Size-limited filter banks for subband image compression

A size-limited filter bank (SLFB) is a maximally decimated filter bank operating on a finite length (duration) input signal resulting in subband components whose total number of independent samples is equal to the number of input samples. A theoretical framework for the design, analysis, and implementation of such filter banks resulting from applying FIR separable filter banks to finite length signals is presented. The concept of maximizing theoretical coding gain (TCG) using optimal bit allocation is generalized for this special case. Using TCG, the relative merits of various different SLFBs are addressed.     

Recursive time-varying filter banks for subband image coding

Filter banks, subband/wavelets, and multiresolution decompositions that employ recursive filters have been considered previously and are recognized for their efficiency in partitioning the frequency spectrum. This paper presents an analysis of a new infinite impulse response (IIR) filter bank in which these computationally efficient filters may be changed adaptively in response to the input. The new filter bank framework is presented and discussed in the context of subband image coding. In the absence of quantization errors, exact reconstruction can be achieved. By the proper choice of an adaptation scheme, it is shown that recursive linear time-varying (LTV) filter banks can yield improvement over conventional ones.     

Theory of optimal orthonormal subband coders

The theory of the orthogonal transform coder and methods for its optimal design have been known for a long time. We derive a set of necessary and sufficient conditions for the coding-gain optimality of an orthonormal subband coder for given input statistics. We also show how these conditions can be satisfied by the construction of a sequence of optimal compaction filters one at a time. Several theoretical properties of optimal compaction filters and optimal subband coders are then derived, especially pertaining to behavior as the number of subbands increases. Significant theoretical differences between optimum subband coders, transform coders, and predictive coders are summarized. Finally, conditions are presented under which optimal orthonormal subband coders yield as much coding gain as biorthogonal ones for a fixed number of subbands     

Computation of the coding gain for subband coders

A procedure to evaluate the coding gain for 2-D subband systems is explicitly presented. The technique operates in the signal domain and requires the knowledge of the input process auto-correlation function. Both the case of uniform subband and pyramid decomposition are considered. In the case of a separable input process spectrum, the evaluation can be performed by considering appropriately defined 1-D systems, thus, making the procedure very convenient in terms of computational complexity. Using a model that has been recently derived for difference images in motion-compensated image sequence coders, we compare the performance of several filter banks and transform coders in terms of coding gain and asymptotic rate-distortion figures. The results for intraframe and interframe coding show that uniform subband coders can have a performance superior to that of transform coders. Pyramidal schemes appear to have a slightly worse performance     

Invertible temporal subband/wavelet filter banks with half-pixel-accurate motion compensation

Three-dimensional (3-D) subband/wavelet coding with motion compensation has been demonstrated to be an efficient technique for video coding applications in some recent research works. When motion compensation is performed with half-pixel accuracy, images need to be interpolated in both temporal subband analysis and synthesis stages. The resulting subband filter banks developed in these former algorithms were not invertible due to image interpolation. In this paper, an invertible temporal analysis/synthesis system with half-pixel-accurate motion compensation is presented. We look at temporal decomposition of image sequences as a kind of down-conversion of the sampling lattices. The earlier motion-compensated (MC) interlaced/progressive scan conversion scheme is extended for temporal subband analysis/synthesis. The proposed subband/wavelet filter banks allow perfect reconstruction of the decomposed video signal while retaining high energy compaction of subband transforms. The invertible filter banks are then utilized in our 3-D subband video coder. This video coding system does not contain the temporal DPCM loop employed in the conventional hybrid coder and the earlier MC 3-D subband coders. The experimental results show a significant PSNR improvement by the proposed method. The generalization of our algorithm for MC temporal filtering at arbitrary subpixel accuracy is also discussed.     

Minimization of quantization noise amplification in biorthogonal subband coders

Quantization noise amplification (QNA) restricts the coding gain (CG) in biorthogonal subband coders. Here, a coloring filter is introduced to color the quantization noise. The optimal coloring filter eliminating/minimizing QNA, with or without order restriction, is found for a given finite-impulse response filter bank (FB). An efficient implementation of the coloring filter is proposed. With the coloring filter, the optimal biorthogonal ideal FB becomes the full whitening coder achieving maximum possible CG. Results verify the CG improvement due to coloring for existing FBs.     

