*Lecturer: Dr N.A. Dodgson*
(`nad@cl.cam.ac.uk`)

*No. of lectures:* 4

*This course is a prerequisite for Computer Vision *(*Part II
and Diploma*)*, Information Theory and Coding *(*Part II*)* and Neural Computing *(*Part II*)*.*

**Aims**

The aims of this course are to review some key concepts and operations defined in continuous mathematics involving real- and complex-valued functions of real variables. Focus is on the use and implementation of these notions in the discrete spaces we enter when computing. Topics include: expansions and basis functions; orthogonality and projections; differential equations and their computational solution; linear operators and their eigenfunctions; wavelets and Fourier analysis.

**Lectures**

*This is the syllabus for 1999-2000; there may be some
changes for 2000-2001.*

**Review of analysis.**Real and complex-valued functions of a real variable. Power series and transcendental functions. Expansions and basis functions. Smoothness, continuity, limits.**Linear vector spaces and decompositions.**Orthogonality, independence, and orthonormality. Linear combinations. Projections, inner products and completeness. Linear subspaces. Useful expansion bases for continuous functions.**Differential and integral operators in computation.**The infinitesimal calculus. Taylor series. Numerical integration. Differential equations and computational ways to solve them. Complex exponentials. Introduction to Fourier analysis in one and two dimensions; useful theorems. Convolution and filtering.**Signals and systems.**Eigenfunctions of linear operators. Fourier analysis and series; continuous Fourier Transforms and their inverses. Representation in non-orthogonal functions, and wavelets. The degrees-of-freedom in a signal. Sampling theorem. How to operate on continuous signals computationally in order to extract their information.

**Objectives**

At the end of the course students should

- understand how data or functions can be represented in terms of
their projections onto other groups of functions
- be fluent in the use of, and properties of, complex variables
- be able to implement and use, in discrete computational form, such
continuous notions as differentiation, integration, and convolution
- grasp key properties and uses of Fourier analysis, transforms,
and wavelets

**Reference books**

Kaplan, W. (1992). *Advanced Calculus*. Addison-Wesley (4th ed.).

Oppenheim, A.V. & Willsky, A.S. (1984). *Signals and Systems*.
Prentice-Hall.