摘要
随着计算机和信息技术的发展，用计算机视觉系统来辅助或代替人的视觉感知系统，减轻人的工作量或完成人类无法完成的任务，如图像数据检索、月球探测等技术领域。当以计算机视觉系统实现人的视觉时，同样会面临辨别和认识所“看”到的物体，人类可以利用不变的信息识别物体。对于计算机而言，则是通过选择和提取物体不变特征来进行模式识别，进而对目标识别。
视点和距离的改变使得对同一场景所获取的图像之间产生差异，从这些图像中提取不受传感器姿态和位置影响的特征，是许多图像智能处理应用中的共性问题，也是实际应用中不可绕开的难题。被测目标在图像传感器上所成的图像之间满足透视变换关系，当目标尺寸远小于目标与传感器的距离时，透视变换可近似为仿射变换模型，仿射变换较好地描述了目标在不同视点和距离下所成图像间的关系。图像检索、图像数字水印、遥感卫星图像处理、目标识别等技术领域中，往往需要从不同视点图像中提取不受几何变形影响的特征量以支持后续其它问题的处理，因此仿射不变特征应用优势明显，也就成为了这些领域的研究热点。
本文以图像处理与目标识别为应用背景，深入研究了基于多尺度理论框架和仿射几何变换提取仿射不变特征的理论和方法，重点研究了区分能力好、抗非仿射变形能力强、计算速度快、适用范围广、实用的不变特征提取方法，论文的主要研究内容：
(1)多尺度自卷积变换快速算法研究
针对多尺度自卷积变换计算复杂度大的问题，分析变换中尺度值对各运算环节复杂度的影响，建立计算复杂度、图像尺寸和尺度值之间关系的数学模型，分析其中傅立叶变换复杂度的比重。研究保持多尺度自卷积变换值的最小变换尺寸要求。针对尺度在[-1,1]区间内以1/Z+（Z+为大于1的整数）间隔取值特点，研究最小化变换次数快速算法；针对尺度在[-1,1]区间外取值特点，研究最小化变换尺寸快速算法；针对尺度变换后部分尺度满足1/Z+间隔取值特点，研究最小化变换尺寸与变换次数串联快速算法。以计算复杂度模型，分析各种尺度下快速算法提升的效率，并与算法实际运行时提升的效率对比。研究快速算法在高维多尺度自卷积变换中的应用，建立三维多尺度自卷积变换计算复杂度模型，在给定尺度下，理论分析算法提高的效率。研究归一化仿射矩不变量的快速算法。实验表明，本文提出的快速算法将各尺度下的变换速度提高到约2~6倍，同时保持了与原方法一致的特征值精度，特别是最小变换次数法推广应用到归一化仿射矩不变量时，可将其速度提高到2倍以上。
(2)多尺度直方图仿射不变量提取算法研究
依据多尺度自卷积变换中密度函数，研究保持密度函数仿射变换关系的归一化方法，构建提取直方图不变特征的算法，利用直方图构造矩和熵不变量，推广算法在高维中应用。推导直方图特征的互换性、对称性和标准形式不变性，分析尺度值与特征值的关系，研究尺度、灰度阈值选择方法和直方图特征实现方法。依托二维坐标在给定的两系数下进行线性变换，构建两尺度密度函数，研究将函数卷积值转化成直方图区间划分方法，提取两尺度直方图特征。设计实验分析尺度取值、灰度阈值对两种特征识别率的影响，实验分析特征对二值图像、灰度图像、视角变换图像的分类效果。实验表明，两种特征的识别率在抗非仿射变形方面优于经典尺度下的MSA特征，其中两尺度直方图特征总体性能最优，计算最快。
(3)扩展质心的仿射几何不变量提取算法研究
扩展质心方法是通过迭代仿射区域划分得到图像中一系列仿射不变点，利用不变点构造几何不变特征。研究仿射区域划分得到的每对扩展质心与图像质心的位置关系，制定无冗余仿射区域划分策略，研究线段长度比仿射不变特征提取方法和扩展图像构造方法，提出新的扩展质心提取方法，通过增加扩展图像数量，减少划分次数，提高特征维数，构造线段长度比不变量。实验分析质心的共线性、所提特征的仿射不变性、扩展函数对特征稳定性的影响、特征的识别能力和计算效率。实验表明，所提划分策略累积误差小，与经典的三角形、四边形面积比不变量相比，在同等特征数量条件下，所提特征稳健性高，分类识别性能更优。
(4)多尺度仿射几何不变量提取算法研究
研究利用多尺度框架构建一系列仿射协变图像方法，研究基于扩展质心构造仿射区域面积比不变特征提取方法，利用协变图像构造多尺度区域面积比不变量，提出组合划分策略，以一次仿射区域划分构造任意数量的不变特征。研究尺度取值对协变图像和特征值不变性的影响，推导特征保持仿射不变性最小定义域。实现不变特征的快速计算。协变图像灰度由原图像灰度和灰度分布信息共同确定，二值图像的协变图像灰度不是单一值，研究特征用于二值图像识别。研究利用协变图像提取不变点配准原图像的方法。实验表明，多尺度仿射几何特征抗噪声、抗照度变化、抗局部遮挡、抗投影变形能力全面优于MSA特征，大多数条件下优于MSA矩特征，运行速度最快，在图像配准方面，精度大大高于原扩展质心方法。
本文提出了一种仿射不变特征快速算法和三种仿射不变特征提取方法，在目标识别和图像自动配准等方面具有重要的应用价值。
With the development of computer and information technology, using computer visionsystem to assist or replace the human visual perception system can reduce the humanworkload or perform tasks that humans can’t complete, such as fields of image data retrieval,lunar exploration. When a computer vision system is used to achieve human one, it will facethe problem of identification and recognition of ‘observed’ Objects like human. Humans canuse invariant information to identify objects, while the computer can recognize targets byselecting and extracting objects for pattern recognition.
