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31
Recovery algorithms for vector valued data with joint sparsity constraints
, 2006
"... 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 ..."
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Cited by 113 (23 self)
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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 nonvanishing 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 updated after each new application of the thresholded Landweber algorithm. The resulting twostep algorithm is interpreted as a doubleminimization scheme for a suitable target functional. We show its ℓ2norm convergence. An implementable version of the algorithm is also formulated, and its norm convergence is proven. Numerical experiments in color image restoration are presented.
An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise
 SIAM J. SCI. COMPUT
, 2009
"... 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 anis ..."
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Cited by 51 (8 self)
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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 L1norm. We derive the algorithm by applying the wellknown quadratic penalty function technique and prove attractive convergence properties including finite convergence for some variables and global qlinear convergence. Under periodic boundary conditions, the main computational requirements of the algorithm are fast Fourier transforms and a lowcomplexity 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.
A fast algorithm for edgepreserving variational multichannel image restoration
"... Abstract. We generalize the alternating minimization algorithm recently proposed in [32] to efficiently solve a general, edgepreserving, variational model for recovering multichannel images degraded by within and crosschannel blurs, as well as additive Gaussian noise. This general model allows th ..."
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Cited by 46 (9 self)
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Abstract. We generalize the alternating minimization algorithm recently proposed in [32] to efficiently solve a general, edgepreserving, variational model for recovering multichannel images degraded by within and crosschannel blurs, as well as additive Gaussian noise. This general model allows the use of localized weights and higherorder 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 halfquadratic 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 periteration 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 qlinear 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 higherorder derivatives to demonstrate improvements in image quality provided by these models over the plain MTV model.
Restoration of color images by vector valued BV functions and variational calculus
 SIAM J. Appl. Math
, 2006
"... 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 incomplet ..."
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Cited by 20 (12 self)
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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 Γconvergence. Then we realize approximations of minimizers of the latter functionals by an iterative procedure to solve the PDE system of the corresponding EulerLagrange 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.
Image restoration via nonstandard diffusion
, 2004
"... 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 l ..."
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Cited by 14 (0 self)
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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.
Deblurring of Color Images Corrupted by Impulsive Noise
, 2006
"... We consider the problem of restoring a multichannel image corrupted by blur and impulsive noise (e.g. saltandpepper noise). Using the variational framework, we consider the L¹ fidelity term and several possible regularizers. In particular, we use generalizations of the MumfordShah functional to ..."
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Cited by 13 (1 self)
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We consider the problem of restoring a multichannel image corrupted by blur and impulsive noise (e.g. saltandpepper noise). Using the variational framework, we consider the L¹ fidelity term and several possible regularizers. In particular, we use generalizations of the MumfordShah functional to color images and Γconvergence approximations to unify deblurring and denoising. Experimental comparisons show that the MumfordShah stabilizer yields better results with respect to Beltrami and Total Variation regularizers. Color edge detection is a beneficial byproduct of our methods.
MumfordShah Regularizer with Contextual Feedback
 JOURNAL OF MATHEMATICAL IMAGING AND VISION
"... 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 no ..."
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Cited by 10 (0 self)
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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 multiscale patterns and textures. Key words variational and PDE methods, feature preserving diffusion, structure preserving diffusion, disambiguation in low level vision. vision. We interpret the AmbrosioTortorelli approximation of the MumfordShah 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
Flux Tensor Constrained Geodesic Active Contours with Sensor Fusion for Persistent Object Tracking
"... 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 requiri ..."
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Cited by 10 (5 self)
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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 tensorbased motion detector when applied to infrared video is more accurate than thresholding ”hotspots”, 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 setbased 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 filterbased 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.
Parameterless discrete regularization on graphs for color image filtering
 in Image Analysis and Recognition, ser. Lecture Notes in Computer Science
, 2007
"... Abstract. A discrete regularization framework on graphs is proposed and studied for color image filtering purposes when images are represented by grid graphs. Image filtering is considered as a variational problem which consists in minimizing an appropriate energy function. In this paper, we propose ..."
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Cited by 4 (4 self)
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Abstract. A discrete regularization framework on graphs is proposed and studied for color image filtering purposes when images are represented by grid graphs. Image filtering is considered as a variational problem which consists in minimizing an appropriate energy function. In this paper, we propose a general discrete regularization framework defined on weighted graphs which can be seen as a discrete analogue of classical regularization theory. With this formulation, we propose a family of fast and simple anisotropic linear and nonlinear filters. The parameters of the proposed discrete regularization are estimated to have a parameterless filtering. 1
Y.Y.: Minimal surfaces, measurebased metric and image segmentation
, 2006
"... One of the primary goals of low level vision is image segmentation: given data g, defined as a function on the ”pixel space ” B, the objective is to deduce an image u which is composed of subdomains, wherein the image is basically homogeneous, separated by a sharp discontinuities (edges). It has be ..."
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Cited by 3 (3 self)
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One of the primary goals of low level vision is image segmentation: given data g, defined as a function on the ”pixel space ” B, the objective is to deduce an image u which is composed of subdomains, wherein the image is basically homogeneous, separated by a sharp discontinuities (edges). It has been shown that a large number of algorithms for image segmentation are closely related to the MumfordShah functional minimization [42]. This functional involves a tradeoff between the image structure, which is a twodimensional surface, and the contours that surround objects or distinct regions in the image, which are onedimensional parametric curves. This functional was first suggested and analyzed in its onedimensional case by Mumford and Shah for gray level images [41]. The above functional was later extensively studied, (see e.g. [40] for an overview). In particular, the Γconvergence framework [46] was invented to overcome the problem of dealing with objects with different dimentionalities in the same functional. The idea is to approximate the functional by a different, parameter dependent functional, that is expected to be more regular. The approximating functional approaches the original one in the limit, while the parameter goes to zero. According to this approach, minimizers of approximating functional approximate the minimizer of original one, while enjoying greater regularity. In this study we propose an alternative functional to MumfordShah’s one. The proposed functional is independent of parameterization; it is a geometric functional which is given in terms of the geometry of surfaces representing the data and image in the feature space. The Γconvergence technique is merged with the minimal surfaces theory in order to yield a global generalization of the MumfordShah segmentation functional. Chapter 1