Multidimensional, paraunitary principal component filter banks

In this correspondence, the one-dimensional (1-D) principal component filter banks (PCFB's) derived by Tsatsatsanis and Giannakis (1995) are generalized to higher dimensions. As presented by Tsatsatsanis and Giannakis, PCFB's minimize the mean-squared error (MSE) when only Q out of P subbands are retained. Previously, two-dimensional (2-D) PCFB's were proposed by Tirakis et al. (1995). The work by Tirakis et al. was limited to 2-D signals and separable resampling operators. The formulation presented here is general in that it can easily accommodate signals of arbitrary (yet finite) dimension and nonseparable sampling. A major result presented in this paper is that in addition to minimizing MSE when reconstructing from Q out of p subbands, the PCFB's result in maximizing theoretical coding gain (TCG) thereby performing optimally in a energy compaction sense     

Multidimensional orthogonal filter bank characterization and design using the Cayley transform.

We present a complete characterization and design of orthogonal infinite impulse response (IIR) and finite impulse response (FIR) filter banks in any dimension using the Cayley transform (CT). Traditional design methods for one-dimensional orthogonal filter banks cannot be extended to higher dimensions directly due to the lack of a multidimensional (MD) spectral factorization theorem. In the polyphase domain, orthogonal filter banks are equivalent to paraunitary matrices and lead to solving a set of nonlinear equations. The CT establishes a one-to-one mapping between paraunitary matrices and para-skew-Hermitian matrices. In contrast to the paraunitary condition, the para-skew-Hermitian condition amounts to linear constraints on the matrix entries which are much easier to solve. Based on this characterization, we propose efficient methods to design MD orthogonal filter banks and present new design results for both IIR and FIR cases.     

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     

Linear phase paraunitary filter banks: theory, factorizations anddesigns

M channel maximally decimated filter banks have been used in the past to decompose signals into subbands. The theory of perfect-reconstruction filter banks has also been studied extensively. Nonparaunitary systems with linear phase filters have also been designed. The authors study paraunitary systems in which each individual filter in the analysis synthesis banks has linear phase. Specific instances of this problem have been addressed by other authors, and linear phase paraunitary systems have been shown to exist. This property is often desirable for several applications, particularly in image processing. They begin by answering several theoretical questions pertaining to linear phase paraunitary systems. Next, they develop a minimal factorization for a large class of such systems. This factorization will be proved to be complete for even M. Further, they structurally impose the additional condition that the filters satisfy pairwise mirror-image symmetry in the frequency domain. This significantly reduces the number of parameters to be optimized in the design process. They then demonstrate the use of these filter banks in the generation of M-band orthonormal wavelets. Several design examples are also given to validate the theory     

Theory and design of signal-adapted FIR paraunitary filter banks

We study the design of signal-adapted FIR paraunitary filter banks, using energy compaction as the adaptation criterion. We present some important properties that globally optimal solutions to this optimization problem satisfy. In particular, we show that the optimal filters in the first channel of the filter bank are spectral factors of the solution to a linear semi-infinite programming (SIP) problem. The remaining filters are related to the first through a matrix eigenvector decomposition. We discuss uniqueness and sensitivity issues. The SIP problem is solved using a discretization method and a standard simplex algorithm. We also show how regularity constraints may be incorporated into the design problem to obtain globally optimal (in the energy compaction sense) filter banks with specified regularity. We also consider a problem in which the polyphase matrix implementation of the filter bank is constrained to be DCT based. Such constraints may also be incorporated into our optimization algorithm; therefore, we are able to obtain globally optimal filter banks subject to regularity and/or computational complexity constraints. Numerous experiments are presented to illustrate the main features that distinguish adapted and nonadapted filters, as well as the effects of the various constraints. The conjecture that energy compaction and coding gain optimization are equivalent design criteria is shown not to hold for FIR filter banks     

Theory of paraunitary filter banks over fields of characteristictwo

Motivated by our wavelet framework for error-control coding, we proceed to develop an important family of wavelet transforms over finite fields. Paraunitary (PU) filter banks that are realizations of orthogonal wavelets by multirate filters are an important subclass of perfect reconstruction (PR) filter banks. A parameterization of PU filter banks that covers all possible PU systems is very desirable in error-control coding because it provides a framework for optimizing the free parameters to maximize coding performance. This paper undertakes the problem of classifying all PU matrices with entries from a polynomial ring, where the coefficients of the polynomials are taken from finite fields. It constructs Householder transformations that are used as elementary operations for the realization of all unitary matrices. Then, it introduces elementary PU building blocks and a factorization technique that is specialized to obtain a complete realization for all PU filter banks over fields of characteristic two. This is proved for the 2 × 2 case, and conjectured for the M × M case, where M ⩾ 3. Using these elementary building blocks, we can construct all PU filter banks over fields of characteristic two. These filter banks can be used to implement transforms which, in turn, provide a powerful new perspective on the problems of constructing and decoding arbitrary-rate error-correcting codes     