As changes of the viewpoint and the distance will make the acquired images of thesame scene differ from each other, extracting features unaffected by the sensor attitude andposition from these images is not only a common problem during many image intelligentprocessing applications, but also a conundrum that cannot bypass in practice. The measuredtargets on the images acquired by image sensors satisfy the perspective transformation, whenthe target size is far smaller than the distance between target and sensor, a perspectivetransformation model can be approximated by an affine transformation model, and the affinetransformation can better describe the relationship between the image target in differentviewpoints and distances. In the technical fields of image retrieval, image digitalwatermarking, satellite remote sensing image processing, and target recognition, it oftenneeds to deal with extraction of features unaffected by geometric deformation from differentview images for the follow-up of other problems, so affine invariant feature has obviousapplication advantages, it has become a hot topic in these areas.
With the application background of image processing and object recognition, this paperstudies the theory and method of affine invariant feature extraction based on multi-scaletheoretical framework and the affine geometry transform. It focuses on practical extractionmethod of affine invariant features that satisfy fine discrimination, strong anti non-affinedeformation ability, quick calculation speed, and wide application range. The main contentsof this paper are as follows:
(1) Research on fast multi-scale auto convolution transform algorithm
Aiming at the great computation complexity of multi-scale auto convolution transform,this paper analyzes the impact of scale value on complexity in every transform computationphase. And a mathematical model that represents the relationship among computationalcomplexity, image size and scale value is established to analyze the proportion of thecomplexity of the Fourier transform. This Paper also studies minimum transform scalerequirements to maintain multi-scale auto convolution transformation value. As scale in therange of [-1,1] keeps an interval of1/Z+(Z+is an integer greater than1), this paperresearches on minimum transform frequency fast algorithm; for scale outside the range of [-1, 1], it studies minimum transform scale fast algorithm; According to the fact that partial scalewill meet1/Z+interval after transformation, it studies on the tandem rapid algorithm ofminimum transform scale and frequency. This paper analyzes the efficiency of themulti-scale fast algorithm and compares it with the improved efficiency of the actualrun-time algorithm by the computational complexity model. Also, application of rapidalgorithm in high-dimensional multi-scale auto convolution transformation has been studiedby establishing the computational complexity model of3-D multi-scale auto convolutiontransformation and analyzing the efficiency improved at a given scale in theory. Additionally,a fast algorithm of normalized affine moment invariants has been studied. Experimentalresults show that the transform speed of the proposed fast algorithm under various scales willbe increased to about2to6times. In the mean time, it still maintains the eigenvalueaccuracy that consistent with the original method. Particularly, the speed is increased to morethan2times when the minimum transform frequency fast algorithm is applied to thenormalized affine moment invariant.
