This paper proposes an epigraphically-relaxed linearly-involved generalized Moreau-enhanced (ER-LiGME) model for layered mixed norm regularization. Group sparse and low-rank (GSpLr)-aware modeling using ℓ1/nuclear-norm-based layered mixed norms has succeeded in precise high dimensional signal recovery, e.g., images and videos. Our previous work significantly expands the potential of the GSpLr-aware modeling by epigraphical relaxation (ER). It enables us to handle a (even non-proximable) deeply-layered mixed norm minimization by decoupling it into a norm and multiple epigraphical constraints (if each proximity operator is available). One problem with typical SpLr modeling is that it suffers from the underestimation effect due to the ℓ1and nuclear norm regularization. To circumvent this problem, LiGME penalty functions, which modify conventional sparsity and low-rankness promoting convex functions to nonconvex ones while keeping overall convexity, have been proposed conventionally. In this work, we integrate the ER technique with the LiGME model to realize deeply-layered (possibly non-proximable) mixed norm regularization and show its effectiveness in denoising and compressed sensing reconstruction.

The past few years have seen a surge of activity around integration of deep learning networks and optimization algorithms for solving inverse problems. Recent work on plug-and-play priors (PnP), regularization by denoising (RED), and deep unfolding has shown the state-of-the-art performance of such integration in a variety of applications. However, the current paradigm for designing such algorithms is inherently Euclidean, due to the usage of the quadratic norm within the projection and proximal operators. We propose to broaden this perspective by considering a non-Euclidean setting based on the more general Bregman distance. Our new Bregman Proximal Gradient Method variant of PnP (PnP-BPGM) and Bregman Steepest Descent variant of RED (RED-BSD) replace the traditional updates in PnP and RED from the quadratic norms to more general Bregman distance. We present a theoretical convergence result for PnP-BPGM and demonstrate the effectiveness of our algorithms on Poisson linear inverse problems.

Convolutional neural networks (CNNs) provide impressive empirical success in various tasks; however, their inner workings generally lack interpretability. In this paper, we interpret shallow CNNs that we have trained for the task of positive sparse signal denoising. We identify and analyze common structures among the trained CNNs. We show that the learned CNN denoisers can be interpreted as a nonlinear locally-adaptive thresholding procedure, which is an empirical approximation of the minimum mean square error estimator. Based on our interpretation, we train constrained CNN denoisers and demonstrate no loss in performance despite having fewer trainable parameters. The interpreted CNN denoiser is an instance of a multivariate spline regression model, and a generalization of classical proximal thresholding operators.

This paper presents a new approach for reconstruction of disconnected digital lines (DDLs) based on a constrained regularization model which ensures connectivity of the digital lines (DLs) in the discrete image plane. The first step in this approach is to determine the order of given pixels of the DDL. To determine connectivity of pixels, we use the usual 8-neighbor connectivity in discrete images. For any neighboring pixels of the DDL that are not connected, we determine a number of new pixel values that need to be reconstructed between these pixels. Next, the integer-valued x - and y -coordinates of the location of the pixels of the DDLs are segregated into two 1D signal vectors. Then the x - and y -coordinates of the missing pixels of the DDLs are estimated using a new constrained regularization. While the solution of this constrained minimization problem provides real values for the x - and y -coordinates of pixels positions, the imposed constraint ensures connectivity of the resulting DLs in the image plane after transforming the computed values from [Formula: see text] to [Formula: see text]. The proposed regularization approach forces connected lines with small curvature. The experimental results demonstrate that the proposed technique improves DL intersection detection, as well. Moreover, this technique has a high potential to be used as a fast approach in binary image inpainting particularly overcoming the shortcomings of conventional methods which cause destruction of thin objects and blurring in the recovered regions.

Fused lasso norm is classically adopted to model sparse piecewise constant signals, however it is not the convex hull of the best representation of such simultaneously structured signal. In this paper, we propose a convex variational norm for better modeling sparse piecewise constant signals. The norm is based on (1) promoting sparsity in first-order difference with total variation norm and (2) exploiting latent group structure in first-order difference with simple linear constraints. We demonstrate the proposed norm outperforms fused lasso norm in a denoising setup with numerical experiments.

