Filter Bank Algorithm Research Articles

Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-Like Transforms

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> We propose a novel method for constructing Hilbert transform (HT) pairs of wavelet bases based on a fundamental approximation-theoretic characterization of scaling functions—the B-spline factorization theorem. In particular, starting from well-localized scaling functions, we construct HT pairs of biorthogonal wavelet bases of <formula formulatype="inline"><tex Notation="TeX">${\rm L}^2({\BBR})$</tex> </formula> by relating the corresponding wavelet filters via a discrete form of the continuous HT filter. As a concrete application of this methodology, we identify HT pairs of spline wavelets of a specific flavor, which are then combined to realize a family of complex wavelets that resemble the optimally-localized Gabor function for sufficiently large orders. Analytic wavelets, derived from the complexification of HT wavelet pairs, exhibit a one-sided spectrum. Based on the tensor-product of such analytic wavelets, and, in effect, by appropriately combining four separable biorthogonal wavelet bases of <formula formulatype="inline"> <tex Notation="TeX">${\rm L}^2({\BBR}^2)$</tex></formula>, we then discuss a methodology for constructing 2-D directional-selective complex wavelets. In particular, analogous to the HT correspondence between the components of the 1-D counterpart, we relate the real and imaginary components of these complex wavelets using a multidimensional extension of the HT—the directional HT. Next, we construct a family of complex spline wavelets that resemble the directional Gabor functions proposed by Daugman. Finally, we present an efficient fast Fourier transform (FFT)-based filterbank algorithm for implementing the associated complex wavelet transform. </para>

Directional Haar wavelet frames on triangles

Traditional wavelets are not very effective in dealing with images that contain orientated discontinuities (edges). To achieve a more efficient representation one has to use basis elements with much higher directional sensitivity. In recent years several approaches like curvelets and shearlets have been studied providing essentially optimal approximation properties for images that are piecewise smooth and have discontinuities along C 2 -curves. While curvelets and shearlets have compact support in frequency domain, we construct directional wavelet frames generated by functions with compact support in time domain. Our Haar wavelet constructions can be seen as special composite dilation wavelets, being based on a generalized multiresolution analysis (MRA) associated with a dilation matrix and a finite collection of ‘shear’ matrices. The complete system of constructed wavelet functions forms a Parseval frame. Based on this MRA structure we provide an efficient filter bank algorithm. The freedom obtained by the redundancy of the applied Haar functions will be used for an efficient sparse representation of piecewise constant images as well as for image denoising.

The Pairing of a Wavelet Basis With a Mildly Redundant Analysis via Subband Regression

A distinction is usually made between wavelet bases and wavelet frames. The former are associated with a one-to-one representation of signals, which is somewhat constrained but most efficient computationally. The latter are over-complete, but they offer advantages in terms of flexibility (shape of the basis functions) and shift-invariance. In this paper, we propose a framework for improved wavelet analysis based on an appropriate pairing of a wavelet basis with a mildly redundant version of itself (frame). The processing is accomplished in four steps: 1) redundant wavelet analysis, 2) wavelet-domain processing, 3) projection of the results onto the wavelet basis, and 4) reconstruction of the signal from its nonredundant wavelet expansion. The wavelet analysis is pyramid-like and is obtained by simple modification of Mallat's filterbank algorithm (e.g., suppression of the down-sampling in the wavelet channels only). The key component of the method is the subband regression filter (Step 3) which computes a wavelet expansion that is maximally consistent in the least squares sense with the redundant wavelet analysis. We demonstrate that this approach significantly improves the performance of soft-threshold wavelet denoising with a moderate increase in computational cost. We also show that the analysis filters in the proposed framework can be adjusted for improved feature detection; in particular, a new quincunx Mexican-hat-like wavelet transform that is fully reversible and essentially behaves the (gamma/2)th Laplacian of a Gaussian.