(2) Research on multi-scale histogram affine invariant extraction
Based on the density function of multi-scale auto convolution transformation, this paperstudies on the normalization method that maintains the affine transformation relations ofdensity function, it proposes a histogram invariant feature extraction algorithm and then useshistogram to construct moments and entropy invariant, moreover, it generalizes thealgorithm to high-dimensional application. This paper derives the interchangeability,symmetry and standard form invariance of histogram feature, it analyzes the relationbetween scale values and eigenvalues and researches on the scales, gray threshold selectionmethod and the histogram features implementation. Relying on a two-dimensionalcoordinates, it makes a linear transformation at the two given factors and builds atwo-dimensional density function, furthermore, it studies how to convert functionconvolution value into histogram interval division for two-dimensional histogram featureextraction. This paper designs experiments to analyze the impact of scale values, graythreshold on two feature recognition rate and put features to test the classification effect ofbinary images, grayscale images and the perspective transformed images. Experiments showthat the recognition rate of two features performs better than MSA features of classic scalesin terms of anti non-affine deformation; especially, the two-dimensional histogram featurehas the best overall performance and fastest computational speed among others.
(3) Research on affine geometry invariant extraction based on extended centroid
The extended centroid method obtains a series of affine invariant points from image byiterative affine zoning; it uses invariant points to construct geometric invariant feature. Thispaper studies the position relation between each pair of extended centroid obtained by affinezoning and the image centroid; it develops non-redundant affine zoning strategy and studiesaffine invariant feature extraction based on line length ratio and an extended image construction method. Also, it proposes a new extended centroid extraction method byincreasing the number of extended images, reducing the number of zoning, improving thefeature dimension and constructing line length ratio invariant. This paper makes experimentsto analyze the impact of the colinearity of the centroid, the affine invariance of extractedfeatures and spread-function on the stability of features, feature recognition ability andcomputational efficiency. Experimental results show that the cumulative error of theproposed zoning strategy is small, compared with the classical triangle, quadrangle area ratioinvariants, in the conditions of same feature numbers, the extracted feature has a betterability of robustness and classification.
(4) Research on multi-scale affine geometry invariant extraction
This paper uses a multi-scale framework to construct a series of affine covariant image,develops an affine area ratio invariant feature extraction method based on extended centroid,utilizes covariant image to construct multi-scale area ratio invariant, proposes a compositepartitioning strategy to construct any number of invariant features by an affine zoning. Itresearches on the impact of scale values on covariant image and feature invariance, derivesthe minimum domain when feature maintains affine invariance. It achieves rapid calculationof invariant features. Covariant image intensity is determined jointly by the original imageintensity and the intensity distribution information, the covariant image intensity of binaryimage is not a single value, study of features can be used for the binary image recognition.This paper also studies on the method of invariant point extraction by using covariant imagefor the original image registration. Experimental results show that multi-scale affinegeometric feature performs better than MSA feature comprehensively in terms of anti-noise,anti-illumination changes, partial occlusion resistant, and anti-deformation. Under mostconditions, it has a better performance and a faster computational speed than MSA momentfeature. In terms of image registration, its accuracy is much higher than the original extendedcentroid method.
This paper presents a fast affine invariant feature algorithm and three affine invariantfeature extraction methods; it has important application value in terms of target recognition,image auto-registration and so on.
引文
[1]孙即祥，等．模式识别中的特征提取与计算机视觉不变量[M]．长沙：国防工业出版社，2001．
[2] Allili M S, Baaziz N. Contourlet-based texture retrieval using a mixture of generalized Gaussiandistributions [M]. Lecture Notes in Computer Science. Heidelberg: Springer,2011,6855.446-454.
[3] Jing F, Li M, et al. An efficient and effective region-based image retrieval framework [J]. IEEETransactions on Image Processing,2004,13(5):699-709.
[4]熊智，刘建业，曾庆化，等．景象匹配辅助导航系统中的图像匹配算法研究[J]．中国图象图形学报，2004，9(1)：29-34．
[5] Hu M K. Visual pattern recognition by moment invariants [J]. IEEE Transactions on InformationTheory,1962,8(2):179-187.
[6] Cassasent D, Psaltis D. Position, rotation, and scale invariant optical correlation [J]. Applied Optics,1976,15(7):1795-1799.
[7] Chen C C. Improved moment invariant for shape discrimination [J]. Pattern Recognition,1993,26(5):683-686.
[8] Unel M, Wolovich W A. A new representation for quartic curves and complete sets geometricinvariants [J]. International Journal of Pattern Recognition and Artificial Intelligence,1999,13(8):1137-1150.
[9] Tzimiropoulos G, Mitianoudis N, Stathaki T. Robust recognition of planar shapes under affinetransforms using principal component analysis [J]. IEEE Signal Processing Letters,2007,14(10):723-726.
[10] Morel J M, Yu G S. ASIFT: A new framework for fully affine invariant image comparison [J]. SIAMJournal on Imaging Sciences,2009,2(2):438-469.
