Fourier transform

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  • F(i,j) – Spatial Domain ImageExponential Function – Basis Functioncorresponding to each point F(k,l) in the Fourier space.basis functions are sine and cosine waves
  • Fourier transform

    1. 1. ₤ Image ╬ A representation of the external form of a person or thing in sculpture, painting, etc.₤ Image Processing ╬ The analysis and manipulation of a digitized image, esp. in order to improve its quality. ╬ Study of any algorithm Image Input Output
    2. 2. ₤ The rate at which image intensity values are changing in the image₤ Its domain over which values of F(u) range. u Freq. of component of Transform₤ Steps: ╬ Transform the image to its frequency representation ╬ Perform image processing ╬ Compute inverse transform.
    3. 3. ₤ Decompose an image into its sine & cosine components.₤ Sinusoidal variations in brightness across the image.₤ Each point represents a particular frequency contained in the spatial domain image. Spatial Domain Freq. Domain (Input) (Output)₤ Applications ╬ Image analysis, ╬ Image filtering, ╬ Image reconstruction ╬ Image compression.
    4. 4. ₤ Functions that are NOT periodic BUT with finite area under the curve can be expressed as the integral of sines and/or cosines multiplied by a weight function₤ The Fourier transform for f(x) exists iff ╬ f(x) is piecewise continuous on every finite interval ╬ f(x) is absolutely integrable
    5. 5. ₤ Fourier Series is the origin.₤ The DFT is the sampled Fourier Transform₤ 2-D DFT of N*N matrix :₤ Complexity of 1-D DFT is N2.₤ Sufficiently accurate
    6. 6. ₤ Multiply the input image by (-1)x+y to center the transform₤ Compute the DFT F(u,v) of the resulting image₤ Multiply F(u,v) by a filter G(u,v)₤ Computer the inverse DFT transform h*(x,y)₤ Obtain the real part h(x,y) of 4₤ Multiply the result by (-1)x+y
    7. 7. ₤ Sinusoidal pattern Single Fourier term that encodes ╬ The spatial frequency, ╬ The magnitude (positive or negative), ╬ The phase.
    8. 8. ₤ The spatial frequency, ╬ Frequency across space₤ The magnitude (positive or negative), ╬ Corresponds to its contrast ╬ A negative magnitude represents a contrast-reversal, i.e. the bright become dark, and vice-versa₤ The phase. ╬ How the wave is shifted relative to the origin
    9. 9. PlausibilityOriginal Magnitude Phase
    10. 10. Magnitude Phase
    11. 11. Brightness Fourier Inverse Image Transform Transformed

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