A stochastic continuation approach to piecewise constant reconstruction

Area of Science:

• Image reconstruction
• Computational imaging
• Inverse problems

Background:

• Reconstructing 3-D objects from limited, noisy 2-D projections is an ill-posed inverse problem.
• Existing methods often struggle with noise and computational complexity.
• The Potts prior model offers regularization but presents optimization challenges.

Purpose of the Study:

• To develop a robust and efficient algorithm for 3-D object reconstruction from noisy line-integral projections.
• To stabilize the ill-conditioned inverse problem using a novel hybrid approach.
• To improve upon existing simulated annealing methods for signal recovery.

Main Methods:

• Introduction of stochastic continuation (SC), a hybrid algorithm combining simulated annealing and deterministic continuation.
• Mathematical proof of SC's finite-time convergence properties under mild assumptions.
• Application of SC to 3-D object reconstruction and analysis of a concave distortion acceleration method.

Main Results:

• SC successfully reconstructs 3-D objects from noisy 2-D projections.
• SC demonstrates superior performance compared to standard simulated annealing in numerical experiments.
• An explicit formula for the cost function's free parameter was derived, enhancing SC's applicability.

Conclusions:

• Stochastic continuation (SC) provides an effective solution for 3-D object reconstruction from noisy projection data.
• SC offers improved accuracy and efficiency over traditional simulated annealing.
• The developed methods are applicable to general n-D signal recovery from indirect, noisy measurements.