Lecture

Image Processing I: Fourier Transform and Optical Systems

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In course

MICRO-511: Image processing I

Introduction to the basic techniques of image processing. Introduction to the development of image-processing software and to prototyping using Jupyter notebooks. Application to real-world examples in industrial vision and biomedical imaging. ...

Description

This lecture covers the basics of image processing, including orientation estimation, Fourier transform, and the importance of phase. It also discusses the characterization of linear shift-invariant (LSI) systems, 2D convolution theorem, transfer functions, modeling of optical systems, and optical transfer functions. The lecture emphasizes the Fourier transform's role in analyzing spatial frequency content and the significance of the Fourier phase in image perception.

Instructors (2)

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,

Dimitri Nestor Alice Van De Ville

Expertise Brain and spinal cord imaging, signal and image processing, network science, dynamic systems Teaching & PhD PhD Students Alexandre Cionca, Chun Hei Michael Chan, Flavia Petruso, Bassam El Rawas, Nikolaos Molochidis, Christophe Pascal Jérémie Chênes, Fabrice Jean Guibert, Alessio Siviglia, Ekansh Sareen, Beatrice Lugli Past EPFL PhD Students Djano Kandaswamy, Fikret Isik Karahanoglu, Nora Leonardi, Zafer Dogan, Jeffrey Adam Kasten, Naghmeh Ghazaleh, Rotem Roza Kopel, Daniela Maria Zöller, Thomas

Official source

Ontological neighbourhood

Information engineering

Signal processing: Fourier analysis

Related concepts (24)

Discrete Fourier transform

In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT (IDFT) is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies.

Fourier analysis

In mathematics, Fourier analysis (ˈfʊrieɪ,_-iər) is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer. The subject of Fourier analysis encompasses a vast spectrum of mathematics.

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.

Fractional Fourier transform

In mathematics, in the area of harmonic analysis, the fractional Fourier transform (FRFT) is a family of linear transformations generalizing the Fourier transform. It can be thought of as the Fourier transform to the n-th power, where n need not be an integer — thus, it can transform a function to any intermediate domain between time and frequency. Its applications range from filter design and signal analysis to phase retrieval and pattern recognition.

Fourier inversion theorem

In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency and phase information about a wave then we may reconstruct the original wave precisely. The theorem says that if we have a function satisfying certain conditions, and we use the convention for the Fourier transform that then In other words, the theorem says that This last equation is called the Fourier integral theorem.

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