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Register a new userAnisotropic Total Variation Regularized L^1-Approximation and Denoising/Deblurring of 2D Bar Codes
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We consider variations of the Rudin-Osher-Fatemi functional which are particularly well-suited to denoising and deblurring of 2D bar codes. These functionals consist of an anisotropic total variation favoring rectangles and a fidelity term which measure the L^1 distance to the signal, both with and without the presence of a deconvolution operator. Based upon the existence of a certain associated vector field, we find necessary and sufficient conditions for a function to be a minimizer. We apply these results to 2D bar codes to find explicit regimes ---in terms of the fidelity parameter and smallest length scale of the bar codes--- for which a perfect bar code is recoverable via minimization of the functionals. Via a discretization reformulated as a linear program, we perform numerical experiments for all functionals demonstrating their denoising and deblurring capabilities.
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Using total variation based energy minimisation we address the recovery of a blurred (convoluted) one dimensional (1D) barcode. We consider functionals defined over all possible barcodes with fidelity to a convoluted signal of a barcode, and regularised by total variation. Our fidelity terms consist of the L^2 distance either directly to the measured signal or preceded by deconvolution. Key length scales and parameters are the X-dimension of the underlying barcode, the size of the supports of the convolution and deconvolution kernels, and the fidelity parameter. For all functionals, we establish regimes (sufficient conditions) wherein the underlying barcode is the unique minimiser. We also present some numerical experiments suggesting that these sufficient conditions are not optimal and the energy methods are quite robust for significant blurring.
Hyperspectral image (HSI) denoising aims to restore clean HSI from the noise-contaminated one. Noise contamination can often be caused during data acquisition and conversion. In this paper, we propose a novel spatial-spectral total variation (SSTV) regularized nonconvex local low-rank (LR) tensor approximation method to remove mixed noise in HSIs. From one aspect, the clean HSI data have its underlying local LR tensor property, even though the real HSI data may not be globally low-rank due to out-liers and non-Gaussian noise. According to this fact, we propose a novel tensor $L_{gamma}$-norm to formulate the local LR prior. From another aspect, HSIs are assumed to be piecewisely smooth in the global spatial and spectral domains. Instead of traditional bandwise total variation, we use the SSTV regularization to simultaneously consider global spatial structure and spectral correlation of neighboring bands. Results on simulated and real HSI datasets indicate that the use of local LR tensor penalty and global SSTV can boost the preserving of local details and overall structural information in HSIs.
Total Generalized Variation (TGV) has recently been proven certainly successful in image processing for preserving sharp features as well as smooth transition variations. However, none of the existing works aims at numerically calculating TGV over triangular meshes. In this paper, we develop a novel numerical framework to discretize the second-order TGV over triangular meshes. Further, we propose a TGV-based variational model to restore the face normal field for mesh denoising. The TGV regularization in the proposed model is represented by a combination of a first- and second-order term, which can be automatically balanced. This TGV regularization is able to locate sharp features and preserve them via the first-order term, while recognize smoothly curved regions and recover them via the second-order term. To solve the optimization problem, we introduce an efficient iterative algorithm based on variable-splitting and augmented Lagrangian method. Extensive results and comparisons on synthetic and real scanning data validate that the proposed method outperforms the state-of-the-art methods visually and numerically.
Several bandwise total variation (TV) regularized low-rank (LR)-based models have been proposed to remove mixed noise in hyperspectral images (HSIs). Conventionally, the rank of LR matrix is approximated using nuclear norm (NN). The NN is defined by adding all singular values together, which is essentially a $L_1$-norm of the singular values. It results in non-negligible approximation errors and thus the resulting matrix estimator can be significantly biased. Moreover, these bandwise TV-based methods exploit the spatial information in a separate manner. To cope with these problems, we propose a spatial-spectral TV (SSTV) regularized non-convex local LR matrix approximation (NonLLRTV) method to remove mixed noise in HSIs. From one aspect, local LR of HSIs is formulated using a non-convex $L_{gamma}$-norm, which provides a closer approximation to the matrix rank than the traditional NN. From another aspect, HSIs are assumed to be piecewisely smooth in the global spatial domain. The TV regularization is effective in preserving the smoothness and removing Gaussian noise. These facts inspire the integration of the NonLLR with TV regularization. To address the limitations of bandwise TV, we use the SSTV regularization to simultaneously consider global spatial structure and spectral correlation of neighboring bands. Experiment results indicate that the use of local non-convex penalty and global SSTV can boost the preserving of spatial piecewise smoothness and overall structural information.
Data clustering is a fundamental problem with a wide range of applications. Standard methods, eg the $k$-means method, usually require solving a non-convex optimization problem. Recently, total variation based convex relaxation to the $k$-means model has emerged as an attractive alternative for data clustering. However, the existing results on its exact clustering property, ie, the condition imposed on data so that the method can provably give correct identification of all cluster memberships, is only applicable to very specific data and is also much more restrictive than that of some other methods. This paper aims at the revisit of total variation based convex clustering, by proposing a weighted sum-of-$ell_1$-norm relating convex model. Its exact clustering property established in this paper, in both deterministic and probabilistic context, is applicable to general data and is much sharper than the existing results. These results provided good insights to advance the research on convex clustering. Moreover, the experiments also demonstrated that the proposed convex model has better empirical performance when be compared to standard clustering methods, and thus it can see its potential in practice.
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