Chapter Summary

Chapter Summary

Key Points

• 1.

  The score function $\nabla_\mathbf{x}\log p(\mathbf{x})$ encodes the data distribution without requiring the normalisation constant. Denoising score matching provides a tractable training objective, and Tweedie's formula connects the score to the posterior mean $\mathbb{E}[\mathbf{x}_0 \mid \mathbf{x}_t]$.

• 2.

  Diffusion Posterior Sampling (DPS) modifies the reverse diffusion process with a likelihood guidance gradient computed via the Tweedie estimate and backpropagation through the score network. The guidance scale $\zeta$ controls the tradeoff between prior fidelity and measurement consistency.

• 3.

  Measurement-consistent methods (DDRM, DDNM, MCG, DiffPIR) offer alternatives to DPS: SVD-based null-space preservation for structured operators, hard projection for exact consistency, and HQS-integrated diffusion for connection to the PnP framework.

• 4.

  The computational bottleneck of diffusion-based reconstruction — hundreds to thousands of NFEs per sample — can be addressed via DDIM ($50$--$200$ steps), DPM-Solver ($10$--$25$ steps), or consistency models ($1$--$4$ steps), with corresponding quality-speed tradeoffs.

• 5.

  Diffusion methods offer the best reconstruction quality among current approaches ($+1$--$3$ dB over PnP) and uniquely enable uncertainty quantification via posterior sampling — but at $2$--$10\times$ higher computational cost.

• 6.

  For RF imaging, the main challenges are limited training data (addressed via transfer learning and simulation), domain gap (addressed via fine-tuning), and computational cost (addressed via acceleration). Physics-constrained diffusion training offers a promising path to domain-adapted priors.