Continuous Mathematics

Lecturer: Dr J.G. Daugman (jgd1000@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).

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.

Reference books:

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

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