Lecture

Signals & Systems I: Cross-Correlation and Convolution

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In course

MICRO-310(b): Signals and systems I (for SV)

Présentation des concepts et des outils de base pour l'analyse et la caractérisation des signaux, la conception de systèmes de traitement et la modélisation linéaire de systèmes pour les étudiants en sciences de la vie. Application de ces techniques au tra ...

Description

This lecture covers the definition of cross-correlation function, intercorrelation between complex or real signals, reversal operator, signal detection by thresholding, optimal detector in the presence of white noise, orthonormal basis representation, Hilbert spaces, orthonormal families of functions, orthogonal approximation theorem, and trigonometric Fourier series. It also discusses image compression using the Discrete Cosine Transform (DCT) and the spectrum of periodic signals.

Instructor

Michaël Unser

Expertise Image Processing, Medical Imaging, Biological Imaging, Wavelets, Splines, Multiresolution Michael Unser is Full Professor with EPFL’s school of Engineering and the academic director of EPFL's Center for Imaging, Lausanne, Switzerland. His primary areas of investigation are signal processing, biomedical imaging and applied functional analysis. He is internationally recognized for his research contributions to sampling theory, wavelets, the use of splines for image processing, stochastic processes,

Official source

Related concepts (32)

Hilbert space

In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space.

Image compression

Image compression is a type of data compression applied to s, to reduce their cost for storage or transmission. Algorithms may take advantage of visual perception and the statistical properties of image data to provide superior results compared with generic data compression methods which are used for other digital data. Image compression may be lossy or lossless. Lossless compression is preferred for archival purposes and often for medical imaging, technical drawings, clip art, or comics.

Fourier series

A Fourier series (ˈfʊrieɪ,_-iər) is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation.

Data compression

In information theory, data compression, source coding, or bit-rate reduction is the process of encoding information using fewer bits than the original representation. Any particular compression is either lossy or lossless. Lossless compression reduces bits by identifying and eliminating statistical redundancy. No information is lost in lossless compression. Lossy compression reduces bits by removing unnecessary or less important information.

Fractal compression

Fractal compression is a lossy compression method for s, based on fractals. The method is best suited for textures and natural images, relying on the fact that parts of an image often resemble other parts of the same image. Fractal algorithms convert these parts into mathematical data called "fractal codes" which are used to recreate the encoded image. Iterated function system Fractal image representation may be described mathematically as an iterated function system (IFS).

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