Multidimensional Filter Banks Research Articles

Algorithm for ocean surface pollution detection based on the results of video surveillance image processing

The problems of timely detection of environmental disturbances caused by the consequences of various man-made disasters on land and on the water surface of water areas of seas and oceans are based on imperfect methods of control and monitoring of large spaces. Therefore, it is relevant to search for approaches for learning procedures used in the construction of monitoring systems. The aim of article to consider approaches to the formation of feature spaces based on the processing of multiple-scale distributions of signals from drone cameras and the application of discrete series of continuous-time wavelets. An approach to the formalization of the recognition task is proposed. The results of using discrete series of continuous-time wavelets are presented, which allows obtaining a result similar to the use of a multi-dimensional filter bank loaded on integrators. Results of a practical experiment are presented, which showed that increasing the dimensionality of feature vectors reduces their contrast. The directions of further research are formed. The presented results open new possibilities for realization of recognition systems at the software level.

A New Approach to Video Coding Using 3D Directional Filter Bank

The high resolution 3-D and higher dimensional volumetric data are increasingly used in a wide variety of applications including biomedical imaging, video processing and compression. Multidimensional filter banks and its associated constructions provide a framework and serve as an efficient computational tool in the formation, representation, and processing of multidimensional data sets. This paper focuses on the construction of 3-D directional filter banks (DFB) and its signal representations for scalable video coding. In this paper we consider five such DFBs and analyse the performance to find the best suitable one for scalable video coding. The five different filters includes: Maxflat filter, Vk filter, kos filter, Sinc filter and Haar filter. The performance evaluation has been done which quantifies the factors like Peak Signal to Noise Ratio (PSNR) and Mean Square Error (MSE). In order to achieve better compression, a comparative study between RLC and Huffman coding schemes has been performed and the average codeword length is measured and tabulated. Among the five filters and two compression scheme it is observed that Kos filters with Huffman coding provide better compression and low Average codeword length.

Higher order feasible building blocks for lattice structure of multidimensional linear phase biorthogonal filter banks

A multidimensional (MD) linear phase biorthogonal filter bank (LPBOFB) with higher order feasible (HOF) building blocks is reported. Basically, there are two ways to design filter banks with large filter supports. One way is to use a cascade of degree-1 building blocks, and the other way is to use a cascade of order-1 building blocks. Unfortunately, both methods have high implementation costs in terms of the number of parameters, especially for the multidimensional case. A previously reported HOF building block has now been applied to MD LPBOFBs. Their generalized structural design supports both an even and odd number of channels. It is shown that the HOF structure cannot be factored into a cascade of order-1 building blocks. The proposed MD LPBOFB has larger filters and uses fewer building blocks than the traditional degree-1 and order-1 structures.

Multi-D Wavelet Filter Bank Design Using Quillen-Suslin Theorem for Laurent Polynomials

In this paper we present a new approach for constructing the wavelet filter bank. Our approach enables constructing nonseparable multidimensional non-redundant wavelet filter banks with FIR filters using the Quillen-Suslin Theorem for Laurent polynomials. Our construction method presents some advantages over the traditional methods of multidimensional wavelet filter bank design. First, it works for any spatial dimension and for any sampling matrix. Second, it does not require the initial lowpass filters to satisfy any additional assumption such as interpolatory condition. Third, it provides an algorithm for constructing a wavelet filter bank from a single lowpass filter so that its vanishing moments are at least as many as the accuracy number of the lowpass filter.

Lattice Structure for Generalized-Support Multidimensional Linear Phase Perfect Reconstruction Filter Bank

Multidimensional linear phase perfect reconstruction filter bank (MDLPPRFB) can be designed and implemented via lattice structure. The lattice structure for the MDLPPRFB with filter support N(MΞ) has been published by Muramatsu , where M is the decimation matrix, Ξ is a positive integer diagonal matrix, and N(N) denotes the set of integer vectors in the fundamental parallelepiped of the matrix N. Obviously, if Ξ is chosen to be other positive diagonal matrices instead of only positive integer ones, the corresponding lattice structure would provide more choices of filter banks, offering better trade-off between filter support and filter performance. We call such resulted filter bank as generalized-support MDLPPRFB (GSMDLPPRFB). The lattice structure for GSMDLPPRFB, however, cannot be designed by simply generalizing the process that Muramatsu employed. Furthermore, the related theories to assist the design also become different from those used by Muramatsu . Such issues will be addressed in this paper. To guide the design of GSMDLPPRFB, the necessary and sufficient conditions are established for a generalized-support multidimensional filter bank to be linear-phase. To determine the cases we can find a GSMDLPPRFB, the necessary conditions about the existence of it are proposed to be related with filter support and symmetry polarity (i.e., the number of symmetric filters ns and antisymmetric filters na). Based on a process (different from the one Muramatsu used) that combines several polyphase matrices to construct the starting block, one of the core building blocks of lattice structure, the lattice structure for GSMDLPPRFB is developed and shown to be minimal. Additionally, the result in this paper includes Muramatsu's as a special case.

