Roughness penalties on finite domains.

Area of Science:

• Image processing and computational mathematics.

Background:

• Traditional estimation and image regularization methods face challenges with complex data.
• Markov random field priors are widely used but can be computationally intensive.

Purpose of the Study:

• To propose a novel class of penalty functions for estimation and image regularization.
• To explore the theoretical underpinnings and practical applications of these new functions.

Main Methods:

• Defining penalty functions based on vector discrepancies across a finite lattice.
• Investigating the relationship between these penalties and Markov random field priors.
• Developing and proving convergence for an iterative algorithm utilizing induced neighborhood structures.

Main Results:

• A divergence roughness penalty is identified as a discretization of a known density estimation penalty.
• The proposed penalty functions demonstrate utility in estimation problems.
• The iterative algorithm shows convergence under specified conditions.

Conclusions:

• The novel penalty functions offer a promising approach for image regularization and estimation.
• The developed iterative algorithm provides an efficient method for applying these penalties.
• Successful application examples in emission tomographic and radar imaging highlight the practical value.