Abstract. In this paper we generalize the iterated refinement method, introduced by the authors in [8], to a time-continuous inverse scale-space formulation. The iterated refinement procedure yields a sequence of convex variational problems, evolving toward the noisy image. The inverse scale space method arises as a limit for a penalization parameter tending to zero, while the number of iteration steps tends to infinity. For the limiting flow, similar properties as for the iterated refinement procedure hold. Specifically, when a discrepancy principle is used as the stopping criterion, the error between the reconstruction and the noise-free image decreases until termination, even if only the noisy image is available and a bound on the variance of the noise is known. The inverse flow is computed directly for one-dimensional signals, yielding high quality restorations. In higher spatial dimensions, we introduce a relaxation technique using two evolution equations. These equations allow accurate, efficient and straightforward implementation. 1

Image restoration is an important and widely studied problem in computer vision and image processing. Various image filtering strategies have been effective, but invariably make strong assumptions about the properties of the signal and/or degradation. Hence, these methods lack the generality to be easily applied to new applications or diverse image collections. This paper describes a novel unsupervised, information-theoretic, adaptive filter (UINTA) that improves the predictability of pixel intensities from their neighborhoods by decreasing their joint entropy. In this way, UINTA automatically discovers the statistical properties of the signal and can thereby restore a wide spectrum of images. The paper describes the formulation to minimize the joint entropy measure and presents several important practical considerations in estimating neighborhood statistics. It presents a series of results on both real and synthetic data along with comparisons with current state-of-the-art techniques, including novel applications to medical image processing.

The restoration of images is an important and widely studied problem in computer vision and image processing. Various image filtering strategies have been effective, but invariably make strong assumptions about the properties of the signal and/or degradation. Therefore, these methods typically lack the generality to be easily applied to new applications or diverse image collections. This paper describes a novel unsupervised, information-theoretic, adaptive filter (UINTA) that improves the predictability of pixel intensities from their neighborhoods by decreasing the joint entropy between them. Thus UINTA automatically discovers the statistical properties of the signal and can thereby restore a wide spectrum of images and applications. This paper describes the formulation required to minimize the joint entropy measure, presents several important practical considerations in estimating image-region statistics, and then presents results on both real and synthetic data. 1.

Optical Flow estimation in noisy image sequences requires a special denoising strategy. Towards this end we introduce a new tensor-driven anisotropic diffusion scheme which is designed to enhance optical-flow-like spatiotemporal structures. This is achieved by selecting diffusivities in a special manner depending on the eigenvalues of the well known structure tensor. We illustrate how the proposed choice differs from edge- and coherence-enhancing anisotropic diffusion. Furthermore we extend a recently discovered discretization scheme for anisotropic diffusion to 3D data. An automatic stop criterion to terminate the diffusion after a suitable time is given. The performance of the introduced method is examined quantitatively using image sequences with a substantial amount of noise added. 1.

This paper presents a novel approach to unsupervised texture segmentation that relies on a very general nonparametric statistical model of image neighborhoods. Themethod models image neighborhoods directly, without the construction of intermediate features. It does not rely on using specific descriptors that work for certain kinds of textures, but is rather based on a more generic approach that tries to adaptively capture the core properties of textures. It exploits the fundamental description of textures as images derived from stationary random fields and models the associated higher-order statistics nonparametrically. This general formulation enables the method to easily adapt to various kinds of textures. The method minimizes an entropy-based metric on the probability density functions of image neighborhoods to give an optimal segmentation. The entropy minimization drives a very fast level-set scheme that uses threshold dynamics, which allows for a very rapid evolution towards the optimal segmentation during the initial iterations. The method does not rely on a training stage and, hence, is unsupervised. It automatically tunes its important internal parameters based on the information content of the data. The method generalizes in a straightforward manner from the two-region case to an arbitrary number of regions and incorporates an efficient multi-phase level-set framework. This paper presents numerous results, for both the two-texture and multiple-texture cases, using synthetic and real images that include electron-microscopy images.