[11] Zhang J G, Tan T N. Affine invariant classification and retrieval of texture images [J]. PatternRecognition,2003,36(3):657-664.
[12] Alferez R, Wang Y F, Jiao L. An affine-invariant tool for retrieving image from homogeneousdatabases [J]. Multimedia Tools and Applications,2005,25(1):133-159.
[13] Lee C F, Wang S C, Wang Y J, et al. Content-based image retrieval based on vector quantizationand affine invariant region [A].20088th International Conference on Intelligent Systems Designand Application [C]. Kaohsuing, Taiwan,2008.287-290.
[14] Qi X J, Qi J. Image content-based geometric transformation resistant watermarking approach [A].Proceedings of the Fifth International Conference on Acoustics, Speech, and Signal Processing [C].Philadelphia, USA,2005.2:829-832.
[15] Kang X G, Huang J W, Shi Y Q, et al. A DWT-DFT composite watermarking scheme robust to bothaffine transform and JPEG compression [J]. IEEE Transactions on Circuits and Systems for VideoTechnology,2003,13(8):776-786.
[16] Huang K, He S W. Feature based affine invariant watermarking robust to geometric distortions [J].Fundamenta Informaticae,2009,92(1/2):131-143.
[17]吕金建．基于特征的多源遥感图像配准技术研究[D]．长沙：国防科学技术大学，2008．
[18] Qin Q M, Chen S J, Wang W J, et al. The building recognition of high resolution satellite remotesensing image based on wavelet analysis [A]. Proceedings of2005International Conference onMachine Learning and Cybernetics [C]. Guangzhou, China,2005.4533-4538.
[19] Lucchese L, Leorin S, Cortelazzo G M. Estimation of two-dimensional affine transformationsthrough Polar curve matching and its application to image mosaicking and remote-sensing dataregistration [J]. IEEE Transactions on Image Processing,2006,15(10):3008-3019.
[20] Flusser J, Kautsky J, roubek F. Object recognition by implicit invariants [A].12th ComputerAnalysis of Images and Patterns [C]. Vienna, Austria,2007.856-863.
[21]吕文先，彭启民，吕玉增，等．一种基于频域特征的仿射不变目标识别方法[J]．计算机研究与发展，2009，46(3)：478-484．
[22] Tuytelaars T, Gool L V. Wide baseline stereo matching based on local affinely invariant regions[A].In: British Machine Vision Conference [C]. Bristol, UK,2000.412-422.
[23] Matas J, Chum O, Urban M, et al. Robust wide baseline stereo from maximally stable extremalregions [A]. In: British Machine Vision Conference [C].Cardiff, Wales,2002.384-393.
[24] Reiss T H. The Revised Fundamental Theorem of Moment Invariants [J]. IEEE Transaction onPattern Analysis and Machine Intelligence,1991,13(8):830-834.
[25] Flusser J, Suk T. Pattern recognition by affine moment invariant [J]. Pattern Recognition,1993,26(1):167-174.
[26] Suk T, Flusser J. Graph method for generating affine moment invariants [A]. In Proc. InternationalConference on Pattern Recognition [C]. Cambridge, England,2004.192-195.
[27] Liu J, Li D, Tao W, et a1. An automatic method for generating affine moment invariants [J]. PatternRecognition Letters,2007,28(16):2295-2304.
[28] Teague M R. Image analysis via the general theory of moments [J]. Journal of the Optical Societyof America,1980,70(8):920-930.
[29] Teh C H. Chin R T. On image analysis by the methods of moments [J]. IEEE Transactions onPattern Analysis and Machine Intelligence,1988,10(4):496-513.
[30]叶斌，彭嘉雄．伪Zernike矩不变性分析及其改进研究[J]．中国图象图形学报，2003，8A(3)：246-252．
[31] Ye B, Peng J X. Invariance analysis of improved Zernike moments [J]. Journal of Optics A: Pureand Applied Optics,2002,4(6):606-614.
[32]杨辰宇．正交仿射不变量的研究应用[D]．南京：东南大学，2009．
[33] Xin Y，Pawlak M，Liao S. Accurate computation of Zernike moments in polar coordinates [J]．IEEETransactions on Image Processing,2007,16(2):581-587．
[34] Yap P T. An efficient method for the computation of Legendre moments [J]. IEEE Transactions onPattern Analysis and Machine Intelligence,2005,27(12):1996-2002.
[35] Fu B, Zhou J Z. Image analysis by modified Legendre moments [J]. Pattern Recognition,2007,40(2):691-704.
[36] Mukundan R, Ramakrishnan K R. Fast computation of Legendre and Zernike moments [J]. PatternRecognition,1995,28(9):1433-1442.
