Signal compression
Compression of images and sound is a fascinating problem that makes use of profound mathematical results. This course seeks to be an introduction to the mathematics used in this problem and the compression problem.
Objectives
1. To familiarize the student with the mathematical language used in the literature on Classic Fourier Analysis
2. Offer a formal version of the basic techniques used in signal analysis (audio or image) as well as their applications.
Syllabus
The first block is mainly focused on providing the correct intuition and formalism necessary to understand Fourier analysis.
Calculus and Linear Algebra
-
Formal definitions and intuition about functions and limits (examples with some loss functions)
-
Geometric and probabilistic intuition of Derivatives and Integrals
-
Intuition approximation of functions and Taylor series.
-
Gradient geometric intuition
-
Vector spaces: examples and basic properties
-
Calculation of distances in vector spaces
The second block concentrates on Fourier analysis and its applications to signal processing (audio or text) as well as its different comprehension methods.
Fourier analysis
-
Discrete Fourier transform
-
Continuous Fourier transform
-
Geometric intuition of the Fourier transform
-
Harmonic functions
-
Applications to signal processing
-
Applications to data compression.