly distrib-uted point defects,such as oxygen vacancies or impurities.The defect peaks area increases in the order NB1ϽNB2ϽNB3.Since the laser spot diameter is larger than the width of all three NBs,but smaller than their length,L ,the NB volume probed by the laser beam is approximated by the product of the width,w ,with the thickness,t .ThevolumeFIG.1.͑Color online ͒Schematic of the experimental setup,showing the path of the laser beam.The ZnO NBs are deposited on a glass slide,which is placed inside a rotating Petridish.FIG.2.͑Color online ͒͑a ͒PR spectra from the c and m planes of a ZnO crystal,shown in blue and green,respectively.The wurtzite structure ͑Zn atoms are brown,O atoms red ͒is also shown,where a ء,b ء,and c ءare the reciprocal lattice vectors.͓͑b ͒–͑d ͔͒AFM images ͑3ϫ3␮m ͒of three NBs labeled NB1,NB2,and NB3and corresponding PR spectra.In ͑d ͒a PR spectrum of the glass substrate is shown at the bottom.All the PR spectra in ͑a ͒–͑d ͒are collected in the copolarized conﬁguration for ␸=0and 90°.The spectra are offset vertically for clarity.TABLE I.Summary of the PR-AFM results for NB1,NB2,and NB3.w ͑nm ͒t ͑nm ͒w /t L ͑␮m ͒␪͑°͒E ͑GPa ͒Defects NB11080875 1.24028Ϯ1562Ϯ5No NB21150710 1.64972Ϯ1538Ϯ5Yes NB315104553.35966Ϯ1517Ϯ5Yesprobed decreases in the order NB1͑wϫt=9.45ϫ103nm2͒ϾNB2͑8.17ϫ103nm2͒ϾNB3͑6.87ϫ103nm2͒.This indi-cates that the density of point defects is highest in NB3,and increases with the width to thickness ratio,w/t,in the order NB1ϽNB2ϽNB3.The PR intensity variations of the438cm−1peak as a function of␸in the various polarization conﬁgurations were ﬁtted by using group theory and crystal symmetry to deter-mine the angle␪between the NB long axis͑or growth di-rection͒and the c-axis͓͑0001͔axis͒of the constituting ZnO wurtzite structure21,23͑see supplementary information25͒.In-tensity variations of the377cm−1peak,when present,are used to conﬁrm the obtained values of␪.The results are shown in Table I and indicate that growth directions other than the most commonly observed c-axis are possible,par-ticularly when point defects are present.Finally,the elastic properties of NB1,NB2,and NB3are characterized by AFM using the modulated nanoindentation method.6,7,22In a previous study,the elastic modulus of ZnO NBs was found to decrease with increasing w/t and this w/t dependence was attributed to the presence of planar defects in NBs with high w/t.6,7By using PR-AFM,we can study the role of randomly distributed defects,morphology,and growth direction on the elastic properties in the same indi-vidual ZnO NB.The measured elastic moduli,E,are62GPa for NB1,38GPa for NB2,and17GPa for NB3.These PR-AFM results conﬁrm the w/t dependence of the elastic modulus in ZnO NBs,but more importantly they reveal that the elastic modulus of ZnO NBs can signiﬁcantly decrease, down by almost one order of magnitude,with the presence of randomly distributed point defects.In summary,a new approach combining polarized Raman spectroscopy and AFM reveals the strong inﬂuence of point defects on the elastic properties of ZnO NBs and their morphology.Based on a scanning probe,PR-AFM pro-vides an in situ and nondestructive tool for the complete characterization of the crystal structure and the physical properties of individual nanostructures that can be in as-fabricated nanodevices.The authors acknowledge theﬁnancial support from the Department of Energy under Grant No.DE-FG02-06ER46293.1Y.Qin,X.Wang,and Z.L.Wang,Nature͑London͒451,809͑2008͒.2X.Wang,J.Song,J.Liu,and Z.L.Wang,Science316,102͑2007͒.3D.J.Müller and Y.F.Dufrêne,Nat.Nanotechnol.3,261͑2008͒.4H.Peng,C.Xie,D.T.Schoen,and Y.Cui,Nano Lett.8,1511͑2008͒. 5U.Diebold,Surf.Sci.Rep.48,53͑2003͒.6M.Lucas,W.J.Mai,R.Yang,Z.L.Wang,and E.Riedo,Nano Lett.7, 1314͑2007͒.7M.Lucas,W.J.Mai,R.Yang,Z.L.Wang,and E.Riedo,Philos.Mag.87, 2135͑2007͒.8M.D.Uchic,D.M.Dimiduk,J.N.Florando,and W.D.Nix,Science305, 986͑2004͒.9D.-S.Yang,o,and A.H.Zewail,Science321,1660͑2008͒.10M.Dienwiebel,G.S.Verhoeven,N.Pradeep,J.W.M.Frenken,J.A. Heimberg,and H.W.Zandbergen,Phys.Rev.Lett.92,126101͑2004͒. 11Ü.Özgür,Ya.I.Alivov,C.Liu,A.Teke,M.A.Reshchikov,S.Doğan,V. Avrutin,S.-J.Cho,and H.Morkoç,J.Appl.Phys.98,041301͑2005͒. 