[37] Belkasim S O, Ahmadi M, Shridhar M. An efficient algorithm for fast computation of Zernikemoments[J]. Journal of the Franklin Institute,1996,333(4):577-581.
[38] Gu J, Shu H Z. A novel algorithm for fast computation of Zernike moments [J]. Pattern Recognition,2002,35(12):2905-2911.
[39] Chong C W, Raveendran P, Mukundan R. A comparative analysis of algorithms for fastcomputation of Zernike moments [J]. Pattern Recognition,2003,36(3):731-742.
[40] Chong C W, et a1. An efficient algorithm for fast computation of Pseudo-Zernike moments [J].International Journal of Pattern Recognition and Artificial Intelligence,2003,17(6):101l-1023.
[41]夏婷，周卫平，李松毅，等．一种新的Pseudo-Zernike矩的快速算法[J]．电子学报，2005，33(7)：1295-1298．
[42]黄荣兵，杜明辉，梁帼英，等．一种改进的伪zernike矩快速计算方法[J]．华南理工大学学报（自然科学版），2009，37(1)：54-58.
[43] Shu H Z, et al. A new fast method for computing Legendre moments [J]. Pattern Recognition,2000,33(2):341-348.
[44] Zhou J D, Shu H Z. Two new algorithms for efficient computation of Legendre moments [J].Pattern Recognition,2002,35(5):1143-1152.
[45]秦磊，舒华忠，於文雪，等．Legendre矩的两种快速算法[J]．电子学报，2004，32(1)：25-28．
[46]杨冠羽，舒华忠，周卫平，等．一种新的灰度图像Legendre矩的快速算法[J]．计算机学报，2004，27(12)：1642-1647．
[47] Mukundan R, Ong S H, Lee P A. Image analysis by Tchebichef moments [J]. IEEE TransactionsImage Processing,2001,10(9):1357-1364.
[48] Yap P T, Raveendran P, Ong S H. Image analysis by Krawtchouk moments [J]. IEEE TransactionsImage Processing,2003,12(11):1367-1377.
[49] Yap P T, Paramesran R. Image analysis using Hahn moments [J]. IEEE Transactions on PatternAnalysis and Machine Intelligence,2007,29(11):2057-2062.
[50] Mukundan R. Radial Tchebichef invariants for pattern recognition [A]. In: Proceedings of IEEETENCON05[C]. Melbourne, Australia,2005.2098-2103.
[51]吴海勇，舒华忠，张辉，等．基于递降阶乘计算的Tchebichef矩的旋转不变量[J]．中国图象图形学报，2010，15(3)：417-421．
[52]李春树，汪西原．基于径向Tchebichef矩的不变性模式识别[J]．兰州大学学报（自然科学版），2011，47(2)：113-116．
[53] Raj P A, Venkataramana A. Radial Krawtchouk moments for rotational invariant pattern recognition
[A].6th International Conference on Information, Communications&Signal Processing [C].Singapore,2007.1-5.
[54]张力，肖薇薇，钱恭斌，等．基于Krawtchouk不变矩的仿射攻击不变性局部水印算法[J]．电子学报，2007，35(7)：1404-1408．
[55]张辉，周健．Tchebichef矩的快速算法[J]．东南大学学报（自然科学版），2006，36(5)：857-862．
[56]朱娜．径向离散正交矩的研究和应用[D]．南京：东南大学，2008．
[57]章品正，王征，徐琴珍，等．二维Tchebichef矩正反变换的快速算法[J]．信号处理，2007，23(1)：69-72．
[58]付波，罗柱，廖家平，等．Krawtchouk矩的误差传递分析及算法改进[J]．计算机与数字工程，2010，38(6)：1-4．
[59] Zahn C T, Roskies R Z. Fourier descriptors for plane closed curves [J]. IEEE Transactions onComputers,1972,21(3):269-281.
[60] Urdiales C, Bandera A, Sandoval F. Non-parametric planar shape representation based on adaptivecurvature functions [J]. Pattern Recognition,2002,35(1):45-53.
[61] Kauppien, Sepp nen T, Pietik inen M. An experiment comparison of autoregressive andFourier-based descriptors in2D shape classification [J]. IEEE Transactions on PAMI,1995,17(2):201-207.
[62] Zhang D S, Lu G J. Study and evaluation of different Fourier methods for image retrieval [J].Image and Vision Computing,2005,23(1):33-49.
[63] Kunttu I, Lepisto L. Shape-based retrieval of industrial surface defects using angular radius Fourierdescriptor. IET Image Processing,2007,1(2):231-236.
