A Survey of Gaussian Convolution Algorithms

Gaussian convolution is a common operation and building block for algorithms in signal and image processing. Consequently, its ecient computation is important, and many fast approximations have been proposed. In this survey, we discuss approximate Gaussian convolution based on nite impulse response lters, DFT and DCT based convolution, box lters, and several recursive lters. Since boundary handling is sometimes overlooked in the original works, we pay particular attention to develop it here. We perform numerical experiments to compare the speed and quality of the algorithms.

[1]  B. Berndt.  Ramanujan’s Notebooks: Part V , 1997 .

[2]  Atsushi Imiya,et al.  Linear Scale-Space has First been Proposed in Japan , 1999, Journal of Mathematical Imaging and Vision.

[3]  Lucas J. van Vliet,et al.  Recursive implementation of the Gaussian filter , 1995, Signal Process..

[4]  Tony Lindeberg,et al.  On the Axiomatic Foundations of Linear Scale-Space , 1997, Gaussian Scale-Space Theory.

[5]  Steven G. Johnson,et al.  The Design and Implementation of FFTW3 , 2005, Proceedings of the IEEE.

[6]  H. Weyl.  Gruppentheorie und Quantenmechanik , 1928 .

[7]  Sovira Tan,et al.  Performance of three recursive algorithms for fast space-variant Gaussian filtering , 2003, Real-time imaging.

[8]  Bill Triggs,et al.  Boundary conditions for Young-van Vliet recursive filtering , 2006, IEEE Transactions on Signal Processing.

[9]  Martin Vetterli,et al.  Fast Fourier transforms: a tutorial review and a state of the art , 1990 .

[10]  Lucas J. van Vliet,et al.  Recursive Gaussian derivative filters , 1998, Proceedings. Fourteenth International Conference on Pattern Recognition (Cat. No.98EX170).

[11]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[12]  Alan L. Yuille,et al.  Scaling Theorems for Zero Crossings , 1987, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[13]  Wesley E. Snyder,et al.  Stacked Integral Image , 2010, 2010 IEEE International Conference on Robotics and Automation.

[14]  Andrew P. Witkin,et al.  Uniqueness of the Gaussian Kernel for Scale-Space Filtering , 1986, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[15]  Michael Unser,et al.  Fast Space-Variant Elliptical Filtering Using Box Splines , 2010, IEEE Transactions on Image Processing.

[16]  Jezekiel Ben-Arie,et al.  Lattice architectures for multiple-scale gaussian convolution, image processing, sinusoid-based transforms and gabor filtering , 1993 .

[17]  Joachim Weickert,et al.  Theoretical Foundations of Gaussian Convolution by Extended Box Filtering , 2011, SSVM.

[18]  Stephen A. Martucci,et al.  Symmetric convolution and the discrete sine and cosine transforms , 1993, IEEE Trans. Signal Process..

[19]  William M. Wells,et al.  Efficient Synthesis of Gaussian Filters by Cascaded Uniform Filters , 1986, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[20]  R. Deriche.  Recursively Implementing the Gaussian and its Derivatives , 1993 .

[21]  Lucas J. van Vliet,et al.  Recursive Gabor filtering , 2000, Proceedings 15th International Conference on Pattern Recognition. ICPR-2000.

[22]  L. Álvarez,et al.  Signal and image restoration using shock filters and anisotropic diffusion , 1994 .

[23]  Alireza Tavakkoli,et al.  Accurate and Efficient Computation of Gabor Features in Real-Time Applications , 2009, ISVC.

[24]  John F. Canny,et al.  A Computational Approach to Edge Detection , 1986, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[25]  Jinhee Yi,et al.  Theta-function identities and the explicit formulas for theta-function and their applications , 2004 .

[26]  W. Heisenberg.  Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik , 1927 .

[27]  Michael Werman,et al.  Efficient and accurate Gaussian image filtering using running sums , 2011, 2012 12th International Conference on Intelligent Systems Design and Applications (ISDA).

[28]  Tony Lindeberg,et al.  Scale-Space for Discrete Signals , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[29]  David G. Lowe,et al.  Object recognition from local scale-invariant features , 1999, Proceedings of the Seventh IEEE International Conference on Computer Vision.