This paper proposes an epigraphical reformulation (ER) technique for non-proximable mixed norm regularization. Various regularization methods using mixed norms have been proposed, where their optimization relies on efficient computation of the proximity operator of the mixed norms. Although the sophisticated design of mixed norms significantly improves the performance of regularization, the proximity operator of such a mixed norm is often unavailable. Our ER decouples a non-proximable mixed norm function into a proximable norm and epigraphical constraints. Thus, it can handle a wide range of non-proximable mixed norms as long as the proximal operator of the outermost norm, and the projection onto the epigraphical constraints can be efficiently computed. Moreover, we prove that our ER does not change the minimizer of the original problem despite using a certain inequality approximation. We also provide a new structure-tensor-based regularization as an application of our framework, which illustrates the utility of ER.

In this paper, a multivariate statistical model that is suitable for describing Optical Coherence Tomography (OCT) images is introduced. The proposed model is comprised of a multivariate Gaussianization function in sparse domain. Such an approach has two advantages, i.e. 1) finding a function that can effectively transform the input - which is often not Gaussian - into normally distributed samples enables the reliable application of methods that assume Gaussianity, 2) although multivariate Gaussianization in spatial domain is a complicated task and rarely results in closed-form analytical model, by transferring data to sparse domain, our approach facilitates multivariate statistical modeling of OCT images. To this end, a proper multivariate probability density function (pdf) which considers all three properties of OCT images in sparse domains (i.e. compression, clustering, and persistence properties) is designed and the proposed sparse domain Gaussianization framework is established. Using this multivariate model, we show that the OCT images often follow a 2-component multivariate Laplace mixture model in the sparse domain. To evaluate the performance of the proposed model, it is employed for OCT image denoising in a Bayesian framework. Visual and numerical comparison with previous prominent methods reveals that our method improves the overall contrast of the image, preserves edges, suppresses background noise to a desirable amount, but is less capable of maintaining tissue texture. As a result, this method is suitable for applications where edge preservation is crucial, and a clean noiseless image is desired.

Non-convex penalties outperform the convex ℓ-norm, but generally sacrifice the cost function convexity. As a middle ground, we propose a framework to design non-convex penalties that induce sparsity more effectively than the ℓ-norm, but without sacrificing the cost function convexity. The non-smooth non-convex regularizers are constructed by subtracting from the non-smooth convex penalty its smoothed version. We propose a generalized infimal convolution smoothing smoothing technique to obtain the smoothed version. We call the proposed framework sharpening sparse regularizers (SSR) to imply its advantages compared to convex and non-convex regularizers. The SSR framework is applicable to any sparsity regularized ill-posed linear inverse problem. Furthermore, it recovers and generalizes several non-convex penalties in the literature as special cases. The SSR-RLS problem can be formulated as a saddle point problem, and solved by a scalable generalized primal-dual algorithm. The effectiveness of the SSR framework is demonstrated by numerical experiments.

A popular strategy for determining solutions to linear least-squares problems relies on using sparsity-promoting regularizers and is widely exploited in image processing applications such as, e.g., image denoising, deblurring, and inpainting. It is well known that, in general, nonconvex regularizers hold the potential for promoting sparsity more effectively than convex regularizers such as, e.g., those involving the $\ell_1$ norm. To avoid the intrinsic difficulties related to non-convex optimization, the convex nonconvex (CNC) strategy has been proposed, which allows the use of nonconvex regularization while maintaining convexity of the total objective function. In this paper, a new CNC variational model is proposed, based on a more general parametric nonconvex nonseparable regularizer. The proposed model is applicable to a greater variety of image processing problems than prior CNC methods. We derive the convexity conditions and related theoretical properties of the presented CNC model, and we analyze existence and uniqueness of its solutions. A primal-dual forward-backward splitting algorithm is proposed for solving the related saddle-point problem. The convergence of the algorithm is demonstrated theoretically and validated empirically. Several numerical experiments are presented which prove the effectiveness of the proposed approach.

The L1 norm is often used as a penalty function to obtain a sparse approximate solution to a system of linear equations, but it often underestimates the true values. This paper proposes a different type of penalty that (1) estimates sparse solutions more accurately and (2) maintains the convexity of the cost function. The new penalty is a multivariate generalization of the minimax-concave (MC) penalty. To define the generalized MC (GMC) penalty we first define a multivariate generalized Huber function. The resulting cost function can be minimized by proximal algorithms comprising simple computations. The effectiveness of the GMC penalty is illustrated in a denoising example.