[64] EI-ghazal A, Basir O, Belkasim S. Farthest point distance: A new shape signature for Fourierdescriptors [J]. Signal Processing: Image Communication,2009,24(7):572-596.
[65]王斌．一种基于多级弦长函数的傅立叶形状描述子[J]．计算机学报，2010，33(12)：2387-2396．
[66] Zhang D S, Lu G J. Generic Fourier descriptor for shape-based image retrieval [A]. In: IEEEInternational Conference on Multimedia and Expo [C]. Lausanne, Switzerland,2002.1:425-428.
[67]范春年，陈建坤，傅德胜．一种二维极坐标傅立叶描述子在图像检索中的应用[J]．计算机工程与应用，2004，40(24)：77-79．
[68] Sajjanhar A, Lu G J, Zhang D S. Image retrieval using modified generic fourier descriptor [A]. InProc. of the19th International Conference on Computers and Their Applications [C]. Washington,USA,2004.32-35.
[69]王涛，刘文印，孙家广，等．傅立叶描述子识别物体的形状[J]．计算机研究与发展，2002，39(12)：1714-1719．
[70] Bartolini L, Ciaccia P, Patella M. WARP: accurate retrieval of shapes using phase of Fourierdescriptors and time warping distance [J]. IEEE Transactions on Pattern Analysis and MachineIntelligence,2005,27(1):142-147.
[71]王斌．一种不变的基于傅立叶变换的区域形状描述子[J]．电子学报，2012，40(1)：84-88．
[72] Kunttu I, Lepist L, Rauhamaa J, et al. Multiscale Fourier descriptor for shape classification [A].Proceedings of the12th International Conference on Image Analysis and Processing [C]. Mantova,Italy,2003.536-541.
[73]章志勇，潘志庚，张明敏，等．基于多尺度通用傅立叶描述子的灰度图像检索[J]．中国图象图形学报，2005，10(5)：611-615．
[74] Kunttu I, Lepist L, Rauhamaa J, et al. Multiscale Fourier descriptor for defect image retrieval [J].Pattern Recognition Letters,2006,27(2):123-132.
[75] Direkoglu C, Mark S N. Shape classification using multiscale Fourier-based description in2-Dspace [A]. Proceedings of the9th International Conference on Signal Processing [C]. Beijing, China,2008.820–823.
[76] Arbter K, Snyder W E, Burkhardt H, et al. Application of affine-invariant Fourier descriptor torecognition of3-D objects [J]. IEEE Transactions Pattern Analysis and Machine intelligence,1990,12(7):640-647.
[77] Oirrak A E, Daoudi M, Aboutajdine D. Affine invariant descriptors using Fourier series [J]. PatternRecognition Letters,2002,23(10):1109-1118.
[78] Chaker F, Bannour M T, Ghorbel F. Contour retrieval and matching by affine invariant Fourierdescriptors [A]. Conference on Machine Vision Application [C]. Tokyo, Japan,2007.291-294.
[79] Lionel Evina Ekombo P, Ennahnahi N, Oumsis M, et al. Application of affine invariant Foruierdescriptor to shape-based image retrieval [J]. International Journal of Computer Science andNetwork Security,2009,9(7):240-247.
[80] Mallat S. A theory for multi-resolution signal decomposition: the wavelet representation [J]. IEEETransactions on Pattern Analysis and Machine Intelligence,1989,11(7):674-693.
[81] Khalil M I, Bayoumi M M. Invariant2D object recognition using the wavelet modulus maxima [J].Pattern Recognition Letters,2000,21(9):863-872.
[82] Quang M T, Boles W W. Recognition of2D object contours using the wavelet transformzero-crossing representation [J]. IEEE Transactions on Pattern Analysis and Machine Intelligence,1997,19(8):910-916.
[83] Saito N et al. Discriminant feature extraction using empirical probability density estimation and alocal basis library [J]. Pattern Recognition,2002,35(12):2841-2852.
[84] Etemad K, Chellappa R. Separability-based multiscale basis selection and feature extraction forsignal and image classification [J]. IEEE Transactions Image Processing,1998,7(10):1453-1465.
[85]熊惠霖，张天序．具有平移和尺度不变性的自适应小波变换[J]．电子学报，2000，28(1)：104-107．
[86] Xiong H L, Zhang T X, Moon Y S. A translation and scale invariant adaptive wavelet transform [J].IEEE Transactions Image Processing,2009,9(12):2100-2108.