A multidimensional wave digital filter bank for video-based motion analysis

Multidimensional wave digital filters (MDWDF) exhibit the same desirable properties as 1D WDFs, most notably including passivity and therefore guaranteed stability as well as high robustness. A possible application for such MDWDFs may be found in motion analysis of image sequences by means of filters with fan-shaped transfer functions, where content with specific movement information can be extracted. For that matter, a parallel filter bank is needed to differentiate object motion into separate classes. In this paper, a new specialized MDWDF fan filter structure is introduced, possessing both reduced computational complexity and memory requirements compared to existing approaches. Additionally, part of the processing can be shared among all bands, further increasing efficiency.

Theory of Sparse Coprime Sensing in Multiple Dimensions

Coprime sampling and coprime sensor arrays have been introduced recently for the one-dimensional (1-D) case, and applications in beamforming and direction finding discussed. A pair of coprime arrays can be used to sample a wide-sense stationary signal sparsely, and then reconstruct the autocorrelation at a significantly denser set of points. All applications based on autocorrelation (e.g., spectrum and DOA estimation) benefit from this property. It was also shown in the past that coprimality can be exploited in the frequency domain by using a pair of coprime DFT filter banks, to produce the effect of a much denser tiling in the frequency domain, compared to what the two filter banks can individually achieve. This paper extends these ideas to multiple dimensions. In the 1-D case the samples or sensors lie on a pair of uniform grids, whereas in the multidimensional case, they lie on a pair of multidimensional lattices, not necessarily rectangular. This makes the developments mathematically more intricate. First several properties of coarrays of lattices are derived. It is shown how one can get dense coarrays from sparse arrays on non rectangular lattices. This requires that the lattice generating matrices M and N be commuting and coprime (to be defined). Multidimensional DFT filter banks for applications such as beamforming, with commuting coprime lattice arrays, are then described, and it is shown that a very dense tiling of the frequency plane can be obtained from the two sparse lattice arrays. A particular family of commuting coprime matrices called adjugate pairs are considered in some detail, and shown to have attractive properties. A brief review of the 1-D case is included at the beginning for convenience.

Image Fusion Algorithm Based on the Quincunx-sampled Discrete Wavelet Frame

A novel image fusion algorithm based on the quincunx-sampled discrete wavelet frame is presented.Firstly, the replaceability of sampling matrix for multidimensional perfect reconstruction filter banks is shown;based on this, the quincunx-sampled discrete wavelet frame is derived, and its characteristics are discussed.Then, the frame is incorporated into the multiscale-based image fusion scheme, and a nearly shift-invariant fusion algorithm with low redundancy is obtained.Finally, the algorithm is tested and compared with existing algorithms.The results show that the algorithm can produce high-quality fused images rapidly.

Design of Regular Wavelets Using a Three-Step Lifting Scheme

We propose structural multidimensional multichannel filter banks with desirable numbers of vanishing moments for the analysis and synthesis banks. For a two-channel filter bank, we use a three-step lifting scheme as opposed to the conventional two-step lifting method in order to provide more symmetry between the analysis and synthesis filters. We show that the resulting filters have more regularity, lower frame bounds ratio, and better frequency selectivity. We also extend our design to a general multichannel framework. Three steps of lifting scheme provides us a degree of freedom which we can benefit from toward a more flexible design.

Adaptive Nonseparable Wavelet Transform via Lifting and its Application to Content-Based Image Retrieval

We present in this paper a novel way to adapt a multidimensional wavelet filter bank, based on the nonseparable lifting scheme framework, to any specific problem. It allows the design of filter banks with a desired number of degrees of freedom, while controlling the number of vanishing moments of the primal wavelet ((~)N moments) and of the dual wavelet ( N moments). The prediction and update filters, in the lifting scheme based filter banks, are defined as Neville filters of order (~)N and N, respectively. However, in order to introduce some degrees of freedom in the design, these filters are not defined as the simplest Neville filters. The proposed method is convenient: the same algorithm is used whatever the dimensionality of the signal, and whatever the lattice used. The method is applied to content-based image retrieval (CBIR): an image signature is derived from this new adaptive nonseparable wavelet transform. The method is evaluated on four image databases and compared to a similar CBIR system, based on an adaptive separable wavelet transform. The mean precision at five of the nonseparable wavelet based system is notably higher on three out of the four databases, and comparable on the other one. The proposed method also compares favorably with the dual-tree complex wavelet transform, an overcomplete nonseparable wavelet transform.

