On the Axioms of Scale Space Theory
Journal of Mathematical Imaging and Vision
International Journal of Computer Vision
Nonlinear diffusion on the 2D Euclidean motion group
SSVM'07 Proceedings of the 1st international conference on Scale space and variational methods in computer vision
Scale-space generation via uncertainty principles
Scale-Space'05 Proceedings of the 5th international conference on Scale Space and PDE Methods in Computer Vision
Coherent States, Wavelets and Their Generalizations
Coherent States, Wavelets and Their Generalizations
Line Enhancement and Completion via Linear Left Invariant Scale Spaces on SE(2)
SSVM '09 Proceedings of the Second International Conference on Scale Space and Variational Methods in Computer Vision
Crossing-Preserving Coherence-Enhancing Diffusion on Invertible Orientation Scores
International Journal of Computer Vision
Nonlinear diffusion on the 2D Euclidean motion group
SSVM'07 Proceedings of the 1st international conference on Scale space and variational methods in computer vision
International Journal of Computer Vision
Group-Valued regularization framework for motion segmentation of dynamic non-rigid shapes
SSVM'11 Proceedings of the Third international conference on Scale Space and Variational Methods in Computer Vision
Fast regularization of matrix-valued images
ECCV'12 Proceedings of the 12th European conference on Computer Vision - Volume Part III
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In the standard scale space approach one obtains a scale space representation u : Rd⋊R+ → R of an image f ∈ L2(Rd) by means of an evolution equation on the additive group (Rd, +). However, it is common to apply a wavelet transform (constructed via a representation U of a Lie-group G and admissible wavelet ψ) to an image which provides a detailed overview of the group structure in an image. The result of such a wavelet transform provides a function g → (Ugψ, f)L2(R2) on a group G (rather than (Rd, +)), which we call a score. Since the wavelet transform is unitary we have stable reconstruction by its adjoint. This allows us to link operators on images to operators on scores in a robust way. To ensure U-invariance of the corresponding operator on the image the operator on the wavelet transform must be left-invariant. Therefore we focus on leftinvariant evolution equations (and their resolvents) on the Lie-group G generated by a quadratic form Q on left invariant vector fields. These evolution equations correspond to stochastic processes on G and their solution is given by a group convolution with the corresponding Green's function, for which we present an explicit derivation in two particular image analysis applications. In this article we describe a general approach how the concept of scale space can be extended by replacing the additive group Rd by a Lie-group with more structure.