[87]潘泓．基于小波矩和小波神经网络的自动目标识别研究[D]．南京：东南大学，2004．
[88] Chen G, Tien D B. Invariant Fourier-wavelet descriptor for pattern recognition [J]. PatternRecognition,1999,32(7):1083-1088.
[89] EI Rube I, Ahmed M, Kamel M. Wavelet approximation-based affine invariant shape representationfunctions [J]. IEEE Transactions on Pattern Analysis and Machine Intelligence,2006,28(2):323-327.
[90] Khalil M I, Bayoumi M M. Affine invariants for object recognition using the wavelet transform [J].Pattern Recognition Letters,2002,23(1-3):57-72.
[91]张华，冯象初，董素媛．小波变换在仿射不变性目标识别中的应用[J]．模式识别与人工智能，2006，19(2)：191-195．
[92] Khalil M I. Bayoumi M M. A dyadic wavelet affine invariant function for2D shape recognition [J].IEEE Transactions on Pattern Analysis and Machine Intelligence,2001,23(10):1152-1164.
[93] Tieng Q M, Boles W W. Wavelet-based affine invariant representation: A Tool for recognitionplanar objects in3D space [J]. IEEE Transactions Pattern Analysis and Machine Intelligence,1997,19(8):846-857.
[94] Bala E. Computationally efficient wavelet affine invariant functions for shape recognition [J]. IEEETransactions Pattern Analysis and Machine Intelligence,2004,26(8):1095-1099.
[95] Chen G Y, Bui T D, Krzy ak A. Invariant pattern recognition using radon, dual-tree complexwavelets and Fourier features [J]. Pattern Recognition,2009,42(9):2013-2019.
[96] Zhang J G, Tan T N. Affine invariant texture analysis based on structural properties [A].·Proceedings of the Fifth Asian Conference on Computer Vision [C]. Melbourne, Australia,2002.216-221.
[97] Yang Z W, Cohen F S. Image registration and object recognition using affine invariants and convexhulls [J]. IEEE Transactions on Image Processing,1999,8(7):934-946.
[98]孙亦南，刘伟军，王越超．基于几何不变量的图像特征识别[J]．计算机应用与软件，2004，21(12)：1-3．
[99]陈涛，粟毅，蒋咏梅，等．利用仿射几何特性提取图像中的仿射不变特征[J]．中国图象图形学报，2007，12(9)：1633-1641．
[100]雷琳，陈涛，李智勇，等．全局仿射变换条件下图像不变量提取新方法[J]．国防科技大学学报，2008，30(4)：64-70．
[101]高峰，文贡坚．利用仿射几何的仿射不变特征提取方法[J]．中国图象图形学报，2011，16(3)：389-397．
[102] Heikkil J. Multi-scale autoconvolution for affine invariant pattern recognition [A]. In Proc.International Conference of Pattern Recognition [C]. Québec, Canada,2002.119-122
[103] Rahtu E, Heikkil J. Object classification with multi-scale autoconvolution [A]. Proc. of the17thInternational Conference on Pattern Recognition [C]. Cambridge, England,2004.37-40.
[104] Rahtu E, Salo M, Heikkil J. Affine invariant pattern recognition using multi-scale autoconvolution[J]. IEEE Transactions on Pattern Analysis and Machine Intelligence,2005,27(6).908-918.
[105] Yang J W, Chen Z R, Chen W S, et al. Robust affine invariant descriptor [J]. MathematicalProblems in Engineering,2011, ID185303.
[106] Rahtu E, Salo M, Heikkil J. Convexity recognition using multi-scale autoconvolution [A]. In Proc.International Conference of Pattern Recognition [C]. Cambridge, England,2004.692-695.
[107] Kannala J, Rahtu E, Heikkil J. Affine registration with Multi-Scale Autoconvolution [A]. Proc.International Conference on Image Processing [C]. Genoa, Italy,2005.1064-1067.
[108] Zhai C M, Du J X. Graphic symbol recognition of engineering drawings based on multi-scaleautoconvolution transform [A]. Proc.4th International Symposium on Neural Networks [C].Nanjing, china,2007.793-800.
[109] Lei L, Su Y, Li Z Y. A new target association algorithm based on invariant features in remotesensing images [A]. The International Society for Optical Engineering [C]. Florence, Italy,2007.67481B.
[110]雷琳，蔡红苹，唐涛，等．基于MSA特征的遥感图像多目标关联算法[J]．遥感学报，2008，12(4)：586-592．
[111]谢锦，蔡自兴，唐琎．基于MSA不变矩的道路导向标线分类[J]．中国图象图形学报，2011，16(8)：1418-1423．
[112] Rahtu E, Salo M, Heikkil J. A new affine invariant image transform based on ridgelets [A]. Proc.16th British Machine Vision Conference [C]. Edinburgh, UK,2006.1059-1068.
