Truong T. Nguyen

Multiresolution direction filterbanks: theory, design, and applications

This paper presents new developments of directional filterbanks (DFBs). The motivation for the paper is the existence of multiresolution and multidirection orthogonal transform for two-dimensional (2-D) discrete signals. Based on the frequency supports of the ideal transform, a new uniformly, maximally decimated DFB with six highpass directional subbands and two lowpass subands is introduced. The uniform DFB (uDFB) can be implemented by a binary tree structure of two-channel filterbanks. The filterbank employed in the tree is shown to be alias-free decimation and permissible. The uDFB is then extended to a nonuniform case (nuDFB), which is still maximally decimated, by combining the two lowpass subbands. The nuDFB yields nonuniform frequency division, which composes of one lowpass filter with a decimation factor of one fourth and six highpass directional filters with a decimation factor of one eighth. The new DFBs offer alternative image decompositions, which overcome the limited directional selectivity of the separable wavelets and the limited multiresolution of the conventional DFB. The lowpass subband of the nuDFB can be used to obtain a multiresolution representation by simply reiterating the same nuDFB decomposition. On the other hand, the directional subbands can also be further refined by simply applying a two-channel conventional DFB at each highpass component. A simple design method yielding near orthogonal uniform and nonuniform multidimensional filterbanks is presented. Finally, the performances of the newly proposed nuDFB are compared with other conventional transforms in nonlinear approximation, image denoising, and texture classification to demonstrate its potential.

The Shiftable Complex Directional Pyramid—Part I: Theoretical Aspects

This paper presents an over-complete multiscale decomposition by combining the Laplacian pyramid and the complex directional filter bank (DFB). The filter bank is constructed in such a way that each complex directional filter is analytical using the dual-tree structure of real fan filters. Necessary and sufficient conditions in order for the resulting multirate filter bank to be shift-invariant in energy sense (shiftability) are derived in terms of the magnitude and phase responses of these filters. Their connection to 2D Hilbert transform relationship is established. The proposed transform possesses several desirable properties including multiresolution, arbitrarily high directional resolution, low redundant ratio, and efficient implementation.

A Class of Multiresolution Directional Filter Banks

In this paper, we introduced a class of directional filter banks (DFBs) having the previously proposed uniform DFB (uDFB) as a special case. Except for the uDFB, each DFB in this class can be used to decompose an image yielding up to 12 directions while maintaining perfect reconstruction and maximal decimation. A multiresolution representation can be obtained by repeating the same decomposition at the lowpass band. The permissible property of the filter banks in cases of being implemented by a tree structure and by direct implementation is discussed. The result shows that only one DFB in the class, called the uniform quincunx DFB (uqDFB), satisfies the permissible property when being implemented directly without using the tree structure. The nonuniform quincunx DFB (nuqDFB) is then constructed from the uqDFB by merging its two lowpass subbands. An alternative structure for constructing the nuqDFB is presented. The new structure, while yielding the same frequency partitioning, allows the DFB to be realized with complexity comparable to that of the separable wavelet filter bank. The connection between the discrete filter bank and the continuous directional wavelet is also established. Numerical experiments on directional feature extractions, image denoising and nonlinear approximation are presented at the end of the paper to demonstrate the potential of the nuqDFB

Multidimensional filter banks design by direct optimization

In this paper, we designed multidimensional filter banks by direct optimization in the frequency domain. The desired properties of a filter bank (such as zero aliasing, perfect reconstruction) are formulated as an objective function of the filter coefficients. The design procedures minimize the objective function by iterative quadratic programming with linear constraints. Our design examples show that the method is flexible towards frequency support configurations, smoothness constraints, and can be used to design multidimensional perfect reconstruction filter banks having up to eight channels.

