A sequential solution for anisotropic total variation image denoising with interval constraints.

Area of Science:

• Optimization
• Image Processing
• Signal Approximation

Background:

• Anisotropic total variation (TV) regularization is crucial for image denoising and signal approximation.
• Interval constraints are often necessary in practical applications, such as non-negative image intensities in X-ray CT.
• Existing methods for constrained problems can be computationally intensive.

Purpose of the Study:

• To introduce a simplified sequential solution for specific constrained anisotropic TV and fused lasso problems.
• To demonstrate the conditions under which this sequential approach is effective.
• To provide a computationally efficient alternative for constrained optimization problems.

Main Methods:

• Derivation of a sequential solution involving unconstrained optimization followed by thresholding.
• Application of Karush-Kuhn-Tucker (KKT) conditions for convex optimization.
• Analysis of uniform and zero-containing interval constraints.

Main Results:

• A simple sequential solution is identified for constrained anisotropic TV denoising (Problem 1).
• The same sequential solution is shown to solve the constrained fused lasso signal approximation (Problem 2) under specific conditions.
• The method is validated for uniform interval constraints and when these constraints include zero.

Conclusions:

• The proposed sequential solution offers a computationally efficient method for specific constrained optimization problems.
• The findings simplify the treatment of interval constraints in anisotropic TV and fused lasso.
• This work provides a novel and practical approach for applications like X-ray CT image denoising.