Symmetric extension for lifted filter banks and obstructions to reversible implementation

Symmetric pre-extension is a standard approach to boundary handling for finite-length input vectors with linear phase filter banks. It works with both conventional linear implementations and the so-called reversible, or integer-to-integer, implementations of odd-length linear phase ( whole-sample symmetric) filter banks. In comparison, significant difficulties arise when using symmetric pre-extension on reversible filter banks with even-length ( half-sample symmetric) linear phase filters. An alternative approach is presented using lifting step extension, in which boundary extensions are performed in each step of a lifting factorization, avoiding some of these difficulties while preserving reversibility and retaining the nonexpansive property of symmetric pre-extension. Another alternative that is capable of preserving both reversibility and subband symmetry for half-sample symmetric filter banks is developed based on ideas from the theory of lattice vector quantization. The practical ramifications of this work are illustrated by describing its influence on the specification of filter bank algorithms in Part 2 of the ISO/IEC JPEG 2000 image coding standard.

From differential equations to the construction of new wavelet-like bases

In this paper, an approach is introduced based on differential operators to construct wavelet-like basis functions. Given a differential operator L with rational transfer function, elementary building blocks are obtained that are shifted replicates of the Green's function of L. It is shown that these can be used to specify a sequence of embedded spline spaces that admit a hierarchical exponential B-spline representation. The corresponding B-splines are entirely specified by their poles and zeros; they are compactly supported, have an explicit analytical form, and generate multiresolution Riesz bases. Moreover, they satisfy generalized refinement equations with a scale-dependent filter and lead to a representation that is dense in L/sub 2/. This allows us to specify a corresponding family of semi-orthogonal exponential spline wavelets, which provides a major extension of earlier polynomial spline constructions. These wavelets are completely characterized, and it is proven that they satisfy the following remarkable properties: 1) they are orthogonal across scales and generate Riesz bases at each resolution level; 2) they yield unconditional bases of L/sub 2/-either compactly supported (B-spline-type) or with exponential decay (orthogonal or dual-type); 3) they have N vanishing exponential moments, where N is the order of the differential operator; 4) they behave like multiresolution versions of the operator L from which they are derived; and 5) their order of approximation is (N-M), where N and M give the number of poles and zeros, respectively. Last but not least, the new wavelet-like decompositions are as computationally efficient as the classical ones. They are computed using an adapted version of Mallat's filter bank algorithm, where the filters depend on the decomposition level.

The contourlet transform: an efficient directional multiresolution image representation

The limitations of commonly used separable extensions of one-dimensional transforms, such as the Fourier and wavelet transforms, in capturing the geometry of image edges are well known. In this paper, we pursue a "true" two-dimensional transform that can capture the intrinsic geometrical structure that is key in visual information. The main challenge in exploring geometry in images comes from the discrete nature of the data. Thus, unlike other approaches, such as curvelets, that first develop a transform in the continuous domain and then discretize for sampled data, our approach starts with a discrete-domain construction and then studies its convergence to an expansion in the continuous domain. Specifically, we construct a discrete-domain multiresolution and multidirection expansion using nonseparable filter banks, in much the same way that wavelets were derived from filter banks. This construction results in a flexible multiresolution, local, and directional image expansion using contour segments, and, thus, it is named the contourlet transform. The discrete contourlet transform has a fast iterated filter bank algorithm that requires an order N operations for N-pixel images. Furthermore, we establish a precise link between the developed filter bank and the associated continuous-domain contourlet expansion via a directional multiresolution analysis framework. We show that with parabolic scaling and sufficient directional vanishing moments, contourlets achieve the optimal approximation rate for piecewise smooth functions with discontinuities along twice continuously differentiable curves. Finally, we show some numerical experiments demonstrating the potential of contourlets in several image processing applications. Index Terms-Contourlets, contours, filter banks, geometric image processing, multidirection, multiresolution, sparse representation, wavelets.

