Multi-scale geometric analysis (Ridgelet, Curvelet, Contourlet, Bandelet, Wedgelet, Beamlet)

The discussion of sparse radicals has been going on for nearly one months, this time to discuss multiscale geometric analysis. However, as the following discussion of these transformations are mainly oriented to the image, and I am now mainly focused on one-dimensional signal processing, so it is not in-depth discussion of these transformations, here only in the conformity of the reference literature to collate some relevant content as their own memo, the concept may not necessarily understand the accuracy, If later into the field of image processing and then carefully study.

first, from wavelet analysis to multi-scale geometric analysis

         Wavelet analysis is one of the key reasons for the great success in multi-disciplinary field, because it is more "sparse" than Fourier analysis to represent one-dimensional piecewise smooth or bounded variation function. Unfortunately, the excellent characteristics of wavelet analysis in one dimension can not be easily extended to two-dimensional or higher-dimensional. This is because the separable wavelet (separable wavelet) of a dimensional wavelet spanned only has a finite direction and cannot be "optimally" represented by a high-dimensional function with a line or a singular surface, but in fact a function with linear or polygon singularity is very common in high-dimensional spaces, for example, The smooth boundary of natural objects makes the discontinuities of natural images often reflect the singularity on the smooth curve, and not just the singularity of the point. In other words, in the case of high-dimensional, wavelet analysis does not take full advantage of the unique geometric features of the data itself, and is not the optimal or "least sparse" function representation, but the multi-scale geometric analysis developed after the wavelet analysis (multiscale geometric analyses, MGA) The aim and motive of development is to develop a new optimal representation method of high-dimensional function, in order to detect, represent and process some high dimensional spatial data, the main characteristic of these spaces is that some important features of the data are concentrated in its low subspace concentration (such as curve, polygon, etc.). For example, for two-dimensional images, the main features can be described by the edge, and in the D-D image, its important features are the filaments (filaments) and tubular (tubes).

The two-dimensional wavelet base with a dimensional wavelet spanned has a square support interval, and its support interval is square with different size in different resolution. The process of approaching the singular curve by the two-dimensional wavelet is ultimately the process of using "point" to approximate the line. At the scale J, the edge length of the wavelet support interval is approximate toJ, the number of the wavelet coefficients is at least O (2J), and when the scale is fine, the numbers of non-zero wavelet coefficients grow exponentially. There are a large number of non-negligible coefficients, which ultimately show that the original function cannot be "sparse". Therefore, in order to make full use of the geometric regularization of the original function, we hope that the support interval of the base should be shown as "long bar" to approximate the singular curve with the least coefficient. The "strip" support interval of the base is actually a manifestation of the "direction", also known as the "anisotropy (anisotropy)" of the base. The transformation we want is "multiscale geometric analysis".

The multi-scale geometric analysis method of image is divided into adaptive and non-adaptive two kinds, the adaptive method usually first edge detection and then use edge information to the original function of the optimal representation, is actually the edge detection and image representation method of combination, such methods with Bandelet and Wdgelet as the representative The non-adaptive method does not need to know the geometrical characteristics of the image, but to decompose the image directly on a fixed set of bases or frames, which is free from the dependence on the structure of the image, which represents Ridgelet, Curvelet and Contourlet transformations.

Two, several multi-scale geometrical analysis 1. Ridge Wave (Ridgelet) Transformation

The Ridge Wave (Ridgelet) theory was proposed by Emmanuelj Candès in his doctoral dissertation in 1998, which is a non-adaptive high-dimensional function representation method with direction selection and recognition ability, which can more effectively represent the singular characteristic with directivity in the signal. The Ridge Wave transformation first radon the image, that is, the straight line in the image is mapped into a point in the Randon domain, and then the singularity is detected by a dimensional wavelet, which effectively solves the problem of the wavelet transform in processing the two-dimensional image. However, the edge lines in natural images are mostly curved, and the ridgelet analysis of the whole image is not very effective. In order to solve the sparse approximation problem of multivariable functions with singular curves, in 1999, Candes proposed a single-scale ridge Wave (Monoscaleridgelet) transformation, and gave its construction method. Another method is to block the image, so that the lines in each block are approximate straight lines, and then the Ridgelet transformation of each block, which is the Multiscale ridgelet. The Ridge Wave transform has good approximation performance to the multivariable function with linear singularity, that is to say, Ridgelet can obtain provide more sparse representation for the rich image of texture (linear singularity), but for the multivariable function with curve singularity, its approximation performance is only equivalent to the wavelet transform. The nonlinear approximation error attenuation order is not optimal.

