A Framework for Fast Image Deconvolution With Incomplete Observations.

Area of Science:

• Image processing
• Computational imaging
• Applied mathematics

Background:

• Image deconvolution often uses Fast Fourier Transform (FFT) for speed, but struggles with incomplete observations and unknown boundaries.
• Standard methods for unknown boundaries are slow or introduce artifacts due to inexact models.
• Existing techniques like edge tapering are used to handle image boundaries.

Purpose of the Study:

• To propose a novel deconvolution framework for images with incomplete observations and unknown boundaries.
• To enable the use of diagonalized convolution operators for faster image deconvolution.
• To provide an efficient and high-quality alternative to current boundary handling methods.

Main Methods:

• A new iterative deconvolution framework alternating estimation of unknown pixels and the deconvolved image.
• Utilizes FFT-based deconvolution methods within the framework.
• Implements the framework using a partial alternating direction method of multipliers (ADMM).

Main Results:

• The proposed framework allows working with diagonalized convolution operators, leading to significant speedups.
• Demonstrates extension of a state-of-the-art method with periodic boundary conditions to unknown boundaries.
• The ADMM-based implementation shows convergence and performs at the state-of-the-art level compared to primal-dual methods.
• Successfully applied to deconvolution, inpainting, superresolution, and demosaicing with unknown boundaries.

Conclusions:

• The novel framework efficiently handles image deconvolution with unknown boundaries.
• It offers a fast and high-quality solution applicable to various image restoration tasks.
• The ADMM-based implementation is robust and converges, matching state-of-the-art performance.