Diffusion Algorithms

This package contains posterior sampling algorithms.

\[- \log p(x|y,A) \propto d(Ax,y) + \reg{x},\]

where \(x\) is the image to be reconstructed, \(y\) are the measurements, \(d(Ax,y) \propto - \log p(y|x,A)\) is the negative log-likelihood and \(\reg{x} \propto - \log p_{\sigma}(x)\) is the negative log-prior.

The negative log likelihood can be set using deepinv.optim.DataFidelity(), which includes Gaussian noise, Poisson noise, etc. The negative log prior can be approximated using deepinv.optim.ScorePrior() with a pretrained denoiser, which leverages Tweedie’s formula, i.e.,

\[- \nabla \log p_{\sigma}(x) \propto \left(x-\denoiser{x}{\sigma}\right)/\sigma^2\]

where \(p_{\sigma} = p*\mathcal{N}(0,I\sigma^2)\) is the prior convolved with a Gaussian kernel, \(\denoiser{\cdot}{\sigma}\) is a (trained or model-based) denoiser with noise level \(\sigma\), which is typically set to a low value.

Note

The approximation of the prior obtained via deepinv.optim.ScorePrior() is also valid for maximum-a-posteriori (MAP) denoisers, but \(p_{\sigma}(x)\) is not given by the convolution with a Gaussian kernel, but rather given by the Moreau-Yosida envelope of \(p(x)\), i.e.,

\[p_{\sigma}(x)=e^{- \inf_z \left(-\log p(z) + \frac{1}{2\sigma}\|x-z\|^2 \right)}.\]

deepinv.sampling.MonteCarlo

Base class for Monte Carlo sampling.

Diffusion

We provide various sota diffusion methods for sampling from the posterior distribution. Diffusion methods produce a sample from the posterior x given a measurement y as x = model(y, physics), where model is the diffusion algorithm and physics is the forward operator.

deepinv.sampling.DDRM	Denoising Diffusion Restoration Models (DDRM).
deepinv.sampling.DiffPIR	Diffusion PnP Image Restoration (DiffPIR).
deepinv.sampling.DPS	Diffusion Posterior Sampling (DPS).

Diffusion methods obtain a single sample per call. If multiple samples are required, the deepinv.sampling.DiffusionSampler can be used to convert a diffusion method into a sampler that obtains multiple samples to compute posterior statistics such as the mean or variance.

deepinv.sampling.DiffusionSampler

Turns a diffusion method into a Monte Carlo sampler

Markov Chain Monte Carlo Langevin

We also provide MCMC methods for sampling from the posterior distribution based on the unadjusted Langevin algorithm.

deepinv.sampling.ULA	Projected Plug-and-Play Unadjusted Langevin Algorithm.
deepinv.sampling.SKRock	Plug-and-Play SKROCK algorithm.