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Introduction to the Fourier Transform

cs.unm.edu · 2,642 words · saved by 1 readers

The definitons of the transform (to expansion coefficients) and the inverse transform are given below: This shows 2 images with their Fourier Transforms directly underneath. The images are a pure horizontal cosine of 8 cycles and a pure vertical cosine of 32 cycles. Notice that the FT for each just has a single component, represented by 2 bright spots symmetrically placed about the center of the FT image. The center of the image is the origin of the frequency coordinate system. The u-axis runs left to right through the center and represents the horizontal component of frequency. The v-axis runs bottom to top through the center and represents the vertical component of frequency. In both cases there is a dot at the center that represents the (0,0) frequency term or average value of the image. Images usually have a large average value (like 128) and lots of low frequency information so FT images usually have a bright blob of components near the center. Notice that high frequencies in the

Introduction to the Fourier Transform INTRODUCTION TO FOURIER TRANSFORMS FOR IMAGE PROCESSING BASIS FUNCTIONS: The Fourier Transform ( in this case, the 2D Fourier Transform ) is the series expansion of an image function ( over the 2D space domain ) in terms of "cosine" image (orthonormal) basis functions. The definitons of the transform (to expansion coefficients) and the inverse transform are given below: F(u,v) = SUM{ f(x,y)*exp(-j*2*pi*(u*x+v*y)/N) } and f(x,y) = SUM{ F(u,v)*exp(+j*2*pi*(u*x+v*y)/N) } where u = 0,1,2,...,N-1 and v = 0,1,2,...,N-1 x = 0,1,2,...,N-1 and y = 0,1,2,...,N-1 j =

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