Boundary operation of 2-D nonseparable linear-phase paraunitary filter banks

This paper proposes a boundary operation technique of 2-D nonseparable linear-phase paraunitary filter banks (NS-LPPUFBs) for size limitation. The proposed technique is based on a lattice structure consisting of the 2-D separable block discrete cosine transform and nonseparable support-extension processes. The bases are allowed to be anisotropic with the fixed critically subsampling, overlapping, orthogonal, symmetric, real-valued, and compact-support properties. First, the blockwise implementation is developed so that the basis images can be locally controlled. The local control of basis images is shown to maintain orthogonality. This property leads a basis termination (BT) technique as a boundary operation. The technique overcomes the drawback of NS-LPPUFBs that the popular symmetric extension method is invalid. Through some experimental results of diagonal texture coding, the significance of the BT is verified.     

Two-dimensional orthogonal filter banks and wavelets with linearphase

Two-dimensional (2-D) compactly supported, orthogonal wavelets and filter banks having linear phase are presented. Two cases are discussed: wavelets with two-fold symmetry (centrosymmetric) and wavelets with four-fold symmetry that are symmetric (or anti-symmetric) about the vertical and horizontal axes. We show that imposing the requirement of linear phase in the case of order-factorable wavelets imposes a simple constraint on each of its polynomial order-1 factors. We thus obtain a simple and complete method of constructing orthogonal order-factorable wavelets with linear phase. This method is exemplified by design in the case of four-band separable sampling. An interesting result that is similar to the one well-known in the one-dimensional (1-D) case is obtained: orthogonal order-factorable wavelets cannot be both continuous and have four-fold symmetry     

A new algorithm for linear-phase paraunitary filter banks with pairwise mirror-image frequency responses

Subband filter banks have attracted much attention during the past few years. In this paper, an efficient design algorithm, which leads to linear-phase paraunitary filter banks with pairwise mirror-image frequency responses, is revisited and further studied. New lattice structures are presented to extend the algorithm to the case where the number (M) of channels is odd. Design examples of filter banks with 3 and 5 channels are presented.     

Results on the factorization of multidimensional matrices for paraunitary filterbanks over the complex field

This paper undertakes the study of multidimensional finite impulse response (FIR) filterbanks. One way to design a filterbank is to factorize its polyphase matrices in terms of elementary building blocks that are fully parameterized. Factorization of one-dimensional (1-D) paraunitary (PU) filterbanks has been successfully accomplished, but its generalization to the multidimensional case has been an open problem. In this paper, a complete factorization for multichannel, two-dimensional (2-D), FIR PU filterbanks is presented. This factorization is based on considering a two-variable FIR PU matrix as a polynomial in one variable whose coefficients are matrices with entries from the ring of polynomials in the other variable. This representation allows the polyphase matrix to be treated as a one-variable matrix polynomial. To perform the factorization, the definition of paraunitariness is generalized to the ring of polynomials. In addition, a new degree-one building block in the ring setting is defined. This results in a building block that generates all two-variable FIR PU matrices. A similar approach is taken for PU matrices with higher dimensions. However, only a first-level factorization is always possible in such cases. Further factorization depends on the structure of the factors obtained in the first level.     

GHM类正交多小波滤波器组的因子化和参数化

本文提出了GHM类正交多小波滤波器组的完备的因子化形式和参数化方法。可用于这类多小波的优化设计和有效实现，同时给出了时频最优准则下的设计实例。     

CORDIC-lifting factorization of paraunitary filter banks based on the quaternionic multipliers for lossless image coding

Quaternions have offered a new paradigm to the signal processing community: to operate directly in a multidimensional domain. We have recently introduced the quaternionic approach to the design and implementation of paraunitary filter banks: four- and eight-channel linear-phase paraunitary filter banks, including those with pairwise-mirror-image symmetric frequency responses. The hypercomplex number theory is utilized to derive novel lattice structures in which quaternion multipliers replace Givens (planar) rotations. Unlike the conventional algorithms, the proposed computational schemes maintain losslessness regardless of their coefficient quantization. Moreover, the one regularity conditions can be expressed directly in terms of the quaternion lattice coefficients and thus easily satisfied even in finite-precision arithmetic. In this paper, a novel approach to realizing CORDIC-lifting factorization of paraunitary filter banks is presented, which is based on the embedding of the CORDIC algorithm inside the lifting scheme. Lifting allows for making multiplications invertible. The 2D CORDIC engine using sparse iterations and asynchronous pipeline processor architecture based on the embedded CORDIC engine as stage of processor is reported. Also it is necessary to notice, that the quaternion multiplier lifting scheme based on the 2D CORDIC algorithm is the structural decision for the lossless digital signal processing. This approach applies to very practical filter banks, which are essential for image processing, and addresses interesting theoretical questions.     

