Anna Jezierska

A Majorize-Minimize Subspace Approach for

In this work, we consider a class of differentiable criteria for sparse image computing problems, where a nonconvex regularization is applied to an arbitrary linear transform of the target image. As special cases, it includes edge-preserving measures or frame-analysis potentials commonly used in image processing. As shown by our asymptotic results, the l2-l0 penalties we consider may be employed to provide approximate solutions to l0-penalized optimization problems. One of the advantages of the proposed approach is that it allows us to derive an efficient Majorize-Minimize subspace algorithm. The convergence of the algorithm is investigated by using recent results in nonconvex optimization. The fast convergence properties of the proposed optimization method are illustrated through image processing examples. In particular, its effectiveness is demonstrated on several data recovery problems.

\ell_2-\ell_0

The Poisson-Gaussian model can accurately describe the noise present in a number of imaging systems. However most existing restoration methods rely on approximations of the Poisson-Gaussian noise statistics. We propose a convex optimization strategy for the reconstruction of images degraded by a linear operator and corrupted with a mixed Poisson-Gaussian noise. The originality of our approach consists of considering the exact, mixed continuous-discrete model corresponding to the data statistics. After establishing the Lipschitz differentiability and convexity of the Poisson-Gaussian neg-log-likelihood, we derive a primal-dual iterative scheme for minimizing the associated penalized criterion. The proposed method is applicable to a large choice of convex penalty terms. The robustness of our scheme allows us to handle computational difficulties due to infinite sums arising from the computation of the gradient of the criterion. We propose finite bounds for these sums, that are dependent on the current image estimate, and thus adapted to each iteration of our algorithm. The proposed approach is validated on image restoration examples. Then, the exact data fidelity term is used as a reference for studying some of its various approximations. We show that in a variational framework the Shifted Poisson and Exponential approximations lead to very good restoration results.

Image Regularization

A Poisson-Gaussian model accurately describes the noise present in many imaging systems such as CCD cameras or fluorescence microscopy. However most existing restoration strategies rely on approximations of the Poisson-Gaussian noise statistics. We propose a convex optimization algorithm for the reconstruction of signals degraded by a linear operator and corrupted with mixed Poisson-Gaussian noise. The originality of our approach consists of considering the exact continuous-discrete model corresponding to the data statistics. After establishing the Lipschitz differentiability of the Poisson-Gaussian log-likelihood, we derive a primal-dual iterative scheme for minimizing the associated penalized criterion. The proposed method is applicable to a large choice of penalty terms. The robustness of our scheme allows us to handle computational difficulties due to infinite sums arising from the computation of the gradient of the criterion. The proposed approach is validated on image restoration examples.

A Convex Approach for Image Restoration with Exact Poisson-Gaussian Likelihood

In this paper, we present a new fully automatic approach for noise parameter estimation in the context of fluorescence imaging systems. In particular, we address the problem of Poisson-Gaussian noise modeling in the nonstationary case. In microscopy practice, the nonstationarity is due to the photobleaching effect. The proposed method consists of an adequate moment based initialization followed by Expectation-Maximization iterations. This approach is shown to provide reliable estimates of the mean and the variance of the Gaussian noise and of the scale parameter of Poisson noise, as well as of the photobleaching rates. The algorithm performance is demonstrated on both synthetic and real macro confocal laser scanning microscope image sequences.

A primal-dual proximal splitting approach for restoring data corrupted with poisson-gaussian noise

The problem of estimating the parameters of a Poisson-Gaussian model from experimental data has recently raised much interest in various applications, for instance in confocal fluorescence microscopy. In this context, a field of independent random variables is observed, which is varying both in time and space. Each variable is a sum of two components, one following a Poisson and the other a Gaussian distribution. In this paper, a general formulation is considered where the associated Poisson process is nonstationary in space and also exhibits an exponential decay in time, whereas the Gaussian component corresponds to a stationary white noise with arbitrary mean. To solve the considered parametric estimation problem, we follow an iterative Expectation-Maximization (EM) approach. The parameter update equations involve deriving finite approximation of infinite sums. Expressions for the maximum error incurred in the process are also given. Since the problem is non-convex, we pay attention to the EM initialization, using a moment-based method where recent optimization tools come into play. We carry out a performance analysis by computing the Cramer-Rao bounds on the estimated variables. The practical performance of the proposed estimation procedure is illustrated on both synthetic data and real fluorescence macroscopy image sequences. The algorithm is shown to provide reliable estimates of the mean/variance of the Gaussian noise and of the scale parameter of the Poisson component, as well as of its exponential decay rate. In particular, the mean estimate of the Poisson component can be interpreted as a good-quality denoised version of the data.

