Showing papers by "Nir Sochen published in 2007"

Deblurring of Color Images Corrupted by Impulsive Noise

Abstract: 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

A Short- Time Beltrami Kernel for Smoothing Images and Manifolds

Abstract: We introduce a short-time kernel for the Beltrami image enhancing flow. The flow is implemented by "convolving" the image with a space dependent kernel in a similar fashion to the solution of the heat equation by a convolution with a Gaussian kernel. The kernel is appropriate for smoothing regular (flat) 2-D images, for smoothing images painted on manifolds, and for simultaneously smoothing images and the manifolds they are painted on. The kernel combines the geometry of the image and that of the manifold into one metric tensor, thus enabling a natural unified approach for the manipulation of both. Additionally, the derivation of the kernel gives a better geometrical understanding of the Beltrami flow and shows that the bilateral filter is a Euclidean approximation of it. On a practical level, the use of the kernel allows arbitrarily large time steps as opposed to the existing explicit numerical schemes for the Beltrami flow. In addition, the kernel works with equal ease on regular 2-D images and on images painted on parametric or triangulated manifolds. We demonstrate the denoising properties of the kernel by applying it to various types of images and manifolds

Prior-based Segmentation and Shape Registration in the Presence of Perspective Distortion

Abstract: Challenging object detection and segmentation tasks can be facilitated by the availability of a reference object. However, accounting for possible transformations between the different object views, as part of the segmentation process, remains difficult. Recent statistical methods address this problem by using comprehensive training data. Other techniques can only accommodate similarity transformations. We suggest a novel variational approach to prior-based segmentation, using a single reference object, that accounts for planar projective transformation. Generalizing the Chan-Vese level set framework, we introduce a novel shape-similarity measure and embed the projective homography between the prior shape and the image to segment within a region-based segmentation functional. The proposed algorithm detects the object of interest, extracts its boundaries, and concurrently carries out the registration to the prior shape. We demonstrate prior-based segmentation on a variety of images and verify the accuracy of the recovered transformation parameters.

Variational Stereo Vision with Sharp Discontinuities and Occlusion Handling

Abstract: This paper addresses the problem of correspondence establishment in binocular stereo vision. We suggest a novel variational approach that considers both the discontinuities and occlusions. It deals with color images as well as gray levels. The proposed method divides the image domain into the visible and occluded regions where each region is handled differently. The depth discontinuities in the visible domain are preserved by use of the total variation term in conjunction with the Mumford-Shah framework. In addition to the dense disparity and the occlusion maps, our method also provides a discontinuity function revealing the location of the boundaries in the disparity map. We evaluate our method on data sets from Middlebury site showing superior performance in comparison to the state of the art variational technique.

Restoration of images with piecewise space-variant blur

Abstract: We address the problem of space-variant image deblurring, where different parts of the image are blurred by different blur kernels. Assuming a region-wise space variant point spread function, we first solve the problem for the case of known blur kernels and known boundaries between the different blur regions in the image. We then generalize the method to the challenging case of unknown boundaries between the blur domains. Using variational and level set techniques, the image is processed globally. The space-variant deconvolution process is stabilized by a unified common regularizer, thus preserving discontinuities between the differently restored image regions. In the case where the blurred subregions are unknown, a segmentation procedure is performed using an evolving level set function, guided by edges and image derivatives.

Fast Invariant Riemannian DT-MRI Regularization

Abstract: We present regularization by invariant denoising/smoothing of Diffusion Tensor MRI (DTI). Our solution to the problem emerges from a pure geometric point of view. The image domain and the image's values are combined together and described as a (mathematical) fiber bundle. The space of all possible DT images is the space of sections of this fiber bundle. DT image is a map that attaches a three-dimensional symmetric and positive-definite (SPD) matrix to each volume element. We treat the more general space Pn of n-dimensional SPD matrices and introduce a natural GL(n)-invariant metric via the underlying algebraic structure. A metric over sections of the fiber bundle is induced then in terms of the natural metric on Pn. This turns P3 tensors, and in general Pn tensors, into a Riemannian symmetric spaces. By means of the Beltrami framework we define a GL(n)-invariant functional over the space of sections. Then, by calculus of variations we derive the invariant equations of motion. We show that by choosing the Iwasawa coordinates the analytical calculations as well as the numerical implementation become simple. These coordinates evolve with respect to the geometry of the section via the induced metric. The numerical implementation of these flows via standard finite difference schemes is straightforward. The result is a full GL(n) invariant algorithm which is at least as fast and efficient as the Log-Euclidean method. Finally, we demonstrate this framework on real DTI data.

Can Born Approximate the Unborn? A New Validity Criterion for the Born Approximation in Microscopic Imaging

Abstract: The Nomarski differential interference contrast (DIC) microscopy is of widespread use for observing live biological specimens. In fertility clinics the DIC microscope is used for evaluating human embryo cells. An image formation model for DIC imaging is needed for reconstruction and quantification of the visualized specimens. This calls for a complicated analysis of the interaction of light waves with biological matter. Most works express the solution via the first Born approximation, yet a theoretical bound is known that limits the validity of such approximation to very small objects. We show in this work that the theoretical bound is not directly relevant to microscopic imaging and is far too limiting. We derive a more realistic bound and show that it may justify in many cases the use of the Born approximation in biological cell microscopic imaging. It also provides limits on the validity of the Born expansion that several works violate.