[113]张洁玉，陈强，白小晶等.仿射不变的多尺度自卷积熵提取方法[J]．计算机辅助设计与图形学报，2009，21(9)：1328-1332．
[114]张洁玉，陈强，白小晶，等．基于广义典型相关分析的仿射不变特征提取方法[J]．电子与信息学报，2009，31(10)：2465-2469．
[115] Rahtu E, Salo M, Heikkil J. Multi-scale autoconvolution histograms for affine invariant patternrecognition [A]. Proc.16th British Machine Vision Conference [C]. Edinburgh, UK,2006.1039-1048.
[116] Rahtu E, Salo M, Heikkil J. A new efficient method for producing global affine invariants [A].Proc.13th International Conference on Image Analysis and Processing [C]. Springer, Germany,2005.407-414.
[117] Rahtu E, et al. Generalized affine moment invariants for object recognition [A]. Proc.18thInternational Conference on Pattern Recognition [C]. Hong Kong,2006.634-637.
[118] Rahtu E. A multi-scale framework for affine invariant pattern recognition and registration [D/OL].http://herkules.oulu.fi/isbn9789514286018/index.html?lang=en,2007-11-2.
[119] Rahtu E, Salo M, Heikkil J. Nonlinear functionals in the construction of multiscale affineinvariants [A]. Proc.15th Scandinavian Conference on Image Analysis [C]. Aalborg, Denmark,2007.482-491.
[120] Hel-Or Y, Teo P C. A common framework for steerability, motion estimation, and invariant featuredetection [A]. In: Proceeding of the IEEE International Symposium on Circuits and System [C].Monterey, CA,1998.337-340.
[121] Ben-Arie J, Wang Z Q. Pictorial recognition of objects employing affine invariance in thefrequency domain [J]. IEEE Transactions on Pattern Analysis and Machine Intelligence,1998,20(6):604-618.
[122] Wang Y, Teoh E K.2D affine-invariant contour matching using B-spline model [J]. IEEETransactions on Pattern Analysis and Machine Intelligence,2007,29(10):1853-1858.
[123]吴刚，李道伦．基于隐含多项式曲线仿射不变量的目标识别[J]．电子学报，2004，32(12)：1987-1991．
[124] Petrou M, Kadyrov A. Features invariant to affine distortions from the Trace transform [A]. In:Proceeding of the IEEE Conference on Image Processing [C]. Thessaloniki, Greece,2001.852-855.
[125] Kadyrov A, Petrou M. The Trace transform and its application [J]. IEEE Transactions on PatternAnalysis and Machine Intelligence,2001,23(8):811-828.
[126] Petrou M, Kadyrov A. Affine invariant features from the Trace transform [J]. IEEE Transactions onPattern Analysis and Machine Intelligence,2004,26(1):30-44.
[127] Kadyrov A, Petrou M. Object descriptors invariant to affine distortions [A]. Proceedings of theBritish Machine Vision Conference [C]. Manchester, UK,2001.391-400.
[128] Columbia University, Coil-100Image Database [OL]. http://www.cs.columbia.edu,2010.
[129]黄波，赵继印，郑蕊蕊，等．基于多尺度自卷积归一化直方图的仿射不变量模式识别[J]．电子学报，2011，39(1)：64-69．
[130]黄波，赵晓晖，曹昕艳，等．基于两尺度自卷积直方图仿射不变量提取方法[J]．通信学报，2012，33(12)：140-146．
[131]贾建华，焦李成，黄文涛．一种基于质心不变特征的仿射不变纹理特征提取算法[J]．电子学报，2008，36(10)：1910-1915．
[132] Khairul H A, Osamu O. Implementation of multi-centroid moment invariants in thermal-based faceidentification system [J]. American Journal of Applied Sciences,2010,7(3):283-289.
[133]黄波，赵晓晖，时公涛，等．线段长度比仿射几何不变量提取方法[J]．吉林大学学报(工学版)，2013，43(2)：497-503．
[134]宋卫东．解析几何[M]．北京：高等教育出版社，2003．
[135]王耀明．图像的矩函数—原理、算法及应用[M]．上海：华东理工大学出版社，2002．
[136]章毓晋．图像工程（上册）—图像处理[M]．北京：清华大学出版社，2006．
[137]陈涛．图像仿射不变特征提取方法研究[D]．长沙：国防科学技术大学，2006．