Path-Based Dictionary Augmentation: A Framework for Improving k-Sparse Image Processing.

Area of Science:

• Signal Processing
• Machine Learning
• Applied Mathematics

Background:

• Orthogonal Matching Pursuit (OMP) is a widely used algorithm for sparse signal reconstruction.
• Baseline OMP can be limited in its reconstruction and denoising performance.
• Augmenting OMP with an additional step in the identification stage has shown promise.

Purpose of the Study:

• To computationally demonstrate improvements in sparse coding and denoising using an augmented OMP algorithm.
• To investigate two methods for constructing a geodesic path between dictionary atoms.
• To prove the existence of a higher-correlation atom in the Euclidean case and introduce algorithmic modifications.

Main Methods:

• Augmenting the orthogonal matching pursuit (OMP) algorithm with a geodesic path generation step.
• Investigating Euclidean geodesics (linear combinations) and 2-Wasserstein geodesics (optimal transport maps).
• Applying the augmented algorithm to canonical datasets with learned and structured dictionaries.

Main Results:

• The augmented OMP algorithm demonstrated improved performance in sparse coding and denoising compared to baseline OMP.
• Both Euclidean and 2-Wasserstein geodesics were investigated for path construction.
• Algorithmic modifications were introduced to enhance the selection of bracketing atoms.

Conclusions:

• The proposed augmentation significantly improves k-sparse reconstruction and denoising performance.
• The augmentation method is generalizable to other reconstruction algorithms relying on atom selection.
• This work provides a robust enhancement for sparse signal processing techniques.