This paper proposes an improved formulation of sparsity-assisted signal smoothing (SASS). The purpose of SASS is to filter/denoise a signal that has jump discontinuities in its derivative (of some designated order) but is otherwise smooth. SASS unifies conventional low-pass filtering and total variation denoising. The SASS algorithm depends on the formulation, in terms of banded Toeplitz matrices, of a zero-phase recursive discrete-time filter as applied to finite-length data. The improved formulation presented in this paper avoids the unwanted end-point transient artifacts which sometimes occur in the original version. For illustration, SASS is applied to ECG signal denoising.

Total variation denoising is a nonlinear filtering method well suited for the estimation of piecewise-constant signals observed in additive white Gaussian noise. The method is defined by the minimization of a particular nondifferentiable convex cost function. This letter describes a generalization of this cost function that can yield more accurate estimation of piecewise constant signals. The new cost function involves a nonconvex penalty (regularizer) designed to maintain the convexity of the cost function. The new penalty is based on the Moreau envelope. The proposed total variation denoising method can be implemented using forward-backward splitting.

This letter proposes to estimate low-rank matrices by formulating a convex optimization problem with nonconvex regularization. We employ parameterized nonconvex penalty functions to estimate the nonzero singular values more accurately than the nuclear norm. A closed-form solution for the global optimum of the proposed objective function (sum of data fidelity and the nonconvex regularizer) is also derived. The solution reduces to singular value thresholding method as a special case. The proposed method is demonstrated for image denoising.

This paper addresses the mitigation of wind turbine clutter (WTC) in weather radar data in order to increase the performance of existing weather radar systems and to improve weather analyses and forecasts. We propose a novel approach for this problem based on signal separation algorithms. We model the weather signal as group sparse in the time-frequency domain; in parallel, we model the WTC signal as having a sparse time derivative. In order to separate WTC and the desired weather returns, we formulate the signal separation problem as an optimization problem. The objective function to be minimized combines total variation regularization and time-frequency group sparsity. We also propose a three-window short-time Fourier transform for the time-frequency representation of the weather signal. To show the effectiveness of the proposed algorithm on weather radar systems, the method is applied to simulated and real data from the next-generation weather radar network. Significant improvements are observed in reflectivity, spectral width, and angular velocity estimates.

Algorithms for signal denoising that combine wavelet-domain sparsity and total variation (TV) regularization are relatively free of artifacts, such as pseudo-Gibbs oscillations, normally introduced by pure wavelet thresholding. This paper formulates wavelet-TV (WATV) denoising as a unified problem. To strongly induce wavelet sparsity, the proposed approach uses non-convex penalty functions. At the same time, in order to draw on the advantages of convex optimization (unique minimum, reliable algorithms, simplified regularization parameter selection), the non-convex penalties are chosen so as to ensure the convexity of the total objective function. A computationally efficient, fast converging algorithm is derived.

This letter considers the problem of signal denoising using a sparse tight-frame analysis prior. The l1 norm has been extensively used as a regularizer to promote sparsity; however, it tends to under-estimate non-zero values of the underlying signal. To more accurately estimate non-zero values, we propose the use of a non-convex regularizer, chosen so as to ensure convexity of the objective function. The convexity of the objective function is ensured by constraining the parameter of the non-convex penalty. We use ADMM to obtain a solution and show how to guarantee that ADMM converges to the global optimum of the objective function. We illustrate the proposed method for 1D and 2D signal denoising.

This letter formulates a convex generalized total variation functional for the estimation of discontinuous piecewise linear signals from corrupted data. The method is based on (1) promoting pairwise group sparsity of the second derivative signal and (2) decoupling the principle knot parameters so they can be separately weighted. The proposed method refines the recent approach by Ongie and Jacob.

Total variation (TV) denoising is an effective noise suppression method when the derivative of the underlying signal is known to be sparse. TV denoising is defined in terms of a convex optimization problem involving a quadratic data fidelity term and a convex regularization term. A non-convex regularizer can promote sparsity more strongly, but generally leads to a non-convex optimization problem with non-optimal local minima. This letter proposes the use of a non-convex regularizer constrained so that the total objective function to be minimized maintains its convexity. Conditions for a non-convex regularizer are given that ensure the total TV denoising objective function is convex. An efficient algorithm is given for the resulting problem.

