Wavelet domain image restoration with adaptive edge-preserving regularization

Microphone array analysis of the first non-axisymmetric mode for the detection of pipe conditions

This paper reports on the use of a circular microphone array to analyze the reflections from a pipe defect with enhanced resolution. A Bayesian maximum a posteriori algorithm is combined with the mode decomposition approach to localize pipe defects with six or fewer microphones. Unlike all previous acoustic reflectometry techniques, which only estimate the location of a pipe defect along the pipe, the proposed method uses the phase information about the wave propagated in the form of the first non-axisymmetric mode to estimate its circumferential position as well as axial location. The method is validated against data obtained from a laboratory measurement in a 150 mm diameter polyvinyl chloride pipe with a 20% in-pipe blockage and 100 mm lateral connection. The accuracy of localization of the lateral connection and blockage attained in this measurement was better than 2% of the axial sensing distance and 9° error in terms of the circumferential position. The practical significance of this approach is that it can be implemented remotely on an autonomous inspection robot so that accurate axial location and circumferential position of lateral connections and small blockages can be estimated with a computationally efficient algorithm.

A Parallel Proximal Algorithm for Anisotropic Total Variation Minimization

Total variation (TV) is a one of the most popular regularizers for stabilizing the solution of ill-posed inverse problems. This paper proposes a novel proximal-gradient algorithm for minimizing TV regularized least-squares cost functionals. Unlike traditional methods that require nested iterations for computing the proximal step of TV, our algorithm approximates the latter with several simple proximals that have closed form solutions. We theoretically prove that the proposed parallel proximal method achieves the TV solution with arbitrarily high precision at a global rate of converge that is equivalent to the fast proximal-gradient methods. The results in this paper have the potential to enhance the applicability of TV for solving very large-scale imaging inverse problems.

Structured penalties for functional linear models-partially empirical eigenvectors for regression

One of the challenges with functional data is incorporating geometric structure, or local correlation, into the analysis. This structure is inherent in the output from an increasing number of biomedical technologies, and a functional linear model is often used to estimate the relationship between the predictor functions and scalar responses. Common approaches to the problem of estimating a coefficient function typically involve two stages: regularization and estimation. Regularization is usually done via dimension reduction, projecting onto a predefined span of basis functions or a reduced set of eigenvectors (principal components). In contrast, we present a unified approach that directly incorporates geometric structure into the estimation process by exploiting the joint eigenproperties of the predictors and a linear penalty operator. In this sense, the components in the regression are 'partially empirical' and the framework is provided by the generalized singular value decomposition (GSVD). The form of the penalized estimation is not new, but the GSVD clarifies the process and informs the choice of penalty by making explicit the joint influence of the penalty and predictors on the bias, variance and performance of the estimated coefficient function. Laboratory spectroscopy data and simulations are used to illustrate the concepts.

Satellite image restoration in the context of a spatially varying point spread function

In this paper, we consider a deconvolution problem where the point spread function (PSF) of the optical imaging system varies between different spatial locations, thus leading to a spatially varying blur. This problem arises, for example, in synthetic aperture instruments and in wide-field optical systems. Unlike the classical deconvolution context where the PSF is assumed to be spatially invariant, the problem cannot be easily solved in the Fourier domain. We propose here an iterative algorithm based on convex optimization techniques and a wavelet frame regularization. This approach allows restoration of the image, taking into account the properties of the blur operator, the latter being known.

Iterative PET Image Reconstruction Using Translation Invariant Wavelet Transform

The present work describes a Bayesian maximum a posteriori (MAP) method using a statistical multiscale wavelet prior model. Rather than using the orthogonal discrete wavelet transform (DWT), this prior is built on the translation invariant wavelet transform (TIWT). The statistical modeling of wavelet coefficients relies on the generalized Gaussian distribution. Image reconstruction is performed in spatial domain with a fast block sequential iteration algorithm. We study theoretically the TIWT MAP method by analyzing the Hessian of the prior function to provide some insights on noise and resolution properties of image reconstruction. We adapt the key concept of local shift invariance and explore how the TIWT MAP algorithm behaves with different scales. It is also shown that larger support wavelet filters do not offer better performance in contrast recovery studies. These theoretical developments are confirmed through simulation studies. The results show that the proposed method is more attractive than other MAP methods using either the conventional Gibbs prior or the DWT-based wavelet prior.

