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We address the problem of blind piecewise space-variant image deblurring where only part of the image is sharp, assuming a shallow depth of field which imposes significant defocus blur.

In this paper concepts from continuum mechanics are used to define geodesic paths in the space of shapes, where shapes are implicitly described as boundary contours of objects. The proposed shape metric is derived from a continuum mechanical notion of viscous dissipation. A geodesic path is defined as the family of shapes such that the total amount of viscous dissipation caused by an optimal material transport along the path is minimized. The approach can easily be generalized to shapes given as segment contours of multi-labeled images and to geodesic paths between partially occluded objects. The proposed computational framework for finding such a minimizer is based on the time discretization of a geodesic path as a sequence of pairwise matching problems, which is strictly invariant with respect to rigid body motions and ensures a 1---1 correspondence along the induced flow in shape space. When decreasing the time step size, the proposed model leads to the minimization of the actual geodesic length, where the Hessian of the pairwise matching energy reflects the chosen Riemannian metric on the underlying shape space. If the constraint of pairwise shape correspondence is replaced by the volume of the shape mismatch as a penalty functional, one obtains for decreasing time step size an optical flow term controlling the transport of the shape by the underlying motion field. The method is implemented via a level set representation of shapes, and a finite element approximation is employed as spatial discretization both for the pairwise matching deformations and for the level set representations. The numerical relaxation of the energy is performed via an efficient multi-scale procedure in space and time. Various examples for 2D and 3D shapes underline the effectiveness and robustness of the proposed approach.

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.

Image restoration and segmentation are both classical problems, that are known to be difficult and have attracted major research efforts. This paper shows that the two problems are tightly coupled and can be successfully solved together. Mutual support of image restoration and segmentation processes within a joint variational framework is theoretically motivated, and validated by successful experimental results. The proposed variational method integrates semi-blind image deconvolution (parametric blur-kernel), and Mumford-Shah segmentation. The functional is formulated using the /spl Gamma/-convergence approximation and is iteratively optimized via the alternate minimization method. While the major novelty of this work is in the unified treatment of the semi-blind restoration and segmentation problems, the important special case of known blur is also considered and promising results are obtained.

Sparse representation theory has been increasingly used in signal processing and machine learning. In this paper we introduce a hierarchical sparse modeling approach which integrates information from the image patch level to derive a mid-level invariant image and pattern representation. The proposed framework is based on a hierarchical architecture of dictionary learning for sparse coding in a cortical (log-polar) space, combined with a novel pooling operator which incorporates the Rapid transform and max pooling to attain rotation and scale invariance. The invariant sparse representation of patterns here presented - can be used in different object recognition tasks. Promising results are obtained for three applications — 2D shapes classification, texture recognition and object detection.

Sparse representation theory has been increasingly used in the fields of signal processing and machine learning. The standard sparse models are not invariant to spatial transformations such as image rotations, and the representation is very sensitive even under small such distortions. Most studies addressing this problem proposed algorithms which either use transformed data as part of the training set, or are invariant or robust only under minor transformations. In this paper we suggest a framework which extracts sparse features invariant under significant rotations and scalings. The algorithm is based on a hierarchical architecture of dictionary learning for sparse coding in a cortical (log-polar) space. The proposed model is tested in supervised classification applications and proved to be robust under transformed data.

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.

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.

The problem of motion estimation and restoration of objects in a blurred video sequence is addressed in this paper. Fast movement of the objects, together with the aperture time of the camera, result in a motion-blurred image. The direct velocity estimation from this blurred video is inaccurate. On the other hand, an accurate estimation of the velocity of the moving objects is critical for restoration of motion-blurred video. Therefore, restoration needs accurate motion estimation and vice versa, and a joint process is called for. To address this problem we derive a novel model of the blurring process and propose a Mumford-Shah type of variational framework, acting on consecutive frames, for joint object deblurring and velocity estimation. The proposed procedure distinguishes between the moving object and the background and is accurate also close to the boundary of the moving object. Experimental results both on simulated and real data show the importance of this joint estimation and its superior performance when compared to the independent estimation of motion and restoration.

The problem of image deblurring in the presence of salt and pepper noise is considered. Standard image deconvolution algorithms, that are designed for Gaussian noise, do not perform well in this case. Median type filtering is a common method for salt and pepper noise removal. Deblurring an image that has been preprocessed by median-type filtering is however difficult, due to the amplification (in the deconvolution stage) of median-induced distortion. A unified variational approach to salt and pepper noise removal and image deblurring is presented. An objective functional that represents the goals of deblurring, noise-robustness and compliance with the piecewise-smooth image model is formulated. A modified L1 data fidelity term integrates deblurring with robustness to outliers. Elements from the Mumford-Shah functional, that favor piecewise smooth images with simple edge-sets, are used for regularization. Promising experimental results are shown for several blur models.

We propose a variational approach for deblurring and impulsive noise removal in multi-channel images. A robust data fidelity measure and edge preserving regularization are employed. We consider several regularization approaches, such as Beltrami flow, Mumford-Shah and Total-Variation Mumford-Shah. The latter two methods are extended to multi-channel images and reformulated using the Γ-convergence approximation. Our main contribution is in the unification of image deblurring and impulse noise removal in a multi-channel variational framework. Theoretical and experimental results show that the Mumford-Shah and Total Variation Mumford Shah regularization methods are superior to other color image restoration regularizers. In addition, these two methods yield a denoised edge map of the image.