Optimal construction of subband coders using Lloyd-Max quantizers

A new method is presented for the analysis of the effects of Lloyd-Max quantization in subband filterbanks and for the optimal design of such filterbanks. A rigorous statistical model of a vector Lloyd-Max quantizer is established first, consisting of a linear time-invariant filter followed by additive noise uncorrelated/with the input. On the basis of this model, an expression for this variance of the error of a subband coder using Lloyd-Max quantizers is explicitly determined. Given analysis filters that statistically separate the subbands, it is shown that this variance is minimized if the synthesis filters are chosen, which mould achieve perfect reconstruction in lossless coding. The globally optimum of such a filterbank, minimizing the coder error variance, is further obtained by proper choice of its analysis filters. An alternative design method is also evaluated and optimized. In this, the errors correlated with the signal are set to zero, leaving a random error residue uncorrelated with the signal. This design method is optimized by choosing the analysis filters so as to minimize the random error variance. The results are evaluated experimentally in the realistic setting of a logarithmically split subband image coding scheme.     

Scheme for subband adaptive beamforming array implementation using quadrature mirror filter banks

A novel scheme for a subband adaptive beamforming array (SABA) implementation is presented. The higher subband output signal is optimised and then multiplied with a factor to generate an output similar to the optimised lower subband. Both outputs are then combined to produce the desired signal. This arrangement reduces circuit complexity as well as the cost of the SABA, while maintaining the same beamforming performance.     

Construction of optimal subband coders using optimized and optimalquantizers

A method is presented for the optimization of arbitrary quantizers by use of a compensating postfilter. It is shown that the resulting optimized quantizers fit the model of a linear time-invariant filter followed by additive noise uncorrelated with the input which also characterizes the optimal (Lloyd-Max) quantizers. On the basis of this model, an expression for the variance of the error of a subband coder using optimized quantizers is explicitly determined. Given analysis filters which statistically separate the subbands, it is shown that this variance is minimized if these synthesis filters are chosen, which would achieve perfect reconstruction in lossless coding. The globally optimum filter bank, minimizing the coder error variance, is further obtained by proper choice of its analysis filters. A novel method for the determination of optimal bit allocation to subbands of the filter banks with optimized quantizers is also developed. The results are evaluated experimentally by comparison of the optimum uniformly split subband image coding scheme to classical logarithmically-split filter bank (wavelet) coding methods.     

On the study of four-parallelogram filter banks

The most commonly used 2-D filter banks are separable filter banks, which can be obtained by cascading two 1-D filter banks in the form of a tree. The supports of the analysis and synthesis filters in the separable systems are unions of four rectangles. The natural nonseparable generalization of such supports are those that are unions of four parallelograms. We study four parallelogram filter banks, which is the class of 2-D filter banks in which the supports of the analysis and synthesis filters consist of four parallelograms. For a given a decimation matrix, there could be more than one possible configuration (the collection of passbands of the analysis filters). Various types of configuration are constructed for four-parallelogram filter banks. Conditions on the configurations are derived such that good design of analysis and synthesis filters are possible. We see that there is only one category of these filter banks. The configurations of four-parallelogram filter banks in this category can always be achieved by designing filter banks of low design cost     

Residual echo signal in critically sampled subband acoustic echocancellers based on IIR and FIR filter banks

The residual echo signal characteristics of critically sampled subband acoustic echo cancellers are analyzed. For finite impulse response (FIR) filter banks, the residual echo signal usually has a relatively broad spectral nature around the subband edges. The residual echo signal of power symmetric infinite impulse response (PS-IIR) filter banks, on the other hand, has very narrowband spectral components around the subband edges. These components can be efficiently removed with PS-IIR notch filters that integrate neatly into the filter banks without introducing perceptually noticeable degradation to the near-end speech. This solution has very low computational complexity and does not impinge on the system performance. Simulation studies with recordings from the cockpit of a car, based on a fast QR least-squares adaptive algorithm, demonstrate the potential of this approach for a practical AEC system     

On orthonormal wavelets and paraunitary filter banks

The known result that a binary-tree-structured filter bank with the same paraunitary polyphase matrix on all levels generates an orthonormal basis is generalized to binary trees having different paraunitary matrices on each level. A converse result that every orthonormal wavelet basis can be generated by a tree-structured filter bank having paraunitary polyphase matrices is then proved. The concept of orthonormal bases is extended to generalized (nonbinary) tree structures, and it is seen that a close relationship exists between orthonormality and paraunitariness. It is proved that a generalized tree structure with paraunitary polyphase matrices produces an orthonormal basis. Since not all phases can be generated by tree-structured filter banks, it is proved that if an orthonormal basis can be generated using a tree structure, it can be generated specifically by a paraunitary tree     

On characterization of linear phase nonuniform filter banks

In this paper, a condition termed as linear phase condition (LPC), which ensures that the filters of a nonuniform filter bank are linear phase, is presented. It is observed that the proposed LPC is also applicable to the uniform filter bank case. Further, the utility of this LPC to find (i) necessary restrictions on the filters lengths, (ii) the number of symmetric and antisymmetric filters in the filter bank and (iii) filter bank decimation factors is also investigated. The results obtained for the different cases are also presented in the form of tables. These tables will facilitate the design of nonuniform filter bank by ruling out the non-solvable cases and by reducing the search space, thus saving the designers’ precious time.     

