This page is under construction, so please ignore the notes I have inserted here and there to remind myself of things still to be done. Also, please note that the revision notes listed below are drafts and may contain more typographical and other errors than you might usually expect.

Mathematics Revision Material for Graduate and Masters Students

The following mathematics revision material for graduate masters and doctoral students is available:

A do-it-yourself test

The test begin with some elementary questions on sets theory and Venn diagrams and progresses via similarly elementary material on functions, differentiation, etc, to some quite tricky stuff on vectors, matrices, eigensolutions, complex variable theory, Fourier series and transorms, probability theory, etc and ends with some, again, more elementary material on graph theory. Some of the questions are genuinely difficult. Some will expand to monstrous amounts of work if tackled in the wrong way by the inexperienced, unskilled or unwary.

Solution hints for the do-it-yourself test

To revise your mathematics, you should obtain (or dig out, if you already have one) an appropriate text book. If you think you need to revise really basic stuff you learnt at school, say the UK A-level material (at least as it was before they started tinkering unnecessarily with the syllabus and organisation of the A-leverl courses), look here. Otherwise, read on.

Some suggested textbooks

Even if you feel confident about your maths, a little refresher and exposure to some of the potentially rather different kinds of mathematics that are important in, for example image processing, computer vision and machine learning may still help. There appears to be no single book covering all that you will need, though the following come close to doing so, in a very rough order of level/size/coverage:

Google can doubtless be used to find lots more. For example, it told me that "There is a new edition of the textbook on mathematical methods by Riley et al for 75.00" and that a review of the first edition by P. Steward says "the book provides scientists who need to use the tool of mathematics for practical purposes with a single, comprehensive book. I recommend this book not only to students in physics and engineering sciences, but also to students in other fields of natural sciences." Older editions of the book by Kreyszig, like many other maths books by him have been translated into other languages.

Unfortunately, these books, especially the ones by Kreyszig and by Riley, also include much extraneous material. Thus, if you feel a need for a more basic treatment of more limited scope, try any of the books in the "Schaum's" Outline Series", distributed in Britain by McGraw-Hill. For example those on "linear algebra", "advanced calculus", and "probability and statistics", cover some of the basic material required in these areas. The one on linear algebra does, however, contain some material that is rather more advanced that you may not have met before.

In addition, you might also like to use the maths revision topics as examples on which to try your Matlab or Mathematica skills. The standard text for Mathematica is Wolfram's book but you may find the shorter book by Shaw and Tigg more useful as an introduction whilst the book by Nancy Blachman is one often favoured by members of the Computer Science Department. Kreysig has also produced one, together with an instructors manual!

Matlab books --- ask Mark Herbster [!! BFB]

Very brief material indicating the kind of thing you should know has been prepared on a number of topics as follows. The links are to typed versions prepared by Matthew Trotter . Topics without links have not (yet) been typed up, but copies of my overheads will be supplied for some during or following the revision lectures. Ask me for copies on notes not supplied if you feel you would benefit from them, but don't be surprised if I judge it in your best interests to turn the request down.

Add the titles of the notes not yet typed. [!!BFB]

Fundamentals: Set theory

Functions of a single variable

More about functions: Limits, continuity, differentiation

Using derivatives

Properties of integrals

Sequences and series

Complex numbers

From cartesian and polar forms, via DeMoivre's theorem, functions of a complex vaiable, Cauchy-Riemann conditions, power series, trigonometric and hyperbolic functions to Cauchy's theorem. Stops just short of Taylor and Laurent series and the residue theorem. It is assumed that you will revise the essential elementary material (eg up to trigonometric and hyperbolic functions) yourself.

Functions of several variables

Using partial derivatives

Constrained optimisation

Lagrange multipliers. No more needs to be said.

Differential geometry

Begins with a fairly standard treatment of plane curves, followed by space curves and Frenet-Serret formulae before getting on to surfaces in 3D. Treatment via usual parametric forms, though in elementary notation so have to stop before reaching Christoffel symbols. Covers: first and second fundamental forms, and their relationship to surface curvature matrix, principal curvatures and directions. Last bit needs rewriting to make the connection with the generalised eigenproblem clear and to explain why this has to be so, which would give a pointer to the really exciting things relating to surface critical points that are more advanced.

Differential geometry was once a topic that was de rigeur in computer vision, but has fallen out of favour now that computer vision researchers are less fanatical about machine stereo vision and structure from motion algorithms. To be useful nowadays, we need to go beyond the above topics to the characterisation of surface shape via their critical points, focal surfaces and their rib and parabolioc lines and the corresponding ridge lines and flexcords on the surface itself. These concepts are used quite extensively in medical image (n-dimensional signal) processing. See Ian Porteous' wonderful book on Geometric Differentiation (2000) for some mathematical details, but beware that treatment of these topics requires mastery of another new notation. A summary was contained in one of our recent masters project dissertations.

