COMPGV08 - Inverse Problems in Imaging

This database contains 2017-18 versions of the syllabuses. For current versions please see here.

CodeCOMPGV08 (Also taught as COMPM078)
PrerequisitesThis course requires good mathematical and programming skills. In particular students should be familiar with :
  • Fourier Theory
    • discrete and continuous
    • sampling
    • convolution
  • Linear Algebra
    • Eigenvalues and Eigenvectors
    • Matrix Algebra
  • Calculus
    • functions of multiple variables
    • calculus of variation
  • Probability
    • Gaussian and Poisson probabilities
    • Bayes Theorem
  • Matlab programming
    • multidimensional arrays
    • image visualisation
  • anonymous functions
If in doubt about your suitability for the course, please review the notes and examples on the course web page, and/or consult the instructor.
Taught BySimon Arridge (100%)
AimsTo introduce the concepts of optimisation, and appropriate mathematical and numerical tools applications in image processing and image reconstruction.
Learning OutcomesTo understand the principles of optimisation and to acquire skills in mathematical methods and programming techniques.



  • Example problems
  • Data Fitting Concepts
  • Existence
  • Uniqueness
  • Stability
  • Bayesian interpretation

Mathematical Tools

  • Linear Algebra
    • Solving Systems of Linear Equations
    • Over and Under Determined Problems
    • Eigen-Analysis and SVD.
  • Variational Methods
    • Calculus of Variation
    • Multivariate Derivatives
    • Frechet and Gateaux Derivatives
  • Regulariation
    • Tikhonov and Generalised Tikhonov
    • Non-Quadratic Regularisation
    • Non-Convex Regularisation
    • Methods for selection of regularisation parameters

Numerical Tools

  • Descent Methods
    • Steepest Descent
    • Conjugate Gradients
    • Line Search

    Newton Methods

    • Gauss Newton and Full Newton
    • Trust-Region and Globalisation
    • Quasi-Newton
    • Inexact Newton

Optimisation Methods

  • Least-Squares Problems
    • Linear Least Squares
    • Non-linear Least Squares
  • Non-Quadratic Problems
    • Poisson Likelihood
    • Kullback-Leibler Divergence
  • Lagrangian penalties and constrained optimisation
  • Proximal methods

Concepts of sparsity

  • L1 and total variation
  • wavelet compression
  • dictionary methods.

Bayesian Approach

  • Maximum Likelihood and Maximum A Posteriori estimates
  • Expectation-Minimisation
  • Posterior Sampling
    • Confidence-Limits
    • Monte Carlo Markov Chain


  • Image Denoising
  • Image Deblurring
  • Inpainting
  • Linear Image Reconstruction
    • Tomographic Reconstruction
    • Reconstruction from Incomplete Data
  • Non-Linear Parameter Estimation
    • General Concepts
    • Direct and Adjoint Differentiation

Other Approaches

  • Learning Based Methods

Method of Instruction

Lecture presentations with associated class coursework and laboratory sessions


The course has the following assessment components:

  • Written Examination (2.5 hours, 75%)
  • Coursework (2 pieces, 25%)

To pass this course, students must:

  • Obtain an overall pass mark of 50% for all sections combined.


Reading list available via the UCL Library catalogue.