Poisson-Gaussian noise parameter estimation in fluorescence microscopy imaging

Quantization, defined as the act of attributing a finite number of levels to an image, is an essential task in image acquisition and coding. It is also intricately linked to image analysis tasks, such as denoising and segmentation. In this paper, we investigate vector quantization combined with regularity constraints, a little-studied area which is of interest, in particular, when quantizing in the presence of noise or other acquisition artifacts. We present an optimization approach to the problem involving a novel two-step, iterative, flexible, joint quantizing-regularization method featuring both convex and combinatorial optimization techniques. We show that when using a small number of levels, our approach can yield better quality images in terms of SNR, with lower entropy, than conventional optimal quantization methods.

An EM Approach for Time-Variant Poisson-Gaussian Model Parameter Estimation

In this paper, we consider a class of differentiable criteria for sparse image recovery problems. The regularization is applied to a linear transform of the target image. As special cases, it includes edge preserving measures or frame analysis potentials. As shown by our asymptotic results, the considered ℓ2 — ℓ0 penalties may be employed to approximate solutions to ℓ0 penalized optimization problems. One of the advantages of the approach is that it allows us to derive an efficient Majorize-Minimize Memory Gradient algorithm. The fast convergence properties of the proposed optimization algorithm are illustrated through image restoration examples.

A Spatial Regularization Approach for Vector Quantization

This paper addresses the problem of minimizing multilabel energies with truncated convex priors. Such priors are known to be useful but difficult and slow to optimize because they are not convex. We propose two novel classes of binary Graph-Cuts (GC) moves, namely the convex move and the quantized move. The moves are complementary. To significantly improve efficiency, the label range is divided into even intervals. The quantized move tends to efficiently put pixel labels into the correct intervals for the energy with truncated convex prior. Then the convex move assigns the labels more precisely within these intervals for the same energy. The quantized move is a modified a-expansion move, adapted to handle a generalized Potts prior, which assigns a constant penalty to arguments above some threshold. Our convex move is a GC representation of the efficient Murotas algorithm. We assume that the data terms are convex, since this is a requirement for Murotas algorithm. We introduce Quantized-Convex Split Moves algorithm which minimizes energies with truncated priors by alternating both moves. This algorithm is a fast solver for labeling problems with a high number of labels and convex data terms. We illustrate its performance on image restoration.

A Memory Gradient algorithm for ℓ 2 — ℓ 0 regularization with applications to image restoration

Tracking instruments in video-assisted minimally invasive surgeries is an attractive and open computer vision problem. A tracker successfully locating instruments would immediately find applications in manual and robotic interventions in the operating theater. We describe a tracking method that uses a rigidly structured model of instrument parts. The rigidly composed parts encode diverse, pose-specific appearance mixtures of the tool. This rigid part mixtures model then jointly explains the evolving structure of the tool parts by switching between mixture components during tracking. We evaluate our approach on publicly available datasets of in-vivo sequences and demonstrate state-of-the-art results.

A fast solver for truncated-convex priors: quantized-convex split moves

Haar-like features are ubiquitous in computer vision, e.g. for Viola and Jones face detection or local descriptors such as Speeded-Up-Robust-Features. They are classically computed in one pass over integral image by reading the values at the feature corners. Here we present a new, general parsing formalism for convolving them more efficiently. Our method is fully automatic and applicable to an arbitrary set of Haar-like features. The parser reduces the number of memory accesses which are the main computational bottleneck during convolution on modern computer architectures. It first splits the features into simpler kernels. Then it aligns and reuses them where applicable forming an ensemble of recursive convolution trees, which can be computed faster. This is illustrated with experiments, which show a significant speed-up over the classic approach.