User Manual - pancroma

to the other. For example a filter is normally convoluted with an image in theneighborhood of a pixel. In order to filter the entire image, one must slide thefilter around and convolve it with the image around every pixel.The convolution theorem states that the Fourier transformation of the convolutionintegral is the product of the transforms of the two functions. That means that wecan separately transform the image and the filter, multiply the two, and inversetransformthe product. The result is the filtered image, without the need tomultiply and sum around every pixel. The price that we pay for this simplificationis the computational and memory expense of transforming the functions backand forth, but with a fast Fourier algorithm this technique can save a great deal ofcomputational time, especially for large convolution kernels. There are manygood descriptions of the Fourier transformation in the literature.The method for calculating the Fourier transform of an image is to take the onedimensionalFFT of each of the rows, followed by the one-dimensional FFT ofeach of the columns. Specifically, start by taking the FFT of the N pixel values inrow 0 of the real array. The real part of the FFT's output is placed back into row 0of the real array, while the imaginary part of the FFT's output is placed into row 0of the imaginary array. After repeating this procedure on rows 1 through N-1,both the real and imaginary arrays contain an intermediate image.Next, the procedure is repeated on each of the columns of the intermediate data.Take the N pixel values from column 0 of the real array, and the N pixel valuesfrom column 0 of the imaginary array, and calculate the FFT. The real part of theFFT's output is placed back into column 0 of the real array, while the imaginarypart of the FFT's output is placed back into column 0 of the imaginary array. Afterthis is repeated on columns 1 through N-1, both arrays have been overwrittenwith the image's frequency spectrum.Note that in this definition the row and column counts are the same. It is possibleand in fact common to transform rectangular arrays using special processingprovisions.The image transform is usually expressed as the magnitude of the vectorconsisting of real and complex (orthogonal) components. The quadrants arequite commonly swapped so that zero frequency appears in the middle of theimage, rather than at coordinate {0,0]. The following figures show graphicallyhow the spatial data and frequency data are stored.284