Sampled Filter Banks Research Articles

Directional mean curvature for textured image demixing

• A demixing method is proposed for decomposing blurred, textured images into cartoon, tex- ture and noise parts. • This high-order partial differential approach is defined by directional mean curvature , G-norm and curvelet space. • A connection between the proposed variational method and filter banks in harmonic analysis is set up. • This article addresses a fundamental problem in image processing: signal is sparse under some transformed domains. • Both theoretical results and examples with natural images are provided to illustrate its per- formance. Approximation theory plays an important role in image processing, especially image deconvolution and decomposition. For piecewise smooth images, there are many methods that have been developed over the past thirty years. The goal of this study is to devise similar and practical methodology for handling textured images. This problem is motivated by forensic imaging, since fingerprints, shoeprints and bullet ballistic evidence are textured images. In particular, it is known that texture information is almost destroyed by a blur operator, such as a blurred ballistic image captured from a low-cost microscope. The contribution of this work is twofold: first, we propose a mathematical model for textured image deconvolution and decomposition into four meaningful components, using a high-order partial differential equation approach based on the directional mean curvature. Second, we uncover a link between functional analysis and multiscale sampling theory, e.g., harmonic analysis and filter banks. Both theoretical results and examples with natural images are provided to illustrate the performance of the proposed model.

M-Channel Graph Filter Banks: Polyphase Analysis and Structures

Two channel critically sampled filter banks for signal over graph domains were first proposed for undirected bipartite graphs by Narang and Ortega. Extension to the M-channel critically sampled case for balanced M-block cyclic graphs was then proposed by Teke and Vaidynathan but the filter bank does not achieve strict perfect reconstruction (PR), only generalized PR. In this letter, we consider the more general case of filter banks on unbalanced M-block cyclic graphs where strict PR is achieved. A polyphase analysis to derive the implementation structures in the downsampled domain is presented here. The relevant system/filter matrices have interesting cyclic properties and projection operators are needed to map signals between subgraphs.

Scalable $M$-Channel Critically Sampled Filter Banks for Graph Signals

We investigate a scalable M-channel critically sampled filter bank for graph signals, where each of the M filters is supported on a different subband of the graph Laplacian spectrum. For analysis, the graph signal is filtered on each subband and downsampled on a corresponding set of vertices. However, the classical synthesis filters are replaced with interpolation operators. For small graphs, we use a full eigendecomposition of the graph Laplacian to partition the graph vertices such that the mth set comprises a uniqueness set for signals supported on the mth subband. The resulting transform is critically sampled, the dictionary atoms are orthogonal to those supported on different bands, and graph signals are perfectly reconstructable from their analysis coefficients. We also investigate fast versions of the proposed transform that scale efficiently for large, sparse graphs. Issues that arise in this context include designing the filter bank to be more amenable to polynomial approximation, estimating the number of samples required for each band, performing nonuniform random sampling for the filtered signals on each band, and using efficient reconstruction methods. We empirically explore the joint vertex-frequency localization of the dictionary atoms, the sparsity of the analysis coefficients for different classes of signals, the reconstruction error resulting from the numerical approximations, and the ability of the proposed transform to compress piecewise-smooth graph signals. The proposed filter bank also yields a fast, approximate graph Fourier transform with a coarse resolution in the spectral domain.

Almost Tight Spectral Graph Wavelets With Polynomial Filters

The construction of spectral filters for graph wavelet transforms is addressed in this paper. Both the undecimated and decimated cases will be considered. The filter functions are polynomials and can be implemented efficiently without the need for any eigendecomposition, which is computationally expensive for large graphs. Polynomial filters also have the advantage of the vertex localization property. The construction is achieved by designing suitable transformations that are used on traditional multirate filter banks. It will be shown how the classical quadrature-mirror-filters and linear phase, critically/over- sampled filter banks can be used to construct spectral graph wavelets that are almost tight. A variety of design examples will be given to show the versatility of the design technique.

LEAST MEAN SQUARED BACKGROUND CALIBRATION FOR OFDM MULTICHANNEL RECEIVERS

This paper presents a data estimation scheme for wide band multichannel charge sampling filter bank receivers together with a complete system calibration algorithm based on the least mean squared (LMS) algorithm. A unified model has been defined for the receiver containing all first order mismatches, offsets, imperfections, and the LMS algorithm is employed to track these errors. The performance of this technique under noisy channel conditions has been verified. Moreover, a detailed complexity analysis of the calibration algorithm is provided which shows that sinc filter banks have much lower complexity than traditional continuous-time filter banks.