On the construction of invertible filter banks on the 2-sphere

The theories of signal sampling, filter banks, wavelets, and "overcomplete wavelets" are well established for the Euclidean spaces and are widely used in the processing and analysis of images. While recent advances have extended some filtering methods to spherical images, many key challenges remain. In this paper, we develop theoretical conditions for the invertibility of filter banks under continuous spherical convolution. Furthermore, we present an analogue of the Papoulis generalized sampling theorem on the 2-Sphere. We use the theoretical results to establish a general framework for the design of invertible filter banks on the sphere and demonstrate the approach with examples of self-invertible spherical wavelets and steerable pyramids. We conclude by examining the use of a self-invertible spherical steerable pyramid in a denoising experiment and discussing the computational complexity of the filtering framework.     

On the design of FIR analysis-synthesis filter banks with highcomputational efficiency

Presents an effective design algorithm for analysis-synthesis filter banks with computationally efficient structures. Although a wide variety of implementation structures can be accommodated, the focus of the paper is on cosine modulated filter banks. The design procedure is based on a time domain formulation of analysis-synthesis filter banks in which each individual channel filter is constrained to be a cosine modulated versions of a baseband filter. The resulting filter banks are very efficient in terms of computational requirements and are relatively easy to design. A unique feature of this approach is that relatively low reconstruction delays can be imposed on the system. A discussion of the associated computational properties of the designed systems and some design examples are included     

On the perfect reconstruction problem in N-band multirate maximallydecimated FIR filter banks

The problem of finding N-K filters of an N-band maximally decimated FIR analysis filter bank, given K filters, so that FIR perfect reconstruction can be achieved, is considered. The perfect reconstruction condition is expressed as a requirement of unimodularity of the polyphase analysis filter matrix. Based on the theory of Euclidean division for matrix polynomials, the conditions the given transfer functions must satisfy are given, and a complete parameterization of the solution is obtained. This approach provides an interesting alternative to the method of the complementary filter in the case of N>2,K相似文献   

On two-channel filter banks with directional vanishing moments.

The contourlet transform was proposed to address the limited directional resolution of the separable wavelet transform. One way to guarantee good approximation behavior is to let the directional filters in the contourlet filter bank have sharp frequency response. This requires filters with large support size. We seek to isolate the key filter property that ensures good approximation. In this direction, we propose filters with directional vanishing moments (DVM). These filters, we show, annihilate information along a given direction. We study two-channel filter banks with DVM filters. We provide conditions under which the design of DVM filter banks is possible. A complete characterization of the product filter is, thus, obtained. We propose a design framework that avoids 2-D factorization using the mapping technique. The filters designed, when used in the contourlet transform, exhibit nonlinear approximation comparable to the conventional filters while being shorter and, therefore, providing better visual quality with less ringing artifacts. Furthermore, experiments show that the proposed filters outperform the conventional ones in image approximation and denoising.     

On reconstruction methods for processing finite-length signals withparaunitary filter banks

New expressions are developed for the perfect reconstruction of the boundary regions of a finite-length signal after subband processing. The time-invariant filter bank is required to be uniform and paraunitary, using FIR filters regardless of phase or symmetry. They accommodate a linear boundary extension in the analysis section, and avoid periodic extensions or storage of extended subband signals. The reconstruction methods are based on the formulation of linear systems that are built as a function of the filters     

Frame-theoretic analysis of oversampled filter banks

We provide a frame-theoretic analysis of oversampled finite impulse response (FIR) and infinite impulse response (FIR) uniform filter banks (FBs). Our analysis is based on a new relationship between the FBs polyphase matrices and the frame operator corresponding to an FB. For a given oversampled analysis FB, we present a parameterization of all synthesis FBs providing perfect reconstruction. We find necessary and sufficient conditions for an oversampled FB to provide a frame expansion. A new frame-theoretic procedure for the design of paraunitary FBs from given nonparaunitary FBs is formulated. We show that the frame bounds of an FB can be obtained by an eigen-analysis of the polyphase matrices. The relevance of the frame bounds as a characterization of important numerical properties of an FB is assessed by means of a stochastic sensitivity analysis. We consider special cases in which the calculation of the frame bounds and synthesis filters is simplified. Finally, simulation results are presented