We propose a novel formulation for relaxed analysis-based sparsity in multiple dictionaries as a general type of prior for images, and apply it for Bayesian estimation in image restoration problems. Our formulation of a ℓ2-relaxed ℓ0 pseudo-norm prior allows for an especially simple maximum a posteriori estimation iterative marginal optimization algorithm, whose convergence we prove. We achieve a significant speedup over the direct (static) solution by using dynamically evolving parameters through the estimation loop. As an added heuristic twist, we fix in advance the number of iterations, and then empirically optimize the involved parameters according to two performance benchmarks. The resulting constrained dynamic method is not just fast and effective, it is also highly robust and flexible. First, it is able to provide an outstanding tradeoff between computational load and performance, in visual and objective, mean square error and structural similarity terms, for a large variety of degradation tests, using the same set of parameter values for all tests. Second, the performance benchmark can be easily adapted to specific types of degradation, image classes, and even performance criteria. Third, it allows for using simultaneously several dictionaries with complementary features. This unique combination makes ours a highly practical deconvolution method.

In this paper, we discuss about applications of different methods for decomposing a signal over elementary waveforms chosen in a family called a dictionary (atomic representations) in optical coherence tomography (OCT). If the representation is learned from the data, a nonparametric dictionary is defined with three fundamental properties of being data-driven, applicability on 3D, and working in multi-scale, which make it appropriate for processing of OCT images. We discuss about application of such representations including complex wavelet based K-SVD, and diffusion wavelets on OCT data. We introduce complex wavelet based K-SVD to take advantage of adaptability in dictionary learning methods to improve the performance of simple dual tree complex wavelets in speckle reduction of OCT datasets in 2D and 3D. The algorithm is evaluated on 144 randomly selected slices from twelve 3D OCTs taken by Topcon 3D OCT-1000 and Cirrus Zeiss Meditec. Improvement of contrast to noise ratio (CNR) (from 0.9 to 11.91 and from 3.09 to 88.9, respectively) is achieved. Furthermore, two approaches are proposed for image segmentation using diffusion. The first method is designing a competition between extended basis functions at each level and the second approach is defining a new distance for each level and clustering based on such distances. A combined algorithm, based on these two methods is then proposed for segmentation of retinal OCTs, which is able to localize 12 boundaries with unsigned border positioning error of 9.22 ±3.05 μm, on a test set of 20 slices selected from 13 3D OCTs.

The fused lasso problem involves the minimization of the sum of a quadratic, a TV term and an ℓ1term. The solution can be obtained by applying a TV denoising filter followed by soft-thresholding. However, soft-thresholding introduces a certain bias to the non-zero coefficients. In order to prevent this bias, we propose to replace the ℓ1penalty with a non-convex penalty. We show that the solution can similarly be obtained by applying a modified thresholding function to the result of the TV-denoising filter.

This study investigates the feasibility of using binaural dichotic presentation of speech components decomposed using a recently proposed resonance-based decomposition method to release listeners from intra-speech masking and yield better perceived sound quality. Resonance-based decomposition is a nonlinear signal analysis method based not on frequency or scale but on resonance. We decomposed different categories of speech stimuli (vowels, consonants, and sentences) into low- and high-resonance component using various combination of low- and high-Q-factors {Q1, Q2}. 10 normal hearing listeners were asked to rate the perceived quality of each individual decomposed component presented diotically, and in pair presented dichotically. We found that the perceived quality rating of these resonance components when presented in pair was higher than the mean of perceived quality ratings of these resonance components when presented individually. Our result suggests that listeners were able to fuse binaural dichotic presentation of high- and low-resonance components and perceived better sound quality.

This paper addresses the problem of smoothing data with additive step discontinuities. The problem formulation is based on least square polynomial approximation and total variation denoising. In earlier work, an ADMM algorithm was proposed to minimize a suitably defined sparsity-promoting cost function. In this paper, an algorithm is derived using the majorization-minimization optimization procedure. The new algorithm converges faster and, unlike the ADMM algorithm, has no parameters that need to be set. The proposed algorithm is formulated so as to utilize fast solvers for banded systems for high computational efficiency. This paper also gives optimality conditions so that the optimality of a result produced by the numerical algorithm can be readily validated.