A Computable Fourier Condition Generating Alias-Free Sampling Lattices

We propose a Fourier analytical condition linking alias-free sampling with the Fourier transform of the indicator function defined on the given frequency support. Our discussions center around how to develop practical computation algorithms based on the proposed analytical condition. We address several issues along this line, including the derivation of simple closed-form expressions for the Fourier transforms of the indicator functions defined on arbitrary polygonal and polyhedral domains; a complete and nonredundant enumeration of all quantized sampling lattices via the Hermite normal forms of integer matrices; and a quantitative analysis of the approximation of the original infinite Fourier condition by using finite computations. Combining these results, we propose a computational testing procedure that can efficiently search for the optimal alias-free sampling lattices for a given polygonal or polyhedral shaped frequency domain. Several examples are presented to show the potential of the proposed algorithm in multidimensional filter bank design, as well as in applications involving the design of efficient sampling patterns for multidimensional band-limited signals.

Constraints of Second-Order Vanishing Moments on Lattice Structures for Non-separable Orthogonal Symmetric Wavelets

In this paper, a design method of two-dimensional (2-D) orthogonal symmetric wavelets is proposed by using a lattice structure for multi-dimensional (M-D) linear-phase paraunitary filter banks (LPPUFB), which the authors have proposed as a previous work and then modified by Lu Gan et al. The derivation process for the constraints on the second-order vanishing moments is shown and some design examples obtained through optimization with the constraints are exemplified. In order to verify the significance of the constraints, some experimental results are shown for Lena and Barbara image.

Eigenfilter Approach to the Design of One-Dimensional and Multidimensional Two-Channel Linear-Phase FIR Perfect Reconstruction Filter Banks

We present an eigenfilter-based approach for the design of two-channel linear-phase FIR perfect-reconstruction (PR) filter banks. This approach can be used to design 1-D two-channel filter banks, as well as multidimensional nonseparable two-channel filter banks. Our method consists of first designing the low-pass analysis filter. Given the low-pass analysis filter, the PR conditions can be expressed as a set of linear constraints on the complementary-synthesis low-pass filter. We design the complementary-synthesis filter by using the eigenfilter design method with linear constraints. We show that, by an appropriate choice of the length of the filters, we can ensure the existence of a solution to the constrained eigenfilter design problem for the complementary-synthesis filter. Thus, our approach gives an eigenfilter-based method of designing the complementary filter, given a ldquopredesignedrdquo analysis filter, with the filter lengths satisfying certain conditions. We present several design examples to demonstrate the effectiveness of the method.

Multidimensional perfect reconstruction filter banks: an approach of algebraic geometry

In image compression and some other applications, multidimensional filter banks are gaining their popularity due to the decrease in implementation cost. In this article, we study these filter banks from a viewpoint of algebraic geometry, where some insights are emerging. Familiar properties, such as perfect reconstruction and linear phase, appear differently when studied from this new angle. For the sake of practicability and a better understanding of problems, our focus in this work is further restricted to the filter banks that can achieve linear phase and perfect reconstruction properties. According to different symmetry nature, the filter banks are categorized into two types. Filters in a multidimensional filter bank represented by multivariate polynomials have common zeros that are different in nature from their 1D counterparts. In this work, the relation between the above-mentioned properties and various zeros is investigated. A criterion to test such filter banks is proposed and this criterion is also interpreted in resultant theory. Based on these results, as well as a proposed conjecture, a factorization of lifting scheme is presented for one type of these filter banks. For the other type of filter banks, we propose a method on perfect reconstruction completion.

A Mapping-Based Design for Nonsubsampled Hourglass Filter Banks in Arbitrary Dimensions

Multidimensional hourglass filter banks decompose the frequency spectrum of input signals into hourglass-shaped directional subbands, each aligned with one of the frequency axes. The directionality of the spectral partitioning makes these filter banks useful in separating the directional information in multidimensional signals. Despite the existence of various design techniques proposed for the 2-D case, to our best knowledge, the design of hourglass filter banks in 3-D and higher dimensions with finite impulse response (FIR) filters and perfect reconstruction has not been previously reported. In this paper, we propose a novel mapping-based design for the hourglass filter banks in arbitrary dimensions, featuring perfect reconstruction, FIR filters, efficient implementation using lifting/ladder structures, and a near-tight frame construction. The effectiveness of the proposed mapping- based design depends on the study of a set of conditions on the frequency supports of the mapping kernels. These conditions ensure that we can still get good frequency responses when the component filters used are nonideal. Among all feasible choices, we then propose an optimal specification for the mapping kernels, which leads to the simplest passband shapes and involves the fewest number of frequency variables. Finally, we illustrate the proposed techniques by a design example in 3-D, and an application in video denoising.