A multiresolution directional filter bank for image applications

We propose a new formulation of directional filter banks (DFBs). By using a non-uniform and non-separable filter bank, a critically sampled multiresolution directional image representation can be obtained efficiently. The resulting DFB yields non-uniform frequency division which is composed of one lowpass channel with a decimation factor of one-fourth and six highpass directional channels with a decimation factor of one-eighth. It overcomes the limited directional selectivity of separable wavelets and the limited resolution of the conventional DFB. The lowpass channel can be used to obtain multiresolution representation by simply re-iterating the same DFB decomposition. On the other hand, the directional subbands can be further refined by simply applying a two-channel DFB at each highpass channel. A simple design method yielding near orthogonal uniform and non-uniform multidimensional filter banks is discussed, and, finally, a numerical experiment is presented to demonstrate the potential of the new image basis.

A directional decomposition: theory, design, and implementation

We propose a new directional decomposition of images. Unlike the conventional original directional filter banks (DFB) with eight directional subbands, our DFB has six directional highpass and two lowpass subbands. The newly constructed DFB is maximally decimated and perfect reconstruction. We also show how to implement the new DFB by cascading three two-channels filter banks and discuss filter design issue related to the structure.

A Shift-Invariant Multiscale Multidirection Image Decomposition

This paper presents a shift-invariant complex directional pyramid transform constructed by a dual-tree pyramidal directional filter banks (DFB). The double filter bank framework consists of a shift-invariant Laplacian pyramid and a dual-tree DFB. The two binary tree structure for the (primal and dual) DFBs employed in the structure are identical except for the filter bank employed at the second level of the dual DFB, where special conditions on the phase of the filters are required. It is proven analytically and experimentally that each pair of corresponding directional filters produced by the primal and dual filter banks are symmetric and anti-symmetric, which can be interpreted as the real and imaginary parts of a complex filter. Therefore, the two subband coefficients can be viewed as the real and imaginary parts of a complex-valued subband image. It is proven that there is no aliasing in the decimated complex-valued signal, which implies that the system is shift-invariant in the energy sense. In addition, the proposed shift-invariant, multiscale, multidirectional image decomposition has two unique characteristics that other shift-invariant decompositions do not possess. First, the directional resolution of the image transform can be arbitrarily high. Secondly, the two-dimensional filter bank is implemented in a separable fashion, which makes the entire structure very computational efficient

Low Bit-Rate Image Coding Based on Pyramidal Directional Filter Banks

This paper presents a novel embedded image coding system based on the pyramidal multidirectional image representation. The multiscale filter bank is a combination of the overcomplete pyramidal directional filter bank at higher scales and the traditional maximally-decimated wavelet filter bank at lower scales, to provide a sparse image representation. The coding algorithm then efficiently clusters the significant coefficients using progressive morphological operations. Context models for arithmetic coding are designed to exploit the intra-band dependency and the correlation existing among the neighboring directional subbands. Experimental results show that the proposed coding algorithm outperforms the current state-of-the-art wavelet based coders, such as JPEG2000, for images with directional features

Integer fast Fourier transform (INTFFT)

The concept of integer fast Fourier transform (IntFFT) for approximating the discrete Fourier transform is introduced. Unlike the fixed-point fast Fourier transform (FxpFFT), the new transform has properties that it is an integer-to-integer mapping, power-adaptable and also reversible. A lifting scheme is used to approximate complex multiplications appearing in the FFT lattice structures. Split-radix FFT is used to illustrate the approach for the case of 2/sup N/-point FFT. The transform can be implemented by using only bit shifts and additions but no multiplication. While preserving the reversibility, the IntFFT is shown experimentally to yield the same accuracy as the FxpFFT when their coefficients are quantized to a certain number of bits. Complexity of the IntFFT is shown to be much lower than that of the FxpFFT in terms of the numbers of additions and shifts.

A class of directional filter banks [image processing applications]

In this paper, we introduced a class of directional filter banks (DFB) having the previously proposed uniform DFB (uDFB) as a special case. The new DFBs implement a multiresolution and multidirection decomposition of images, while remaining perfect reconstruction (PR) and maximally decimated. We also discussed the permissible property of the filter banks in cases of being implemented by a tree structure and by direct implementation. The result shows that only one DFB in the class, which we call the quincunx uDFB, has the permissibility when being implemented directly by an eight-channel filter bank. Finally, an efficient tree structure for implementation of the quincunx uDFB is proposed.