MULTIRIDGELETS FOR TEXTURE ANALYSIS

Directional wavelets have orientation selectivity and thus are able to efficiently represent highly anisotropic elements such as line segments and edges. Ridgelet transform is a kind of directional multi-resolution transform and has been successful in many image processing and texture analysis applications. The objective of this research is to develop multi-ridgelet transform by applying multiwavelet transform to the Radon transform so as to attain attractive improvements. By adapting the cardinal orthogonal multiwavelets to the ridgelet transform, it is shown that the proposed cardinal multiridgelet transform (CMRT) possesses cardinality, approximate translation invariance, and approximate rotation invariance simultaneously, whereas no single ridgelet transform can hold all these properties at the same time. These properties are beneficial to image texture analysis. This is demonstrated in three studies of texture analysis applications. Firstly a texture database retrieval study taking a portion of the Brodatz texture album as an example has demonstrated that the CMRT-based texture representation for database retrieval performed better than other directional wavelet methods. Secondly the study of the LCD mura defect detection was based upon the classification of simulated abnormalities with a linear support vector machine classifier, the CMRT-based analysis of defects were shown to provide efficient features for superior detection performance than other competitive methods. Lastly and the most importantly, a study on the prostate cancer tissue image classification was conducted. With the CMRT-based texture extraction, Gaussian kernel support vector machines have been developed to discriminate prostate cancer Gleason grade 3 versus grade 4. Based on a limited database of prostate specimens, one classifier was trained to have remarkable test performance. This approach is unquestionably promising and is worthy to be fully developed

Balanced Multiwavelets Theory and Design

This paper deals with multiwavelets which are a recent generalization of wavelets in the context of multirate filter banks and with their applications to signal processing and especially compression. By their inherent structure, multiwavelets are fit for processing multi-channel signals. First, we will recall some general results on multifilters by looking at them as time-varying filters. Then, we will link this to multiwavelets, looking closely at the convergence of the iterated matrix product leading to them and the typical properties we can expect. Then, we will define under what conditions we can apply systems based on multiwavelets to one-dimensional signals in a simple way. That means we will give some natural and simple conditions that should help in the design of new multiwavelets for signal processing. Finally, we will provide some tools in order to construct multiwavelets with the required properties, the so-called `balanced multiwavelets'

MULTIWAVELET NEURAL NETWORK PREPROCESSING OF IRREGULARLY SAMPLED DATA

Abstract. Multiwavelets are briefly reviewed and preprocessing and postprocessing for such wavelets are introduced. Least squares curve fitting of irregularly sampled data is achieved by means of unshifted and shifted multiscaling functions. This preprocessing procedure combined with multiwavelet neural networks for data-adaptive curve fitting is shown to perform well in the case of high resolution. In the case of low resolution it is more accurate than numerical integration and cheaper than matrix inversion

Sampling—50 Years After Shannon

This paper presents an account of the current state of sampling, 50 years after Shannon's formulation of the sampling theorem. The emphasis is on regular sampling where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbert-space formulation, we re-interpret Shannon's sampling procedure as an orthogonal projection onto the subspace of bandlimited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of "shift-invariant" functions spaces, including splines and wavelets. Practically, this allows for simpler—and possibly more realistic—interpolation models, which can be used in conjunction with a much wider class of (anti-aliasing) pre-filters that are not necessarily ideal lowpass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., non-bandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multi-wavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned

