2D Filter Banks Research Articles

Synthesis of High-Selectivity Two-Dimensional Filter Banks Using Sigmoidal Function

This paper explores an application of the sigmoidal function—the complementary integral Gaussian error function (erfc(.))—in two-dimensional (2D) uniform and nonuniform filter bank synthesis. The complementary integral Gaussian error function graph represents a smooth low-pass filter magnitude response. A parameter changes the function slope and increases the magnitude response selectivity. The theory is applied to 2D band-pass filter banks. Exact expressions for the magnitude response parameters are determined. As a result, 2D uniform and nonuniform filter banks with very high selectivity and exact shapes are obtained. Three synthesis examples of 2D filter banks with circular and fan-shaped magnitude responses are provided. The theoretical exposition is supplemented with two examples of image analysis using 2D uniform and nonuniform filter banks. A procedure to reduce the computations in image analysis is proposed. A comparison of filter synthesis between Parks–McLellan’s 2D filters and the erfc(.) demonstrates the significantly shorter calculation time of the proposed method.

Analytic Design Technique for 2D FIR Circular Filter Banks and Their Efficient Implementation Using Polyphase Approach.

This paper proposes an analytical design procedure for 2D FIR circular filter banks and also a novel, computationally efficient implementation of the designed filter bank based on a polyphase structure and a block filtering approach. The component filters of the bank are designed in the frequency domain using a specific frequency transformation applied to a low-pass, band-pass and high-pass 1D prototype with a specified Gaussian shape and imposed specifications (peak frequency, bandwidth). The 1D prototype filter frequency response is derived in a closed form as a trigonometric polynomial with a specified order using Fourier series, and then it is factored. Since the design starts from a 1D prototype with a factored transfer function, the frequency response of the designed 2D filter bank components also results directly in a factored form. The designed filters have an accurate shape, with negligible distortions at a relatively low order. We present the design of two types of circular filter banks: uniform and non-uniform (dyadic). An example of image analysis with the uniform filter bank is also provided, showing that the original image can be accurately reconstructed from the sub-band images. The proposed implementation is presented for a simpler case, namely for a smaller size of the filter kernel and of the input image. Using the polyphase and block filtering approach, a convenient implementation at the system level is obtained for the designed 2D FIR filter, with a relatively low computational complexity.

Multi-Resolution Common Fate Transform

The multi-resolution common fate transform (MCFT) is an audio signal representation useful for representing mixtures of multiple audio signals that overlap in both time and frequency. The MCFT combines the invertibility of a state-of-the-art representation, the common fate transform (CFT), and the multi-resolution property of the cortical stage output of an auditory model. Since the MCFT is computed based on a fully invertible complex time-frequency representation, separation of audio sources with high time-frequency overlap may be performed directly in the MCFT domain, where there is less overlap between sources than in the time-frequency domain. The MCFT circumvents the resolution issue of the CFT by using a multi-resolution two-dimensional (2D) filter bank instead of fixed-size 2D windows. This enables higher quality separation without the need to hand-tune the window size to the specific case. In this work, we describe the MCFT, discuss the properties of the MCFT with the aid of illustrative examples, and provide definitions and objective measures for two desirable representation properties: separability of source signals and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">clusterability</i> of components of each signal. The utility of the MCFT for source separation is illustrated by performing ideal masking on a comprehensive dataset of audio mixtures of musical tones played in unison, including audio samples from a wide pitch range and a variety of instruments/playing techniques. Results show that the ideal masks made in the MCFT domain yield better separability than those made in commonly used time-frequency signal representations as well as the CFT. The use of the MCFT also results in more reliable clusterability than the CFT in most cases.

Theory and design of two-dimensional DFT modulated filter bank with arbitrary modulation and decimation matrices

It is well known that two-dimensional (2D) filter bank is far removed from a straightforward extension of one-dimensional (1D) filter bank. There are many challenging problems on the theory and design methods for the 2D filter bank. Among these problems, the perfect-reconstruction (PR) theory of the 2D DFT modulated filter bank with arbitrary modulation and decimation matrices remains an unsolved difficulty, which is the focus of this paper. The necessary and sufficient condition for perfect reconstruction (PR) is derived by using the polyphase decomposition of the analysis and synthesis filters, as well as the fast implementation structure of the filter bank. Then, the PR condition in frequency domain is transformed into a set of quadratic equations with respect to the prototype filter (PF), which is utilized to formulate the design problem into an unconstrained optimization problem. An efficient iterative algorithm is proposed to solve the problem. Numerical examples are included to verify the validity of the PR condition and the effectiveness of the design method.

