An efficient GEM model for image inpainting using a new directional sparse representation: Discrete Shearlet Transform

Shearlets

“…This approach was derived within a larger class of affine-like systems-the so-called composite wavelets [39,40,41]-as a truly multivariate extension of the wavelet framework. One of the distinctive features of shearlets is the use of shearing to control directional selectivity, in contrast to rotation used by curvelets.…”

“…By removing the assumption that ψ is well localized in Definition 5, one can construct discrete shearlet systems which form not only tight frames but also orthonormal bases, as indicated in [39,41]. This naturally raises the question whether well-localized shearlet orthonormal bases do exit.…”

“…Due to the simplicity of the mathematical structure of shearlets, their extensions to higher dimensions are very natural. Indeed, some basic ideas were already introduced in [41], where it was observed that there exist several ways to extend the shearing matrix to higher dimensions. A new construction yielding smooth Parseval frames of discrete shearlets in any dimensions was recently introduced in [37].…”

“…Furthermore, several attempts were made to develop a multiresolution analysis similar to the one associated with wavelets, in an effort to develop MRA-based implementations [41,46,58]. We also refer to [28] for useful observations about shearlet-based numerical shearlet decompositions.…”

Image Processing Using Shearlets

“…In [19], another shearlet-based image fusion method was developed for panchromatic high resolution images and multispectral images, and it was shown that it provides superior performance in terms of spatial resolution and preservation of spectral information. For the applications of inpainting, which can be described as an interpolation or estimation problem with missing data, a shearletbased techique was recently presented in [41]. Fig.…”