12Z.L.Wang,J.Phys.:Condens.Matter16,R829͑2004͒.13G.R.Li,T.Hu,G.L.Pan,T.Y.Yan,X.P.Gao,and H.Y.Zhu,J.Phys. Chem.C112,11859͑2008͒.14Z.W.Pan,Z.R.Dai,and Z.L.Wang,Science291,1947͑2001͒.15P.Poncharal,Z.L.Wang,D.Ugarte,and W.A.De Heer,Science283, 1513͑1999͒.16A.M.Minor,J.W.Morris,and E.A.Stach,Appl.Phys.Lett.79,1625͑2001͒.17B.Varghese,Y.Zhang,L.Dai,V.B.C.Tan,C.T.Lim,and C.-H.Sow, Nano Lett.8,3226͑2008͒.18M.Lucas and R.J.Young,Phys.Rev.B69,085405͑2004͒.19I.Calizo,A.A.Balandin,W.Bao,F.Miao,and u,Nano Lett.7, 2645͑2007͒.20H.Zhong,J.Wang,X.Chen,Z.Li,W.Xu,and W.Lu,J.Appl.Phys.99, 103905͑2006͒.21T.Livneh,J.Zhang,G.Cheng,and M.Moskovits,Phys.Rev.B74, 035320͑2006͒.22I.Palaci,S.Fedrigo,H.Brune,C.Klinke,M.Chen,and E.Riedo,Phys. Rev.Lett.94,175502͑2005͒.23C.A.Arguello,D.L.Rousseau,and S.P.S.Porto,Phys.Rev.181,1351͑1969͒.24H.M.Fan,X.F.Fan,Z.H.Ni,Z.X.Shen,Y.P.Feng,and B.S.Zou, J.Phys.Chem.C112,1865͑2008͒.25See EPAPS supplementary material at /10.1063/ 1.3177065for more information on the PR spectra.Growth direction and morphology of ZnO nanobelts revealed by combining in situ atomic forcemicroscopy and polarized Raman spectroscopyMarcel Lucas,1,*Zhong Lin Wang,2and Elisa Riedo1,†1School of Physics,Georgia Institute of Technology,Atlanta,Georgia30332-0430,USA 2School of Materials Science and Engineering,Georgia Institute of Technology,Atlanta,Georgia30332-0245,USA ͑Received26June2009;revised manuscript received28September2009;published14January2010͒Control over the morphology and structure of nanostructures is essential for their technological applications,since their physical properties depend signiﬁcantly on their dimensions,crystallographic structure,and growthdirection.A combination of polarized Raman͑PR͒spectroscopy and atomic force microscopy͑AFM͒is usedto characterize the growth direction,the presence of point defects and the morphology of individual ZnOnanobelts.PR-AFM data reveal two growth modes during the synthesis of ZnO nanobelts by physical vapordeposition.In the thermodynamics-controlled growth mode,nanobelts grow along a direction close to͓0001͔,their morphology is growth-direction dependent,and they exhibit no point defects.In the kinetics-controlledgrowth mode,nanobelts grow along directions almost perpendicular to͓0001͔,and they exhibit point defects.DOI:10.1103/PhysRevB.81.045415PACS number͑s͒:61.46.Ϫw,61.72.Dd,78.30.Ly,81.10.ϪhI.INTRODUCTIONControl over the morphology and structure of nanostruc-tured materials is essential for the development of future de-vices,since their physical properties depend on their dimen-sions and crystallographic structure.1–15In particular,the growth direction of single-crystal nanostructures affects their piezoelectric,1,2transport,3catalytic,4mechanical,5–9 optoelectronic,10and tribological properties.11ZnO nano-structures with various morphologies͑wires,belts,helices, rings,tubes,…͒have been successfully synthesized in solu-tion and in the vapor phase,14–19but little is known about their growth mechanism,particularly in a process not involv-ing catalyst particles.17Understanding the growth mecha-nism and determining the decisive parameters directing the growth of nanostructures and tailoring their morphology is essential for the use of ZnO nanobelts as power generators or electromechanical systems.1,2,5,6From a theoretical stand-point,a shape-dependent thermodynamic model showed that the morphology of ZnO nanobelts grown in equilibrium con-ditions depends on their growth direction,but the role of defects was not considered.20Experimentally,it was shown that the growth direction of ZnO nanostructures can be di-rected by the synthesis conditions,such as the oxygen con-tent in the furnace.19A previous study combining scanning electron microscopy and x-ray diffraction suggested a growth-direction-dependent morphology.20An atomic force microscopy͑AFM͒combined with transmission electron mi-croscopy also suggested that the morphology of ZnO nano-belts is correlated with their growth direction and highlighted the potentially important role of planar defects.5 Growth modes out of thermodynamic equilibrium and the role of point defects5,17are particularly challenging to inves-tigate experimentally,21due to the lack of appropriate experi-mental techniques.Electron microscopy can determine