http:// Exploiting a priori known structural information lies at the core of many image reconstruction methods that can be stated as inverse problems. The synthesis model, which assumes that images can be decomposed into a linear combination of very few atoms of some dictionary, is now a well established tool for the design of image reconstruction algorithms. An interesting alternative is the analysis model, where the signal is multiplied by an analysis operator and the outcome is assumed to be sparse. This approach has only recently gained increasing interest. The quality of reconstruction methods based on an analysis model severely depends on the right choice of the suitable operator.In this work, we present an algorithm for learning an analysis operator from training images. Our method is based on p -norm minimization on the set of full rank matrices with normalized columns. We carefully introduce the employed conjugate gradient method on manifolds, and explain the underlying geometry of the constraints. Moreover, we compare our approach to state-of-the-art methods for image denoising, inpainting, and single image super-resolution. Our numerical results show competitive performance of our general approach in all presented applications compared to the specialized state-of-the-art techniques.

Many techniques in computer vision, machine learning, and statistics rely on the fact that a signal of interest admits a sparse representation over some dictionary. Dictionaries are either available analytically, or can be learned from a suitable training set. While analytic dictionaries permit to capture the global structure of a signal and allow a fast implementation, learned dictionaries often perform better in applications as they are more adapted to the considered class of signals. In imagery, unfortunately, the numerical burden for (i) learning a dictionary and for (ii) employing the dictionary for reconstruction tasks only allows to deal with relatively small image patches that only capture local image information.The approach presented in this paper aims at overcoming these drawbacks by allowing a separable structure on the dictionary throughout the learning process. On the one hand, this permits larger patch-sizes for the learning phase, on the other hand, the dictionary is applied efficiently in reconstruction tasks. The learning procedure is based on optimizing over a product of spheres which updates the dictionary as a whole, thus enforces basic dictionary properties such as mutual coherence explicitly during the learning procedure. In the special case where no separable structure is enforced, our method competes with state-of-the-art dictionary learning methods like K-SVD.

Abstract

Many modern tools in machine learning and signal processing, such as sparse dictionary learning, principal component analysis, non-negative matrix factorization, K -means clustering, and so on, rely on the factorization of a matrix obtained by concatenating high-dimensional vectors from a training collection. While the idealized task would be to optimize the expected quality of the factors over the underlying distribution of training vectors, it is achieved in practice by minimizing an empirical average over the considered collection. The focus of this paper is to provide sample complexity estimates to uniformly control how much the empirical average deviates from the expected cost function. Standard arguments imply that the performance of the empirical predictor also exhibit such guarantees. The level of genericity of the approach encompasses several possible constraints on the factors (tensor product structure, shift-invariance, sparsity…), thus providing a unified perspective on the sample complexity of several widely used matrix factorization schemes. The derived generalization bounds behave proportional to (log(n)/n) 1/2 with respect to the number of samples n for the considered matrix factorization techniques.Index Terms-Dictionary learning, sparse coding, principal component analysis, K -means clustering, non-negative matrix factorization, structured learning, sample complexity.

High-resolution depth maps can be inferred from low-resolution depth measurements and an additional high-resolution intensity image of the same scene. To that end, we introduce a bimodal co-sparse analysis model, which is able to capture the interdependency of registered intensity and depth information. This model is based on the assumption that the co-supports of corresponding bimodal image structures are aligned when computed by a suitable pair of analysis operators. No analytic form of such operators exist and we propose a method for learning them from a set of registered training signals. This learning process is done offline and returns a bimodal analysis operator that is universally applicable to natural scenes. We use this to exploit the bimodal co-sparse analysis model as a prior for solving inverse problems, which leads to an efficient algorithm for depth map super-resolution.

No abstract

An increasing number of methods for background subtraction use Robust PCA to identify sparse foreground objects. While many algorithms use the 1 -norm as a convex relaxation of the ideal sparsifying function, we approach the problem with a smoothed p -norm and present pROST, a method for robust online subspace tracking. The algorithm is based on alternating minimization on manifolds. Implemented on a graphics processing unit it achieves realtime performance. Experimental results on a state-of-the-art benchmark for background subtraction on real-world video data indicate that the method succeeds at a broad variety of background subtraction scenarios, and it outperforms competing approaches when video quality is deteriorated by camera jitter.

Motivated by applications in quantum information and quantum control, a new type of C-numerical range, the relative C-numerical range denoted W K (C, A), is introduced. It arises upon replacing the unitary group U (N ) in the definition of the classical C-numerical range by any of its compact and connected subgroups K ⊂ U (N ).The geometric properties of the relative C-numerical range are analysed in detail. Counterexamples prove its geometry is more intricate than in the classical case: e.g. W K (C, A) is neither star-shaped nor simply-connected. Yet, a well-known result on the rotational symmetry of the classical C-numerical range extends to W K (C, A), as shown by a new approach based on Lie theory. Furthermore, we concentrate on the subgroup SU loc (2 n ) := SU (2) ⊗ • • • ⊗ SU (2), i.e. the n-fold tensor product of SU (2), which is of particular interest in applications. In this case, sufficient conditions are derived for W K (C, A) being a circular disc centered at origin of the complex plane. Finally, the previous results are illustrated in detail for SU (2) ⊗ SU (2).