Vector integration

From ordinary (Riemann) integrals, via scalar annd vector line integrals of scalar and vector functions, conservative fields, surface integrals of various types, the divergence (Gauss) theorem, Stokes theorem, derivatives of scalar and vector fields, generalisations of Gauss and Stokes theorems, to vector integration by parts, otherwise known as Green's theorems (not to be confused with Green's functions - see below under differential equations). This is regarded as more specialised material not covered in the revision sessions. The handwritten notes are available to help those who can convince me they need it. The generalisation of Gauss and Stokes theorem seems rarely to be covered in books, which is frustrating if you are unfortunate enough to discover you need it.

Differential equations

Ordinary differential equations (ODEs). From types of differential equations, order etc, and their solution, including specification of initial/boundary conditions, via: first order, separable and linear equations, homogeneous and inhomogeneous equations, complementary fuctions and particular integrals to the idea of a Green's function. Worked example of a Green's function for a first order, linear ordinary differential equation.

More complicated differential equations, including: exact first order equations and integrating factors; second order ordinary differential equations, homogeneous and inhomogeneous equations, complementary fuctions and particular integrals. The method of variation of constants and the Wronskian leading to the Greens function again, but in its more usual context. Complex solutions of linear second order ordinary differential equations.

The main point of the above is to show that Green's functions can be introduced in an elementary and very concrete way (courtesy of Eugene Butkov's book, but here shorn of the unnecessary physics) and therefore something not be feared, yet can be easily generalised by judicious use of the Dirac delta-function, to provide a very general, powerful concept with wide application.

Eigenfunctions

Example from second order lineasr ordinary differential equation as an introduction; linear independence, orthogonality, and analogy with vectors, basis functions and least squares convergence of eigenfunction expansions, Fourier examples. Convergence and Dirichlet conditions. Sturm-Liouville theory and importance of and types of boundary conditions. Relationship to Green's functions. Fourier series and complex Fourier series, differentiation and integration of Fourier series. The delta-function, mention of the notion of a generalised function.

I could and should have put more in the notes about good functions and fairly good functions using the averaging concepts that the delta-function example forces on us. Hopefully in the future, I willbe able to do so.

Partial Differential Equations

Commences with some remarks on the difficulty of solving partial differential equations (PDEs) and the concomitant importance of understanding their behaviour. Illustration of integration of a partial differential equation by a simple example. Homogeneous and inhomogeneous PDEs and use of Green's functions. Surface terms and second Green's theorem. Eigenfunctions of PDEs. Separation of variables. Degeneracy and eiggenfunction expansions. General solution of homogeneous PDEs by separation of variables. Eigenfunction expansion of the Green's function. Multidimensional Fourier series, notion of k-space (reciprocal space) and mention of Fourier-Bessel series.

Would be better if I had included discussion of characteristics and a little on first order PDEs. Perhaps in the future...

Matrices

Idea of a matrix, types of matrices, row and column vectors and matrix algebra including: matrix multiplication, the unit matrix, matrix-vector multiplication, transpose of a matrix, special matrices, diagonal matrices, inverse matrix, and singular matrices.

Matrices and linear algebra. Formal solution of simultaneous linear equations, practical solution via Gaussian elimination procedures, triangular matrices, idea of LU factorisation. Equivalence relationships including row and column operations, partitioning, block diagonal matrices, congruence of square matrices, similarity of square matrices. Rank of a matrix. Solution of simultaneous linear equations, homogeneous and inhomogeneous equations.

Matrix-vector transformations, active and passive interpretations. Examples: reflection in a line in 2D and idea of invariance and of characteristic (eigen) vectors, a natural basis set and diagonalisation of a matrix. Projection onto a line in 2D, idempotency, and projection on a plane in 3D. One-dimensional shear in 2D, and a conundrum re the number of characteristic vectors.

Eigenvalues and eigenvectors of a matrix

Definition, some properties and notion of generalised eigenvalues and eigenvectors. Relationship of eigenvalues and eigenvectors of a matrix to linear algebra. Characteristic equation, the eigen solutions, and distinct eigenvalues. Matrix diagonalisation, uses, some matrix functions and polynomials, cautionary note re effects of non-commutativity. Example of rotations in 2D. Cayley-Hamilton theorem, geometric interpretation and repeated roots of characteristic equation, algebraic and geometric multiplicities.

Real symmetric matrices and orthonormal eigen/basis vectors. Householder reflections, QR factorisation and Schur decomposition. Complex matrices, Hermitian, unitary and normal matrices.

Singular value decomposition of a real matrix

Extension of ideas of eigensolutions of a matrix to the singular value decomposition (SVD) of a real matrix and relevance of real symmetric matrices. Use of the SVD. Matrix norms, relationship to SVD, sensitivity analysis of solution of linear simultaneous equations, condition number. Least squares problems, application of SVD and alternative formal solutions in over, under and well-determined solutions.

That's all for now; I'm afraid we don't have time to type up more of the notes at the moment.

BFB 10/10/2002