Abstract. Vector valued data appearing in concrete applications often possess sparse expansions with respect to a preassigned frame for each vector component individually. Additionally, different components may also exhibit common sparsity patterns. Recently, there were introduced sparsity measures that take into account such joint sparsity patterns, promoting coupling of non-vanishing components. These measures are typically constructed as weighted ℓ1 norms of componentwise ℓq norms of frame coefficients. We show how to compute solutions of linear inverse problems with such joint sparsity regularization constraints by fast thresholded Landweber algorithms. Next we discuss the adaptive choice of suitable weights appearing in the definition of sparsity measures. The weights are interpreted as indicators of the sparsity pattern and are iteratively up-dated after each new application of the thresholded Landweber algorithm. The resulting two-step algorithm is interpreted as a double-minimization scheme for a suitable target functional. We show its ℓ2-norm convergence. An implementable version of the algorithm is also formulated, and its norm convergence is proven. Numerical experiments in color image restoration are presented.

Abstract. We analyze a variational problem for the recovery of vector valued functions and we compute its numerical solution. The data of the problem are a small set of complete samples of the vector valued function and a significant incomplete information where the former are missing. The incomplete information is assumed as the result of a distortion, with values in a lower dimensional manifold. For the recovery of the function we minimize a functional which is formed by the discrepancy with respect to the data and total variation regularization constraints. We show existence of minimizers in the space of vector valued BV functions. For the computation of minimizers we provide a stable and efficient method. First we approximate the functional by coercive functionals on W 1,2 in terms of Γ-convergence. Then we realize approximations of minimizers of the latter functionals by an iterative procedure to solve the PDE system of the corresponding Euler-Lagrange equations. The numerical implementation comes naturally by finite element discretization. We apply the algorithm to the restoration of color images from a limited color information and gray levels where the colors are missing. The numerical experiments show that this scheme is very fast and robust. The reconstruction capabilities of the model are shown, also from very limited (randomly distributed) color data. Several examples are included from the real restoration problem of the A. Mantegna’s art frescoes in Italy.

We extend the alternating minimization algorithm recently proposed in [38, 39] to the case of recovering blurry multichannel (color) images corrupted by impulsive rather than Gaussian noise. The algorithm minimizes the sum of a multichannel extension of total variation (TV), either isotropic or anisotropic, and a data fidelity term measured in the L1-norm. We derive the algorithm by applying the well-known quadratic penalty function technique and prove attractive convergence properties including finite convergence for some variables and global q-linear convergence. Under periodic boundary conditions, the main computational requirements of the algorithm are fast Fourier transforms and a low-complexity Gaussian elimination procedure. Numerical results on images with different blurs and impulsive noise are presented to demonstrate the efficiency of the algorithm. In addition, it is numerically compared to an algorithm recently proposed in [20] that uses a linear program and an interior point method for recovering grayscale images.

Abstract. We generalize the alternating minimization algorithm recently proposed in [32] to efficiently solve a general, edge-preserving, variational model for recovering multichannel images degraded by within- and cross-channel blurs, as well as additive Gaussian noise. This general model allows the use of localized weights and higher-order derivatives in regularization, and includes a multichannel extension of total variation (MTV) regularization as a special case. In the MTV case, we show that the model can be derived from an extended half-quadratic transform of Geman and Yang [14]. For color images with three channels and when applied to the MTV model (either locally weighted or not), the per-iteration computational complexity of this algorithm is dominated by nine fast Fourier transforms. We establish strong convergence results for the algorithm including finite convergence for some variables and fast q-linear convergence for the others. Numerical results on various types of blurs are presented to demonstrate the performance of our algorithm compared to that of the MATLAB deblurring functions. We also present experimental results on regularization models using weighted MTV and higher-order derivatives to demonstrate improvements in image quality provided by these models over the plain MTV model.

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 L 1 fidelity term and several possible regu-larizers. In particular, we use generalizations of the Mumford-Shah functional to color images and Γ-convergence approximations to unify deblurring and denoising. Experimental comparisons show that the Mumford-Shah stabilizer yields better results with respect to Beltrami and Total Variation regularizers. Color edge detection is a beneficial by-product of our methods.

We present a simple and robust feature preserving image regularization by letting local region measures to modulate the diffusivity. The purpose of this modulation is to disambiguate low level cues in early both gray and color natural images demonstrate the potential of the method under difficult noise types, nonuniform contrast, existence of multi-scale patterns and textures. Key words variational and PDE methods, feature preserving diffusion, structure preserving diffusion, disambiguation in low level vision. vision. We interpret the Ambrosio-Tortorelli approximation of the Mumford-Shah model as a system with modulatory feedback and utilize this interpretation to integrate high level information into the regularization process. The method does not require any prior model or learning; the high level information is extracted from local regions and fed back to the regularization step. An important characteristic of the method is that both negative and positive feedback can be simultaneously used without creating oscillations. Experiments performed with

We present a functional of nonstandard growth for which the corresponding minimization problem provides a model for image denoising, enhancement, and restoration. The diffusion resulting from the proposed model is a combination of isotropic and anisotropic diffusion. Isotropic diffusion is used at locations with low gradient and total variation based diffusion is used along likely edges. At all other locations, the type of anisotropy varies according to the local image information. Experimental results illustrate the effectiveness of the model in removing noise and retaining sharp edges while avoiding the ’staircasing effect’. Existence and uniqueness of the proposed model are also established. 1.

Abstract. In this article we present the results of an unsupervised segmentation algorithm based on a multiresolution method. The algorithm uses color and edge information in an iterative minimization process of an energy function. The process has been applied to fruit images to distinguish the different areas of the fruit surface in fruit quality assessment applications. Due to the unsupervised nature of the procedure, it can adapt itself to the huge variability of colors and shapes of the regions in fruit inspection applications. 1

In this article, we present an unsupervised segmentation algorithm through a multiresolution approach which uses both color and edge information with a quadtree structure, through as well as an iterative minimization process of an energy function. The algorithm has been applied to fruit images in order to distinguish the different areas of the fruit surface in fruit quality assessment applications. Due to the unsupervised nature of the method, it can adapt itself to the huge variability of colors and shapes of the regions in fruit inspection tasks. 1.

Abstract — This paper makes new contributions in motion detection, object segmentation and trajectory estimation to create a successful object tracking system. A new efficient motion detection algorithm referred to as the flux tensor is used to detect moving objects in infrared video without requiring background modeling or contour extraction. The flux tensor-based motion detector when applied to infrared video is more accurate than thresholding ”hot-spots”, and is insensitive to shadows as well as illumination changes in the visible channel. In real world monitoring tasks fusing scene information from multiple sensors and sources is a useful core mechanism to deal with complex scenes, lighting conditions and environmental variables. The object segmentation algorithm uses level set-based geodesic active contour evolution that incorporates the fusion of visible color and infrared edge informations in a novel manner. Touching or overlapping objects are further refined during the segmentation process using an appropriate shapebased model. Multiple object tracking using correspondence graphs is extended to handle groups of objects and occlusion events by Kalman filter-based cluster trajectory analysis and watershed segmentation. The proposed object tracking algorithm was successfully tested on several difficult outdoor multispectral videos from stationary sensors and is not confounded by shadows or illumination variations. Index Terms — Flux tensor, sensor fusion, object tracking, active contours, level set, infrared images. I.