DESIGN AND REALIZATION OF CIRCULAR CONTOURLET TRANSFORM

In this paper, circular contourlet transform (CCT) is proposed, designed and realized. As in the classical contourlet transform (CT), a double filter bank structure is also considered in this work but in different manners. A circularly-decomposed filter bank is first used to capture the points of discontinuities in the image edges, and then followed by a directional filter bank to obtain smoothed contours. The resulting CCT contains a critically sampled filter bank that decomposes images into any power of two's number of directional subbands at multiple scales. The designed CCT is realized by 2-D lattice allpass sections with separable and non-separable 2-D functions of z1 and z2. The resulting structure preserves both modularity and regularity properties which are suitable for VLSI implementations. Objectively, the performances of the realized CCT are tested and proved to be better than the classical CT in detail image preservation. The resulting subband images also indicate the superiority of the proposed CCT. Keywords: Circular contourlet transform, Contourlet transform, Laplacian pyramid, Directional filter bank, 2-D lattice allpass sections, Multiresolution (multiscale & multidirection) analysis.

Orthonormal FBs With Rational Sampling Factors and Oversampled DFT-Modulated FBs: A Connection and Filter Design

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Methods widely used to design filters for uniformly sampled filter banks (FBs) are not applicable for FBs with rational sampling factors and oversampled discrete Fourier transform (DFT)-modulated FBs. In this paper, we show that the filter design problem (with regularity factors/vanishing moments) for these two types of FBs is the same. Following this, we propose two finite-impulse-response (FIR) filter design methods for these FBs. The first method describes a parameterization of FBs with a single regularity factor/vanishing moment. The second method, which can be used to design FBs with an arbitrary number of regularity factors/vanishing moments, uses results from frame theory. We also describe how to modify this method so as to obtain linear phase filters. Finally, we discuss and provide a motivation for iterated DFT-modulated FBs. </para>

General design algorithm for sparse frame expansions

Signal expansions using frames may be considered as generalizations of signal representations based on transforms and filter banks. Frames, or dictionaries, for sparse signal representations may be designed using an iterative algorithm with two main steps: (1) Frame vector selection and expansion coefficient determination for signals in a training set, selected to be representative of the signals for which compact representations are desired, using the frame designed in the previous iteration. (2) Update of frame vectors with the objective of improving the representation of step (1). This method for frame design was used by [Engan et al., Signal Processing 80 (2000) 2121-2140] for block-oriented signal expansions, i.e. generalizations of block-oriented transforms and by [Aase et al., IEEE Trans. Signal Process. 49(5) (2001) 1087-1096] for non-block-oriented frames--for short overlapping frames, that may be viewed as generalizations of critically sampled filter banks. Here we give the solution to the general frame design problem using the compact notation of linear algebra. This makes the solution both conceptually and computationally easier, especially for the overlapping frame case. Also, the solution is more general than those presented earlier, facilitating the imposition of constraints, such as symmetry, on the designed frame vectors.

Overlap-save convolution applied to wavelet analysis

The discrete wavelet transform (DWT), wavelet packet transform (WPT), and M-band wavelet techniques are often implemented as a cascade of critically sampled filter banks. Many applications apply these techniques to finite frames of samples, leading to distortion in the filtered coefficients near the frame boundaries. Overlap-save convolution (OSC) eliminates boundary distortion for a single filtering process. This article reports an application of OSC for cascaded filter banks, eliminating boundary distortion in the frame-based application of the DWT, WPT, and M-band wavelet techniques.

Rational sampling filter banks based on IIR filters

The problem of splitting the spectrum of a digital signal by using nonuniform infinite impulse response (IIR) filter banks is addressed. Near perfect reconstruction (NPR) is considered. The method uses the modulation of different IIR prototypes. The cancellation of the main aliasing components constrains the prototypes to be dependent on each other. By using this approach, linear-phase prototypes are needed, and noncausal filtering is required. Numerical examples of filter bank design are given, and the computational complexity is compared with the finite impulse response (FIR) case.

Frequency-warped filter banks and wavelet transforms: a discrete-time approach via Laguerre expansion

We introduce a new generation of perfect-reconstruction filter banks that can be obtained from classical critically sampled filter banks by means of frequency transformations. The novel filters are Laguerre type IIR filters that can be directly derived and designed from ordinary orthogonal or biorthogonal filter banks. Generalized downsampling and upsampling operators based on dispersive delay lines are the building blocks of our structures. By iterating the filter banks, we construct new orthogonal and complete sets of wavelets whose passbands are not octave spaced and may be designed by selecting a single parameter.