High Performance Techniques for Face Recognition

The identification of individuals using face recognition techniques is a challenging task. This is due to the variations resulting from facial expressions, makeup, rotations, illuminations, gestures, etc. Also, facial images contain a great deal of redundant information, which negatively affects the performance of the recognition system. The dimensionality and the redundancy of the facial features have a direct effect on the face recognition accuracy. Not all the features in the feature vector space are useful. For example, non-discriminating features in the feature vector space not only degrade the recognition accuracy but also increase the computational complexity. In the field of computer vision, pattern recognition, and image processing, face recognition has become a popular research topic. This is due to its wide spread applications in security and control, which allow the identified individual to access secure areas, personal information, etc. The performance of any recognition system depends on three factors: 1) the storage requirements, 2) the computational complexity, and 3) the recognition rates. Two different recognition system families are presented and developed in this dissertation. Each family consists of several face recognition systems. Each system contains three main steps, namely, preprocessing, feature extraction, and classification. Several preprocessing steps, such as cropping, facial detection, dividing the facial image into sub-images, etc. are applied to the facial images. This reduces the effect of the irrelevant information (background) and improves the system performance. In this dissertation, either a Neural Network (NN) based classifier or Euclidean distance is used for classification purposes. Five widely used databases, namely, ORL, YALE, FERET, FEI, and LFW, each containing different facial variations, such as light condition, rotations, facial expressions, facial details, etc., are used to evaluate the proposed systems. The experimental results of the proposed systems are analyzed using K-folds Cross Validation (CV). In the family-1, Several systems are proposed for face recognition. Each system employs different integrated tools in the feature extraction step. These tools, Two Dimensional Discrete Multiwavelet Transform (2D DMWT), 2D Radon Transform (2D RT), 2D or 3D DWT, and Fast Independent Component Analysis (FastICA), are applied to the processed facial images to reduce the dimensionality and to obtain discriminating features. Each proposed system produces a unique representation, and achieves less storage requirements and better performance than the existing methods. For further facial compression, there are three face recognition systems in the second family. Each system uses different integrated tools to obtain better facial representation. The integrated tools, Vector Quantization (VQ), Discrete cosine Transform (DCT), and 2D DWT, are applied to the facial images for further facial compression and better facial representation. In the systems using the tools VQ/2D DCT and VQ/ 2D DWT, each pose in the databases is represented by one centroid with 4*4*16 dimensions. In the third system, VQ/ Facial Part Detection (FPD), each person in the databases is represented by four centroids with 4*Centroids (4*4*16) dimensions. The systems in the family-2 are proposed to further reduce the dimensions of the data compared to the systems in the family-1 while attaining comparable results. For example, in family-1, the integrated tools, FastICA/ 2D DMWT, applied to different combinations of sub-images in the FERET database with K-fold=5 (9 different poses used in the training mode), reduce the dimensions of the database by 97.22% and achieve 99% accuracy. In contrast, the integrated tools, VQ/ FPD, in the family-2 reduce the dimensions of the data by 99.31% and achieve 97.98% accuracy. In this example, the integrated tools, VQ/ FPD, accomplished further data compression and less accuracy compared to those reported by FastICA/ 2D DMWT tools. Various experiments and simulations using MATLAB are applied. The experimental results of both families confirm the improvements in the storage requirements, as well as the recognition rates as compared to some recently reported methods

Highly Symmetric Multiple Bi-Frames for Curve and Surface Multiresolution Processing

Wavelets and wavelet frames are important and useful mathematical tools in numerous applications, such as signal and image processing, and numerical analysis. Recently, the theory of wavelet frames plays an essential role in signal processing, image processing, sampling theory, and harmonic analysis. However, multiwavelets and multiple frames are more flexible and have more freedom in their construction which can provide more desired properties than the scalar case, such as short compact support, orthogonality, high approximation order, and symmetry. These properties are useful in several applications, such as curve and surface noise-removing as studied in this dissertation. Thus, the study of multiwavelets and multiple frames construction has more advantages for many applications. Recently, the construction of highly symmetric bi-frames for curve and surface multiresolution processing has been investigated. The 6-fold symmetric bi-frames, which lead to highly symmetric analysis and synthesis bi-frame algorithms, have been introduced. Moreover, these multiple bi-frame algorithms play an important role on curve and surface multiresolution processing. This dissertation is an extension of the study of construction of univariate biorthogonal wavelet frames (bi-frames for short) or dual wavelet frames with each framelet being symmetric in the scalar case. We will expand the study of biorthogonal wavelets and bi-frames construction from the scalar case to the vector case to construct biorthogonal multiwavelets and multiple bi-frames in one-dimension. In addition, we will extend the study of highly symmetric bi-frames for triangle surface multiresolution processing from the scalar case to the vector case. More precisely, the objective of this research is to construct highly symmetric biorthogonal multiwavelets and multiple bi-frames in one and two dimensions for curve and surface multiresolution processing. It runs in parallel with the scalar case. We mainly present the methods of constructing biorthogonal multiwavelets and multiple bi-frames in both dimensions by using the idea of lifting scheme. On the whole, we discuss several topics include a brief introduction and discussion of multiwavelets theory, multiresolution analysis, scalar wavelet frames, multiple frames, and the lifting scheme. Then, we present and discuss some results of one-dimensional biorthogonal multiwavelets and multiple bi-frames for curve multiresolution processing with uniform symmetry: type I and type II along with biorthogonality, sum rule orders, vanishing moments, and uniform symmetry for both types. In addition, we present and discuss some results of two-dimensional biorthogonal multiwavelets and multiple bi-frames and the multiresolution algorithms for surface multiresolution processing. Finally, we show experimental results on curve and surface noise-removing by applying our multiple bi-frame algorithms

