Fourier Transform and Applications By Njegos Nincic Fourier

Overview w Transforms n Mathematical Introduction w Fourier Transform n n Time-Space Domain and Frequency Domain Discret Fourier Transform l n Fast Fourier Transform Applications w Summary w References

Transforms w Transform: n In mathematics, a function that results when a given function is multiplied by a so-called kernel function, and the product is integrated between suitable limits. (Britannica) w Can be thought of as a substitution

Transforms w Example of a substitution: w Original equation: x 4 + 4 x² – 8 = 0 w Familiar form: ax² + bx + c = 0 w Let: y = x² w Solve for y w x = ±√y

Transforms w Transforms are used in mathematics to solve differential equations: n Original equation: Apply Laplace Transform: n Take inverse Transform: y = Lˉ¹(y) n

Fourier Transform w Property of transforms: n They convert a function from one domain to another with no loss of information w Fourier Transform: converts a function from the time (or spatial) domain to the frequency domain

Time Domain and Frequency Domain w Time Domain: n Tells us how properties (air pressure in a sound function, for example) change over time: l l Amplitude = 100 Frequency = number of cycles in one second = 200 Hz

Time Domain and Frequency Domain w Frequency domain: n Tells us how properties (amplitudes) change over frequencies:

Time Domain and Frequency Domain w Example: n Human ears do not hear wave-like oscilations, but constant tone w Often it is easier to work in the frequency domain

Time Domain and Frequency Domain w In 1807, Jean Baptiste Joseph Fourier showed that any periodic signal could be represented by a series of sinusoidal functions In picture: the composition of the first two functions gives the bottom one

Time Domain and Frequency Domain

Fourier Transform w Because of the property: w Fourier Transform takes us to the frequency domain:

Discrete Fourier Transform w In practice, we often deal with discrete functions (digital signals, for example) w Discrete version of the Fourier Transform is much more useful in computer science: w O(n²) time complexity

Fast Fourier Transform w Many techniques introduced that reduce computing time to O(n log n) w Most popular one: radix-2 decimation-in-time (DIT) FFT Cooley-Tukey algorithm: (Divide and conquer)

Applications w In image processing: n n Instead of time domain: spatial domain (normal image space) frequency domain: space in which each image value at image position F represents the amount that the intensity values in image I vary over a specific distance related to F

Applications: Frequency Domain In Images w If there is value 20 at the point that represents the frequency 0. 1 (or 1 period every 10 pixels). This means that in the corresponding spatial domain image I the intensity values vary from dark to light and back to dark over a distance of 10 pixels, and that the contrast between the lightest and darkest is 40 gray levels

Applications: Frequency Domain In Images w Spatial frequency of an image refers to the rate at which the pixel intensities change w In picture on right: n High frequences: l n Near center Low frequences: l Corners

Applications: Image Filtering w

Other Applications of the DFT w Signal analysis w Sound filtering w Data compression w Partial differential equations w Multiplication of large integers

Summary w Transforms: n Useful in mathematics (solving DE) w Fourier Transform: n n n Lets us easily switch between time-space domain and frequency domain so applicable in many other areas Easy to pick out frequencies Many applications

References w w w w Concepts and the frequency domain n http: //www. spd. eee. strath. ac. uk/~interact/fourier/concepts. html THE FREQUENCY DOMAIN Introduction n http: //www. netnam. vn/unescocourse/computervision/91. htm JPNM Physics Fourier Transform n http: //www. med. harvard. edu/JPNM/physics/didactics/improc/intro/fourier 2. html Introduction to the Frequency Domain n http: //zone. ni. com/devzone/conceptd. nsf/webmain/F 814 BEB 1 A 040 CDC 6862568460 0508 C 88 Fourier Transform Filtering Techniques n http: //www. olympusmicro. com/primer/java/digitalimaging/processing/fouriertransfor m/index. html Fourier Transform (Efunda) n http: //www. efunda. com/math/fourier_transform/ Integral Transforms n http: //www. britannica. com/ebc/article? toc. Id=9368037&query=transform&ct=