Fourier Trans and applications University f Central Florida傅立葉變換及應用中佛羅里達大學

1、fourier transform and applicationsby njegos nincic fourierovervieww transformsnmathematical introductionw fourier transformntime-space domain and frequency domainndiscret fourier transformlfast fourier transformnapplicationsw summaryw referencestransformsw transform:nin mathematics, a function that

2、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 substitutiontransformsw example of a substitution:w original equation: x + 4x 8 = 0w familiar form: ax + bx + c = 0w let: y = xw sol

3、ve for yw x = y4transformsw transforms are used in mathematics to solve differential equations:noriginal equation: napply laplace transform: ntake inverse transform: y = l(y)y9y15e2ts2ly9ly15s2l y15s 2 s29fourier transformw property of transforms:nthey convert a function from one domain to another w

4、ith no loss of informationw fourier transform: converts a function from the time (or spatial) domain to the frequency domaintime domain and frequency domainw time domain:ntells us how properties (air pressure in a sound function, for example) change over time:lamplitude = 100lfrequency = number of c

5、ycles in one second = 200 hztime domain and frequency domainw frequency domain:ntells us how properties (amplitudes) change over frequencies:time domain and frequency domainw example:nhuman ears do not hear wave-like oscilations, but constant tonew often it is easier to work in the frequency domaint

6、ime domain and frequency domainw 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 onetime domain and frequency domainfourier transformw because of the

7、 property:w fourier transform takes us to the frequency domain:discrete fourier transformw 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(n) time complexityfast fourier transform

8、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)applicationsw in image processing:ninstead of time domain: spatial domain (normal image space)nfrequency domain: space in which ea

9、ch 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 imagesw 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

10、 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 imagesw spatial frequency of an image refers to the rate at whi

11、ch the pixel intensities change w in picture on right:nhigh frequences:lnear centernlow frequences:lcornersapplications: image filteringw other applications of the dftw signal analysisw sound filteringw data compressionw partial differential equationsw multiplication of large integerssummaryw transf

12、orms:nuseful in mathematics (solving de)w fourier transform:nlets us easily switch between time-space domain and frequency domain so applicable in many other areasneasy to pick out frequenciesnmany applicationsreferenceswconcepts and the frequency domainnhttp:/www.spd.eee.strath.ac.uk/interact/fourier/concepts.htmlwthe frequency domain introductionnhttp:/nam.vn/unescocourse/computervision/91.htmwjpnm physics fourier transform nhttp:/