Abstract
Gaussian convolution and its discrete analogue, Gauss transform, have many science and engineering applications, such as mathematical statistics, thermodynamics and machine learning, and are widely applied to computer vision and image processing tasks. Due to its computational expense (quadratic and exponential complexities with respect to the number of points and dimensionality, respectively) and rapid spreading of high quality data (bit depth/dynamic range), accurate approximation has become important in practice compared with conventional fast methods, such as recursive or box kernel methods. In this paper, we propose a novel approximation method for fast Gaussian convolution of two-dimensional uniform point sets, such as 2D images. Our method employs L1 distance metric for Gaussian function and domain splitting approach to achieve fast computation (linear computational complexity) while preserving high accuracy. Our numerical experiments show the advantages over conventional methods in terms of speed and precision. We also introduce a novel and effective joint image filtering approach based on the proposed method, and demonstrate its capability on edge-aware smoothing and detail enhancement. The experiments show that filters based on the proposed L1 Gauss transform give higher quality of the result and are faster than the original filters that use box kernel for Gaussian convolution approximation.
Highlights
Gaussian convolution is a core tool in mathematics and many related research areas, such as probability theory, physics, and signal processing
Gauss transform is a discrete analogue to the Gaussian convolution, and has been widely used for many applications including kernel density estimation [1] and image filtering [2]
We compared the multipole version of our algorithm with box kernel (Box) using moving average method [7], the 1D domain splitting (YY14) with separable implementations [11], and Fast Discrete Cosine Transform (FDCT) via the FFT package [13] well-known for its efficiency
Summary
Gaussian convolution is a core tool in mathematics and many related research areas, such as probability theory, physics, and signal processing. One of the highly accurate methods is called fast L1 Gauss transform approximation [11] based on using L1 distance instead of conventional L2 Euclidean metric This L1 metric preserves most of the properties of the L2 Gaussian, and is separable, it allows to perform computations along each dimension separately, which is very beneficial in terms of computational complexity. L1 Gaussian has only one peak in Fourier domain at the coordinate origin, and its convolution does not have some undesirable artefacts that box kernels and truncation methods usually have This algorithm works only on one-dimensional (1D) point sets, it can be extended to uniformly distributed points in higher dimensions by performing it separately in each dimension. Труды ИСП РАН, том 29, вып. 4, 2017 г., стр. 55-72
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