Runyi Yu

Hilbert transform pairs of orthogonal wavelet bases: necessary and sufficient conditions

The condition on scaling filters of two orthogonal wavelet bases that render the corresponding wavelets as Hilbert transform pairs is re-examined in this note. Without making any pre-assumption on the relationship between the two scaling filters, the authors derive necessary and sufficient conditions for forming Hilbert transform pairs. They lead to new magnitude conditions and Selesnicks phase condition. Unique solutions to these conditions are concluded. It is shown that orthogonal wavelet bases form Hilbert transform pairs if and only if the two scaling filters are offset from one another by half a sample.

On the phase condition and its solution for Hilbert transform pairs of wavelet bases

In this paper, the phase condition on the scaling filters of two wavelet bases that renders the corresponding wavelets as Hilbert transform pairs is studied. An alternative and equivalent phase condition is derived. With the equivalent condition and using Fourier series expansions, we show that the solution for which the corresponding scaling filters are offset from one another by a half sample is the only solution satisfying the phase condition.

Hilbert transform pairs of biorthogonal wavelet bases

The forming of Hilbert transform pairs of biorthogonal wavelet bases of two-band filter banks is studied in this paper. We first derive necessary and sufficient conditions on the scaling filters that render two Hilbert transform pairs: one decomposition pair and one reconstruction pair. We show that the Hilbert transform pairs are achieved if and only if the decomposition scaling filter of one filter bank is half-sample delayed from that of the other filter bank; and the reconstruction scaling filter of the former is half-sample advanced from that of the latter. Hilbert transform pairs of wavelet bases are also characterized by equivalent relationships on the wavelet filters and the scaling functions associated with the two filter banks. An illustrative example is provided.

Theory of Dual-Tree Complex Wavelets

We study analyticity of the complex wavelets in Kingsburys dual-tree wavelet transform. A notion of scaling transformation function that defines the relationship between the primal and dual scaling functions is introduced and studied in detail. The analyticity property is examined and dealt with via the transformation function. We separate analyticity from other properties of the wavelet such as orthogonality or biorthogonality. This separation allows a unified treatment of analyticity for general setting of the wavelet system, which can be dyadic or M-band; orthogonal or biorthogonal; scalar or multiple; bases or frames. We show that analyticity of the complex wavelets can be characterized by scaling filter relationship and wavelet filter relationship via the scaling transformation function. For general orthonormal wavelets and dyadic biorthogonal scalar wavelets, the transformation function is shown to be paraunitary and has a linear phase delay of omega/2 in (0, 2pi).

A New Shift-Invariance of Discrete-Time Systems and Its Application to Discrete Wavelet Transform Analysis

This work is motivated by the search for discrete wavelet transform (DWT) with near shift-invariance. After examining the elements of the property, we introduce a new notion of shift-invariance, which is particularly informative for multirate systems that are not shift-invariant in the strict sense. Briefly speaking, a discrete-time system is mu-shift-invariant if a shift in input results in the output being shifted as well. However, the amount of the shift in output is not necessarily identical to that in input. A fractional shift is also acceptable and can be properly specified in the Fourier domain. The mu-shift-invariance can be interpreted as invariance of magnitude spectrum with linear phase offset of output with respect to shift in input. It is stronger than the shiftability in position, which is equivalent to insensitivity of energy to shift in input. Under this generalized notion, the expander is always mu-shift-invariant. The M-fold decimator is mu-shift-invariant for input with width of frequency support not more than 2pi/M ; equivalently, the output contains no aliasing term in some frequency band with length of 2pi . We generalize the transfer function description of linear shift-invariant systems for mu -shift-invariant systems. We then perform mu-shift-invariance analysis of 2-band orthogonal DWT and of the 2-band dual-tree complex wavelet transform (DT-CWT). The analysis in each case provides clarifications to early understanding of near shift-invariance. We show that the DWT is mu-shift-invariant if and only if the conjugate quadrature filter (CQF) is analytic or antianalytic. For the DT-CWT, the CQFs must have supports included within [-2pi/3, 2pi/3], in addition to the well-know half-sample delay condition at higher levels and the one-sample delay condition at the first level.

Characterization and Sampled-Data Design of Dual-Tree Filter Banks for Hilbert Transform Pairs of Wavelet Bases

Characterization and design of dual-tree filter banks for forming Hilbert transform pairs of wavelet bases are studied. The characterization extends the existing results for quadrature mirror filter banks to general prefect reconstruction filter banks that satisfy only a mild technical assumption regarding the ratio of determinants of the two filter banks. We establish equivalent relationships of Hilbert transform pairs on scaling filters, wavelet filters, or scaling functions. The design of scaling filters of a dual filter bank is formulated as a sampled-data Hinfin optimization problem. The wavelet filters are then determined using the relationship on the determinants of the filter banks. We convert the sampled-data problem into an equivalent discrete-time Hinfin control problem, which can be solved by standard Hinfin control theory. An analytical solution to the sampled-data design problem is obtained for a special case. The sampled-data design approach usually gives infinite impulse response filter. In the case where the primal filter bank is of finite impulse response (FIR), we may truncate the impulse responses to get FIR approximations. They also lead to approximate Hilbert transform pairs. Design examples are presented

Sampled-Data Design of FIR Dual Filter Banks for Dual-Tree Complex Wavelet Transforms via LMI Optimization

Starting from a given finite-impulse-response (FIR) primal filter bank, we design a dual filter bank such that the complex wavelets associated with the dual-tree filter bank are (almost) analytic. The dual filter bank is required to be FIR and have a prescribed number of zeros at . We formulate a sampled-data optimization problem based on the half-sample delay condition on scaling filters. A discrete-time filter is introduced in the formulation to specify the number of the zeros. The optimization problem is converted into an equivalent discrete-time control problem; the latter is further reduced to an LMI optimization problem. We then present a procedure for design of FIR dual filter banks. Illustrative examples are provided; the results compare favorably to early designs.

Design of Halfband Filters for Orthogonal Wavelets via Sum of Squares Decomposition

This letter is concerned with design of halfband product filters for orthogonal wavelets. We first remark that the recent zero-pinning technique for orthogonal wavelet design cannot always guarantee the nonnegativity of the filter. We then propose to use sum of squares (SOS) decomposition to ensure its nonnegativity. The use of SOS decomposition also allows us to solve two optimization problems on the halfband filter via semidefinite programming. For a given length with pre-specified number of zeros at , we obtain halfband filters with maximal passband width. Design examples are provided.

A Dual-Tree Complex Wavelet with Application in Image Denoising

This paper introduces a recently designed dual-tree complex wavelet and studies its application in image denoising. The primal filter bank is selected to be the Daubechies 9/7 filter bank, and the dual filter bank is designed to have length of 10/8; both filter banks are biorthogonal and symmetric. The wavelets of the dual-tree filter bank form (almost) Hilbert transform pairs, allowing nearly shift-invariance and good directionality of the dual-tree complex wavelet transform. The transform is then used in image denoising. We employ the bivariate shrinkage algorithm for wavelet coefficient modeling and thresholding. Various images are tested. The experimental results compare favorably to some other dual-tree complex wavelets.

On the phase condition and its solution for Hilbert transform pairs of wavelets bases

In this work, the phase condition on the scaling filters of two wavelet bases which ensures that the corresponding wavelets are Hilbert transform pairs of each other is studied. An alternative and equivalent phase condition is derived. With the equivalent condition and Fourier series expansions, it is shown that the solution for which the corresponding scaling filters are offset from one another by a half sample is the only solution satisfying the phase condition which results in Hilbert transform pairs of wavelet bases.