Optimally sparse representations of cartoon-like cylindrical data
Sparse wavelet approximations have been successfully used in one dimensional signal analysis and have been extended to higher dimensions. However, the efficiency of conventional wavelets in dealing with higher dimensional discontinuities is limited. In attempts to overcome these difficulties, tools such as curvelets, shearlets etc have been introduced to take care of multiple orientations and shapes. In dimension 2, these have been very successful in optimally sparse approximations of what are called `cartoon-like' data (bivariate functions that are \(C^2\) away from \(C^2\) edges). Extensions of this to dimension 3 worked well in many applications. But when the data had surface singularities orthogonal to the \(xy\)-plane (e.g. remote sensing), this method is not effective. This paper deals with a model in which discontinuities are independent of the \(z\)-variable. For this purpose, the authors employ what they call `cylindrical shearlets' introduced recently. These are designed to specifically deal with the geometry of data having surface singularities orthogonal to the \(xy\)-plane. Unlike the usual shearlets in 3 dimensions, cylindrical shearlets have only one orientation parameter. The paper relies on microlocal properties of shearlets and the main result is that cylindrical shearlets provide optimally sparse representations over a suitable class of bounded, compactly supported functions on \(\mathbb R^3\) that are \(C^2\) off surface discontinuities perpendicular to the \(xy\)-plane. This method achieves superior approximation properties in comparison with the usual multiscale representations and three dimensional shearlets and curvelets. The \(N\)-term approximation obtained has a decay rate of \(N^{-2}\) (ignoring the logarithmic factor) compared to \(N^{-1/2}\) and \(N^{-1}\) achieved by earlier methods. Numerical implementation of the method is also provided.
scientific article; zbMATH DE number 7388947
scientific article