arXiv subjects

Van Tiep Do

Publications and source records attributed to Van Tiep Do.

3 recordsLinked to original sources

arXiv2022

Multi-component separation, inpainting and denoising with recovery guarantees

In image processing, problems of separation and reconstruction of missing pixels from incomplete digital images have been far more advanced in past decades. Many empirical results have produced very good results, however, providing a theoretical analysis for the success of algorithms is not an easy task, especially, for inpainting and separating multi-component signals. In this paper, we propose two main algorithms based on $l_1$ constrained and unconstrained minimization for separating $N$ distinct geometric components and simultaneously filling-in the missing part of the observed image. We then present a theoretical guarantee for these algorithms using compressed sensing technique, which is based on a principle that each component can be sparsely represented by a suitably chosen dictionary. Those sparsifying systems are extended to the case of general frames instead of Parseval frames which have been typically used in the past. We finally prove that the method does indeed succeed in separating point singularities from curvilinear singularities and texture as well as inpainting the missing band contained in curvilinear singularities and texture.

math.FA↗

arXiv2021

An asymptotic analysis of separating pointlike and $C^β$-curvelike singularities

In this paper, we present a theoretical analysis of separating images consisting of pointlike and $C^{ β}$-curvelike structures, where $β\in (1,2] $. Our approach is based on $l_1$-minimization, in which the sparsity of the desired solution is exploited by two sparse representation systems. It is well known that for such components wavelets provide an optimally sparse representation for point singularities, whereas $α$-shearlet type with $α$=$\frac{2}β$ might be best adapted to the $C^β$-curvilinear singularities. In our analysis, we first propose a reconstruction framework with a theoretical guarantee on convergence, which is extended to use general frames instead of Parseval frames. We then construct a dual pair of bandlimited $α$-shearlets which possesses a good time and frequency localization. Finally, we apply the result to derive an asymptotic accuracy of the reconstructions. In addition, we show that it is possible to separate these two components as long as $α<2$, i.e., bandlimited $α$-shearlets which range from wavelet to shearlet type do not coincide with wavelets in the sense of isotropic fashion.

math.FA↗

arXiv2021

Analysis of simultaneous inpainting and geometric separation based on sparse decomposition

Natural images are often the superposition of various parts of different geometric characteristics. For instance, an image might be a mixture of cartoon and texture structures. In addition, images are often given with missing data. In this paper, we develop a method for simultaneously decomposing an image to its two underlying parts and inpainting the missing data. Our separation inpainting method is based on and $l_1$ minimization approach, using two dictionaries, each sparsifying one of the image parts but not the other. We introduce a comprehensive convergence analysis of our method, in a general setting, utilizing the concepts of joint concentration, clustered sparsity, and cluster coherence. As the main application of our theory, we consider the problem of separating and inpainting an image to a cartoon and texture parts.

math.FA↗