2, Qu Bo (curvelet) Transformation

         because of the large redundancy of Multiscale ridgelet analysis, Candès and Donoho in 1999 on the basis of the Ridgelet transformation, the continuous Qu Bo (curvelet) Transformation, the Curvelet year in the first generation curvelet99; 2002 transformation, Strack, Candès and Donoho proposed the Curvelet02 in the first generation Curvelet transformation. The first generation of Curvelet transforms is essentially derived from the Ridgelet theory and is based on Ridgelet transformation theory, Multiscale Ridgelet transformation theory and bandpass filter theory. The basic scale of the single-scale ridge wave transformation is fixed, and the Curvelet transformation is not, it is decomposed on all possible scales, in fact curvelet transformation is a special filtering process and multi-scale Ridge Wave transformation (multiscale ridgelet Transform) Combination: Firstly, the image is decomposed by sub-band, then the sub-band images of different scales are divided into different sizes, and the Ridgelet analysis is carried out for each block at last. Like the definition of calculus, a curve can be seen as a straight line at a sufficiently small scale, and the singularity of the curve can be represented by a linear singularity, so the Curvelet transformation can be called the integral of the Ridgelet transformation.

The first generation of Curvelet digital implementation is more complex, need sub-band decomposition, smoothing block, regularization and ridgelet analysis, and so on a series of steps, and Curvelet pyramid decomposition also brought a huge amount of data redundancy, so candès and other people in 2002 proposed to achieve more simple, A faster Curvelet transformation algorithm that is easier to understand, i.e. the second generation of Curvelet (Fastcurvelet transform). The second generation of Curvelet and the first generation of curvelet have been completely different in structure. The construction idea of the first generation of Curvelet is that the curve is approximated to a straight line in each block by a small enough block, then the characteristics are analyzed by the local ridgelet, and the second generation Curvelet and Ridgelet theory are not related, and the implementation process does not need to use Ridgelet , the same point between the two is only the abstract mathematical meaning of tight support and frame. In 2005, Candès and Donoho proposed two fast discrete curvelet transformation methods based on the second generation Curvelet transformation theory: two-dimensional FFT algorithm for non-uniform spatial sampling (unequally-spaced Fastfourier TRANSFORM,USFFT) and The Wrap algorithm (wrapping-basedtransform). For the Curvelet transform, the MATLAB program and the C + + program with Curvelet fast discrete algorithm can be downloaded on the Web in the Curvlab;curvlab package.

3. Contour Wave (contourlet) transform

In 2002, MN do and Martin Vetterli proposed a "real" two-dimensional representation of the image: The Contourlet transformation, also known as the tower-directional filter Bank (pyramidal directional filter banks, PDFB). The Contourlet transform is another multi-resolution, local and directional image representation method that is realized by using the Pull tower decomposition (LP) and directional filter set (DFB).

Contourlet transform inherits the anisotropic scale relation of Curvelet transform, so it can be considered as another fast and effective way to realize curvelet transformation in a certain sense. The support interval of Contourlet base is a "long strip" structure with scale varying aspect ratio, with directionality and anisotropy, contourlet coefficients, the coefficients of image edges are more concentrated, or the Contourlet transformation has more "sparse" expression for the curve. Contourlet transforms the Multiscale analysis and the direction analysis, first by the LP (Laplacian Pyramid) transform the image Multiscale decomposition to "capture" the point singular, and then by the direction of the filter bank (directional filter banks, DFB) The singular points that are distributed in the same direction are combined into a factor. The final result of the Contourlet transformation is to approximate the original image with a base structure similar to the contour segment (Contour segment), which is why it is called the Contourlet transformation. The wavelet is constructed from a dimensional wavelet tensor product, its base lacks directionality, and it does not have anisotropy. can only be limited to the square support interval to describe the contour, different size of the square corresponding to the multi-resolution structure of the wavelet. When the resolution becomes fine enough, the wavelet becomes a point to capture the contour.

