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Differential Methods for Multidimensional Visual Data Analysis

“…Predefining the smoothing geometry T of each applied PDE iteration is the first stage of most of the PDE-based regularization algorithms. Most of the differences between existing regularization methods (as in [2,3,10,18,19,49,52,58,59,[67][68][69]) lie first on the definition of T, but also on the kind of the diffusion PDE that will be used indeed to perform the desired smoothing.…”

“…• Global interpretation: The problem of image regularization can be regarded as the minimization of the energy functional E(I) by a gradient descent (i.e., a PDE), coming from the EulerLagrange equations of E(I) [3,19,49,51,78]:…”

An Affine Invariant Patch Similarity

“…In (1) the similarity measure is given in a general form, since G 1 and G 2 can be any Riemannian metrics on the respective image domains of u and v. It is well known, that the structure tensor (second-moment matrix) can be seen as a metric in the image domain (e.g. [35,36,15,4,3]). A novel iterative scheme was presented in [9,7] to define and compute the structure tensors.…”

PET-MRI Joint Reconstruction by Joint Sparsity Based Tight Frame Regularization

“…As the function cannot be independent from structure [26,45], it is reasonable to assume that the images to be reconstructed have structural correlation [26]. In the literature, the structure of an image is mathematically modeled as a singularity such as edge and ridge, and the idea of structural correlation based joint reconstruction was first explored in the color image processing as the color channels in general share the singularities [8,25,29,36,47,50,53].…”