the crystallographic structure and morphology of conductive nanomaterials,3,17,22–24but is not suitable for the character-ization of point defects,especially when their distribution is disordered.17,22–24Raman spectroscopy has been used for the characterization of the structure of carbon nanotubes,25,26the identiﬁcation of impurities,27and the determination of the crystal structure28and growth direction of individual single-crystal nanowires.29Recently,polarized Raman͑PR͒spec-troscopy has been coupled to AFM to study in situ the inter-play between point defects and mechanical properties of ZnO nanobelts.30Here,PR-AFM is used to study the growth mechanism and the relationship between growth direction,point defects, and morphology of individual ZnO nanobelts.The morphol-ogy of an individual ZnO nanobelt is determined by AFM, while the growth direction and randomly distributed defects in the same individual nanobelt are characterized by polar-ized Raman spectroscopy.II.EXPERIMENTALThe ZnO nanobelts were prepared by physical vapor deposition͑PVD͒without catalysts following the method de-scribed in Ref.17.The ZnO nanobelts were deposited on a glass cover slip,which was glued to a Petri dish.The rotat-able Petri dish was then placed on a sample plate under an Agilent PicoPlus AFM equipped with a scanner of100ϫ100␮m2range.Topography images of the ZnO nanobelts were collected in the contact mode with CONTR probes͑NanoWorld AG,Neuchâtel,Switzerland͒of normal spring constant0.21N/m at a set point of2nN.The AFM was placed on top of an Olympus IX71inverted optical micro-scope that is coupled to a Horiba Jobin-Yvon LabRam HR800.PR spectra were recorded in the backscattering ge-ometry using a40ϫ͑0.6NA͒objective focusing a laser beam of wavelength785nm on the sample to a power den-sity of about105W/cm2and a spot size of about2␮m. The incident polarization direction can be rotated continu-ously with a half-wave plate.The scattered light was ana-lyzed along one of two perpendicular directions by a polar-izer at the entrance of the spectrometer.The intensity,center, and width of the Raman bands were obtained byﬁtting the spectra with Lorentzian lines.The polarization dependence of the quantum efﬁciency of the Raman spectrometer was tested by measuring the intensity variations of the377,409,PHYSICAL REVIEW B81,045415͑2010͒1098-0121/2010/81͑4͒/045415͑5͒©2010The American Physical Society045415-1and 438cm −1bands from two bulk ZnO crystals ͑c -plane and m -plane ZnO crystals,MTI Corporation ͒.The PR data from bulk crystals were successfully ﬁtted using group theory and crystal symmetry 28without further calibration of the spectrometer or data correction.III.RESULTS AND DISCUSSIONAFM images and PR data of two individual ZnO nano-belts are presented in Fig.1.These nanobelts have different cross-sections,1320ϫ1080nm 2͑nanobelt labeled NB A͒FIG.1.͑Color online ͒PR-AFM results on individual ZnO nanobelts.͑a ͒AFM topography image,͑b ͒typical PR spectra for different sample orientations ␸and polarization conﬁgurations,and ͑c ͒–͑f ͒polar plots of the angular dependence of the Raman intensities for the nanobelt NB A.͑g ͒AFM topography image,͑h ͒typical PR spectra,and ͑i ͒–͑l ͒polar plots of the angular dependence of the Raman intensities for the nanobelt NB B.The Raman spectra in ͑h ͒exhibit peaks centered at 224and 275cm −1͑triangles ͒that are characteristic of defects in the nanobelt NB B.The Raman spectra are offset vertically for clarity.In ͑c ͒,͑d ͒,͑i ͒,and ͑j ͒,the nanobelt axis is rotated in a ﬁxed polarization conﬁguration ͑solid squares:copolarized;open squares:cross polarized ͒and is parallel to the incident polarization for ␸=0°.In ͑e ͒,͑f ͒,͑k ͒,and ͑l ͒,the incident polarization is rotated,while the analyzed polarization and the nanobelt axis are ﬁxed.In ͑e ͒,͑f ͒,͑k ͒,and ͑l ͒,at the angle 0°,the nanobelt is perpendicular to the incident polarization and the incident and analyzed polarizations are parallel ͑solid squares ͒or perpendicular ͑open squares ͒.Typical Raman spectra of the glass cover slip in the copolarized and cross-polarized conﬁgurations are shown as a reference in ͑b ͒and ͑h ͒,respectively.LUCAS,WANG,AND RIEDO PHYSICAL REVIEW B 81,045415͑2010͒045415-2。