This paper describes an extension to total variation denoising wherein it is assumed the first-order difference function of the unknown signal is not only sparse, but also that large values of the first-order difference function do not generally occur in isolation. This approach is designed to alleviate the staircase artifact often arising in total variation based solutions. A convex cost function is given and an iterative algorithm is derived using majorization-minimization. The algorithm is both fast converging and computationally efficient due to the use of fast solvers for banded systems.

Imaging of cardiac perfusion with MR is a challenging area of research especially due to the motion of the heart and limited time of data acquisition. Compressed sensing is a popular signal estimation method recently adopted by researchers in MRI which can improve the spatial and/or temporal resolution of the acquired images by reducing the number of necessary samples for image reconstruction. This paper focuses on performance of temporal regularization with total variation and wavelets in compressed sensing. The impact of the choice of regularization parameters on the image quality and the temporal variation of intensity in region of interests (ROIs) are discussed. It is found that selecting the regularization parameter so as to optimize the quality of the reconstructed image sequence as a whole, leads to erroneous reconstruction of certain regions due to over regularization.

This paper proposes the separation of signal components based on resonance. The method relies on several recent developments in sparse signal processing: morphological component analysis (MCA), the rational-dilation wavelet transform (RADWT), and fast algorithms for ℓ1-norm regularized linear inverse problems (for example, SALSA). The RADWT allows one to extract signal components according to resonance characteristics because the RADWT allows the Q-factor (frequency resolution) of the wavelet transform to be varied. The sought decomposition can not be accomplished by frequency-based filtering. An example illustrates the method.

The detection of moving objects on the ground by airborne radar is one application of space-time adaptive processing (STAP). The goal is to estimate the position and velocity of objects. This paper considers the problem as a linear inverse problem and uses ℓ1-norm regularization to promote sparsity in the solution. It is proposed that the angle-Doppler plane be explicitly segmented into the clutter ridge component and a non-clutter-ridge component. We propose that the second component be modeled as sparse - as the moving objects are assumed to be well isolated in the angle-Doppler plane.

In this letter, we expand upon the method of Tay for the design of orthonormal Q-shift filters for the dual-tree complex wavelet transform. The method of Tay searches for good Hilbert-pairs in a one-parameter family of conjugate-quadrature filters that have one vanishing moment less than the Daubechies conjugate-quadrature filters (CQFs). In this letter, we compute feasible sets for one- and two-parameter families of CQFs by employing the trace parameterization of nonnegative trigonometric polynomials and semidefinite programming. This permits the design of CQF pairs that define complex wavelets that are more nearly analytic, yet still have a high number of vanishing moments.

We present new Bayesian estimators for spherically-contoured Bessel K form (BKF) random vectors in additive white Gaussian noise (AWGN). The derivations are an extension of existing results for the scalar BKF and multivariate Laplace (MLAP) densities. MAP and MMSE estimators are derived. We show that the MMSE estimator can be written in exact form in terms of the generalized incomplete Gamma function. Computationally efficient approximations are given. We compare the proposed exact and approximate MMSE estimators with recent results using the BKF density, both in terms of the shrinkage rules and the associated mean-square error.

In this paper, we present a new image denoising algorithm. We assume a mixture of bivariate circular symmetric Laplacian probability density functions (pdfs) where for each wavelet coefficients may have different local parameter. This pdf characterizes simultaneously 1) the heavy-tailed nature, 2) the interscale dependencies of the wavelet coefficients and also 3) the empirically observed correlation between the coefficient amplitudes. We employ this local bivariate mixture model to derive a Bayesian image denoising technique. This proposed pdf, potentially can fits better the statistical properties of the wavelet coefficients than several other existing models. Our simulation results reveal that the proposed denoising method is among the best reported in the literature. This is justified since the accuracy of the employed distribution for noise-free data determines the denoising performance.

This paper describes a new wavelet-based denoising algorithm based on a non-stationary noise assumption. Even though stationary noise models can simplify the development and implementation of denoising algorithms, they do not always accurately describe the statistical properties of the noise. The proposed algorithm has been developed to address signal processing problems under environments where the noise is spatially varying. We illustrate two signal denoising examples in order to show the performance of the proposed algorithm.

