Inversion of large-support ill-posed linear operators using a piecewise Gaussian MRF

PIRPLE: a penalized-likelihood framework for incorporation of prior images in CT reconstruction

Over the course of diagnosis and treatment, it is common for a number of imaging studies to be acquired. Such imaging sequences can provide substantial patient-specific prior knowledge about the anatomy that can be incorporated into a prior-image-based tomographic reconstruction for improved image quality and better dose utilization. We present a general methodology using a model-based reconstruction approach including formulations of the measurement noise that also integrates prior images. This penalized-likelihood technique adopts a sparsity enforcing penalty that incorporates prior information yet allows for change between the current reconstruction and the prior image. Moreover, since prior images are generally not registered with the current image volume, we present a modified model-based approach that seeks a joint registration of the prior image in addition to the reconstruction of projection data. We demonstrate that the combined prior-image- and model-based technique outperforms methods that ignore the prior data or lack a noise model. Moreover, we demonstrate the importance of registration for prior-image-based reconstruction methods and show that the prior-image-registered penalized-likelihood estimation (PIRPLE) approach can maintain a high level of image quality in the presence of noisy and undersampled projection data.

Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction

Nonconvex nonsmooth regularization has advantages over convex regularization for restoring images with neat edges. However, its practical interest used to be limited by the difficulty of the computational stage which requires a nonconvex nonsmooth minimization. In this paper, we deal with nonconvex nonsmooth minimization methods for image restoration and reconstruction. Our theoretical results show that the solution of the nonconvex nonsmooth minimization problem is composed of constant regions surrounded by closed contours and neat edges. The main goal of this paper is to develop fast minimization algorithms to solve the nonconvex nonsmooth minimization problem. Our experimental results show that the effectiveness and efficiency of the proposed algorithms.

Joint NDT image restoration and segmentation using Gauss-Markov-Potts prior models and variational Bayesian computation

In this paper, we propose a method to simultaneously restore and to segment piecewise homogeneous images degraded by a known point spread function (PSF) and additive noise. For this purpose, we propose a family of nonhomogeneous Gauss-Markov fields with Potts region labels model for images to be used in a Bayesian estimation framework. The joint posterior law of all the unknowns (the unknown image, its segmentation (hidden variable) and all the hyperparameters) is approximated by a separable probability law via the variational Bayes technique. This approximation gives the possibility to obtain practically implemented joint restoration and segmentation algorithm. We will present some preliminary results and comparison with a MCMC Gibbs sampling based algorithm. We may note that the prior models proposed in this work are particularly appropriate for the images of the scenes or objects that are composed of a finite set of homogeneous materials. This is the case of many images obtained in nondestructive testing (NDT) applications.

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.

Deblurring of color images corrupted by impulsive noise

We consider the problem of restoring a multichannel image corrupted by blur and impulsive noise (e.g., salt-and-pepper noise). Using the variational framework, we consider the L1 fidelity term and several possible regularizers. In particular, we use generalizations of the Mumford-Shah (MS) functional to color images and gamma-convergence approximations to unify deblurring and denoising. Experimental comparisons show that the MS stabilizer yields better results with respect to Beltrami and total variation regularizers. Color edge detection is a beneficial by-product of our methods.

Statistical regularization in linearized microwave imaging through MRF-based MAP estimation: hyperparameter estimation and image computation

The application of a Markov random fields (MRF) based maximum a posteriori (MAP) estimation method for microwave imaging is presented in this paper. The adopted MRF family is the so-called Gaussian-MRF (GMRF), whose energy function is quadratic. In order to implement the MAP estimation, first, the MRF hyperparameters are estimated by means of the expectation-maximization (EM) algorithm, extended in this case to complex and nonhomogeneous images. Then, it is implemented by minimizing a cost function whose gradient is fully analytically evaluated. Thanks to the quadratic nature of the energy function of the MRF, well posedness and efficiency of the proposed method can be simultaneously guaranteed. Numerical results, also performed on real data, show the good performance of the method, also when compared with conventional techniques like Tikhonov regularization.

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.

Template matching based object recognition with unknown geometric parameters

In this paper, we examine the problem of locating an object in an image when size and rotation are unknown. Previous work has shown that with known geometric parameters, an image restoration method can be useful by estimating a delta function at the object location. When the geometric parameters are unknown, this method becomes impractical because the likelihood surface to be minimized across size and rotation has numerous local minima and areas of zero gradient. We propose a new approach where a smooth approximation of the template is used to minimize a well-behaved likelihood surface. A coarse-to-fine approximation of the original template using a diffusion-like equation is used to create a library of templates. Using this library, we can successively perform minimizations which are locally well-behaved. As detail is added to the template, the likelihood surface gains local minima, but previous estimates place us within a well-behaved "bowl" around the global minimum, leading to an accurate estimate. Numerical experiments are shown which verify the value of this approach for a wide range of values of the geometric parameters.

Exact distribution of edge-preserving MAP estimators for linear signal models with Gaussian measurement noise

We derive the exact statistical distribution of maximum a posteriori (MAP) estimators having edge-preserving nonGaussian priors. Such estimators have been widely advocated for image restoration and reconstruction problems. Previous investigations of these image recovery methods have been primarily empirical; the distribution we derive enables theoretical analysis. The signal model is linear with Gaussian measurement noise. We assume that the energy function of the prior distribution is chosen to ensure a unimodal posterior distribution (for which convexity of the energy function is sufficient), and that the energy function satisfies a uniform Lipschitz regularity condition. The regularity conditions are sufficiently general to encompass popular priors such as the generalized Gaussian Markov random field prior and the Huber prior, even though those priors are not everywhere twice continuously differentiable.

Markovian reconstruction using a GNC approach

This paper is concerned with the reconstruction of images (or signals) from incomplete, noisy data, obtained at the output of an observation system. The solution is defined in maximum a posteriori (MAP) sense and it appears as the global minimum of an energy function joining a convex data-fidelity term and a Markovian prior energy. The sought images are composed of nearly homogeneous zones separated by edges and the prior term accounts for this knowledge. This term combines general nonconvex potential functions (PFs) which are applied to the differences between neighboring pixels. The resultant MAP energy generally exhibits numerous local minima. Calculating its local minimum, placed in the vicinity of the maximum likelihood estimate, is inexpensive but the resultant estimate is usually disappointing. Optimization using simulated annealing is practical only in restricted situations. Several deterministic suboptimal techniques approach the global minimum of special MAP energies, employed in the field of image denoising, at a reasonable numerical cost. The latter techniques are not directly applicable to general observation systems, nor to general Markovian prior energies. This work is devoted to the generalization of one of them, the graduated nonconvexity (GNC) algorithm, in order to calculate nearly-optimal MAP solutions in a wide range of situations. In fact, GNC provides a solution by tracking a set of minima along a sequence of approximate energies, starting from a convex energy and progressing toward the original energy. In this paper, we develop a common method to derive efficient GNC-algorithms for the minimization of MAP energies which arise in the context of any observation system giving rise to a convex data-fidelity term and of Markov random field (MRF) energies involving any nonconvex and/or nonsmooth PFs. As a side-result, we propose how to construct pertinent initializations which allow us to obtain meaningful solutions using local minimization of these MAP energies. Two numerical experiments-an image deblurring and an emission tomography reconstruction-illustrate the performance of the proposed technique.