Journal Articles
Permanent URI for this collectionhttps://idr.nitk.ac.in/handle/123456789/19884
Browse
279 results
Search Results
Item An iterative regularization method for ill-posed Hammerstein type operator equation(Walter de Gruyter GmbH, 2009) George, S.; Kunhanandan, M.A combination of Newton's method and a regularization method has been considered for obtaining a stable approximate solution for ill-posed Hammerstein type operator equation. By choosing the regularization parameter according to an adaptive scheme considered by Pereverzev and Schock (2005) an order optimal error estimate has been obtained. Moreover the method that we consider gives quadratic convergence compared to the linear convergence obtained by George and Nair (2008). © de Gruyter 2009.Item On convergence of regularized modified Newton's method for nonlinear ill-posed problems(Walter de Gruyter GmbH and Co. KG, 2010) George, S.In this paper we consider regularized modified Newton's method for approximately solving the nonlinear ill-posed problem F(x) = y, where the right hand side is replaced by noisy data y?? Y with y - y?? ? and F : D(F) ? X ? Y is a nonlinear operator between Hilbert spaces X and Y. Under the assumption that Fréchet derivative F? of F is Lipschitz continuous, a choice of the regularization parameter and a stopping rule based on a majorizing sequence are presented. We prove that under a general source condition on , the error between the regularized approximation and the solution of optimal order. © de Gruyter 2010.Item Iterative regularization methods for ill-posed hammerstein type operator equation with monotone nonlinear part(2010) George, S.; Kunhanandan, M.We considered a procedure for solving an ill-posed Hammerstein type operator equation KF (x) = y, by solving the linear equation Kz = y first for z and then solving the nonlinear equation F (x) = z. Convergence analysis is carried out by means of suitably constructed majorizing sequences. The derived error estimate using an adaptive method proposed by Perverzev and Schock (2005) in relation to the noise level and a stopping rule based on the majorizing sequences are shown to be of optimal order with respect to certain assumptions on F (x?), where x? is the solution of KF (x) = y.Item An iteratively regularized projection method with quadratic convergence for nonlinear Ill-posed problems(2010) George, S.; Elmahdy, A.I.An iteratively regularized projection method, which converges quadratically, has been considered for obtaining stable approximate solution to nonlinear ill-posed operator equations F(x) = y where F : D(F) ? X ? X is a nonlinear monotone operator defined on the real Hilbert space X: We assume that only a noisy data y? with y-y? ? ? are available. Under the assumption that the Fréchet derivative F? of F is Lipschitz continuous, a choice of the regularization parameter using an adaptive selection of the parameter and a stopping rule for the iteration index using a majorizing sequence are presented. We prove that under a general source condition on x0 - x?, the error between the regularized approximation where Ph is an orthog-onal projection on to a nite dimensional subspace Xh of X) and the solution x? is of optimal order.Item Curvature driven diffusion coupled with shock for image enhancement/reconstruction(Inderscience Publishers, 2011) Padikkal, P.; George, S.Curvature driven diffusion is widely used for image denoising and inpainting. Among the curvature driven diffusion techniques Gauss Curvature Driven Diffusion (GCDD) became a prominent image denoising method due to its capability to retain some important structures with non zero curvatures, like curved edges, corners etc. Unlike many other non-linear diffusion techniques, the curvature driven diffusion hardly has any inverse diffusion characteristics. In this work we propose to introduce a shock term along with the GCDD term to enhance the edges while smoothing-out the noise. This technique will preserve some important structures and enhance them while denoising the image. The experiments clearly demonstrates the efficiency of the method. Copyright © 2011 Inderscience Enterprises Ltd.Item Fourth-order variational model with local-constraints for denoising images with textures(2011) Padikkal, P.; George, S.A fourth-order partial differential equation-based approach with a set of local constraints is proposed in this paper, to denoise the images without losing much of the semantically important features like edges and textures. The results provided both in terms of qualitative and quantitative measures substantially endorse the capability of the method. © 2011 Inderscience Enterprises Ltd.Item Reconstruction of signals by standard Tikhonov method(2011) George, S.; Padikkal, P.In this work we propose a standard Tikhonov regularization approach for obtaining the signal f from the observed signal ye. The observed signal is distorted by an additive noise or error e. Deviating from the usual assumption on the bound onItem Shock coupled fourth-order diffusion for image enhancement(Elsevier Ltd, 2012) Padikkal, P.; George, S.In this paper a shock coupled fourth-order diffusion filter is proposed for image enhancement. This filter converges at a faster rate while preserving and enhancing edges, ramps and textures present in the images. The proposed filter diffuses with varying magnitudes in the directions normal to the level-curve and along it. The magnitude of the directional diffusion is controlled by a diffusion function, meant to provide a good response in the direction along the level-curves, than across them. The proposed filter can still preserve the planar approximation of the image, thereby avoiding the discrepancy caused due to the staircase effect, as in the second-order counterparts. The anisotropic property of the filter is thoroughly studied, analyzed and demonstrated with perspective and quantitative results. The performance of the proposed filter is compared with the state-of-the-art methods for image enhancement. The quantitative and perspective measures provided endorse the capability of the method to enhance various kinds of images. © 2012 Elsevier Ltd. All rights reserved.Item A quadratic convergence yielding iterative method for nonlinear ill-posed operator equations(2012) George, S.; Elmahdy, A.I.In this paper, we consider an iterative method for the approximate solution of the nonlinear ill-posed operator equation Tx = y; where the right hand side is replaced by noisy data y? ? X with ?y - y ?? ? ? and T : D(T) ? X ? X is a nonlinear monotone operator defined on a Hilbert space X: The iteration x ?n,? converges quadratically to the unique solution x?? of the equation T(x) + ?(x - x0) = y? (x0 := x 0,??). It is known that (Tautanhahn (2002)) x?? converges to the solution x? of Tx = y: The convergence analysis and the stopping rule are based on a suitably constructed majorizing sequence. Under a general source condition on x 0 - x? we proved that the error ?x? - x n, ??;? is of optimal order. We show that the adaptive scheme considered by Perverzev and Schock (2005) for choosing the regularization parameter can be effectively used here for obtaining an optimal order error estimate. © 2012 Institute of Mathematics, NAS of Belarus.Item A time-dependent switching anisotropic diffusion model for denoising and deblurring images(2012) Padikkal, P.; George, S.A conditionally anisotropic diffusion based deblurring and denoising filter is introduced in this paper. This is a time-dependent curvature based model and the steady state can be attained at a faster rate, using the explicit time-marching scheme. The filter switches between isotropic and anisotropic diffusion depending on the local image features. The switching of the filter is controlled by a binary function, which returns either zero or one, based on the underlying local image gradient features. The parameters in the proposed filter can be fine-tuned to get the desired output image. The filter is applied to various kinds of input test images and the response is analyzed. The filter is found to be effective in the reconstruction of partially textured, textured, constant-intensity and color images, as is evident from the results provided. © 2011 Copyright Taylor and Francis Group, LLC.