Wavelet domain statistical modeling of images has focused on modeling the peaked heavy-tailed behavior of the marginal distribution and on modeling the dependencies between coefficients that are adjacent (in location and/or scale). In this paper we describe the extension of the Laplace marginal model to the multivariate case so that groups of wavelet coefficients can be modeled together using Laplace marginal models. We derive the nonlinear MAP and MMSE shrinkage functions for a Laplace vector in Gaussian noise and provide computationally efficient approximations to them. The development depends on the generalized incomplete Gamma function.

In this paper we introduce a class of multivariate quasi-Laplacian models as a generalization of the single-variable Laplacian distribution to multi-dimensions. A mixture model is used as the wavelet coefficient prior for the wavelet-based Bayesian image denoising algorithm. As a multivariate probability model, it is able to capture the intra-scale or inter-scale dependencies among wavelet coefficients. Two special cases are studied for orthogonal transform based image denoising. Efficient parameter estimation methods and denoising rules are derived for the two cases. Denoising results are compared with existing techniques in both PSNR values and visual qualities.

Recently, it has been shown that algorithms exploiting dependencies between coefficients for modeling probability density function (pdf) of wavelet coefficients, could achieve better results for image denoising in wavelet domain compared with the ones based on the independence assumption. In this context, we design a bivariate maximum a posteriori (MAP) estimator which relies on a mixture of bivariate Laplacian models. This model not only is bivariate but also is mixture and therefore, using this new statistical model, we are able to better capture heavy-tailed natures of the data as well as the interscale dependencies of wavelet coefficients. The simulation results show that our proposed technique achieves better performance than several published methods both visually and in terms of peak signal-to-noise ratio (PSNR)

The paper explores the use of a recently introduced 3D dual-tree discrete wavelet transform (DDWT) for video coding. The 3D DDWT is an attractive video representation because it isolates motion along different directions in separate subbands. However, it is an overcomplete transform with 8:1 or 4:1 redundancy. Based on the effectiveness of the iterative projection-based noise shaping scheme proposed by Kingsbury in reducing the number of coefficients, and our prior investigation about the correlation between subbands at the same spatial/temporal location, both in the significance map and in actual coefficient values, a new video coding scheme using 3D DDWT is proposed. The proposed video codec does not require motion compensation and provides better performance than the 3D SPIHT codec, both objectively and subjectively, despite the fact that the raw number of coefficients resulting from the 3D DDWT is much more than that of the conventional 3D DWT. The proposed coder allows full scalability in spatial, temporal and quality dimensions.

Video processing techniques using true 3D transforms are largely unexploited, partly because of the drawbacks of traditional separable 3D transforms. In this paper, we use a new type of non-separable 3D wavelet transform for video denoising and overcome the motion-mixture problem by using oriented complex wavelets. This wavelet transform is a 3D version of Kingsbury's 1D and 2D dual-tree wavelet transforms. We also investigate video denoising techniques using a combination of both 2D and 3D oriented wavelet transforms. The results are compared with those obtained by separable wavelet transforms.

This paper examines the properties of a recently introduced 3D dual-tree discrete wavelet transform (DDWT) for video coding. The 3D DDWT is an attractive video representation because it isolates motion along different directions in separate subbands. However, it is an overcomplete transform with 8:1 redundancy. We examine the effectiveness of the iterative projection-based noise shaping scheme proposed by Kingsbury et al., (2002) on reducing the number of coefficients. We also investigate the correlation between subbands at the same spatial/temporal location, both in the significance map and in actual coefficient values.

Connexions is a new approach to authoring, teaching, and learning that aims to fully exploit modern information technology. Available free of charge to anyone under open-content and open-source licenses, Connexions offers custom-tailored, current course material, is adaptable to a wide range of learning styles, and encourages students to explore the links among courses and disciplines. In contrast to the traditional process of textbook writing and publishing, Connexions fosters world-wide, cross-institution communities of authors, instructors, and students, who collaboratively and dynamically fashion “modules” from which courses are constructed. We believe the ideas and philosophy embodied by Connexions have the potential to change the very nature of textbook writing and publishing, producing a dynamic, interconnected educational environment that is pedagogically sound, both time and cost efficient, and fun. This paper overviews the philosophy and technology behind Connexions and describes a nascent community developing material for DSP education.

Most simple nonlinear thresholding rules for wavelet-based denoising assume the wavelet coefficients are independent. However, wavelet coefficients of natural images have significant dependency. In this paper, a new heavy-tailed bivariate pdf is proposed to model the statistics of wavelet coefficients, and a simple nonlinear threshold function (shrinkage function) is derived from the pdf using Bayesian estimation theory. The new shrinkage function does not assume the independence of wavelet coefficients.

