Detecting Regular Patterns Using Frequency Domain Self-Filtering

Donald G. Bailey Physics Department, Massey University Palmerston North, New Zealand D.G.Bailey @ massey . ac.nz Filtering is often used in image processing to smooth noise, and to enhance or detect features within an image. Images which have regular patterns in the spatial domain have peaks in the frequency domain corresponding to the spatial frequencies of the regular patterns. When processing such images, it is often desirable to keep such peaks, enhancing the pattern and removing noise or irregularities. This is effectively a bandpass filtering operation. The problem with such filtering is that it requires a priori knowledge of the contents of the image so that the filter can be 'tuned' to select the appropriate frequencies. Self-filtering overcomes this by multiplying the frequency domain image with its own magnitude. This gives a bandpass filter that is automatically tuned to the frequency content of the image. Applications include: detecting and enhancing regular patterns; interpolating or extrapolating regular patterns; and smoothing or reducing noise. [ 11. This convolution in the spatial domain may be readily implemented as a multiplication in the frequency domain. 1.1 Filtering in the frequency domain Let g(x, y ) be the image being filtered, and G(u, v ) be its Fourier transformation. Linear filtering may be represented as F ( u , v ) = G(u, v)W(u, v ) (1) where W(u ,v ) is the frequency domain filter function, and F(u , v ) is the Fourier transform of the filtered image. Filtering in the frequency domain therefore corresponds to weighting the relative importance of the different spatial frequency components of an image. The particular application of the filter governs what information is important, and should be kept, and what is not, and should be discarded. The selection of an appropriate filter function requires some knowledge of how the image information is represented in the frequency domain [ 11.