[Paper Review] A Parallel Douglas Rachford Algorithm for Restoring Images with Values in Symmetric Hadamard Manifolds

Ronny Bergmann, Johannes Persch+1 more|arXiv (Cornell University)|Dec 9, 2015

Sparse and Compressive Sensing Techniques43 references3 citations

TL;DR

This paper proposes a parallel Douglas-Rachford algorithm for restoring images with values in symmetric Hadamard manifolds by minimizing a total variation-like functional. It extends the algorithm to Riemannian manifolds using nonexpansive reflections of distance and indicator functions, enabling efficient proximal mapping via exponential/logarithmic maps, with demonstrated superior convergence and performance over existing methods on manifolds like symmetric positive definite matrices.

ABSTRACT

We are interested in restoring images having values in a symmetric Hadamard manifold by minimizing a functional with a total variation like regularizing term. To solve the convex minimization problem, we extend the Douglas-Rachford algorithm and its parallel version to symmetric Hadamard manifolds. For the convergence proof we investigate the corresponding reflection operators. We prove that the reflections of certain distance functions on the manifold are nonexpansive which is an interesting result on its own. Furthermore, the reflection of the involved indicator function of a special closed convex set is nonexpansive on manifolds with constant curvature. The performance of the generalized Douglas-Rachford algorithm for our model is based on analytic expressions for the proximal mappings. It requires the evaluation of exponential and logarithmic functions which can be done efficiently. Several numerical examples demonstrate the advantageous performance of the suggested algorithm compared to other existing methods as the cyclic proximal point algorithm or half-quadratic minimization. Numerical convergence is also observed for the manifold of symmetric positive definite matrices with the affine invariant metric which does not have a constant curvature.

Motivation & Objective

• To address image restoration for data lying in symmetric Hadamard manifolds, such as symmetric positive definite matrices, where standard Euclidean methods fail.
• To develop a scalable and convergent optimization algorithm tailored for non-Euclidean image data with total variation regularization.
• To extend the Douglas-Rachford algorithm to Riemannian manifolds with non-constant curvature, particularly focusing on convergence via nonexpansive reflections.
• To provide analytic expressions for proximal mappings using exponential and logarithmic maps, enabling efficient computation on symmetric Hadamard manifolds.
• To demonstrate the algorithm’s superiority over cyclic proximal point and half-quadratic minimization methods in numerical experiments.

Proposed method

• The algorithm extends the classical Douglas-Rachford splitting to symmetric Hadamard manifolds using reflection operators derived from distance functions and indicator functions of closed convex sets.
• Nonexpansiveness of reflections for distance functions and indicator functions is proven on manifolds with constant curvature, ensuring convergence under appropriate conditions.
• Proximal mappings are computed using closed-form expressions involving exponential and logarithmic maps, which are efficiently evaluated on symmetric Hadamard manifolds.
• A parallel variant of the Douglas-Rachford algorithm is proposed to accelerate convergence, especially for large-scale image restoration problems.
• The method is applied to the manifold of symmetric positive definite matrices with the affine-invariant metric, even though this manifold lacks constant curvature.
• Convergence is numerically observed despite the absence of constant curvature, indicating robustness of the approach.

Experimental results

Research questions

• RQ1 Can the Douglas-Rachford algorithm be generalized to symmetric Hadamard manifolds for image restoration with total variation regularization?
• RQ2 Are the reflection operators of distance and indicator functions nonexpansive on symmetric Hadamard manifolds, particularly those with constant curvature?
• RQ3 Can efficient proximal mappings be derived using exponential and logarithmic maps for practical implementation on Riemannian manifolds?
• RQ4 How does the performance of the proposed parallel Douglas-Rachford algorithm compare to existing methods like cyclic proximal point and half-quadratic minimization?
• RQ5 Does the algorithm converge numerically on manifolds without constant curvature, such as the symmetric positive definite matrix manifold with affine-invariant metric?

Key findings

• The reflection of the distance function on a symmetric Hadamard manifold is nonexpansive, a novel result with implications for optimization on Riemannian manifolds.
• The reflection of the indicator function of a closed convex set is nonexpansive on symmetric Hadamard manifolds with constant curvature, supporting convergence guarantees.
• The proposed algorithm achieves faster convergence and better restoration quality than the cyclic proximal point algorithm and half-quadratic minimization in numerical experiments.
• Efficient computation of proximal mappings via exponential and logarithmic maps enables practical deployment on symmetric Hadamard manifolds, including the symmetric positive definite matrix manifold.
• Numerical convergence is observed even on the symmetric positive definite matrix manifold with the affine-invariant metric, despite its non-constant curvature, indicating robustness.
• The parallel variant of the algorithm demonstrates improved scalability and performance for large-scale image restoration tasks on non-Euclidean domains.