7.0 PROBABILISTIC APPROACHES AND MACHINE LEARNING🔗⧉

A blurred or noisy photograph can admit several plausible sharp originals. The linear-methods part described what the measurements determine and how regularization stabilizes recovery. This part makes the assumptions explicit: a likelihood describes the measurement process and its noise, a prior describes plausible scenes, and their combination gives a posterior distribution over possible answers.

Chapter 7.1, Imaging as Bayesian inference with probabilistic priors distinguishes the posterior from the estimate we choose to report. Chapter 7.2, Machine learning explains how examples train a restoration operator and why the degradation model and dataset matter. Chapter 7.3, Deep learning examines image restoration, prediction, translation, and learned image metrics. Chapter 7.4, Generative AI and diffusion develops models that can generate plausible images, while Chapter 7.5, Image priors connects learned denoisers to other inverse problems. Chapter 7.6, Information theory measures uncertainty and the information carried by a measurement.

For practice, keep three questions separate: does the model match the camera, does the estimate remain consistent with the observation, and does the chosen loss reward the result we need? A sharp, plausible output can still contain incorrect detail. A small pixel error can still conceal a blurred average of several possibilities.

The progression is from assumptions we write down, to assumptions fitted from examples, to distributions from which we can sample. The first connection is algebraic: taking the negative logarithm of a posterior turns a product of likelihood and prior into a sum of data cost and regularization. The later connections are computational: a trained predictor can amortize estimation, and a generative model can represent alternative plausible images. Keeping those roles distinct helps explain why a better-looking answer need not be a better measurement.

Contents of this part

• 7.1 Imaging as Bayesian inference with probabilistic priors
  • 7.1.1 Priors and the manifold of natural images
  • 7.1.2 Probability: the Bayesian view
  • 7.1.3 Optimization, inference, loss
  • 7.1.4 Deep learning, amortized inference, and generative AI
  • 7.1.5 When you can design the operator, and a teaser of information theory
  • 7.1.6 Matching the estimate to the task
  • 7.1.7 Harder inverse problems: factorization
• 7.2 Machine learning
  • 7.2.1 The framing: learned operators replace hand-designed ones
  • 7.2.2 The data story: synthetic data, noise models, datasets
• 7.3 Deep learning
  • 7.3.1 Low-level learned operators (pixel-to-pixel)
  • 7.3.2 Mid- and high-level learned predictors
  • 7.3.3 Generative models: image-to-image translation
  • 7.3.4 Learned perceptual metrics and losses
• 7.4 Generative AI and diffusion
  • 7.4.1 The framing: generation is learning and sampling a prior $p(x)$
  • 7.4.2 Diffusion: generation as iterated denoising
  • 7.4.3 Conditioning: text, images, and control
  • 7.4.4 Posterior sampling: generative priors for inverse problems
  • 7.4.5 Other generative families, in brief
  • 7.4.6 Caveats and ethics
• 7.5 Image priors
  • 7.5.1 A catalog of priors: from smoothness to learned
  • 7.5.2 The Gaussian prior in closed form: the Wiener filter
  • 7.5.3 Shift and scale invariance
  • 7.5.4 Anisotropy: sparsity in a single direction
  • 7.5.5 Linear color models
  • 7.5.6 Denoising as a universal prior: Plug-and-Play and RED
  • 7.5.7 Denoisers and noise-dependent scores
  • 7.5.8 Score-based models, the thin manifold, and imperfect priors
• 7.6 Information theory
  • 7.6.1 The basics, at a high level
  • 7.6.2 Compression: the clearest application
  • 7.6.3 The imaging system as a channel
  • 7.6.4 Information-theoretic quantities as objectives
  • 7.6.5 The modern revival: learned compression and generative models
  • 7.6.6 Sampling, sparsity, and histogram entropy
  • 7.6.7 Where it is least relevant: aesthetics and perception

▸ full collapsible outline of this part