Arindama Singh

A hybrid convex variational model for image restoration

We propose a new hybrid model for variational image restoration using an alternative diffusion switching non-quadratic function with a parameter. The parameter is chosen adaptively so as to minimize the smoothing near the edges and allow the diffusion to smooth away from the edges. This model belongs to a class of edge-preserving regularization methods proposed in the past, the @f-function formulation. This involves a minimizer to the associated energy functional. We study the existence and uniqueness of the energy functional of the model. Using real and synthetic images we show that the model is effective in image restoration.

Well-Posed Inhomogeneous Nonlinear Diffusion Scheme for Digital Image Denoising

We study an inhomogeneous partial differential equation which includes a separate edge detection part to control smoothing in and around possible discontinuities, under the framework of anisotropic diffusion. By incorporating edges found at multiple scales via an adaptive edge detector-based indicator function, the proposed scheme removes noise while respecting salient boundaries. We create a smooth transition region around probable edges found and reduce the diffusion rate near it by a gradient-based diffusion coefficient. In contrast to the previous anisotropic diffusion schemes, we prove the well-posedness of our scheme in the space of bounded variation. The proposed scheme is general in the sense that it can be used with any of the existing diffusion equations. Numerical simulations on noisy images show the advantages of our scheme when compared to other related schemes.

Multispectral image denoising by well-posed anisotropic diffusion scheme with channel coupling

A novel way to denoise multispectral images is proposed via an anisotropic diffusion based partial differential equation (PDE). A coupling term is added to the divergence term and it facilitates the modelling of interchannel relations in multidimensional image data. A total variation function is used to model the intrachannel smoothing and gives a piecewise smooth result with edge preservation. The coupling term uses weights computed from different bands of the input image and balances the interchannel information in the diffusion process. It aligns edges from different channels and stops the diffusion transfer using the weights. Well-posedness of the PDE is proved in the space of bounded variation functions. Comparison with the previous approaches is provided to demonstrate the advantages of the proposed scheme. The simulation results show that the proposed scheme effectively removes noise and preserves the main features of multispectral image data by taking channel coupling into consideration.

Ringing Artifact Reduction in Blind Image Deblurring and Denoising Problems by Regularization Methods

Image deblurring and denoising are the main steps in early vision problems. A common problem in deblurring is the ringing artifacts created by trying to restore the unknown point spread function (PSF). The random noise present makes this task even harder. Variational blind deconvolution methods add a smoothness term for the PSF as well as for the unknown image. These methods can amplify the outliers correspond to noisy pixels. To remedy these problems we propose the addition of a first order reaction term which penalizes the deviation in gradients. This reduces the ringing artifact in blind image deconvolution. Numerical results show the effectiveness of this additional term in various blind and semi-blind image deblurring and denoising problems.

Tikhonov regularization of an elliptic PDE

This paper considers an elliptic PDE with a small parameter multiplied with one of the derivatives. Recognizing the equation as an ill-posed equation, Tikhonov’s regularization is used to cast a related well-posed problem. A-priori estimates of the regularized solution and of the difference between the original and the regularized one are also derived. It is suggested that the regularized solution may be computed by using any well-known numerical method.

AN ADAPTIVE DIFFUSION SCHEME FOR IMAGE RESTORATION AND SELECTIVE SMOOTHING

Anisotropic partial differential equation (PDE)-based image restoration schemes employ a local edge indicator function typically based on gradients. In this paper, an alternative pixel-wise adaptive diffusion scheme is proposed. It uses a spatial function giving better edge information to the diffusion process. It avoids the over-locality problem of gradient-based schemes and preserves discontinuities coherently. The scheme satisfies scale space axioms for a multiscale diffusion scheme; and it uses a well-posed regularized total variation (TV) scheme along with Perona-Malik type functions. Median-based weight function is used to handle the impulse noise case. Numerical results show promise of such an adaptive approach on real noisy images.

Lavrentiev regularization of a singularly perturbed elliptic PDE

This paper considers a reaction-diffusion equation with a small parameter multiplied with the derivatives. Considering this as an ill-posed problem, Lavrentiev regularization method is used to formulate the related well-posed problem. Pointwise a priori estimates of the regularized solutions and of the regularization error are also derived. The regularized equations are solved numerically via the central difference scheme. It is suggested that any other suitable discretization might also be used to solve the regularized problems. Also it is found that with a comparatively small number of grid points, the numerical solution of the regularized equation comes up as a good approximation to the original solution.

Lardy's regularization of a singularly perturbed elliptic PDE

In this paper, we consider an elliptic partial differential equation where a small parameter is multiplied with one or both of the second derivatives. Viewing it as an ill-posed problem, Lardys regularization method is applied to approximate the solution. Convergence of the regularized solution to the original is proved. Numerical examples have been included for illustrating the method.

Cantor’s Little Theorem

This article discusses two theorems of Georg Cantor: Cantor’s Little Theorem and Cantor’s Diagonal Theorem. The results are obtained by generalizing the method of proof of the well known Cantor’s theorem about the cardinalities of a set and its power set. As an application of these, Gödel’s first incompleteness theorem is proved. Hints are given as to how to derive other deeper results including the existence of Parikh’s sentence.

Block-Diagonal Representation

In this chapter, we are concerned with special types of matrix representations of linear operators on finite dimensional spaces. The simplest matrix representation of a linear operator that one would like to have is a diagonal matrix. In such a case, the eigenvalues can be read off the diagonal.