Lecture

Fourier Transform Applications: Image Processing Basics

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In course

PHYS-468: Physics of life

Life has emerged on our planet from physical principles such as molecular self-organization, thermodynamics, stochastics and iterative refinement. This course will introduce the physical methods to study life and will discuss the quantitative and physical ...

Description

This lecture covers the basics of image processing, including Fourier filters, low-pass and high-pass filters, band-pass filters, Fourier transformation, convolution, cross-correlation, and 3D reconstruction using the Radon projection theorem. It explains how to apply Fourier equations, couple conjugation, and the convolution of images. The instructor demonstrates how to speed up convolution by multiplying FFTs and calculating the inverse FFT. The lecture also discusses the relationship between contrast in images and FFT amplitude, as well as the process of deconvolution. Additionally, it explores the Fourier transform applications in analyzing microscope performance, image resolution, correcting image artifacts, extracting crystal structures, and 3D reconstruction from multiple image orientations.

Instructor

Henning Paul-Julius Stahlberg

Mission We use high-resolution cryo-electron microscopy to study neurodegeneration and other protein systems. We also develop novel imaging technology in the areas of electron microscopy hardware and applications, sample preparation, data collection schemes, and data analysis software systems. The Stahlberg group is supported by the University of Lausanne, Faculty of Medicine and Biology, and by the EPFL, Faculty of Basic Sciences, Physics Institute. The LBEM is a research laboratory of both UNIL and EPFL.

Official source

Ontological neighbourhood

Information engineering

Signal processing: Fourier analysis

Related concepts (32)

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.

Correlation

In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics it usually refers to the degree to which a pair of variables are linearly related. Familiar examples of dependent phenomena include the correlation between the height of parents and their offspring, and the correlation between the price of a good and the quantity the consumers are willing to purchase, as it is depicted in the so-called demand curve.

Discrete-time Fourier transform

In mathematics, the discrete-time Fourier transform (DTFT), also called the finite Fourier transform, is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function. The term discrete-time refers to the fact that the transform operates on discrete data, often samples whose interval has units of time. From uniformly spaced samples it produces a function of frequency that is a periodic summation of the continuous Fourier transform of the original continuous function.

Cross-correlation

In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a sliding dot product or sliding inner-product. It is commonly used for searching a long signal for a shorter, known feature. It has applications in pattern recognition, single particle analysis, electron tomography, averaging, cryptanalysis, and neurophysiology. The cross-correlation is similar in nature to the convolution of two functions.

Crystal structure

In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions, or molecules in a crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter. The smallest group of particles in the material that constitutes this repeating pattern is the unit cell of the structure.

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