This paper describes a new complex-directional expansive perfect reconstruction two-dimensional wavelet transform. Each complex wavelet is oriented along one of six possible directions, and the magnitude of each complex wavelet has a smooth bell-shape. The transform is based both on the complex dual-tree wavelet transform introduced by Kingsbury (see Phil. Trans. Royal Society London, A, September 1999 and Applied and Computational Harmonic Analysis, vol.10, no.3, p.234-53, 2001) and on the double-density DWT. It is designed so as to possess simultaneously the properties of the complex dual-tree DWT and the double-density DWT. The paper also describes a simple subband-dependent data-driven denoising algorithm for use with this transform. An example is shown to illustrate the performance of the denoising algorithm and the transform.

It is well known that the wavelet coefficients of natural images have significant statistical dependencies. To model the non-Gaussian nature of these statistics, a new bivariate PDF is proposed in this paper and applied to the image denoising problem. For this purpose, the corresponding new bivariate shrinkage function is derived using the MAP estimator. Using this function, a subband dependent data-driven system is described and applied to both orthogonal and dual-tree complex wavelet coefficients. Also, some comparisons to the other effective data-driven techniques are given.

The performance of image-denoising algorithms using wavelet transforms can be improved significantly by taking into account the statistical dependencies among wavelet coefficients as demonstrated by several algorithms presented in the literature. In two earlier papers by the authors, a simple bivariate shrinkage rule is described using a coefficient and its parent. The performance can also be improved using simple models by estimating model parameters in a local neighborhood. This letter presents a locally adaptive denoising algorithm using the bivariate shrinkage function. The algorithm is illustrated using both the orthogonal and dual tree complex wavelet transforms. Some comparisons with the best available results are given in order to illustrate the effectiveness of the proposed algorithm.

The FIR 2-band wavelets have found wide applications in practice. One of their disadvantages, however, is that they cannot be made both symmetric and orthogonal. There have been some works on filters which are orthogonal and nearly symmetric. Grobner methods are used to design orthogonal conjugate quadrature filters (CQF) with a sub-set of exactly symmetric coefficients of various lengths, as opposed to nearly symmetric coefficients.

This paper describes a simple procedure, based on spectral factorization, for the design of a pair of orthonormal wavelet bases where the two wavelets form a Hilbert transform pair. The two scaling filters respectively have the numerator and denominator of the flat delay all-pass filter as factors. The design procedure allows for an arbitrary number of zero wavelet moments to be specified. A Matlab program for the procedure is given, and examples are also given to illustrate the results.

This paper considers the design of pairs of wavelet bases where the wavelets form a Hilbert transform pair. The derivation is based on the limit functions defined by the infinite product formula. It is found that the scaling filters should be offset from one another by a half sample. This gives an alternative derivation and explanation for the result by Kingsbury (1999), that the dual-tree DWT is (nearly) shift-invariant when the scaling filters satisfy the same offset.

This paper considers the design and application of wavelet tight frames based on iterated oversampled filter banks. The greater design freedom available makes possible the construction of wavelets with a high degree of smoothness, in comparison with orthonormal wavelet bases. Grobner bases are used to obtain the solutions to the nonlinear design equations. Following the dual-tree DWT of Kingsbury (see Proceedings of the Eighth IEEE DSP Workshop, Utah, 1998, and Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing (ICASSP), Phoenix, 1999), one goal is to keep the redundancy-factor bounded by 2, instead of allowing it to grow as it does for the undecimated DWT (which is exactly shift-invariant). For the tight frame presented here, optimal-tree based denoising algorithms can be directly applied.

This paper considers the classical Shannon sampling theorem in multiresolution spaces with scaling functions as interpolants. As discussed by Xia and Zhang (1993), for an orthogonal scaling function to support such a sampling theorem, the scaling function must be cardinal. They also showed that the only orthogonal scaling function that is both cardinal and of compact support is the Haar function, which has only 1 vanishing moment and is not continuous. This paper addresses the same question, but in the multiwavelet context, where the situation is different. This paper presents the construction of orthogonal multiscaling functions that are simultaneously cardinal, of compact support, and have more than one vanishing moment. The scaling functions thereby support a Shannon-like sampling theorem. Such wavelet bases are appealing because the initialization of the discrete wavelet transform (prefiltering) is the identity operator-the projection of a function onto the scaling space is given by its samples.

