209 research outputs found
Well-posedness of a nonlinear integro-differential problem and its rearranged formulation
We study the existence and uniqueness of solutions of a nonlinear
integro-differential problem which we reformulate introducing the notion of the
decreasing rearrangement of the solution. A dimensional reduction of the
problem is obtained and a detailed analysis of the properties of the solutions
of the model is provided. Finally, a fast numerical method is devised and
implemented to show the performance of the model when typical image processing
tasks such as filtering and segmentation are performed.Comment: Final version. To appear in Nolinear Analysis Real World Applications
(2016
Novel integro-differential schemes for multiscale image representation
Multiscale representation of a given image is the problem of constructing a family of images, where each image in this family represents a scaled version of the given image. This finds its motivation from biological vision studies. Using the hierarchical multiscale image representation proposed by Tadmor et. al. [32],
an image is decomposed into sums of simpler `slices', which extract more refined information from the previous scales. This approach motivates us to propose a novel integro-differential equation (IDE), for a multiscale image representation. We examine various properties of this IDE.
The advantage of formulating the IDE this way is that, although this IDE is motivated by variational approach, we no longer need to be associated with any minimization problem and can modify the IDE, suitable to our image processing needs. For example, we may need to find different scales in the image, while retaining or enhancing prominent edges, which may define boundaries of objects. We propose some edge preserving modifications to our IDE.
One of the important problems in image processing is deblurring a blurred image. Images get blurred due to various reasons, such as unfocused camera lens, relative motion between the camera and the object pictured, etc. The blurring can be modeled with a continuous, linear operator. Recovering a clean image from a blurry image, is an ill-posed problem, which is solved using Tikhonov-like regularization. We propose a different IDE to solve the deblurring problem.
We propose hierarchical multiscale scheme based on (BV; L1) decomposition, proposed by Chan, Esedoglu, Nikolova and Alliney [12, 25, 3]. We finally propose another hierarchical multiscale representation based on a novel weighted (BV;L1) decomposition
Error estimates of residual minimization using neural networks for linear PDEs
We propose an abstract framework for analyzing the convergence of
least-squares methods based on residual minimization when feasible solutions
are neural networks. With the norm relations and compactness arguments, we
derive error estimates for both continuous and discrete formulations of
residual minimization in strong and weak forms. The formulations cover recently
developed physics-informed neural networks based on strong and variational
formulations
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Dirichlet Form Theory and its Applications
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Regularization Methods in Banach Spaces applied to Inverse Medium Scattering Problems
This work handles inverse scattering problems for both acoustic and electromagnetic waves. That is to reconstruct the irradiated media from measurements of the scattered felds by regularization methods. As a particular feature, the contrasts of the scattering objects are assumed to be supported within a small region, hence called sparse. To apply sparsity regularization schemes it becomes crucial to model the problems in Banach spaces. Traditionally, they are given in a Hilbert space setting, such that reformulation in an L p-sense becomes a key point. Contrasts are linked to the data by forward operators, basing on beforehand stated solution operators and their continuity properties. Thereby, appropriate regularization techniques providing sparsity are given. As the case of scalar-valued contrast functions is already covered in the literature, mainly inverse scattering problems for anisotropic media are shown. In the case where electromagnetic waves are considered, a distinction is made between magnetic and non-magnetic media, since the latter is less complex. Finally, the case of inverse acoustic backscattering is handled, which is rarely seen in literature
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