What does the fourier transform do

In summary, the conversation discusses the discovery of a program that performs Fourier transforms on pictures. The individual has tried it on a lattice of dots and is seeking an explanation for the resulting Fourier transform of the picture. They mention understanding the mathematical concept of the Fourier transform and how it represents a function in frequency domain. They are curious about the mathematical idea behind the Fourier transform and its connection to the original function and frequency. The conversation then delves into a related topic of scattering of electromagnetic waves on objects and references Kirchhoff's scalar theory of diffraction. The individual is directed to a Wikipedia page for further understanding.
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Today I found a program, which does Fourier transforms on pictures and tried it on some basic patterns. One of those was a lattice of dots and I have attached this and its Fourier transform to the thread.
I would very much like if someone in basic details could explain what is going on. Why the Fourier transform of the picture looks as it does. Mathematically I understand what there is to know about the Fourier transform and how we represent a function in its frequency domain by writing it as a superposition of sinusoidal functions with different frequencies.
In terms of the mathematical idea behind the FT what is it then, that I observe on the picture? What is my original function (the brightness?) and what does the frequency represent? Please don't hesitate to write a long answer.
 

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  • #2
This is more a physics than a mathematics question. However, it's a very interesting one.

A pretty complicated topic in theoretical electromagnetism and optics is the scattering of electromagnetic waves on objects. The most simple approximation is Kirchhoff's scalar theory of diffraction. The idea is close to Huygen's principle, i.e., you shine with light on an opaque screen with some openings (like the one with regular holes shown on the left in your picture). Then from any point of the openings a spherical wave is emitted, and the total wave for an observer is given by the superposition of all these elementary spherical waves. The superposition of the spherical waves leads to partial destruction or amplifying, resulting in characteristic interference patterns.

In a certain limit, the socalled Fraunhofer-diffraction, where you observe the diffraction picture of your object in a distance very far from the extension of the object, the amplitude on this screen is given by the Fourier transform of the object.

Have a look at Wikipedia:

http://en.wikipedia.org/wiki/Fraunhofer_diffraction
 
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What is the Fourier transform?

The Fourier transform is a mathematical operation that decomposes a function of time (or space) into its constituent frequencies. It is commonly used in signal processing, image processing, and other areas of science and engineering.

What does the Fourier transform do?

The Fourier transform takes a function in the time (or space) domain and converts it into a representation in the frequency domain. This allows us to analyze the different frequencies present in a signal or image, which can provide valuable insights and information.

How does the Fourier transform work?

The Fourier transform works by breaking down a function into a series of sine and cosine waves of different frequencies. These waves are then combined to recreate the original function. The Fourier transform can be performed using mathematical equations or with the help of specialized algorithms.

What are the applications of the Fourier transform?

The Fourier transform has many applications in various fields such as signal processing, image processing, digital communications, audio and video compression, and more. It is also used in the analysis of physical phenomena, such as vibrations and waves.

What is the difference between the Fourier transform and the inverse Fourier transform?

The Fourier transform converts a function from the time (or space) domain to the frequency domain, while the inverse Fourier transform does the opposite - it converts a function from the frequency domain back to the time (or space) domain. Essentially, the inverse Fourier transform is the reverse operation of the Fourier transform.

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