4. Stripe wave (Bandelet) transform

        2000 year, ELe Pennec and Stephane Mallat in the literature "EL Pennec, S Mallat." Image compression with geometrical wavelets[a]. In Proc. Oficip ' 2000[c]. Bandelet transformation was proposed in Vancouver, Canada, september,2000.661-664. Bandelet transform is an edge-based image representation method that can adaptively track the geometric regularization direction of an image. Pennec and Mallat that: in the image processing task, if we can know the geometric regularization of the image in advance and make full use of it, it will undoubtedly improve the approximation performance of the image transformation method. Pennec and Mallat first defines a geometric vector line that can characterize the local regular direction of the image, and then the support interval s of the image is S=∪ i Ω i when the subdivision is sufficiently thin, Ω i for each section Contains only one contour line (edge) of the image. In all local areas without footprints, ω i , the change in image grayscale values is consistent, so the direction of the geometric vector lines is not defined within these areas. For the local area containing the contour line, the direction of the geometric regularization is the tangent direction of the contour. According to the local geometric regular direction, the vector line of the region Ω i is computed under the global optimal constraints, and the vectors of the field Tau ( x 1, x 2) are then defined in the Ω The interval wavelet of i is Bandelet (bandeletization) to generate a bandelet base to take advantage of the local geometric regularization of the image itself. The process of Bandelet is actually the process of wavelet transform along the vector line, which is called the curved wavelet transform (warped wavelet transform). The set of Bandelet on the Ω i of all the split regions then forms the standard orthogonal base on a set of L 2 (S).

         Based on the image edge effect, the Bandelet transform adaptively constructs a local bending wavelet transform, which transforms the singular of the curve in the local region into a straight line singularity in the vertical or horizontal direction, then uses the ordinary two-dimensional tensor wavelet processing, and the two-dimensional tensor wavelets can effectively deal with the singularity in the horizontal and vertical directions. So, the crux of the problem boils down to the analysis of the image itself, that is, how to extract the prior information of the image itself, how to split the image, how to "track" the singular direction in the local area, and so on. However, in the natural image, the gray value mutation does not always correspond to the edge of the object, on the one hand, the diffraction effect makes the edge of the object in the image may not be obvious gray-scale mutation, on the other hand, many times the gray value of the image changes drastically, not by the edge of the object but due to changes in texture. A common problem that all edge-based adaptive methods need to solve is how to determine whether the regions with drastic changes in gray values in the image correspond to the object edges or texture changes, which is actually a very difficult problem. Most Edge-based adaptive algorithms in practical applications, when the image appears more complex geometric features, such as Lena image, in the sense of approximation error, the performance can not exceed the separable orthogonal wavelet analysis. In low bit rate encoding of images, the cost of representing the location of a non-0 coefficient is far greater than the cost of representing a non-0 coefficient value. Bandelet has two advantages compared with wavelet: (1) Make full use of the geometrical regularization, the high frequency sub-band energy is more concentrated, under the same quantization step, the non-0 coefficient is relatively reduced; (2) The Bandelet coefficients can be rearranged by the quadtree structure and the geometrical flow information, and the coefficient scanning mode is more flexible when encoded. The potential advantages of Bandelet transform in image compression are illustrated.

The central idea of constructing the Bandelet transformation is to define the geometric features in the image as vector fields rather than as normal edge sets. The vector field represents the local regularization direction of the change of the gray value of the spatial structure of the image. Bandelet is not predetermined, but rather to optimize the final application results from the adaptive selection of the specific base composition. Pennec and Mallat give the optimal base fast searching algorithm of Bandelet transform, and the preliminary experimental results show that Bandelet has some advantages and potential in denoising and compressing compared with ordinary wavelet transform.

5. Wedge Wave (Wedgelet) transform

In the multi-scale geometric analysis tool, the Wedgelet transform has good "line" and "polygon" characteristics.

Wedgelet is a direction information detection model proposed by Professor Davidl.donoho when he studies the problem of recovering the original image from the noise-containing data. Wedgelet transform is a concise method of image contour representation. The segmented linear representation of the image using Multiscale Wedgelet can automatically determine the block size according to the content of the image, and better capture the characters of line and polygon in the image. Overcomes the shortcomings of the sliding window method.

Multi-scale Wedgelet transform consists of two parts: Multiscale wedgelet decomposition and multiscale wedgelet representation. Multi-scale Wedgelet decomposition divides the image into different scale image blocks, and each image block is projected into the wedgelet of each allowable azimuth, and the multi-scale wedgelet representation chooses the best partition of the image according to the decomposition result, and selects the optimal Wedgelet representation for each image block. Thus, the region segmentation of the image is accomplished.

What is Wedgelet? To put it bluntly, it is to draw a line segment in an image sub-block (dyadic Square), dividing it into two wedges, each of which is represented by a unique grayscale value. The position of the line, two gray values, approximates the nature of the sub-block.

