Adaptive regularization with the structure tensor

Area of Science:

• Computer Vision
• Image Processing
• Applied Mathematics

Background:

• Natural images possess geometric structures crucial for scene understanding.
• Existing image processing methods use regularizers that capture natural image statistics, like Total Variation (TV).
• Gradient magnitude alone is insufficient; structure tensors offer richer directional and scale information.

Purpose of the Study:

• To propose a variational model for image reconstruction using structure tensors.
• To adapt regularization functionals to local image geometry.
• To develop robust and efficient algorithms for image restoration tasks.

Main Methods:

• A two-step minimization procedure: robust structure tensor estimation (semidefinite program) and adaptive regularization.
• Extension of anisotropic diffusion into the convex setting.
• Incorporation of nonlocal regularization leveraging local self-similarity.

Main Results:

• Development of robust, efficient, and easy-to-code algorithms for denoising, deblurring, and compressed sensing.
• Consistent accuracy improvements over classic regularization methods.
• Effective extension to nonlocal regularization, enhancing nonlocal TV and diffusion.

Conclusions:

• The proposed structure tensor-based regularization effectively captures local image geometry.
• The method provides significant accuracy improvements for various image reconstruction tasks.
• The approach is versatile, extending to nonlocal regularization and various image restoration problems.