Ridgelets and the Representation of Mutilated Sobolev Functions

Ridgelets and the Representation of Mutilated Sobolev Functions

Ridgelets and Their Derivatives: Representation of Images with Edges

This paper reviews the development of several recent tools from computational harmonic analysis. These new systems are presented under a coherent perspective, namely, the representation of bivariate functions that are singular along smooth curves (edges). First, the representation of functions that are smooth away from straight edges is presented, and ridgelets will be shown to provide near opt...

Ridgelets: Estimating with Ridge Functions

Feedforward neural networks, projection pursuit regression, and more generally, estimation via ridge functions have been proposed as an approach to bypass the curse of dimensionality and are now becoming widely applied to approximation or prediction in applied sciences. To address problems inherent to these methods – ranging from the construction of neural networks to their efficiency and capab...

Monoscale Ridgelets for the Representation of Images with Edges

In a previous paper [1], the author introduced a new system for representing multivariate functions, namely, the ridgelets. In a following article [3], ridgelets were shown to be optimal for representing functions that are smooth away from hyperplanes, e.g. in two dimensions ridgelets provide optimally sparse representations of smooth functions that are discontinuous along lines, i.e. straight ...

PNAS Inaugural Article: Tight Frames of k-Plane Ridgelets and the Problem of Representing Objects Which Are Smooth Away from d-Dimensional Singularities in R

For each pair (n, k) with 1 ≤ k < n, we construct a tight frame (ρλ : λ ∈ Λ) for L(R), which we call a frame of k-plane ridgelets. The intent is to efficiently represent functions which are smooth away from singularities along k-planes in R. We also develop tools to help decide whether in fact k-plane ridgelets provide the desired efficient representation. We first construct a wavelet-like tigh...