Preservation of 2-D signal symmetries in quincunx filter-banks

An important problem in the analysis of symmetric extension methods is to determine the conditions under which signal symmetries are preserved by the filtering (convolution) and downsampling operations. In this letter, we provide a complete characterization of four-fold two-dimensional signal symmetries viz. quadrantal symmetry, diagonal symmetry, and 90deg rotational symmetry. We then consider Quincunx filter-banks and determine the conditions under which the four-fold signal symmetries are preserved by the filtering and downsampling operations.IEE

Design of near-perfect reconstruction two-parallelogram filter-banks

The design of nonseparable 2-D filter banks presents several difficulties, which do not arise in the 1-D counterparts. One particular aspect to be taken into consideration is the shape of the passbands of the filters. In the work of Lin and Vaidyanathan, a formulation of perfect-reconstruction (PR) cosine-modulated filter-banks (CMFBs) is proposed, with filters having a "two-parallelogram" (Two-P) support. The design problem is considerably simplified to the design of a single prototype-filter. In this letter, we extend the design method of Lin and Vaidyanathan to design near-perfect reconstruction (NPR) Two-P CMFBs, with the intention of trading off the PR property for filters with better stopband attenuation. We achieve NPR by optimization of the polyphase components of the prototype, imposing a "near power-complementary" constraint. We then demonstrate the better stopband attenuation achieved by the NPR design method, by comparing a PR and NPR design of a three-channel filter-bank.IEE

Design of 2-D Mth band lowpass FIR eigenfilters with symmetries

In this letter, we extend the eigenfllter filter design method for the design of two-dimensional Mth band lowpass filters, with the filter impulse response having quadrantal or diagonal symmetry. We show that imposing the Mth band and symmetry constraints puts restrictions on the possible choices of the matrix M. We identify sufficient conditions on M, and show a class of matrices satisfying those conditions, so that Mth band lowpass filters with quadrantal or diagonal symmetry can be designed.IEE

On Filter symmetries in a class of tree-structured 2-D nonseparable filter-banks

In one-dimensional (1-D) filter-banks (FBs), symmetries (or anti-symmetries) in the filter impulse-responses (which implies linear-phase filters) are required for symmetric signal extension schemes for finite-extent signals. In two-dimensional (2-D) separable FBs, essentially, 1-D processing is done independently along each dimension. When 2-D nonseparable FBs are considered, the 2-D filters (2-D signals in general) can have a much larger variety of symmetries (anti-symmetries) than the 1-D case. Some examples of 2-D symmetries possible are quadrantal, diagonal, centro, 4-fold rotational, etc. In this letter, we analyze the filter symmetries in a subclass of tree-structured 2-D nonseparable FBs, whose sampling matrices can be factored as a product of Quincunx sampling matrix and a diagonal matrix. Within this subclass, we distinguish between two types and show that we can have diagonally symmetric filters in Type-I FBs and quadrantally symmetric filters in Type-II FBs. We then discuss how these FBs with quadrantally and diagonally symmetric filters can be used with a symmetric signal extension scheme on finite-extent signals.IEE

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 I-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 "predesigned" analysis filter, with the filter lengths satisfying certain conditions. We present several design examples to demonstrate the effectiveness of the method

On Parallelepiped-Shaped Passbands for Multidimensional Nonseparable Mth-Band Low-Pass Filters

The “shape” of the desired frequency passband is an important consideration in the design of nonseparable multidimensional ($M$ -D) filters in $M$-D multirate systems. For $M$-D ${bf M}$th-band filters, the passband shape should be chosen such that the ${bf M}$th-band constraint is satisfied. The most commonly used shape of the passband for $M$-D ${bf M}$ th-band low-pass filters is the so-called symmetric parallelepiped (SPD) ${rm SPD}(pi {bf M}^{- {rm T}})$ . In this paper, we consider the more general parallelepiped passband ${rm SPD}(pi {bf L} ^{rm T})$, and derive conditions on ${bf L}$ such that the ${bf M}$ th-band constraint is satisfied. This result gives some flexibility in designing $M$-D ${bf M}$th-band filters with parallelepiped shapes other than the commonly used case of ${bf L} = {bf M}^{- 1}$. We present design examples of 2-D ${bf M}$th-band filters to illustrate this flexibility in the choice of ${bf L}$.© IEE

On the design of FIR wavelet filter banks using factorization of a halfband polynomial

A classic method of designing two-channel biorthogonal wavelet FIR filter banks is by the factorization of a halfband polynomial. Most of the popular biorthogonal filter banks are designed by the factorization of the Lagrange halfband polynomial, which has the maximum number of zeros at z = -1. However, the Lagrange halfband polynomial does not have any "free parameter," and thus, there is no direct control over the frequency response of the filters obtained by factorization. In this letter, our aim is have some control over the frequency response of the filters designed by factorization. To achieve this, we start with a general halfband filter (not the Lagrange halfband filter) whose coefficients are parameters to be designed. We then impose the regularity constraint by imposing zeros at z = -1. The number of zeros we impose is less than the maximum possible. Imposing the zeros gives a set of constraints on the coefficients of the halfband polynomial. We then factorize the halfband polynomial by expressing it in terms of the independent (free) parameters after imposing the regularity constraints. We use the free parameters to control the frequency response of the filters. We present design examples to illustrate this method

Design of 3D Nonseparable Mth Band Eigenfilters with the 48-Hedral Symmetry

We present a design method for designing 3D Mth band eigenfilters with 48-hedral symmetry, along with their application to sampling structure conversion of 3D signals defined on the face centered cubic (FCC) or body centered cubic (BCC) lattice to cartesian cubic (CC) lattice. While we present experimental results for FCC and BCC lattices, since these are commonly used 3D lattices, the design formulations we present are valid for other 3D lattices with 48-hedral symmetry as well. The sampling structure conversion of a signal sampled on FCC (or BCC) lattice to the CC lattice is a 3D interpolation problem, with the upsampling operation defined using the FCC (or BCC) lattice. In the design formulation for 3D eigenfilters, we impose the Mth band constraint, the so-called zero direct component (DC) leakage constraint, and the 48-hedral symmetry of the 3D filter impulse response. The Mth band constraint ensures that the original input samples are preserved at the output and the zero DC leakage results in the suppression of the zero (DC) frequency in the aliases due to the upsampling operation, thus improving the quality of the interpolated output. The symmetry results in the reduction of independent parameters in the filter design