Sparsity constrained regularization for multiframe image restoration

In this paper we present a new algorithm for restoring an object from multiple undersampled low-resolution (LR) images that are degraded by optical blur and additive white Gaussian noise. We formulate the multiframe superresolution problem as maximum a posteriori estimation. The prior knowledge that the object is sparse in some domain is incorporated in two ways: first we use the popular l(1) norm as the regularization operator. Second, we model wavelet coefficients of natural objects using generalized Gaussian densities. The model parameters are learned from a set of training objects, and the regularization operator is derived from these parameters. We compare the results from our algorithms with an expectation-maximization (EM) algorithm for l(1) norm minimization and also with the linear minimum-mean-squared error (LMMSE) estimator. Using only eight 4 x 4 pixel downsampled LR images the reconstruction errors of object estimates obtained from our algorithm are 5.5% smaller than by the EM method and 14.3% smaller than by the LMMSE method.

Wavelet domain Bayesian method for high noise level PET image reconstruction

In this paper, a new maximum a posterior(MAP) method for PET image reconstruction defined in wavelet domain is proposed. Compared to the conventional MAP methods with Markov Random Field (MRF) prior models, the proposed method, named WD-MAP method, has better performance in characterize both local and global feature of reconstructed image due to the wavelet transform. Wavelet packet decomposition strategy is applied to further improve the reconstruction quality. The convergence speed of WD-MAP method is accelerated by adopting conjugate gradient(CG) technique. Simulated experiment suggests that the proposed method offers competitive performance in PET image reconstruction.

Majorization-minimization algorithms for wavelet-based image restoration

Standard formulations of image/signal deconvolution under wavelet-based priors/regularizers lead to very high-dimensional optimization problems involving the following difficulties: the non-Gaussian (heavy-tailed) wavelet priors lead to objective functions which are nonquadratic, usually nondifferentiable, and sometimes even nonconvex; the presence of the convolution operator destroys the separability which underlies the simplicity of wavelet-based denoising. This paper presents a unified view of several recently proposed algorithms for handling this class of optimization problems, placing them in a common majorization-minimization (MM) framework. One of the classes of algorithms considered (when using quadratic bounds on nondifferentiable log-priors) shares the infamous "singularity issue" (SI) of "iteratively reweighted least squares" (IRLS) algorithms: the possibility of having to handle infinite weights, which may cause both numerical and convergence issues. In this paper, we prove several new results which strongly support the claim that the SI does not compromise the usefulness of this class of algorithms. Exploiting the unified MM perspective, we introduce a new algorithm, resulting from using l1 bounds for nonconvex regularizers; the experiments confirm the superior performance of this method, when compared to the one based on quadratic majorization. Finally, an experimental comparison of the several algorithms, reveals their relative merits for different standard types of scenarios.

A stochastic continuation approach to piecewise constant reconstruction

We address the problem of reconstructing a piecewise constant 3-D object from a few noisy 2-D line-integral projections. More generally, the theory developed here readily applies to the recovery of an ideal n-D signal (n > or =1) from indirect measurements corrupted by noise. Stabilization of this ill-conditioned inverse problem is achieved with the Potts prior model, which leads to a challenging optimization task. To overcome this difficulty, we introduce a new class of hybrid algorithms that combines simulated annealing with deterministic continuation. We call this class of algorithms stochastic continuation (SC). We first prove that, under mild assumptions, SC inherits the finite-time convergence properties of generalized simulated annealing. Then, we show that SC can be successfully applied to our reconstruction problem. In addition, we look into the concave distortion acceleration method introduced for standard simulated annealing and we derive an explicit formula for choosing the free parameter of the cost function. Numerical experiments using both synthetic data and real radiographic testing data show that SC outperforms standard simulated annealing.

