Shapes From Pixels.

Area of Science:

• Computer Vision
• Image Processing
• Computational Geometry

Background:

• Digital images are discrete approximations of continuous visual signals, often leading to information loss.
• Reconstructing original continuous signals from discrete images is generally non-invertible without specific constraints like bandlimitedness.

Purpose of the Study:

• To develop a method for recovering shape images with smooth boundaries from discrete samples.
• To ensure reconstructed images are consistent with original samples and form valid shapes.

Main Methods:

• Formulating reconstruction by minimizing shape perimeter within the set of consistent binary shapes.
• Relaxing the non-convex shape constraint to minimize total variation over consistent non-negative-valued images.
• Introducing a 'reducibility' property to ensure equivalence between the perimeter minimization and total variation minimization problems.

Main Results:

• Demonstrating that the reducibility property establishes a minimum sampling density requirement for accurate reconstruction.
• Evaluating the performance of the relaxed total variation minimization approach through numerical experiments.
• Showing that the relaxed method effectively reconstructs shape images while maintaining sample consistency.

Conclusions:

• The proposed method provides a robust approach for shape image reconstruction from samples.
• The reducibility property is crucial for guaranteeing the accuracy and efficiency of the reconstruction process.
• The relaxed formulation offers a computationally tractable alternative for recovering smooth shape images.