Coordinate-free diffusion over compact Lie-groups

Abstract: We have seen in recent years a need for regularization of complicated feature spaces: Vector fields, orientation fields, color perceptual spaces, the structure tensor and Diffusion Weighted Images (DWI) are few examples. In most cases we represent the feature space as a manifold. In the proposed formalism, the image is described as a section of a fiber bundle where the image domain is the base space and the feature space is the fiber. In some distinguished cases the feature space has algebraic structure as well. In the proposed framework we treat fibers which are compact Lie-group manifolds (e.g., O(N), SU(N)). We study here this case and show that the algebraic structure can help in defining a sensible regularization scheme. We solve the parameterization problem of compact manifold that is responsible for singularities anytime that one wishes to describe in one coordinate system a compact manifold. The proposed solution defines a coordinate-free diffusion process accompanied by an appropriate numerical scheme. We demonstrate this framework in an example of S1 feature space regularization which is known also as orientation diffusion.

Convergence of an Iterative Method for Variational Deconvolution and Impulsive Noise Removal

Abstract: Image restoration, i.e., the recovery of images that have been degraded by blur and noise, is a challenging inverse problem. A unified variational approach to edge-preserving image deconvolution and impulsive noise removal has recently been suggested by the authors and shown to be effective. It leads to a minimization problem that is iteratively solved by alternate minimization for both the recovered image and the discontinuity set. The variational formulation yields a nonlinear integro-differential equation. This equation was linearized by fixed point iteration. In this paper, we analyze and prove the convergence of the iterative method.

On the Discrete Maximum Principle for the Beltrami Color Flow

Abstract: We analyze the discrete maximum principle for the Beltrami color flow. The Beltrami flow can display linear as well as nonlinear behavior according to the values of a parameter β, which represents the ratio between spatial and color distances. In general, the standard schemes fail to satisfy the discrete maximum principle. In this work we show that a nonnegative second order difference scheme can be built for this flow only for small β, i.e. linear-like diffusion. Since this limitation is too severe, we construct a novel finite difference scheme, which is not nonnegative and satisfies the discrete maximum principle for all values of β. Numerical results support the analysis.

Propagating distributions for segmentation of brain atlas

Abstract: We present a novel method for segmentation of anatomical structures in histological data. Segmentation is carried out slice-by-slice where the successful segmentation of one section provides a prior for the subsequent one. Intensities and spatial locations of the region of interest and the background are modeled by three-dimensional Gaussian mixtures. This information adaptively propagates across the sections. Segmentation is inferred by minimizing a cost functional that enforces the compatibility of the partitions with the corresponding models together with the alignment of the boundaries with the image gradients. The algorithm is demonstrated on histological images of mouse brain. The segmentation results compare well with manual segmentation.

Regularization of diffusion tensor MRI via local coordinates

Abstract: We propose a novel framework for regularization of symmetric positive-definite (SPD) tensors (e.g., diffusion tensors). This framework is based on differential geometry. The space of SPD matrices, Pn, is described as a Riemannian manifold that is parameterized via the Iwasawa coordinate system. Then, distances on Pn are measured in terms of a natural GL (n)-invariant Riemannian metric. Using the Beltrami framework we construct a set of coupled geometric PDEs with respect to the Iwasawa coordinates. Then, by means of the gradient descent method these equations define the regularization flow over Pn. It appears to be that the local coordinate approach via that coordinate system results in very simple numerics that leads to fast convergence of the algorithm. We demonstrate the efficiency of this algorithm on real volumetric DTI datasets. Results of fibers tractography before and afterthe regularization process arepresented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Full affine wavelets are scale-space with a twist

Abstract: In this work we study the relation between the Gabor-Morlet wavelet transform and scale-space theory. It is shown that the usual wavelet transform is a projection of scale-space on a specific frequency component. This result is then generalized to the full two-dimensional affine group. A close relation between this generalized wavelet transform and a family of scale-spaces of images that are related by SL(2) is established. Using frame theory we show that sampling from different images in this family, and from different scales enables a complete reconstruction of the image.

Anisotropic α-kernels and associated flows

Abstract: The Laplacian raised to fractional powers can be used to generate scale spaces as was shown in recent literature by Duits et al In this paper, we study the anisotropic diffusion processes by defining new generators that are fractional powers of an anisotropic scale space generator This is done in a general framework that allows us to explain the relation between a differential operator that generates the flow and the generators that are constructed from its fractional powers We then generalize this to any other function of the operator We discuss important issues involved in the numerical implementation of this framework and present several examples of fractional versions of the Perona-Malik flow along with their properties

Anisotropic alpha -Kernels and Associated Flows.

Abstract: The Laplacian raised to fractional powers can be used to generate scale spaces as was shown in recent literature by Duits, Felsberg, Florack, and Platel [$\alpha$ scale spaces on a bounded domain, in Scale Space Methods in Computer Vision, L. D. Griffin and M. Lillholm, eds., Lecture Notes in Comput. Sci. 2695, Springer, Berlin, Heidelberg, 2003, pp. 494-510] and Duits, Florack, de Graaf, and ter Haar Romeny [J. Math. Imaging Vision, 20 (2004), pp. 267-298]. In this paper, we study the anisotropic diffusion processes by defining new generators that are fractional powers of an anisotropic scale space generator. This is done in a general framework that allows us to explain the relation between a differential operator that generates the flow and the generators that are constructed from its fractional powers. We then generalize this to any other function of the operator. We discuss important issues involved in the numerical implementation of this framework and present several examples of fractional versions of the Perona-Malik and Beltrami flows along with their properties.