[Paper Review] Shearlet frames with short support

Li Song, Yi Shen|arXiv (Cornell University)|Jan 25, 2011

Mathematical Analysis and Transform Methods3 references3 citations

TL;DR

This paper constructs symmetric, compactly supported shearlet frames using B-splines, achieving explicit analytical forms and optimal sparsity for cartoon-like images. By leveraging pseudo splines of type II and B-spline generators, the authors establish shearlet systems that provide almost optimally sparse approximations with guaranteed frame bounds and improved spatial localization.

ABSTRACT

Compactly supported shearlets have been studied in both theory and applications. In this paper, we construct symmetric compactly supported shearlet systems based on pseudo splines of type II. Specially, using B-splines, we construct shearlet frame having explicit analytical forms which is important for applications. The shearlet systems based on B-splines also provide optimally sparse approximation within cartoon-liked image.

Motivation & Objective

• To address the lack of symmetric, compactly supported shearlet frames with explicit analytical forms in the spatial domain.
• To overcome the limitations of existing compactly supported shearlets, which are neither symmetric nor analytically explicit.
• To develop shearlet systems based on B-splines that ensure optimal sparsity for cartoon-like images.
• To establish tight frame bounds and prove the frame property for shearlet systems derived from B-spline generators.
• To provide a practical construction with minimal support and explicit formulas for use in image processing applications.

Proposed method

• Constructs shearlet systems using B-spline functions of order $ m $ as generators, ensuring explicit analytical expressions in the spatial domain.
• Employs a parabolic scaling law via matrices $ A_{2^j} $ and $ \tilde{A}_{2^j} $, and directional encoding via shear matrices $ S_k $.
• Defines cone-adapted discrete shearlet systems $ SH(\phi, \psi, \tilde{\psi}; c) $ with separate frequency coverage for $ \mathcal{C}_1 \cup \mathcal{C}_3 $ and $ \mathcal{C}_2 \cup \mathcal{C}_4 $.
• Derives the Fourier transform of the shearlet $ \hat{\psi}(\xi) = \hat{b}(\xi_1)\hat{\phi}(\xi_1)\hat{\phi}(\xi_2) $, where $ \hat{b} $ is derived from pseudo-spline masks.
• Establishes lower and upper bounds for the Fourier transforms of pseudo-splines of type II, crucial for frame bound analysis.
• Proves that the resulting shearlet systems satisfy the frame condition with explicit frame bounds, ensuring stable reconstruction.

Experimental results

Research questions

• RQ1 Can symmetric, compactly supported shearlet frames with explicit analytical forms be constructed using B-splines?
• RQ2 Do shearlet systems based on B-splines achieve almost optimal sparsity for cartoon-like images?
• RQ3 What are the frame bounds for shearlet systems derived from B-spline generators, and how do they depend on the order of the B-spline?
• RQ4 How do the decay properties of the Fourier transforms of B-spline-based shearlets compare to those of other constructions?
• RQ5 Can the construction be generalized to any $ 0 < \alpha < \pi/2 $, ensuring frame stability for a wide range of sampling parameters?

Key findings

• Shearlet systems based on B-splines achieve explicit analytical forms in the spatial domain, unlike most wavelet and shearlet constructions.
• The constructed shearlets are symmetric or anti-symmetric, addressing a key limitation of prior compactly supported shearlet systems.
• The shearlet systems provide almost optimally sparse approximations for cartoon-like images, matching theoretical limits of sparsity.
• For B-spline order 3, the shearlet system $ \Psi(\psi;c) $ forms a frame for $ \{f \in L_2(\mathbb{R}^2) : \mathrm{supp}\,\hat{f} \in \mathcal{C}_1(\alpha) \cup \mathcal{C}_3(\alpha)\} $ with $ c_2 \leq c_1 \leq \hat{c} $, where $ \hat{c} > 0 $.
• For B-spline order 4, the system $ \Psi(\phi, \psi_1, \psi_2; c) $ provides almost optimally sparse approximations for functions in $ \mathscr{E}^2(v) $, with a positive sampling constant $ c > 0 $.