Sparse stochastic processes and discretization of linear inverse problems.

Area of Science:

• Statistical Signal Processing
• Inverse Problems
• Computational Imaging

Background:

• Ill-conditioned linear inverse problems are prevalent in scientific imaging and data analysis.
• Existing regularization methods like Tikhonov and l1-regularization have limitations in capturing complex signal properties.
• Sparse stochastic processes offer a theoretical framework for modeling continuous-domain signals.

Purpose of the Study:

• To develop a novel statistically-based discretization paradigm for ill-conditioned linear inverse problems.
• To derive a class of maximum a posteriori (MAP) estimators grounded in sparse stochastic process theory.
• To extend regularization techniques beyond current limitations, including nonconvex sparsity-promoting methods.

Main Methods:

• Formulating continuous-domain signals as solutions to linear stochastic differential equations.
• Confining admissible priors for discretized signals to infinitely divisible distributions.
• Developing a computational algorithm to handle nonconvex regularization problems.
• Applying the formalism to deconvolution, magnetic resonance imaging (MRI), and X-ray tomographic reconstruction.

Main Results:

• The derived MAP estimators encompass Tikhonov and l1-regularization as special cases.
• The framework supports a broader range of sparsity-promoting regularization schemes, including nonconvex ones.
• Demonstrated successful application and performance comparison across various inverse problems.
• Showcased improved performance with increasing signal sparsity.

Conclusions:

• The proposed statistically-based discretization paradigm offers a unified framework for inverse problems.
• This approach expands the toolkit for sparsity-promoting regularization, particularly for nonconvex problems.
• The method shows significant promise for enhancing image reconstruction and deconvolution in scientific applications.