Multidimensional Directional Filter Banks and Surfacelets

In 1992, Bamberger and Smith proposed the directional filter bank (DFB) for an efficient directional decomposition of 2-D signals. Due to the nonseparable nature of the system, extending the DFB to higher dimensions while still retaining its attractive features is a challenging and previously unsolved problem. We propose a new family of filter banks, named NDFB, that can achieve the directional decomposition of arbitrary N-dimensional (N > or =2) signals with a simple and efficient tree-structured construction. In 3-D, the ideal passbands of the proposed NDFB are rectangular-based pyramids radiating out from the origin at different orientations and tiling the entire frequency space. The proposed NDFB achieves perfect reconstruction via an iterated filter bank with a redundancy factor of N in N-D. The angular resolution of the proposed NDFB can be iteratively refined by invoking more levels of decomposition through a simple expansion rule. By combining the NDFB with a new multiscale pyramid, we propose the surfacelet transform, which can be used to efficiently capture and represent surface-like singularities in multidimensional data.

Translation-Invariant Contourlet Transform and Its Application to Image Denoising

Most subsampled filter banks lack the feature of translation invariance, which is an important characteristic in denoising applications. In this paper, we study and develop new methods to convert a general multichannel, multidimensional filter bank to a corresponding translation-invariant (TI) framework. In particular, we propose a generalized algorithme à trous, which is an extension of the algorithme à trous introduced for 1-D wavelet transforms. Using the proposed algorithm, as well as incorporating modified versions of directional filter banks, we construct the TI contourlet transform (TICT). To reduce the high redundancy and complexity of the TICT, we also introduce semi-translation-invariant contourlet transform (STICT). Then, we employ an adapted bivariate shrinkage scheme to the STICT to achieve an efficient image denoising approach. Our experimental results demonstrate the benefits and potential of the proposed denoising approach. Complexity analysis and efficient realization of the proposed TI schemes are also presented.

Comments on “Multi-dimensional filter banks and wavelets—A system theoretic perspective”

In [S. Basu, Multi-dimensional filter banks and wavelets—a system theoretic perspective, J. Franklin Inst. 335B(8) (1998) 1367–1409], the problems of constructing and parameterizing multi-dimensional (MD) filter banks (FB) and wavelets were studied extensively, especially in the situation of two-dimensional (2D) and quincunx-downsampled systems. Finding the other filter complementary to a given 2D filter for quincunx sampling and linear phase perfect reconstruction (LPPR) system is based on the fact that the construction is possible if and only if the polyphase components of the given filter do not share any common zero. The construction is elaborated in another book chapter [S. Basu, Multidimensional signals, circuits and systems, in: K. Galkowski, J. Wood (Eds.), On the Structure of Linear Phase Perfect Reconstruction Quincunx Filter Banks, Taylor & Francis, London, 2001, pp. 193–208 (Chapter 10)] by the same author. This approach further relies on the fact that the resultant of the polyphase components is monomial. In this comment, we show that both above mentioned facts are not necessarily true for the quincunx FB as well as for general MD FB.

Oversampled filter banks from extended perfect reconstruction filter banks

Oversampled filter banks are currently being proposed for robust transmission applications. In this paper, we completely characterize multidimensional doubly finite-impulse-response (FIR) filter banks, that is, oversampled filter banks whose dual is FIR. Then, we consider the problem of extending perfect reconstruction critically sampled multidimensional filter banks in order to obtain doubly FIR (DFIR) filter banks. As a result, very simple criteria for constructing DFIR filter banks as extensions of orthogonal filter banks are obtained. This paper also analyzes the problem of constructing totally FIR filter banks, i.e., DFIR filter banks that remain DFIR even when some channels are removed. It is shown that any totally FIR filter bank can be implemented as the cascade of a critically sampled DFIR filter bank whose number of channels is equal to the subsampling factor, a redundant finite-dimensional transform, and a suitable set of delays

Special paraunitary matrices, Cayley transform, and multidimensional orthogonal filter banks

We characterize and design multidimensional (MD) orthogonal filter banks using special paraunitary matrices and the Cayley transform. Orthogonal filter banks are represented by paraunitary matrices in the polyphase domain. We define special paraunitary matrices as paraunitary matrices with unit determinant. We show that every paraunitary matrix can be characterized by a special paraunitary matrix and a phase factor. Therefore, the design of paraunitary matrices (and thus of orthogonal filter banks) becomes the design of special paraunitary matrices, which requires a smaller set of nonlinear equations. Moreover, we provide a complete characterization of special paraunitary matrices in the Cayley domain, which converts nonlinear constraints into linear constraints. Our method greatly simplifies the design of MD orthogonal filter banks and leads to complete characterizations of such filter banks.