1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。

arXiv:astro-ph/0401307v1 15 Jan 2004Searchfornon-Gaussianityinpixel,harmonicandwaveletspace:comparedandcombined

PaoloCabella,1,∗FrodeHansen,1,†DomenicoMarinucci,2,‡DanielePagano,1andNicolaVittorio11DipartimentodiFisica,Universit`adiRoma‘TorVergata’,

ViadellaRicercaScientiﬁca1,I-00133Roma,Italy2DipartimentodiMatematica,Universit`adiRoma‘TorVergata’,

ViadellaRicercaScientiﬁca1,I-00133Roma,Italy(Dated:February2,2008)AbstractWepresentacomparisonbetweenthreeapproachestotestnon-Gaussianityofcosmicmicrowavebackgrounddata.TheMinkowskifunctionals,theempiricalprocessmethodandtheskewnessofwaveletcoeﬃcientsareappliedtomapsgeneratedfromnon-standardinﬂationarymodelsandtoGaussianmapswithpointsourcesincluded.Wediscussthediﬀerentpowerofthepixel,harmonicandwaveletspacemethodsonthesesimulatedalmostfull-skydata(withPlancklikenoise).Wealsosuggestanewprocedureconsistingofacombinationofstatisticsinpixel,harmonicandwaveletspace.

PACSnumbers:02.50.Ng,95.75.Pq,02.50.Tt,98.80.EsI.INTRODUCTIONTheﬂuctuationsofthecosmicmicrowavebackground(CMB)areexpectedtobeclosetoGaussiandistributed.InviewoftheincreasingquantityofCMBexperiments,itisnowpossibletocheckthisassumptionondatawithgrowingresolutionandskycoverage.MostmodelsfortheearlyuniversepredictsomesmalldeviationsfromGaussianity;non-standardmodelsofinﬂation[1,2,3,4,5,6,7],cosmicstrings(SeeRef.[8]forareview)andpointsources.Detectingthesesmalldeviationswouldbeofgreatimportancefortheunderstandingofthephysicsoftheearlyuniverse.Alsosystematiceﬀectslikeanon-symmetricbeamandnoisecouldgiverisetonon-Gaussianfeatures.Forthisreasonanon-Gaussianitycheckcouldrevealwhethertheimpactoftheinstrumentaleﬀectsonthedataoftheexperimentiswellunderstood.