Simulating deformations of MR brain images for validation of atlas-based segmentation and registration algorithms

Simulated deformations and images can act as the gold standard for evaluating various template-based image segmentation and registration algorithms. Traditional deformable simulation methods, such as the use of analytic deformation fields or the displacement of landmarks followed by some form of interpolation, are often unable to construct rich (complex) and/or realistic deformations of anatomical organs. This paper presents new methods aiming to automatically simulate realistic inter- and intra-individual deformations. The paper first describes a statistical approach to capturing inter-individual variability of high-deformation fields from a number of examples (training samples). In this approach, Wavelet-Packet Transform (WPT) of the training deformations and their Jacobians, in conjunction with a Markov random field (MRF) spatial regularization, are used to capture both coarse and fine characteristics of the training deformations in a statistical fashion. Simulated deformations can then be constructed by randomly sampling the resultant statistical distribution in an unconstrained or a landmark-constrained fashion. The paper also describes a model for generating tissue atrophy or growth in order to simulate intra-individual brain deformations. Several sets of simulated deformation fields and respective images are generated, which can be used in the future for systematic and extensive validation studies of automated atlas-based segmentation and deformable registration methods. The code and simulated data are available through our Web site.

A segmentation-based regularization term for image deconvolution

This paper proposes a new and original inhomogeneous restoration (deconvolution) model under the Bayesian framework for observed images degraded by space-invariant blur and additive Gaussian noise. In this model, regularization is achieved during the iterative restoration process with a segmentation-based a priori term. This adaptive edge-preserving regularization term applies a local smoothness constraint to pre-estimated constant-valued regions of the target image. These constant-valued regions (the segmentation map) of the target image are obtained from a preliminary Wiener deconvolution estimate. In order to estimate reliable segmentation maps, we have also adopted a Bayesian Markovian framework in which the regularized segmentations are estimated in the maximum a posteriori (MAP) sense with the joint use of local Potts prior and appropriate Gaussian conditional luminance distributions. In order to make these segmentations unsupervised, these likelihood distributions are estimated in the maximum likelihood sense. To compute the MAP estimate associated to the restoration, we use a simple steepest descent procedure resulting in an efficient iterative process converging to a globally optimal restoration. The experiments reported in this paper demonstrate that the discussed method performs competitively and sometimes better than the best existing state-of-the-art methods in benchmark tests.

Statistical representation and simulation of high-dimensional deformations: application to synthesizing brain deformations

This paper proposes an approach to effectively representing the statistics of high-dimensional deformations, when relatively few training samples are available, and conventional methods, like PCA, fail due to insufficient training. Based on previous work on scale-space decomposition of deformation fields, herein we represent the space of "valid deformations" as the intersection of three subspaces: one that satisfies constraints on deformations themselves, one that satisfies constraints on Jacobian determinants of deformations, and one that represents smooth deformations via a Markov Random Field (MRF). The first two are extensions of PCA-based statistical shape models. They are based on a wavelet packet basis decomposition that allows for more accurate estimation of the covariance structure of deformation or Jacobian fields, and they are used jointly due to their complementary strengths and limitations. The third is a nested MRF regularization aiming at eliminating potential discontinuities introduced by assumptions in the statistical models. A randomly sampled deformation field is projected onto the space of valid deformations via iterative projections on each of these subspaces until convergence, i.e. all three constraints are met. A deformation field simulator uses this process to generate random samples of deformation fields that are not only realistic but also representative of the full range of anatomical variability. These simulated deformations can be used for validation of deformable registration methods. Other potential uses of this approach include representation of shape priors in statistical shape models as well as various estimation and hypothesis testing paradigms in the general fields of computational anatomy and pattern recognition.

Bayesian wavelet-based image deconvolution: a GEM algorithm exploiting a class of heavy-tailed priors

Image deconvolution is formulated in the wavelet domain under the Bayesian framework. The well-known sparsity of the wavelet coefficients of real-world images is modeled by heavy-tailed priors belonging to the Gaussian scale mixture (GSM) class; i.e., priors given by a linear (finite of infinite) combination of Gaussian densities. This class includes, among others, the generalized Gaussian, the Jeffreys, and the Gaussian mixture priors. Necessary and sufficient conditions are stated under which the prior induced by a thresholding/shrinking denoising rule is a GSM. This result is then used to show that the prior induced by the "nonnegative garrote" thresholding/shrinking rule, herein termed the garrote prior, is a GSM. To compute the maximum a posteriori estimate, we propose a new generalized expectation maximization (GEM) algorithm, where the missing variables are the scale factors of the GSM densities. The maximization step of the underlying expectation maximization algorithm is replaced with a linear stationary second-order iterative method. The result is a GEM algorithm of O(N log N) computational complexity. In a series of benchmark tests, the proposed approach outperforms or performs similarly to state-of-the art methods, demanding comparable (in some cases, much less) computational complexity.

