Uniqueness of the Gaussian Kernel for Scale-Space Filtering

Linear spatio-temporal scale-space

“…Formulations closely related to this have been used when proving the uniqueness of Gaussian filtering for generating scale-space representations for continuous or discrete signals (Koenderink 1984, Babaud et al 1986, Yuille & Poggio 1986, Lindeberg 1990, Lindeberg 1994, Lindeberg 1996 with extension to non-linear diffusion schemes by (Weickert 1998). While scale invariance may at first also be regarded as a very desirable and thus a primary property of a multi-scale representation, this criterion has the limitation that it can only be formulated for signals defined on a continuous domain.…”

“…This insight is a major motivation for the development of multi-scale representations such as pyramids (Burt 1981, Crowley 1981 and scale-space representation (Iijima 1962, Witkin 1983, Koenderink 1984, Babaud et al 1986, Yuille & Poggio 1986, Koenderink & van Doorn 1992, Lindeberg 1994, Pauwels et al 1995, Florack 1997); see also (Alvarez et al 1993, ter Haar Romeny 1994, Sporring et al 1996, ter Haar Romeny et al 1997, Weickert 1998, Nielsen et al 1999, Kerckhove 2001.…”

On the Number of Modes of a Gaussian Mixture

“…In order to track the maxima from small to large scales, and generally to have a non-committed visual front-end, it is desirable to use a convolution kernel that does not introduce new maxima as the scale increases-since these would be spurious detail due to the kernel rather than reflecting true structure in the image. In this context, several researchers (including among others 1 Koenderink, 1984;Babaud et al, 1986;Yuille and Poggio, 1986;Roberts, 1997) proved that the Gaussian kernel never creates new maxima in 1D and, further, is the only kernel to do so. Their proofs are typically based on the following points.…”

Morphological scale-space in image processing

“…The original Gaussian scale-space theory, proposed by Iijima [5] and later by Witkin [6], has been shown to be the only viable linear scale-space in image processing [7]. Non-linear scale-space research has generally focused on nonlinear partial differential equations (PDEs), such as those used in anisotropic diffusion techniques, and on mathematical morphology.…”

“…It has been proven that the Gaussian is the only linear convolution kernel that satisfies the scale-space monotone requirement in one dimension (1-D), with zero-crossing of the second derivative defined as features [7,8]. However, in two dimensions (2-D), with zerocrossings of the Laplacian as features, "a closed zero-crossing contour can split into two as the scale increases .…”