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
₤ 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
₤ 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.
₤ 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.
₤ 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
₤ 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
₤ 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
₤ Sinusoidal pattern Single Fourier term that encodes ╬ The spatial frequency, ╬ The magnitude (positive or negative), ╬ The phase.
₤ 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