Local Structure Analysis by Isotropic Hilbert Transforms Lennart Wietzke 1 , Oliver Fleischmann 2 , Anne Sedlazeck 2 , and Gerald Sommer 2 1 Raytrix GmbH, Germany 2 Cognitive Systems Group, Department of Computer Science, Kiel University Abstract. This work presents the isotropic and orthogonal decomposition of 2D signals into local geometrical and structural components. We will present the so- lution for 2D image signals in four steps: signal modeling in scale space, signal extension by higher order generalized Hilbert transforms, signal representation in classical matrix form, followed by the most important step, in which the matrix- valued signal will be mapped to a so called multivector. We will show that this novel multivector-valued signal representation is an interesting space for com- plete geometrical and structural signal analysis. In practical computer vision ap- plications lines, edges, corners, and junctions as well as local texture patterns can be analyzed in one unified algebraic framework. Our novel approach will be applied to parameter-free multilayer decomposition. 1 Introduction Low level image analysis is often the first step of many computer vision tasks. There- fore, local signal features determine the quality of subsequent higher level processing steps. In this work we present a general 2D image analysis theory which is accurate and less time consuming (seven 2D convolutions are required), either because of its rotationally invariance. The first step of low level signal analysis is the designation of a reasonable signal model. Based on the fact that signals f ∈ L 2 (Ω) ∩ L 1 (Ω) with Ω ⊆ R 2 can be decomposed into their corresponding Fourier series, we assume that each frequency component of the original image signal consists locally of a superposi- tion of intrinsically 1D (i1D) signals f ν (z ) with z =(x, y) ∈ R 2 , and ν ∈{1, 2}, see Equation (3). Each of them is determined by its individual amplitude a ν ∈ R, phase φ ν ∈ [0,π) [1,2], and orientation θ ν ∈ [0,π). To access each one of those frequency components, an appropriate filter must be applied to the original signal. Although any scale space concept can be used, in this work we will choose the Poisson low pass filter kernel [3] p(z ; s) instead of the Gaussian kernel. The Poisson scale space is naturally related to the generalized Hilbert transform by the Cauchy kernel [4]. In Fourier space F{·} [5] it can be seen that the well known derivative operator of order m F{D (m) }(u) = [2πu i] m with u ∈ R 2 (1) is closely related to the generalized Hilbert transform operator of m concatenations F{H (m) }(u) = [2π ¯ u i] m with ¯ u = u ‖u‖ . (2) M. Goesele et al. (Eds.): DAGM 2010, LNCS 6376, pp. 131–140, 2010. c  Springer-Verlag Berlin Heidelberg 2010