2021/07/05 by Van Tiep, Do, Van Tiep, Alex Goeßmann +1
Mathematics · #65J22 #65T60 #68U10 #90C25 #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2107.01949
openalex publication_date 2021/07/05 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
In this paper, we present a theoretical analysis of separating images consisting of pointlike and C β-curvelike structures, where β∈ (1,2] . Our approach is based on l1-minimization, in which the sparsity of the desired solution is exploited by two sparse representation systems. It is well known that for such components wavelets provide an optimally sparse representation for point singularities, whereas α-shearlet type with α=\frac2β might be best adapted to the Cβ-curvilinear singularities. In our analysis, we first propose a reconstruction framework with a theoretical guarantee on convergence, which is extended to use general frames instead of Parseval frames. We then construct a dual pair of bandlimited α-shearlets which possesses a good time and frequency localization. Finally, we apply the result to derive an asymptotic accuracy of the reconstructions. In addition, we show that it is possible to separate these two components as long as α<2, i.e., bandlimited α-shearlets which range from wavelet to shearlet type do not coincide with wavelets in the sense of isotropic fashion.