Convex set theoretic image recovery by extrapolated iterations of parallel subgradient projections

Area of Science:

• Image Processing
• Optimization
• Applied Mathematics

Background:

• Image recovery problems often involve finding points in the intersection of convex sets within a Hilbert space.
• The Projection Onto Convex Sets (POCS) algorithm is a common method but suffers from slow convergence and limitations in parallel processing.
• POCS requires exact projections at each iteration, hindering its practical application.

Purpose of the Study:

• To introduce a novel algorithm, the Extended MOnotonic Projection Scheme (EMOPSP), for convex set theoretic image recovery.
• To overcome the limitations of the POCS algorithm, including slow convergence and lack of parallelizability.
• To develop a method that allows for extrapolated relaxation parameters beyond the conventional [0,2] range.

Main Methods:

• EMOPSP employs a convex combination of subgradient projections onto multiple sets in each iteration.
• The algorithm utilizes a relaxation step with an iteration-dependent parameter that can extend beyond [0,2].
• Theoretical convergence is established, and numerical simulations are conducted to validate performance.

Main Results:

• EMOPSP demonstrates significantly more efficient convergence compared to existing projection-based methods.
• The proposed method is well-suited for parallel processing, addressing a key POCS limitation.
• Extrapolated relaxations contribute to the enhanced convergence speed of EMOPSP.

Conclusions:

• EMOPSP provides a generalized and more efficient framework for solving convex set theoretic image recovery problems.
• The algorithm's parallel nature and faster convergence make it a valuable advancement in the field.
• The study validates the effectiveness of extrapolated relaxations in projection-based optimization.