The near shift-invariance of the dual-tree complex wavelet transform revisited

The dual-tree complex wavelet transform (DTCWT) is an enhancement of the conventional discrete wavelet transform (DWT) due to a higher degree of shift-invariance and a greater directional selectivity, finding its applications in signal and image processing. This paper presents a quantitative proof of the superiority of the DTCWT over the DWT in case of modulated wavelets.Comment: 15 page

On the Hilbert transform of wavelets

A wavelet is a localized function having a prescribed number of vanishing moments. In this correspondence, we provide precise arguments as to why the Hilbert transform of a wavelet is again a wavelet. In particular, we provide sharp estimates of the localization, vanishing moments, and smoothness of the transformed wavelet. We work in the general setting of non-compactly supported wavelets. Our main result is that, in the presence of some minimal smoothness and decay, the Hilbert transform of a wavelet is again as smooth and oscillating as the original wavelet, whereas its localization is controlled by the number of vanishing moments of the original wavelet. We motivate our results using concrete examples.Comment: Appears in IEEE Transactions on Signal Processing, vol. 59, no. 4, pp. 1890-1894, 201

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

Hilbert pairs of M-band orthonormal wavelet bases

International audienceRecently, there has been a growing interest for wavelet frames corresponding to the union of an orthonormal wavelet basis and its dual Hilbert transformed wavelet basis. However, most of the existing works specifically address the dyadic case. In this paper, we consider orthonormal M-band wavelet decompositions, since we are motivated by their advantages in terms of frequency selectivity and symmetry of the analysis functions, for M > 2. More precisely, we establish phase conditions for a pair of critically subsampled M-band filter banks. The conditions we obtain generalize a previous result given in the two-band case. We also show that, when the primal filter bank and its wavelets have symmetry, it is inherited by their duals. Furthermore, we give a design example where the number of vanishing moments of the approximate dual wavelets is imposed numerically to be the same as for the primal ones

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

Steerable filters generated with the hypercomplex dual-tree wavelet transform

The use of wavelets in the image processing domain is still in its infancy, and largely associated with image compression. With the advent of the dual-tree hypercomplex wavelet transform (DHWT) and its improved shift invariance and directional selectivity, applications in other areas of image processing are more conceivable. This paper discusses the problems and solutions in developing the DHWT and its inverse. It also offers a practical implementation of the algorithms involved. The aim of this work is to apply the DHWT in machine vision. Tentative work on a possible new way of feature extraction is presented. The paper shows that 2-D hypercomplex basis wavelets can be used to generate steerable filters which allow rotation as well as translation.</p