Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-like Transforms

We propose a novel method for constructing Hilbert transform (HT) pairs of wavelet bases based on a fundamental approximation-theoretic characterization of scaling functions--the B-spline factorization theorem. In particular, starting from well-localized scaling functions, we construct HT pairs of biorthogonal wavelet bases of L^2(R) by relating the corresponding wavelet filters via a discrete form of the continuous HT filter. As a concrete application of this methodology, we identify HT pairs of spline wavelets of a specific flavor, which are then combined to realize a family of complex wavelets that resemble the optimally-localized Gabor function for sufficiently large orders. Analytic wavelets, derived from the complexification of HT wavelet pairs, exhibit a one-sided spectrum. Based on the tensor-product of such analytic wavelets, and, in effect, by appropriately combining four separable biorthogonal wavelet bases of L^2(R^2), we then discuss a methodology for constructing 2D directional-selective complex wavelets. In particular, analogous to the HT correspondence between the components of the 1D counterpart, we relate the real and imaginary components of these complex wavelets using a multi-dimensional extension of the HT--the directional HT. Next, we construct a family of complex spline wavelets that resemble the directional Gabor functions proposed by Daugman. Finally, we present an efficient FFT-based filterbank algorithm for implementing the associated complex wavelet transform.Comment: 36 pages, 8 figure

各種の性質を改善した直交DTCWTの設計に関する研究

The Dual tree complex wavelet transforms (DTCWTs) have been found to be successful in many applications of signal and image processing. DTCWTs employ two real wavelet transforms, where one wavelet corresponds to the real part of complex wavelet and the other is the imaginary part. Two wavelet bases are required to be a Hilbert transform pair. Thus, DTCWTs are nearly shift invariant and have a good directional selectivity in two or higher dimensions with limited redundancies. In this dissertation, we propose two new classes of DTCWTs with improved properties. In Chapter 2, we review the Fourier transform at first and then introduce the fundamentals of dual tree complex wavelet transform. The wavelet transform has been proved to be a successful tool to express the signal in time and frequency domain simultaneously. To obtain the wavelet coefficients efficiently, the discrete wavelet transform has been introduced since it can be achieved by a tree of two-channel filter banks. Then, we discuss the design conditions of two-channel filter banks, i.e., the perfect reconstruction and orthonormality. Additionally, some properties of scaling and wavelet functions including orthonormality, symmetry and vanishing moments are also given. Moreover, the structure of DTCWT is introduced, where two wavelet bases are required to form a Hilbert transform pair. Thus, the corresponding scaling lowpass filters must satisfy the half-sample delay condition. Finally, the objective measures of quality are given to evaluate the performance of the complex wavelet. In Chapter 3, we propose a new class of DTCWTs with improved analyticity and frequency selectivity by using general IIR filters with numerator and denominator of different degree. In the common-factor technique proposed by Selesnick, the maximally at allpass filter was used to satisfy the halfsample delay condition, resulting in poor analyticity of complex wavelets. Thus, to improve the analyticity of complex wavelets, we present a method for designing allpass filters with the specified degree of flatness and equiripple phase response in the approximation band. Moreover, to improve the frequency selectivity of scaling lowpass filters, we locate the specified number of zeros at z = -1 and minimize the stopband error. The well-known Remez exchange algorithm has been applied to approximate the equiripple response. Therefore, a set of filter coefficients can be easily obtained by solving the eigenvalue problem. Furthermore, we investigate the performance on the proposed DTCWTs and dedicate how to choose the approximation band and stopband properly. It is shown that the conventional DTCWTs proposed by Selesnick are only the special cases of DTCWTs proposed in this dissertation. In Chapter 4, we propose another class of almost symmetric DTCWTs with arbitrary center of symmetry. We specify the degree of flatness of group delay, and the number of vanishing moments, then apply the Remez exchange algorithm to minimize the difference between two scaling lowpass filters in the frequency domain, in order to improve the analyticity of complex wavelets. Therefore, the equiripple behaviour of the error function can be obtained through a few iterations. Moreover, two scaling lowpass filters can be obtained simultaneously. As a result, the complex wavelets are orthogonal and almost symmetric, and have the improved analyticity. Since the group delay of scaling lowpass filters can be arbitrarily specified, the scaling functions have the arbitrary center of symmetry. Finally, several experiments of signal denoising are carried out to demonstrate the efficiency of the proposed DTCWTs. It is clear that the proposed DTCWTs can achieve better performance on noise reduction.電気通信大学201

A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity

The richness of natural images makes the quest for optimal representations in image processing and computer vision challenging. The latter observation has not prevented the design of image representations, which trade off between efficiency and complexity, while achieving accurate rendering of smooth regions as well as reproducing faithful contours and textures. The most recent ones, proposed in the past decade, share an hybrid heritage highlighting the multiscale and oriented nature of edges and patterns in images. This paper presents a panorama of the aforementioned literature on decompositions in multiscale, multi-orientation bases or dictionaries. They typically exhibit redundancy to improve sparsity in the transformed domain and sometimes its invariance with respect to simple geometric deformations (translation, rotation). Oriented multiscale dictionaries extend traditional wavelet processing and may offer rotation invariance. Highly redundant dictionaries require specific algorithms to simplify the search for an efficient (sparse) representation. We also discuss the extension of multiscale geometric decompositions to non-Euclidean domains such as the sphere or arbitrary meshed surfaces. The etymology of panorama suggests an overview, based on a choice of partially overlapping "pictures". We hope that this paper will contribute to the appreciation and apprehension of a stream of current research directions in image understanding.Comment: 65 pages, 33 figures, 303 reference

NOVEL OFDM SYSTEM BASED ON DUAL-TREE COMPLEX WAVELET TRANSFORM

The demand for higher and higher capacity in wireless networks, such as cellular, mobile and local area network etc, is driving the development of new signaling techniques with improved spectral and power efficiencies. At all stages of a transceiver, from the bandwidth efficiency of the modulation schemes through highly nonlinear power amplifier of the transmitters to the channel sharing between different users, the problems relating to power usage and spectrum are aplenty. In the coming future, orthogonal frequency division multiplexing (OFDM) technology promises to be a ready solution to achieving the high data capacity and better spectral efficiency in wireless communication systems by virtue of its well-known and desirable characteristics. Towards these ends, this dissertation investigates a novel OFDM system based on dual-tree complex wavelet transform (D

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

Wavelet Theory

The wavelet is a powerful mathematical tool that plays an important role in science and technology. This book looks at some of the most creative and popular applications of wavelets including biomedical signal processing, image processing, communication signal processing, Internet of Things (IoT), acoustical signal processing, financial market data analysis, energy and power management, and COVID-19 pandemic measurements and calculations. The editor’s personal interest is the application of wavelet transform to identify time domain changes on signals and corresponding frequency components and in improving power amplifier behavior