A simplified lattice factorization for linear-phase paraunitary filter banks with pairwise mirror image frequency responses

In this paper, by extending our previous work on general linear-phase paraunitary filter banks even-channel (LPPUFBs), we develop a new structure for LPPUFBs with the pairwise mirror image (PMI) frequency responses, which is a simplified version of the lattice proposed by Nguyen et al. Our simplification is achieved through trivial matrix manipulations and the cosine-sine (C-S) decomposition of a general orthogonal matrix. The resulting new structure covers the same class of PMI-LPPUFBs as the original lattice, while substantially reducing the number of free parameters involved in the nonlinear optimization. A design example is presented to demonstrate the effectiveness of the new structure.     

Design of IIR orthogonal wavelet filter banks using lifting scheme

The lifting scheme is well known to be an efficient tool for constructing second generation wavelets and is often used to design a class of biorthogonal wavelet filter banks. For its efficiency, the lifting implementation has been adopted in the international standard JPEG2000. It is known that the orthogonality of wavelets is an important property for many applications. This paper presents how to implement a class of infinite-impulse-response (IIR) orthogonal wavelet filter banks by using the lifting scheme with two lifting steps. It is shown that a class of IIR orthogonal wavelet filter banks can be realized by using allpass filters in the lifting steps. Then, the design of the proposed IIR orthogonal wavelet filter banks is discussed. The designed IIR orthogonal wavelet filter banks have approximately linear phase responses. Finally, the proposed IIR orthogonal wavelet filter banks are applied to the image compression, and then the coding performance of the proposed IIR filter banks is evaluated and compared with the conventional wavelet transforms.     

A group theoretic approach to multidimensional filter banks: theoryand applications

In this paper, we provide a new method for analyzing multidimensional filter banks. This method enables us to solve various open problems in multidimensional filter bank characterization and design. The essential element in this new approach is the redefinition of polyphase components. It will be shown that a rich set of mathematical tools, in particular algebraic group theory, will become available for use in the analysis of filter banks. We demonstrate the elegance and power of the tool set by employing it for the characterization of multidimensional filter banks and applying it to two open problems. The first problem is concerned with the development of a general method to design multichannel (⩾2), multidimensional filter banks using transformations, while the second problem is concerned with the derivation of general restrictions on group delays in linear phase filter banks. The treatment of these problems is only an illustration of the power of the tool set of algebraic group theory, employed for the first time in the context of multidimensional filter banks     

Antipodal paraunitary matrices and their application to OFDM systems

Paraunitary (PU) matrices and filterbanks have played an important role in many applications. This paper studies a special class of PU matrices, called antipodal paraunitary (APU) matrices. APU matrices are PU matrices whose coefficient matrices consist of /spl plusmn/1 only. Several new methods will be introduced for the construction of APU matrices. In particular, a new method based on the butterfly structure that has a low cost implementation is proposed. Moreover, one application of APU matrices in precoded orthogonal frequency division multiplexing (OFDM) systems will be considered. By using an APU precoding matrix, the complexity will be very low, and experiments show that the use of APU matrices can greatly enhance the performance.     

Filters and filter banks for periodic signals, the Zak transform,and fast wavelet decomposition

We present a new approach to filtering and reconstruction of periodic signals. The tool that proves to handle these tasks very efficiently is the discrete Zak transform. The discrete Zak transform can be viewed as the discrete Fourier transform performed on the signal blocks. It also can be considered the polyphase representation of periodic signals. Fast filtering-decimation-interpolation-reconstruction algorithms are developed in the Zak transform domain both for the undersampling and critical sampling cases. The technique of finding the optimal biorthogonal filter banks, i.e., those that would provide the best reconstruction even in the undersampling case, is presented. An algorithm for orthogonalization of nonorthogonal filters is developed. The condition for perfect reconstruction for the periodic signals is derived. The generalizations are made for the nonperiodic sequences, and several ways to apply the developed technique to the nonperiodic sequences are considered. Finally, the developed technique is applied to recursive filter banks and the discrete wavelet decomposition     

A MC-CDMA system based on orthogonal filter banks of wavelet transforms and partial combining

Multi Carrier Code Division Multiple Access (MC-CDMA) is an appropriate modulation technique for high data rates. In this modulation scheme, employing of combining techniques are unavoidable for restoring orthogonality between different user signals. In this paper, the combining techniques of a MC-CDMA based on wavelet transform are studied. The partial combining of the wavelet based MC-CDMA system is suggested and its performance is compared with the maximum ratio combining (MRC), equal gain combining (EGC), orthogonality restoring combining (ORC) and the minimum mean square error (MMSE) techniques. Orthogonal filter banks of the wavelet transforms provide orthogonality between the wavelets and scaling functions, subcarriers and spreading sequences. Therefore, they provide new dimensions in combatting multipath fading, inter carrier interference (ICI) and narrowband interference in MC-CDMA systems. Furthermore, analysis of partial combining shows that coefficients of this technique can be computed in a simple manner. Simulations results indicate that the bit error rate (BER) performance of the proposed scheme is almost comparable with the MMSE combining scheme and the proposed system has a proper performance in terms of BER versus the signal to noise ratio (SNR) in frequency selective fading channels.     

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.