Finite representation of finite energy signals

Ankara : The Department of Electrical and Electronics Engineering and the Institute of Engineering and Sciences of Bilkent University, 2011.Thesis (Master's) -- Bilkent University, 2011.Includes bibliographical references leaves 81-93.In this thesis, we study how to encode finite energy signals by finitely many bits. Since such an encoding is bound to be lossy, there is an inevitable reconstruction error in the recovery of the original signal. We also analyze this reconstruction error. In our work, we not only verify the intuition that finiteness of the energy for a signal implies finite degree of freedom, but also optimize the reconstruction parameters to get the minimum possible reconstruction error by using a given number of bits and to achieve a given reconstruction error by using minimum number of bits. This optimization leads to a number of bits vs reconstruction error curve consisting of the best achievable points, which reminds us the rate distortion curve in information theory. However, the rate distortion theorem are not concerned with sampling, whereas we need to take sampling into consideration in order to reduce the finite energy signal we deal with to finitely many variables to be quantized. Therefore, we first propose a finite sample representation scheme and question the optimality of it. Then, after representing the signal of interest by finite number of samples at the expense of a certain error, we discuss several quantization methods for these finitely many samples and compare their performances.Gülcü, Talha CihadM.S

Multidimensional Wavelets and Computer Vision

This report deals with the construction and the mathematical analysis of multidimensional nonseparable wavelets and their efficient application in computer vision. In the first part, the fundamental principles and ideas of multidimensional wavelet filter design such as the question for the existence of good scaling matrices and sensible design criteria are presented and extended in various directions. Afterwards, the analytical properties of these wavelets are investigated in some detail. It will turn out that they are especially well-suited to represent (discretized) data as well as large classes of operators in a sparse form - a property that directly yields efficient numerical algorithms. The final part of this work is dedicated to the application of the developed methods to the typical computer vision problems of nonlinear image regularization and the computation of optical flow in image sequences. It is demonstrated how the wavelet framework leads to stable and reliable results for these problems of generally ill-posed nature. Furthermore, all the algorithms are of order O(n) leading to fast processing

Development of Some Novel Spatial-Domain and Transform-Domain Digital Image Filters

Some spatial-domain and transform-domain digital image filtering algorithms have been developed in this thesis to suppress additive white Gaussian noise (AWGN). In many occasions, noise in digital images is found to be additive in nature with uniform power in the whole bandwidth and with Gaussian probability distribution. Such a noise is referred to as Additive White Gaussian Noise (AWGN). It is difficult to suppress AWGN since it corrupts almost all pixels in an image. The arithmetic mean filter, commonly known as Mean filter, can be employed to suppress AWGN but it introduces a blurring effect. Image denoising is usually required to be performed before display or further processing like segmentation, feature extraction, object recognition, texture analysis, etc. The purpose of denoising is to suppress the noise quite efficiently while retaining the edges and other detailed features as much as possible