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Non-creation of local extrema and semi-group structure.

Lindeberg [] considered the problem of characterizing those kernels in one dimension that share the property of not introducing new local extrema in a signal under convolution. Such kernels have to be non-negative and unimodal. Moreover, they can be completely classified. He also imposed a semi-group structure on the family of kernels, which means that if two such kernels are convolved with each other, then the resulting kernel will be a member of the same family
 
In particular, this condition ensures that the transformation from a fine scale to any coarse scale should be of the same type as the transformation from the original signal to any scale in the scale-space representation,
 
If this semi-group structure is combined with non-creation of local extrema and the existence of a continuous scale parameter, and if the kernels are required to be symmetric and satisfy a mild degree of smoothness in the scale direction, then it can be shown that the family is uniquely determined to consist of Gaussian kernels.