# INVERSE PROBLEMS IN IMAGE PROCESSING

# Analysis Operator Learning and Its Application to Image Reconstruction

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 the 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 an lp-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.

*S. Hawe, M. Kleinsteuber, and K. Diepold Analysis Operator Learning and Its Application to Image ReconstructionIEEE Trans. on Image Processing 22(6) 2138-2150, (2013).*[Preprint] Matlab Implementation: [Code] Citation: [BibTeX]

# Separable Dictionary Learning

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 image processing, 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.

*S. Hawe, M. Seibert, and M. Kleinsteuber Separable Dictionary LearningIEEE Conference on Computer Vision and Pattern Recognition (2013).*[Preprint] Matlab Implementation: [Code]

# Analysis Based Blind Compressive Sensing

In this work we address the problem of blindly reconstructing compressively sensed signals by exploiting the co-sparse analysis model. In the analysis model it is assumed that a signal multiplied by an analysis operator results in a sparse vector. We propose an algorithm that learns the operator adaptively during the reconstruction process. The arising optimization problem is tackled via a geometric conjugate gradient approach. Different types of sampling noise are handled by simply exchanging the data fidelity term. Numerical experiments are performed for measurements corrupted with Gaussian as well as impulsive noise to show the effectiveness of our method.

# Disparity Map Reconstruction via Convex Optimization

In this work we propose a method for estimating dense disparity maps from very few disparity measurements. Based on the theory of Compressive Sensing, our algorithm accurately reconstructs disparity maps only using about 5% of the entire map. We propose a conjugate subgradient method for the arising optimization problem that is applicable to large scale systems and recovers the disparity map efficiently. Our experiments show the effectiveness of the proposed approach and its robust behavior under noisy conditions.

**Code:**

*S. Hawe, M. Kleinsteuber, and K. Diepold*

**Dense Disparity Maps from Sparse Disparity Measurements**

Proceedings of the ICCV 2011

Citation: [bibTeX] Preprint: [PDF]

# Non-convex Image Reconstruction

This paper considers the problem of reconstructing images from only a few measurements. A method is proposed that is based on the theory of Compressive Sensing. We introduce a new prior that combines an lp-pseudo-norm approximation of the image gradient and the bounded range of the original signal. Ultimately, this leads to a reconstruction algorithm that works particularly well for Cartoon-like images that commonly occur in medical imagery. The arising optimization task is solved by a Conjugate Gradient method that is capable of dealing with large scale problems and easily adapts to extensions of the prior. To overcome the none differentiability of the lp-pseudo-norm we employ a Huber-loss term like approximation together with a continuation of the smoothing parameter. Numerical results and a comparison with the stateof-the-art methods show the effectiveness of the proposed algorithm

**Code:**

*S. Hawe, M. Kleinsteuber, and K. Diepold*

**Carton-like image reconstruction via constrained lp-minimization**

Proceedings of the ICASSP 2012, pp. 717-720, March 2012.

Citation: [bibTeX] Preprint: [PDF]