Generalized axiomatic scale-space theory

Tony Lindeberg

Advances in Imaging and Electron Physics, volume 178, pages 1-96.

Digitally published with doi:10.1016/B978-0-12-407701-0.00001-7

Abstract

A fundamental problem in vision is what types of image operations should be used at the first stages of visual processing. I discuss a principled approach to this problem by describing a generalized axiomatic scale-space theory that makes it possible to derive the notions of linear scale-space, affine Gaussian scale-space, and linear spatio-temporal scale-space using similar sets of assumptions (scale-space axioms).

Based on a requirement that new image structures should not be created with increasing scale formalized into a condition of non-enhancement of local extrema, a complete classification is given of the linear (Gaussian) scale-space concepts that satisfy these conditions on isotropic spatial, non-isotropic spatial, and spatio-temporal domains, which results in a general taxonomy of Gaussian scale-spaces for continuous image data. The resulting theory allows filter shapes to be tuned from specific context information and provides a theoretical foundation for the recently exploited mechanisms of affine shape adaptation and Galilean velocity adaptation, with highly useful applications in computer vision. It is also shown how time-causal and time-recursive spatio-temporal scale-space concepts can be derived from similar or closely related assumptions.

The receptive fields arising from the spatial, spatiochromatic, and spatio-temporal derivatives resulting from these scale-space concepts can be used as a general basis for expressing image operations for a large class of computer vision or image analysis methods. The receptive field profiles generated by necessity from these theories also have close similarities to receptive fields measured by cell recordings in biological vision, specifically regarding space-time separable cells in the retina and the lateral geniculate nucleus (LGN), as well as both space-time separable and non-separable cells in the striate cortex (V1) of higher mammals.

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Background and related material: (Underlying mathematical theory for linear, affine and spatio-temporal scale-space) (Underlying theory for scale invariant, affine invariant and Galilean invariant receptive fields with relations to receptive fields in biological vision) (Extension of this theory for computing invariant visual representations under scaling transformations, local affine image deformations, local Galilean transformations and local multiplicative illumination transformations)

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