A method of designing filter banks with alias‐free characteristics at equally spaced frequency points

AbstractIt is known that the quadrature mirror filter (QMF) bank used for subband coding cannot avoid the aliasing effect for adaptive signal processing or spectral estimation, etc. A new class of filter banks is desired for these applications where the effect of aliasing should also be considered.In this paper, a new class of filter bank is reported which has alias‐free characteristics at equally spaced frequency points. This filter bank is called frequency sampling filter bank, and it consists of nonrecursive FIR filters. This paper also reports design methods for two types of the filter banks, keeping in mind the reduction in amount of calculation for practical implementation. In one method, all channel filters of the filter bank are determined by complex exponential modulation of a prototype filter. the other method determines the channel filters by cosine or sine modulation of a prototype filter. Both designs have the forementioned aliasing characteristics, and an efficient structure based on polyphase configuration is possible.

Perfect reconstruction with critically sampled filter banks and linear boundary conditions

This work is concerned with the (image) boundary conditions involved in processing a finite discrete-time signal with a critically sampled perfect reconstruction filter bank. It is desirable that the boundary conditions reduce edge effects and define a transformation into a space having the same dimensionality as the original signal. The complication that arises is in the computation of the inverse transform. Although it is straightforward to reconstruct the signal values that were not influenced by the boundary conditions, recovering those values on the boundaries is nontrivial. The solution of this problem is discussed for general linear boundary conditions. No symmetry assumptions are made on the boundary conditions or on the impulse responses of the analysis filters. A low-rank linear transform is derived that expresses the boundary values in terms of the transform coefficients, which in turn provides a method for inverting the subband decomposition. The application of the results in the case of two-channel orthonormal wavelet filters is discussed, and the effects of the filter support on the conditioning of the inverse problem are investigated.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A subband adaptive filter allowing maximally decimation

Conventional subband adaptive filters (ADF) using filter banks have shown degradation in performance because of the nonideal nature of filter banks. For this problem, the authors propose an alias free subband structure for adaptive filtering using polyphase frequency sampling filter (FSF) banks. As a preliminary, they make it clear that the conventional polyphase discrete Fourier transform (DFT) bank is equivalent to a class of the FSF bank. Then, they propose a new class of ADF using the FSF banks based on the frequency sampling theorem. As a result, the proposed technique enables subband adaptive filtering without the degradation effect of both the aliasing and cross terms, even if one chooses critical subsampling.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

LINC: a common theory of transform and subband coding

A common theory of lapped orthogonal transforms (LOTs) and critically sampled filter banks, called L into N coding (LINC), is presented. The theory includes a unified analysis of both coding methods and identity relations between the transform, inverse transform, analysis filter bank, and synthesis filter bank. A design procedure for LINC analysis/synthesis systems, which satisfy the conditions for perfect reconstruction, is developed. The common LINC theory is used to define an ideal LINC system which is used, together with the power spectral density of the input signal, to calculate theoretical bounds for the coding gain. A generalized overlapping block transform (OBT) with time domain aliasing cancellation (TDAC) is used to approximate the ideal LINC. A generalization of the OBT includes multiple block overlap and additional windowing. A recursive design procedure for windows of arbitrary lengths is presented. The coding gain of the generalized OBT is higher than that of the Karhunen-Loeve transform (KLT) and close to the theoretical bounds for LINC. In the case of image coding, the generalized OBT reduces the blocking effects when compared with the DCT. >

Application of the Princen-Bradley filter bank to speech and image compression

The authors provide an efficient fast Fourier transform implementation, a set of optimal windows, and a new proof of the perfect reconstruction property for a critically sampled filter bank introduced by Princen and Bradley. Based on the new proof, necessary conditions for perfect reconstruction are also derived for the oversampled case. This filter bank is well suited to compression of speech and images due to its good frequency resolution, overlapped output, computational efficiency, and low delay. The authors evaluated its performance in a 32 band, 16-kb/s adaptive speech compression system, and found both its objective and subjective performance to be comparable to a more complex quadrature mirror filter bank. In image coding experiments, the Princen-Bradley filter bank demonstrated quality similar to DCT-based systems at high rates and reduced blocking artifacts at rates of 0.25-0.5 bpp. A psychophysically based bit allocation algorithm that provides a significant perceptual improvement over a related algorithm that seeks to minimize the mean-square error is introduced.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>