A Self-Reconfigurable Platform for the Implementation of 2D Filterbanks with Real and Complex-Valued Inputs, Outputs, and Filter Coefficients

We introduce a dynamically reconfigurable 2D filterbank that supports both real and complex-valued inputs, outputs, and filter coefficients. This general purpose filterbank allows for the efficient implementation of 2D filterbanks based on separable 2D FIR filters that support all possible combinations of input and output signals. The system relies on the use of dynamic reconfiguration of real/complex one-dimensional filters to minimize the required hardware resources. The system is demonstrated using an equiripple and a Gabor filterbank and the results using both real and complex-valued input images. We summarize the performance of the system in terms of the required processing times, energy, and accuracy.

Design of 2D oversampled linear phase DFT modulated filter banks via modified Newton's method

In this paper, the structure of the 2D oversampled DFT modulated filter banks is analyzed and a spatial-domain condition of a filter bank without transfer function distortion is derived. Based upon the spatial-domain condition, a modified Newton's method is presented for fast design of 2D oversampled linear phase (LP) DFT modulated filter banks with nearly perfect reconstruction (NPR). We formulate the design problem into an unconstrained optimization with a fourth-order objective function, which is the weighted sum of the transfer function distortion of the filter bank and the stopband energy of the prototype filter (PF). The optimization is solved by the modified Newton's method, where each of iterations updates the PF by a set of linear equations. It is proved that the iteration process fast converges to a stationary point of the objective function. Compared with the existing methods, the new method is fast in computation and can design 2D filter banks with a large number of subbands.

Finite Discrete, Shift-Invariant, Directional Filterbanks for Visual Information Processing, I : Construction

As is well known in neuroscience, simple cells of the mammalian’s striate cortex possess both orientation and spatial-frequency selectivity, and are similar to the Gabor filters or Gaussian derivative filters in shape. The purpose of this paper is to propose a method of designing perfect reconstruction 2D filterbanks which act on finite dimensional linear spaces consisting of 2D signals of a certain size, and have several analogous features to simple cells: (1) the filterbanks consist of several spatial-frequency channels with orientation selectivity, (2) the filterbanks have shift-invariant multiresolution (multiscale) structures, (3) filters contained in them are FIR, and are similar in appearance to not only Gaussian derivatives of 1st and 2nd order, but also ones of higher order. Moreover, they are constructed by finite linear combinations of separable filters. As is described in the text, by virtue of these properties, our 2D filterbanks can become bases of constructing computational nonlinear models of visual information processing. In this paper we construct the 2D filterbanks, and discuss them from the viewpoint of vision science. For example we disclose a possible role of “Gaussian-derivative-like” filters of higher order in our filterbanks. Practical applications of our 2D filterbanks to vision science and image processing will be given in our subsequent papers.

Design of 2D filter banks based on 1D lattice structure

AbstractThis paper proposes a design procedure for the 2D linear‐phase para‐unitary filter bank, based on the 1D lattice structure. As the first step, the perfect reconstruction conditions of 1D and 2D cases are compared, and it is shown that the 2D filter bank satisfying the perfect reconstruction condition can be realized by the 1D lattice structure with a constraint. It is known, on the other hand, that the 1D perfect reconstruction filter bank can be converted to the lattice structure, using only half of the filter coefficients, and a simple optimization procedure can derive the whole filter coefficients. Based on those investigations, it is shown that the 2D filter bank can be designed by directly designing only half of the channels of the 2D filter bank in the time domain. © 2002 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 85(12): 38–46, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.10005

Region of Interest Reconstruction of Tomographic Images using Filter Banks

A new technique for ROI image reconstruction from projections using filter banks is reported in this paper. It is known that the convolution back projection operator, used for reconstruction of images from 1D-projections, has a non-local filter which requires global projection data. In X-ray tomography, the exposure time of the object to harmful radiation is thereby increased. Recently, it has been proved that this filtering operation can be done on a chosen wavelet function instead of the projections. This localizes the filter and leads to ROI image reconstruction with reduced projections. This concept was extended using 2D and 1D MRA filter banks, in which the filters are combined with the non-local CBP filter to get short length filters. In this paper it is proved that the presence of unwanted information in all previous algorithms is due to aliasing and a new technique is proposed for reconstruction without any aliasing. The proposed scheme is implemented using Shepp-Logan phantom head image and the performance is compared.

