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A new twIst: two-step iterative shrinkage/thresholding algorithms for image restoration.

José M Bioucas-Dias¹, Mario A T Figueiredo

¹Instituto de Telecomunicações and the Instituto Superior Técnico, Technical University of Lisbon, 1049-001 Lisboa, Portugal. jose.bioucas@lx.it.pt

IEEE Transactions on Image Processing : a Publication of the IEEE Signal Processing Society

|December 21, 2007

Summary

Two-step iterative shrinkage/thresholding (TwIST) algorithms accelerate convergence for ill-posed inverse problems in image restoration. These methods offer significant speedups over standard IST algorithms, even for non-invertible operators.

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Area of Science:

Applied Mathematics
Image Processing
Optimization

Background:

Iterative shrinkage/thresholding (IST) algorithms are used for convex optimization in image restoration and linear inverse problems.
The convergence rate of IST algorithms is limited by the condition number of the linear observation operator, especially for ill-posed problems.

Purpose of the Study:

Introduce two-step IST (TwIST) algorithms to improve convergence speed for ill-conditioned inverse problems.
Develop a monotonic version (MTwIST) for non-invertible operators.

Main Methods:

Developed TwIST algorithms by modifying the iterative steps of standard IST.
Introduced a monotonic variant (MTwIST) to handle non-invertible observation operators.
Analyzed convergence properties for various nonquadratic convex regularizers, including total variation and Besov norms.

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Main Results:

TwIST algorithms demonstrate significantly faster convergence rates compared to IST for ill-conditioned problems.
TwIST algorithms are shown to converge to a minimizer for a range of parameters and regularizers.
MTwIST shows comparable speed gains to IST experimentally, despite theoretical convergence not being proven for non-invertible operators.

Conclusions:

TwIST algorithms provide a substantial improvement in convergence speed for image restoration and linear inverse problems.
MTwIST offers a promising approach for handling non-invertible operators in these applications.
The proposed methods are validated through experimental results in image deconvolution and restoration with missing samples.