Inverse Problems in Optical Imaging

Inverse problems in optical imaging are concerned with recovering internal properties of a medium (absorption, scattering, refractive index) from external measurements of light. These problems are almost always ill-posed in the sense of Hadamard: solutions may not exist, may not be unique, or may not depend continuously on the data.

The Regularization Imperative

Because direct inversion amplifies noise, we must impose additional structure on the solution. The standard approach is Tikhonov regularization, which adds a penalty term to the least-squares objective. But the choice of regularization — and its strength — encodes our prior knowledge about the solution.

For biological tissue, physically motivated priors include:

• Spectral smoothness — chromophore absorption spectra vary slowly with wavelength
• Spatial continuity — tissue properties don’t change discontinuously (usually)
• Non-negativity — optical properties are positive

These connect directly to the work in Optical Coherence Tomography Fundamentals, where we exploit spectral smoothness as a regularization prior for S-OCT inverse algorithms.

Physics-Informed Approaches

Recent work incorporates the governing physics (e.g., the radiative transfer equation or diffusion approximation) directly into the inversion, either as hard constraints or as penalty terms in a neural network loss function. This connects to the broader trend of physics-informed neural networks (PINNs).

Open Questions

• How do we select regularization parameters automatically in the clinical setting, where ground truth is unavailable?
• Can learned regularization (via neural networks) outperform hand-crafted priors while maintaining physical interpretability?
• What is the fundamental information-theoretic limit on what can be recovered from a given measurement geometry?