Cosine-modulated two-dimensional time-varying filter banks

Two-dimensional (2D) filter banks are widely used for analysis and synthesis systems applied to subband coding of images. In recent years, time-varying filter banks that vary their structure appropriately for a local property of signals have been studied for application to the compression of image data. In time-varying filter banks, each different part of the signals is processed with different filter coefficients or different numbers of channels (a different sampling matrix). Those signals must be perfectly reconstructed even across transitions of the filter banks. In this paper, we consider time-varying systems of cosine-modulated 2D filter banks for arbitrary sampling lattices. We show perfect reconstruction (PR) conditions for time-varying filter banks that vary filter coefficients or a sampling matrix. Some applications are presented to show the effectiveness of this method. © 1997 Scripta Technica, Inc. Electron Comm Jpn Pt 3, 80(11): 46–54, 1997

Cosine‐modulated two‐dimensional time‐varying filter banks

Two-dimensional (2D) filter banks are widely used for analysis and synthesis systems applied to subband coding of images. In recent years, time-varying filter banks that vary their structure appropriately for a local property of signals have been studied for application to the compression of image data. In time-varying filter banks, each different part of the signals is processed with different filter coefficients or different numbers of channels (a different sampling matrix). Those signals must be perfectly reconstructed even across transitions of the filter banks. In this paper, we consider time-varying systems of cosine-modulated 2D filter banks for arbitrary sampling lattices. We show perfect reconstruction (PR) conditions for time-varying filter banks that vary filter coefficients or a sampling matrix. Some applications are presented to show the effectiveness of this method. © 1997 Scripta Technica, Inc. Electron Comm Jpn Pt 3, 80(11): 46–54, 1997

Theory and design of two-dimensional filter Banks: A review

There has been considerable interest in the design of multidimensional (MD) filter banks. MD filter banks find application in subband coding of images and video data. MD filter banks can be designed by cascading one-dimensional (1D) filter banks in the form of a tree structure. In this case, the individual analysis and synthesis filters are separable and the filter bank is called a separable filter bank. MD filter banks with nonseparable filters offer more flexibility and usually provide better performance. Nonetheless, their design is considerably more difficult than separable filter banks. The purpose of this paper is to provide an overview of developments in this field on the design techniques for MD filter banks, mostly two-dimensional (2D) filter banks. In some image coding applications, the 2D two-channel filter banks are of great importance, particularly the filter bank with diamond-shaped filters. We will present several design techniques for the 2D two-channel nonseparable filter banks. As the design of MD filters are not as tractable as that of 1D filters, we seek design techniques that do not involve direct optimization of MD filters. To facilitate this, transformations that turn a separable MD filter bank into a nonseparable one are developed. Also, transformations of 1D filter banks to MD filter banks are investigated. We will review some designs of MD filter banks using transformations. In the context of 1D filter bank design, the cosine modulated filter bank (CMFB) is well-known for its design and implementation efficiency. All the analysis filters are cosine modulated versions of a prototype filter. The design cost of the filter bank is equivalent to that of the prototype and the implementation complexity is comparable to that of the prototype plus a low-complexity matrix. The success with 1D CMFB motivate the generalization to the 2D case. We will construct the 2D CMFB by following a very close analogy of 1D case. It is well-known that the 1D lossless systems can be characterized by state space description. In 1D, the connection between the losslessness of a transfer matrix and the unitariness of the realization matrix is well-established. We will present the developments on the study of 2D lossless systems. As in 1D case, the 2D FIR lossless systems can be characterized in terms of state space realizations. We will review this, and then address the factorizability of 2D FIR lossless systems by using the properties of state space realizations.

Two‐dimensional filter banks containing prefilter

AbstractTwo‐dimensional (2D) filter banks are used in subband coding of images, and are systems which split a signal into the narrow bands and/or resynthesize it. This paper presents a method of constructing 2D filter banks using 2D prefilter and 2D polyphase filter banks. First, the input signal is divided into four kinds of band on the 2D plane by 2D prefilter with IFIR (Interpolated FIR) transfer function. Then the 2D prefilter is designed so that the four kinds of band become 1. These resulting output signals are band‐split separately and synthesized by 2D DFT polyphase filter banks. Hence, its design method becomes very simple and both the required order and the number of multiplications can be reduced.