Consider the problem of image deblurring in the presence of impulsive noise. Standard image deconvolution methods rely on the Gaussian noise model and do not perform well with impulsive noise. The main challenge is to deblur the image, recover its discontinuities and at the same time remove the impulse noise. Median-based approaches are inadequate, because at high noise levels they induce nonlinear distortion that hampers the deblurring process. Distinguishing outliers from edge elements is difficult in current gradient-based edge-preserving restoration methods. The suggested approach integrates and extends the robust statistics, line process (half quadratic) and anisotropic diffusion points of view. We present a unified variational approach to image deblurring and impulse noise removal. The objective functional consists of a fidelity term and a regularizer. Data fidelity is quantified using the robust modified L 1 norm, and elements from the Mumford-Shah functional are used for regularization. We show that the Mumford-Shah regularizer can be viewed as an extended line process. It reflects spatial organization properties of the image edges, that do not appear in the common line process or anisotropic diffusion. This allows to distinguish outliers from edges and leads to superior experimental results. 1

We propose a new technique for edge-suppressing operations on images. We introduce cross projection tensors to achieve affine transformations of gradient fields. We use these tensors, for example, to remove edges in one image based on the edge-information in a second image. Traditionally, edge suppression is achieved by setting image gradients to zero based on thresholds. A common application is in the Retinex problem, where the illumination map is recovered by suppressing the reflectance edges, assuming it is slowly varying.

Abstract. In this paper, we present a novel framework to carry out computations on tensors, i.e. symmetric positive definite matrices. We endow the space of tensors with an affine-invariant Riemannian metric, which leads to strong theoretical properties: The space of positive definite symmetric matrices is replaced by a regular and geodesically complete manifold without boundaries. Thus, tensors with non-positive eigenvalues are at an infinite distance of any positive definite matrix. Moreover, the tools of differential geometry apply and we generalize to tensors numerous algorithms that were reserved to vector spaces. The application of this framework to the processing of diffusion tensor images shows very promising results. We apply this framework to the processing of structure tensor images and show that it could help to extract low-level features thanks to the affine-invariance of our metric. However, the same affine-invariance causes the whole framework to be noise sensitive and we believe that the choice of a more adapted metric could significantly improve the robustness of the result. 1

Three-dimensional rotational angiography (3DRA) is a new and promising technique for obtaining high-resolution isotropic 3D images of vascular structures. However, due to the relatively high noise level and the presence of other background structures in clinical 3DRA images, noise reduction is inevitable. In this paper, we evaluate a number of linear and nonlinear diffusion techniques for this purpose. Specifically, we analyze the effects of these techniques on the thresholdbased visualization and quantification of vascular anomalies in 3DRA images. The results of invitro experiments indicate that edge-enhancing anisotropic diffusion filtering is most suitable: the increase in the user-dependency of visualizations and quantifications is considerably less with this technique compared to linear filtering techniques, and it is better at reducing noise near edges than isotropic nonlinear diffusion. However, in view of the memory and computation-time requirements of this technique, the latter scheme may be considered a useful alternative.

The problem of denoising images is one of the most important and widely studied problems in image processing and computer vision. Various image filtering strategies based on linear systems, statistics, information theory, and variational calculus, have been effective, but invariably make strong assumptions about the properties of the signal and/or noise. Therefore, they lack the generality to be easily applied to new applications or diverse image collections. This paper describes a novel unsupervised, information-theoretic, adaptive filter (UINTA) that improves the predictability of pixel intensities from their neighborhoods by decreasing the joint entropy between them. In this way UINTA automatically discovers the statistical properties of the signal and can thereby reduce noise in a wide spectrum of images and applications. The paper describes the formulation required to minimize the joint entropy measure, presents several important practical considerations in estimating image-region statistics, and then presents a series of results and comparisons on both real and synthetic data. Image Denoising with Unsupervised,