Differential Geometry of Monogenic Signal Representations
This paper presents the fusion of monogenic signal processing and differential geometry to enable monogenic analyzing of local intrinsic 2D features of low level image data. New rotational invariant features such as structure and geometry (angle of inters
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Institute of Computer Science, Chair of Cognitive Systems, Christian-Albrechts-University, 24118 Kiel, Germany Mathematical Image Analysis Group, Faculty of Mathematics and Computer Science, Building E1 1, Saarland University, 66041 Saarbr¨ ucken, Germany
Abstract. This paper presents the fusion of monogenic signal processing and differential geometry to enable monogenic analyzing of local intrinsic 2D features of low level image data. New rotational invariant features such as structure and geometry (angle of intersection) of two superimposed intrinsic 1D signals will be extracted without the need of any steerable filters. These features are important for all kinds of low level image matching tasks in robot vision because they are invariant against local and global illumination changes and result from one unique framework within the monogenic scale-space. Keywords: Low-level Image Analysis, Differential Geometry, Geometric (Clifford) Algebra, Monogenic Curvature Tensor, Monogenic Signal, Radon Transform, Riesz Transform, Hilbert Transform, Local Phase Based Signal Processing.
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Introduction
Local image analysis is the first step and therefore very important for detecting key points in robot vision. One aim is to decompose a given signal into as many features as possible. In low level image analysis it is necessary not to loose any information but to be invariant against certain features. Two similar key points (i.e. two crossing lines 1) can have different main orientations in two images but they should have the same structure (phase) and apex angle (angle of intersection) in both images. The local phase ϕ(t) of an assumed signal model cos(ϕ(t)) and the apex angle of two superimposed signals are rotational invariant features of images and therefore very important for matching tasks in robot vision. Two superimposed patterns are of big interest being analyzed because they are the most frequently (after intrinsic 0D and 1D signals) appearing 2D structures in images. In this paper we present how to decompose 2D image signals which consist locally of two intrinsical 1D signals. To extract the essential features like
We acknowledge funding by the German Research Foundation (DFG) under the projects SO 320/4-2 and We 2602/5-1.
G. Sommer and R. Klette (Eds.): RobVis 2008, LNCS 4931, pp. 454–465, 2008. c Springer-Verlag Berlin Heidelberg 2008
Differential Geometry of Monogenic Signal Representations
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the main orientation, the apex angle and the phase of those signals no heuristics will be applied in this paper like in many other works [Stuke2004] but a fundamental access to the local geometry of two superimposed 1D signals will be given. This paper will be the first step and provides also the grasp to analyze an arbitrary number of superimposed signals in future work. In the following 2D
Minimum curvature N1
Normal vector Relation
Maximum curvature N 2 Fig. 1. This figure illustrates the relation of signal processing and differential geometry in Ê3 . Images will be visualized in their Monge patch embedding with t
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