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Fourier transform in SVG graphics

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Fourier transform in SVG graphics
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Paris, France

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In this paper author proposes one efficient method of raster to vector conversion, based on the Fast Fourier Transform (FFT). Let’s consider a problem of conversion of photorealistic raster images to vectors and back transform- rasterization of vectors. A raster image to be converted to vector file can be represented as a combination of three components: (sharp) contours, which separate more or less smoothly colored regions with low color gradients, and hi-frequency noise or texture. This paper focuses on highly efficient method of raster-to-vector conversion of contours and gradient fills. Hi-frequency noise or texture probably should be converted to vector objects with the aid of fractals. On the one hand, the goal is to create a vector file looking similar to its raster prototype, and on the other, this file must be as small and compact as possible. Let’s speak for simplicity about grayscale images. The first step is to replace the original raster with the matrix of the same size, containing color gradients. For the grayscale image, this gradient is a vector, one component of which is a derivative of brightness along X-axis, and the other is a derivative of a brightness along Y-axis. This matrix can be converted to raster image by convolution with Cauchy kernel. The best way to perform this operation is by using FFT. The essence is that this matrix of gradients can be efficiently compressed with only minor losses. Such a matrix, obtained from a general raster, will be filled mostly with low gradients, and only small part of it would contain high gradients, corresponding to contours of the source raster image. These high gradients will be located along narrow lines, like fences between plots of land.