6, Small line (beamlet) Transformation

The Small Line Transformation (beamletstransform) was first proposed by Professor David L.donoho of Stanford University in 1999, and has been used in a preliminary application. The small line analysis introduced by the Small Line transformation (Beamlets analysis) is also a multiscale one, but it is different from the multi-scale concept of wavelet analysis, which can be understood as the extension of Multiscale concept of wavelet analysis, and the small line analysis is based on small line segments of various directions, scales and positions to set up small wire library. The small line transformation coefficients are generated in the image and the library, and the transformation coefficients are organized by the small line pyramid, and the small line transformation coefficients are extracted from the pyramid by the graph to achieve multiscale analysis. This is a better tool for two-dimensional or higher-dimensional singularity analysis.

According to the theory of small line and its research results, it has an incomparable advantage in dealing with images with strong noise background. But the early preparation of small line transformation, such as small line dictionary, small line pyramid scan these parts of the workload is too large, not conducive to research. If this part can be simplified, or made into a fixed module reference, it is believed that the small line analysis can quickly expand its application field. In general, the research of small-line analysis is still in the preliminary stage, the related research results are not much, and the applied research field needs to be further expanded.

In Beamlet analysis, the segment is similar to the position of point in wavelet analysis. Beamlet can provide the local scale, position and direction of the segment based on the binary organization, the precise location of the line is easy to realize, and the algorithm is not complicated, so the extraction of line feature based on Beamlet is worth studying.

Beamlet is a set of multi-scale, directional segments with a binary feature, and the binary features represent the always-point coordinates of the line segments being binary, and the scale being the binary.

Donoho proposes a continuous Beamlet transform and its application in Multiscale analysis, Xiaoming Huo proposes a discrete beamlet transformation for reducing computational capacity and more suitable for computer processing.

From the framework of the Beamlet, each beamlet is divided into two parts, each of which is called Wedgelet, and the two parts are complementary wedgelet, thus each beamlet corresponds to two complementary wedgelet, So that the Beamlet base and the wedgelet correspond to the Wedgelet transformation has multi-scale characteristics, you can also see that the Wedgelet base is a flake base, unlike the linear base of Beamlet.

Reference Documents:

"1" Engineering, Tan Son nhat. Multi-scale geometric analysis of images: review and Prospect [J]. Journal of Electronics, 2003,31 (12A): 1975-1981.

"2" engineering, Tan Son Nhat, Liu Fang. Ridge wave Theory: from Ridge Wave to Curvelet transformation [J]. Chinese Journal of Engineering Mathematics, 2005,22 (5): 761-773.

"3" Longgang, Xiao Lei, Chen Xuahing. The application of Curvelet transform in image processing [J]. Computer research and Development, 2005,42 (8): 1331-1337.

"4" Liu Yunxia. Research on the technology of intelligent traffic monitoring system based on finite ridge Wave Transform and compression perception theory [D]. Shandong University, 2012.

"5"hongbo01. Chapter III Ridge Wave and Qu Bo transformation, Baidu Library.

"6"hongbo01. 7th Chapter Bandelet Transformation and its application, Baidu Library.

"7"hongbo01. 8th Chapter Beamlet and its application, Baidu Library.

"8"hongbo01. Nineth Chapter Contourlet Transformation and its application, Baidu Library.

"9" Hu 镠. Curvelet study notes, NetEase blog.

"10" Quiqingchun, Qun. Image edge detection based on Wedgelet transform [J]. Biomedical engineering Research, 2005,24 (1): 8-10.

"11" Yang Ming, Guoji, Qun, Zhou Wenhong. Beamlet transformation and multi-scale line feature extraction [J]. Journal of Electronics, 2007,35 (1): 100-103.

"12" Houbiao, Liu Pei, engineering. SAR image edge detection based on improved Wedgelet transform [J]. Journal of Infrared and Millimeter waves, 2009,28 (5): 396-400.

"13" Han Min, Jerry Lamb. A remote sensing image classification algorithm based on Wedgelet transform [J]. Journal of Infrared and Millimeter waves, 2008,27 (4): 280-284.

"14"wy19811007. Nova Wedgelet, Baidu Library.

"15"hongbo01. The 2nd chapter of multi-resolution analysis and tower algorithm, Baidu Library.

"16"hongbo01. 4th Chapter 3D-DFB and Surfacelet transformation, Baidu Library.

Source: http://blog.csdn.net/jbb0523/article/details/42689465

From for notes (Wiz)

Multi-scale geometric analysis (Ridgelet, Curvelet, Contourlet, Bandelet, Wedgelet, Beamlet)