Reformulating Optical Flow to Solve Image-Based Inverse Problems and Quantify Uncertainty.

Area of Science:

• Multidisciplinary applications in scientific imaging, including meteorology, medical imaging, and cell mechanics.
• Focus on the development of novel computational frameworks for image analysis and physical measurement extraction.

Background:

• Traditional pipelines combine optical flow (OF) and inverse problems (IPs) to derive physical measurements from image sequences.
• This combined OFIP approach amplifies inherent ill-posedness, leading to error propagation and hindering uncertainty quantification.

Purpose of the Study:

• To introduce a Bayesian PDE-constrained framework for direct transformation of visual information into physical measurements.
• To address the limitations of existing OFIP methods by providing accurate reconstructions and enabling uncertainty quantification.

Main Methods:

• Development of a Bayesian framework integrating partial differential equation (PDE) constraints.
• Directly processing visual information into probability distributions representing physical measurements.
• Utilizing posterior mean for constrained inverse problems and posterior covariance for deriving measurement error from image noise.

Main Results:

• The proposed framework yields more accurate reconstructions compared to traditional OFIP methods.
• Demonstrates unprecedented flexibility in experimental design, including the use of arbitrary boundary conditions.
• Successfully derives measurement error exclusively from image noise, a crucial aspect for empirical science.

Conclusions:

• The Bayesian PDE-constrained framework offers a robust alternative to conventional OFIP pipelines for extracting physical measurements from image data.
• Enables accurate reconstructions, flexible experimental design, and reliable uncertainty quantification in scientific imaging.
• Provides a method for isolating measurement error, enhancing the reliability of scientific findings derived from imaging techniques.