Smooth Projections and Optimally Sparse Representation of Cartoon-like Cylindrical Solids

In this dissertation, we explore two mathematical structures: projections and sparse representations. Projections are self-adjoint and idempotent linear operators that can be used to decompose a function space into smaller components thus making it easier to analyze. First, we extend the construction of smooth orthogonal projections on L²(\R) to higher dimensions. To achieve this goal, we introduce a novel approach based on cyclic permutation operators onto $n$-dimensional real space by defining smooth orthogonal projections associated with certain partitions of the space. Second, we construct a sparse function system capable of accurately approximating functions in a given class using a linear combination of relatively few representation terms. Since a well designed sparse system has the ability to efficiently capture relevant information, they have applications in numerous areas such as data restoration, feature extraction and compression. Inspired by the success of shearlets -- a multiscale sparse representation framework for analysing multivariate data -- we introduce a new variant, called cylindrical shearlets. This system is especially designed to provide sparse representations of functions in the class \cC \subset L²(\R³) of 3-dimensional data dominated by surface singularities that meet perpendicularly at the $xy$-plane. The construction of this new shearlet system relies on the application of the smooth projections we described earlier. In addition, we prove that this new representation achieves superior approximation properties compared to 3-dimensional shearlets and wavelets for functions in $\cC$. Specifically, the $N$-term approximation f_N^S, obtained by selecting the $N$ largest coefficients of the cylindrical shearlet expansion of a function $f \in \cC$, satisfies the asymptotic estimate $ \norm{f - f_N^S}₂² \le c N^-2 (\ln N)³, \quad \text{as } N \to \infty. $ The new system outperforms conventional 3D wavelet and 3D shearlet approximation rates, which are of order $N^{-\frac{1}{2}}$ and $N^{-1}$ on the same type of data. More importantly, we establish that the decay rate $O(N^{-2})$ is optimal for the data type considered.

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Houston

Year dc:date.issued

2020

Author dc:creator

• Pahari, Basanta Raj

Advisor dc:contributor.advisor

• Labate, Demetrio

Committee members dc:contributor.committeemember

• Bodmann, Bernhard G.
• Mang, Andreas
• Prasad, Saurabh

• Sparse Representations, Smooth Projections, Smooth Cylindrical Shearlets

Statement dc:rights

• The author of this work is the copyright owner. UH Libraries and the Texas Digital Library have their permission to store and provide access to this work. UH Libraries has secured permission to reproduce any and all previously published materials contained in the work. Further transmission, reproduction, or presentation of this work is prohibited except with permission of the author(s).

Language dc:language.iso

eng

Handle dc:identifier.uri

https://hdl.handle.net/10657/10271

OAI identifier oai:identifier

oai:uh-ir.tdl.org:10657/10271