Quantitative real-time blood flow estimation with intravascular ultrasound in the presence of in-plane flow

Previously, we showed a source of error in blood flow estimation introduced by in-plane flow using a slow-time finite-impulse response (FIR) filter-bank method measuring blood flow through the image plane of an intravascular ultrasound (IVUS) catheter array. There is a monotonic relationship between flow velocity and the normalized second moment of the slow-time spectrum when flow is orthogonal to the image plane of a side-looking catheter array. However, this relationship changes in the presence of in-plane flow, as slow-time spectra shift and spread with varying in-plane and out-of-plane components. These two effects increase the normalized spectral second moment, resulting in flow overestimates. However, by resampling the received signal with variable time delay from pulse to pulse (i.e., tilting the slow-time signals), the slow-time spectrum shifts back to direct current (DC), and the orthogonal estimation method can be used. We present a method to correct this overestimation and accurately estimate blood flow through the image plane in real time. Initially, the tilt delay needed to shift the slow-time spectrum back to DC at each point within the flow field is calculated. Knowing this tilt delay, a tilted slow-time signal is obtained for the velocity component normal to the image plane, and its spectrum is estimated using a filter-bank. That spectrum then is used to estimate the flow speed using a mapping function closely related to the monotonic relationship between the slow-time spectrum and flow speed observed for orthogonal flow. To accurately estimate flow angles, we modified the filter-bank algorithm, applying slow-time filter coefficients in a tilted arrangement and studying the slow-time spectral energy as a function of tilt. The slow-time spectral estimate is constructed with the tilted output of eight narrow, band-pass filters from a filter-bank. Independent simulations show that, for blood slowing at angles between +/-6 degrees and +/-15 degrees at a speed of 300 mm/s, flow velocity would be overestimated by as much as 38.79% and 249%, respectively, using the direct filter-bank approach. However, this error can be corrected using the modified method presented here, reducing the maximum overestimation error by a factor of 2.69 and 10.88 for those angles, respectively. Although the remaining error is not negligible, the volume flow rate, calculated by integrating the flow velocity over the entire vessel lumen, differs by only 3% or less from the true value over the angular range considered here. This represents an improvement of a factor of 40 over uncompensated estimates at maximum flow angles. Consequently, the modified real-time method can quantitatively measure flow in most IVUS applications in which the catheter's image plane is not precisely orthogonal to the flow direction.

Constituent subband allocation for system modeling nonuniform subband adaptive filters

Subband adaptive filters (SAFs) have gained widespread use in key applications such as echo cancellation, system identification, and equalization. SAFs are normally constrained to utilize uniform filterbanks. However, it has been recently proposed that nonuniform filterbanks can give additional flexibility to SAFs that may be used to increase performance. A rigorous mean square error (MSE) analysis has also been completed for standard system modeling nonuniform SAF configurations. This paper completes the next step by theoretically deriving an adaptive algorithm to control the nonuniform filterbank for a system modeling SAF. The algorithm is constructed so that MSE, convergence time, and computational complexity are minimized. Two numerical simulations compare the nonuniform SAF adaptive filterbank algorithm to standard, equivalent uniform SAF performance.

Conditioning of Distorted Nonlinear Sensor Measurements in Non-Stationary Noisy Environments

In this paper, an effective signal reconstruction method is proposed for signals that are distorted by sensor nonlinearity under non-stationary noisy environments. When a bandlimited signal is measured through a nonlinear sensor, the high amplitude part of a signal will be distorted by sensor nonlinearity and the low amplitude part of the signal may be indistinguishable from sensor noise. A distortion compensation technique that includes efficient noise removal procedures is necessary in order to reconstruct the true sensor input from the distorted and possibly corrupted sensor output. It is shown that the effects of sensor nonlinearity may be effectively compensated by a few step recursive signal recovery scheme and the denoising performance of this signal recovery method can be increased by incorporating a wavelet based denoising procedure. In order to treat sensor measurement problems that involves non-stationary noise, the traditional techniques such as low pass filtering may not be efficient. It is shown that wavelet filter bank algorithms can be utilized to achieve a better noise performance and a more accurate signal reconstruction. Simulation and experimental results are presented to validate the proposed recovery and denoising schemes.