The Geronimo-Bardin-Massopust (see J. Approx. Theory, vol.78, p.373-401, 1994) multiwavelet basis exhibits symmetry, orthogonality, short support, and approximation order K=2, which is not possible for wavelet bases based on a single scaling-wavelet function pair. However, the filterbank associated with this basis does not inherit the zero moment properties of the basis. This work describes a version of the GHM multiscaling functions (constructed with Grobner bases) for which the zero moment properties do carry over to the associated filterbank. That is, the basis is balanced up to its approximation order K=2.

This paper introduces a new set of exchange rules for a Remez-like algorithm for the Chebyshev design of IIR digital filters. It is explained that the essential difficulty, in applying the Remez algorithm to rational functions, is that on some iterations, there may be no solution to the interpolation problem for which the denominator is strictly non-zero in the interval of approximation. Then the usual procedure for updating the interpolation points cannot be applied. The new rules for updating the interpolation points address precisely this problem for the two-pole case. It is shown with examples that, when the Remez-like algorithm of Hofstetter et al. (1971) is applied to rational functions, there is a way to update the interpolation points so that the algorithm converges rapidly, even when poles arise in the interval of approximation.

This paper presents a formula-based method for the design of IIR filters having more zeros than (nontrivial) poles. The filters are designed so that their square magnitude frequency responses are maximally-flat at /spl omega/=0 and at /spl omega/=/spl pi/ and are thereby generalizations of classical digital Butterworth filters. A main result of the paper is that, for a specified half-magnitude frequency and a specified number of zeros, there is only one valid way in which to split the zeros between z=-1 and the passband. IIR filters having more zeros than poles are of interest, because often, to obtain a good trade-off between performance and the expense of implementation, just a few poles are best.

Describes a set of programs for circular convolution and prime length FFTs that are short, possess great structure, share many computational procedures, and cover a large variety of lengths. The programs make clear the structure of the algorithms and clearly enumerate independent computational branches that can be performed in parallel. Moreover, each of these independent operations is made up of a sequence of sub-operations which can be implemented as vector/parallel operations. This is in contrast with previously existing programs for prime length FFTs: they consist of straight line code, no code is shared between them, and they cannot be easily adapted for vector/parallel implementations. The authors have also developed a program that automatically generates these programs for prime length FFTs.

We consider the design of digital filters and discuss the inclusion of explicitly specified transition bands in the frequency domain design of FIR filters. We put forth the notion that explicitly specified transition bands have been introduced in the filter design literature as an indirect and sometimes inadequate approach for dealing with discontinuities in the desired frequency response. We also present a rapidly converging, robust, simple algorithm for the design of optimal peak constrained least square lowpass FIR filters that does not require the use of transition bands. This versatile algorithm will design linear and minimum phase FIR filters and gives the best L/sub 2/ filter and a continuum of Chebyshev filters as special cases.

The paper introduces a novel speckle reduction method based on thresholding the wavelet coefficients of the logarithmically transformed image. The method is computational efficient and can significantly reduce the speckle while preserving the resolution of the original image. Both soft and hard thresholding schemes are studied and the results are compared. When fully polarimetric SAR images are available, the authors propose several approaches to combine the data from different polarizations to achieve even better performance. Wavelet processed imagery is shown to provide better detection performance for the synthetic-aperture radar (SAR) based automatic target detection/recognition (ATD/R) problem.>

For short data sequences, Winograd's convolution algorithms attaining the minimum number of multiplications also attain a low number of additions, making them very efficient. However, for longer lengths they require a larger number of additions. Winograd's approach is usually extended to longer lengths by using a nesting approach such as the Agarwal-Cooley (1977) or Split-Nesting algorithms. Although these nesting algorithms are organizationally quite simple, they do not make the greatest use of the factorability of the data sequence length. The algorithm we propose adheres to Winograd's original approach more closely than do the nesting algorithms. By evaluating polynomials over simple matrices we retain, in algorithms for longer lengths, the basic structure and strategy of Winograd's approach, thereby designing computationally refined algorithms. This tactic is arithmetically profitable because Winograd's approach is based on a theory of minimum multiplicative complexity.>