Detection of heart murmurs using wavelet analysis and artificial neural networks

This paper presents the algorithm and technical aspects of an intelligent diagnostic system for the detection of heart murmurs. The purpose of this research is to address the lack of effectively accurate cardiac auscultation present at the primary care physician office by development of an algorithm capable of operating within the hectic environment of the primary care office. The proposed algorithm consists of three main stages. First; denoising of input data (digital recordings of heart sounds), via Wavelet Packet Analysis. Second; input vector preparation through the use of Principal Component Analysis and block processing. Third; classification of the heart sound using an Artificial Neural Network. Initial testing revealed the intelligent diagnostic system can differentiate between normal healthy heart sounds and abnormal heart sounds (e.g., murmurs), with a specificity of 70.5% and a sensitivity of 64.7%.

Reconstruction of nonuniformly sampled images in spline spaces

This paper presents a novel approach to the reconstruction of images from nonuniformly spaced samples. This problem is often encountered in digital image processing applications. Nonrecursive video coding with motion compensation, spatiotemporal interpolation of video sequences, and generation of new views in multicamera systems are three possible applications. We propose a new reconstruction algorithm based on a spline model for images. We use regularization, since this is an ill-posed inverse problem. We minimize a cost function composed of two terms: one related to the approximation error and the other related to the smoothness of the modeling function. All the processing is carried out in the space of spline coefficients; this space is discrete, although the problem itself is of a continuous nature. The coefficients of regularization and approximation filters are computed exactly by using the explicit expressions of B-spline functions in the time domain. The regularization is carried out locally, while the computation of the regularization factor accounts for the structure of the nonuniform sampling grid. The linear system of equations obtained is solved iteratively. Our results show a very good performance in motion-compensated interpolation applications.

An EM algorithm for wavelet-based image restoration

This paper introduces an expectation-maximization (EM) algorithm for image restoration (deconvolution) based on a penalized likelihood formulated in the wavelet domain. Regularization is achieved by promoting a reconstruction with low-complexity, expressed in the wavelet coefficients, taking advantage of the well known sparsity of wavelet representations. Previous works have investigated wavelet-based restoration but, except for certain special cases, the resulting criteria are solved approximately or require demanding optimization methods. The EM algorithm herein proposed combines the efficient image representation offered by the discrete wavelet transform (DWT) with the diagonalization of the convolution operator obtained in the Fourier domain. Thus, it is a general-purpose approach to wavelet-based image restoration with computational complexity comparable to that of standard wavelet denoising schemes or of frequency domain deconvolution methods. The algorithm alternates between an E-step based on the fast Fourier transform (FFT) and a DWT-based M-step, resulting in an efficient iterative process requiring O(N log N) operations per iteration. The convergence behavior of the algorithm is investigated, and it is shown that under mild conditions the algorithm converges to a globally optimal restoration. Moreover, our new approach performs competitively with, in some cases better than, the best existing methods in benchmark tests.

Stochastic nonlinear image restoration using the wavelet transform

The dominant methodology for image restoration is to stabilize the problem by including a roughness penalty in addition to faithfulness to the data. Among various choices, concave stabilizers stand out for their boundary detection capabilities, but the resulting cost function to be minimized is generally multimodal. Although simulated annealing is theoretically optimal to take up this challenge, standard stochastic algorithms suffer from two drawbacks: i) practical convergence difficulties are encountered with second-order prior models and ii) it remains computationally demanding to favor the formation of smooth contour lines by taking the discontinuity field explicitly into account. This work shows that both weaknesses can be overcome in a multiresolution framework by means of the 2-D discrete wavelet transform (DWT). We first propose to improve convergence toward global minima by single-site updating on the wavelet domain. For this purpose, a new restricted DWT space is introduced and a theoretically sound updating mechanism is constructed on this subspace. Next, we suggest to incorporate the smoothness of the discontinuity field via an additional penalty term defined on the high frequency subbands. The resulting increase in complexity is small and the approach requires the specification of a unique extra parameter for which an explicit selection formula is derived.