A shearlet-based fast thresholded Landweber algorithm for deconvolution

Image deconvolution is an important problem, which has seen plenty of progress in the last decades. Due to its ill-posedness, a common approach is to formulate the reconstruction as an optimization problem[Formula: see text] regularized by an additional sparsity-enforcing term. This term is often modeled as an [Formula: see text] norm measured in the domain of a suitable signal transform. The resulting optimization problem can be solved by an iterative approach via Landweber iterations with soft thresholding of the transform coefficients. Previous approaches focused on thresholding in the wavelet-domain. In particular, recent work [C. Vonesch and M. Unser, A fast thresholded Landweber algorithm for wavelet-regularized multidimensional deconvolution, IEEE Trans. Image Process. 17(4) (2008) 539–549.] has shown that the use of Shannon wavelets results in particularly efficient reconstruction algorithms. The present paper extends this approach to Shannon shearlets, which we also introduce in this work. We show that for anisotropic blurring filters, such as the motion blur, the novel shearlet-based approach allows for further a improvement in efficiency. In particular, we observe that for such kernels using shearlets instead of wavelets improves the quality of image restoration and SERG, when compared after the same number of iterations.

Signorini Cylindrical Waves and Shannon Wavelets

Hyperelastic materials based on Signorini’s strain energy density are studied by using Shannon wavelets. Cylindrical waves propagating in a nonlinear elastic material from the circular cylindrical cavity along the radius are analyzed in the following by focusing both on the main nonlinear effects and on the method of solution for the corresponding nonlinear differential equation. Cylindrical waves’ solution of the resulting equations can be easily represented in terms of this family of wavelets. It will be shown that Hankel functions can be linked with Shannon wavelets, so that wavelets can have some physical meaning being a good approximation of cylindrical waves. The nonlinearity is introduced by Signorini elastic energy density and corresponds to the quadratic nonlinearity relative to displacements. The configuration state of elastic medium is defined through cylindrical coordinates but the deformation is considered as functionally depending only on the radial coordinate. The physical and geometrical nonlinearities arising from the wave propagation are discussed from the point of view of wavelet analysis.

Fractional Calculus and Shannon Wavelet

An explicit analytical formula for the any order fractional derivative of Shannon wavelet is given as wavelet series based on connection coefficients. So that for anyL2(ℝ)function, reconstructed by Shannon wavelets, we can easily define its fractional derivative. The approximation error is explicitly computed, and the wavelet series is compared with Grünwald fractional derivative by focusing on the many advantages of the wavelet method, in terms of rate of convergence.