Themethodstosearchfornon-Gaussianityintheliteraturemainlyconcentrateonim-plementingthetestinthreediﬀerentspaces:(1)Inpixelspace:theMinkowskifunctionals[9,10](whichwereusedtosetlimitsonthenon-GaussianityintheWMAPdata[11]),temperaturecorrelationfunctions[12],thepeaktopeakcorrelationfunction[13],skewnessandkurtosisofthetemperatureﬁeld[14]andcurvatureproperties[15,16],tomentionafew.(2)Inharmonicspace:analysisofthebispectrumanditsnormalizedversion[17,18,19,20]andthebispectrumintheﬂatskyapproximation[21].TheexplicitformofthetrispectrumforCMBdatawasderivedin[22,23].Phasemapping[24].ApplicationstoCOBE,MaximaandBoomerangdatahavealsodrawnenormousattentionandraisedwidedebate[25,26,27,28].Theempiricalprocessmethod[29,30,31].Finally,(3)waveletspace:[32,33,34,35].Traditionally,thesetestsareperformedseparatelyineachspace.Inthisarticle,wewilltakemethodsinpixel-(theMinkowskifunctionals),harmonic-(theempiricalprocess)andwavelet-space(skewness),andwewillmakeacomparisonfortwodiﬀerentmodelsofnon-Gaussianity.Wewillalsocombinethemethodsinordertoimprovethetotalpower.Itshouldbenotedthatalltheproceduresweconsiderarenon-parametric,thatistheydonotassumeanyaprioriknowledgeaboutthenatureofnon-Gaussianity.

Wewillusethesemethodson100mapsgeneratedfromanon-standardinﬂationarymodel[6]andon100mapswherewehaveincludedpointsources.Weassesstheperformance

2ofthemethodsinthediﬀerentspacesforthediﬀerenttypesofnon-Gaussianity.Wealsoproposeacombinedtestwhichturnsouttobemorerobust.

InsectionII,wereviewthemethodofMinkowskifunctionals,insectionIIIwedescribeourimplementationofthemethodandinsectionIVwedeﬁneourproposedstatistic.InsectionV,wereviewtheempiricalprocessmethodwhilesectionVIisdevotedtothewavelets.InsectionVIIthemethodsarecomparedandappliedtonon-standardinﬂationarymodels,insectionVIIItomapswithpointsources.FinallyinsectionIXwesummarizeandcommentonourresults.

II.MINKOWSKIFUNCTIONALSToanalyzeasphericalmapintermsofMinkowskifunctionals,weconsidertheexcursionsets,thatis,themapsubsetswhichexceedagiventhresholdvalue.Thethresholdislabelledν,anditistreatedasanindependentvariable,onwhichthesefunctionalsdepend.Moreprecisely,consideringthenormalizedrandomﬁeldoftemperatureﬂuctuations,u=∆T/σ(∆T);wecandeﬁnethe’hotregion’Qastheensembleofpixelsuihigherthantheνlevel:Q≡Q(ν)={i|u(θi,ϕi)>ν}.(1)

Thethreefunctionalsofinterestthenare,uptoconstantfactors[36]:1)Area:M0(ν)isthetotalareaofallhotregions.2)Boundarylength:M1(ν)isproportionaltothetotallengthoftheboundarybetweencoldandhotregions3)Eulercharacteristicorgenus:M2(ν),apurelytopologicalquantity,countsthenum-berofisolatedhotregionsminusthenumberofisolatedcoldregions,i.e.thenumberofconnectedcomponentsinQminusthenumberof’holes’.TherationalebehindthesestatisticscanbeexplainedfrommathematicalresultsinHad-wiger(1959);inparticular,theseresultscanbeinterpretedbystatingthatallthemorpho-logicalinformationofaconvexbodyiscontainedintheMinkowskifunctionals(WinitzkiandKosowsky,1997));here,bymorphologicalwemeanthepropertieswhichareinvariantundertranslationsandrotationsandwhichareadditive[37,38,39].Thethreestatistics,