Blood flow estimation error with intravascular ultrasound due to in-plane component of flow.

Previously, we presented a real-time method to measure blood flow perpendicular to the image plane of an intravascular ultrasound (IVUS) imaging system using a slow-time FIR (finite impulse response) filter bank. Any in-plane flow introduces error in the flow measurement using the filter bank algorithm. Simulations show that for a flow angle of +/- 10 degrees and velocities between 200 mm/s and 300 mm/s, the energy within the lowest frequency band filter is 6.92 to 7.80 times higher than for perpendicular flow in the worst case. We present a variation of the FIR filter bank algorithm, applying filter coefficients in a tilted fashion to slow-time signals (i.e., combining slow-time and fast-time). An appropriate tilt, which depends on the flow angle and velocity, corrects for the increased energy under the frequency bands. In this case, the energy under the lowest frequency band filter for an angle of +/- 10 degrees and velocities ranging from 200 mm/s to 300 mm/s is 2.09 to 2.94 times higher than for perpendicular flow, yielding greater than a factor of three improvement in the worst case over the original slow-time method. Moreover, the average energy over the vessel determined with the appropriate tilt is within 2-3% of the true value.

In-Place Computation of Block Discrete Wavelet Transform

A Novel algorithm for in-place computation of Discrete Wavelet Transform (DWT) and its inverse (IDWT), suitable for real-time and adaptive processing, is presented. The scheme presented here, reduces the memory requirement and latency of the filter bank algorithm by incorporating the principle of block convolution into DWT computation.

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.

Filter bank algorithms for piecewise linear prewavelets on arbitrary triangulations

This paper studies algorithms for decomposition, reconstruction, and approximation based on piecewise linear prewavelets on bounded triangulations of arbitrary topology. Our key mathematical result is showing that the Schur complement of the associated two scale matrix is symmetric, positive definite, and well conditioned. Numerical examples suggest that thresholding based on prewavelets yields a smaller approximation error than when based on the simple ‘Faber’ decomposition scheme.

A new design algorithm for two-band orthonormal rational filter banks and orthonormal rational wavelets

We present a new algorithm for the design of orthonormal two-band rational filter banks. Owing to the connection between iterated rational filter banks and rational wavelets, this is also a design algorithm for orthonormal rational wavelets. It is basically a simple iterative procedure, which explains its exponential convergence and adaptability under various linear constraints (e,g., regularity). Although the filters obtained from this algorithm are suboptimally designed, they show excellent frequency selectivity. After an in-depth account of the algorithm, we discuss the properties of the rational wavelets generated by some designed filters. In particular, we stress the possibility to design "almost" shift error-free wavelets, which allows the implementation of a rational wavelet transform.

A generalized algorithm for linear-phase paraunitary filter banks

A new algorithm for a subclass of linear-phase paraunitary filter banks with generalized filter length and symmetry polarity is reported. New properties in the polyphase matrix are derived, and the lattice factorizations for filter banks with an even and odd number of channels are examined in a generalized algorithm.

On the design of FIR analysis-synthesis filter banks with high computational efficiency

Presents an effective design algorithm for analysis-synthesis filter banks with computationally efficient structures. Although a wide variety of implementation structures can be accommodated, the focus of the paper is on cosine modulated filter banks. The design procedure is based on a time domain formulation of analysis-synthesis filter banks in which each individual channel filter is constrained to be a cosine modulated versions of a baseband filter. The resulting filter banks are very efficient in terms of computational requirements and are relatively easy to design. A unique feature of this approach is that relatively low reconstruction delays can be imposed on the system. A discussion of the associated computational properties of the designed systems and some design examples are included.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>