Given a frame for a subspace W of a Hilbert space H, we consider a class of oblique dual frame sequences. These dual frame sequences are not constrained to lie in W . Our main focus is on shift-invariant frame sequences of the form {φ(· − k)}k∈Z in subspaces of L2(R); for such frame sequences we are able to characterize the set of shift-invariant oblique dual Bessel sequences. Given frame seque...

Recently introduced inpainting algorithms using a combination of applied harmonic analysis and compressed sensing have turned out to be very successful. One key ingredient is a carefully chosen representation system which provides (optimally) sparse approximations of the original image. Due to the common assumption that images are typically governed by anisotropic features, directional represen...

Recently orthonormal Wilson bases with good time–frequency localization have been constructed by Daubechies, Jaffard, and Journé. We extend this construction to Wilson sets and frames with arbitrary oversampling (or redundancy). We state conditions under which dual Weyl–Heisenberg (WH) sets induce dual Wilson sets, and we formulate duality conditions in the time domain and frequency domain. We ...

The continuous curvelet and shearlet transforms have recently been shown to be much more effective than the traditional wavelet transform in dealing with the set of discontinuities of functions and distributions. In particular, the continuous shearlet transform has the ability to provide a very precise geometrical characterization of general discontinuity curves occurring in images. In this pap...

Abstract. We study efficient and reliable methods of capturing and sparsely representing anisotropic structures in 3D data. As a model class for multidimensional data with anisotropic features, we introduce generalized three-dimensional cartoon-like images. This function class will have two smoothness parameters: one parameter β controlling classical smoothness and one parameter α controlling a...

It is now widely acknowledged that traditional wavelets are not very effective in dealing with multidimensional signals containing distributed discontinuities. Shearlet Transform is a new discrete multiscale directional representation, which combines the power of multiscale methods with a unique ability to capture the geometry of multidimensional data and is optimally efficient in representing ...

Description: The process is a method of classifying prostate tumors as cancerous or benign. It classifies the tumors according to the Gleason grading scale to determine the cancerous nature of the tumor. The process utilizes a shearlet transform, as well as three other features, and combines them via multiple kernel learning. The shearlet transform is used to represent the local structure of im...