This paper provides novel results for the recovery of signals from undersampled measurements based on analysis @math -minimization, when the analysis operator is given by a frame. We both provide so-called uniform and nonuniform recovery guarantees for cosparse (analysis-sparse) signals using Gaussian random measurement matrices. The nonuniform result relies on a recovery condition via tangent cones and the uniform recovery guarantee is based on an analysis version of the null space property. Examining these conditions for Gaussian random matrices leads to precise bounds on the number of measurements required for successful recovery. In the special case of standard sparsity, our result improves a bound due to Rudelson and Vershynin concerning the exact reconstruction of sparse signals from Gaussian measurements with respect to the constant and extends it to stability under passing to approximately sparse signals and to robustness under noise on the measurements.
Also, we note that the optimization problems ) and ) often appear in image processing @cite_16 @cite_20 .
@cite_16 @cite_20
2092663520 2465405228 2007872281 2119088708
In this paper we study a first-order primal-dual algorithm for non-smooth convex optimization problems with known saddle-point structure. We prove convergence to a saddle-point with rate O(1 N) in finite dimensions for the complete class of problems. We further show accelerations of the proposed algorithm to yield improved rates on problems with some degree of smoothness. In particular we show that we can achieve O(1 N 2) convergence on problems, where the primal or the dual objective is uniformly convex, and we can show linear convergence, i.e. O(? N ) for some ??(0,1), on smooth problems. The wide applicability of the proposed algorithm is demonstrated on several imaging problems such as image denoising, image deconvolution, image inpainting, motion estimation and multi-label image segmentation. Several important tasks in medical image analysis can be stated in the form of an optimization problem whose feasible solutions are connected subgraphs. Examples include the reconstruction of neural or vascular structures under connectedness constraints. This paper deals with the problem of finding images that contain a given query image, the so-called content-based sub-image retrieval. We propose an approach based on a hierarchical tree that encodes the color feature of image tiles which are in turn stored as an index sequence. The index sequences of both candidate images and the query sub-image are then compared in order to rank the database images suitability with respect to the query. In our experiments, using 10,000 images and disk-resident metadata, for 60Σ (80Σ) of the queries the relevant image, i.e., the one where the query sub-image was extracted from, was found among the first 10 (50) retrieved images in about 0.15 sec. Recently, there has been interest in formulating various geometric problems in Computer Vision as L optimization problems. The advantage of this approach is that under L norm, such problems typically have a single minimum, and may be efficiently solved using Second-Order Cone Programming (SOCP). This paper shows that such techniques may be used effectively on the problem of determining the track of a camera given observations of features in the environment. The approach to this problem involves two steps: determination of the orientation of the camera by estimation of relative orientation between pairs of views, followed by determination of the translation of the camera. This paper focusses on the second step, that of determining the motion of the camera. It is shown that it may be solved effectively by using SOCP to reconcile translation estimates obtained for pairs or triples of views. In addition, it is observed that the individual translation estimates are not known with equal certainty in all directions. To account for this anisotropy in uncertainty, we introduce the use of covariances into the L optimization framework.