The 2D Analytic Signal

This technical report covers a fundamental problem of 2D local phase based signal processing: the isotropic generalization of the analytic signal (D. Gabor) for two dimensional signals. The analytic signal extends a real valued 1D signal to a complex valued signal by means of the classical 1D Hilbert transform. This enables the complete analysis of local phase and amplitude information. Local phase, amplitude and additional orientation information can be extracted by the monogenic signal (M. Felsberg and G. Sommer) which is always restricted to the subclass of intrinsically one dimensional signals. In case of 2D image signals the monogenic signal enables the rotationally invariant analysis of lines and edges. In contrast to the 1D analytic signal the monogenic signal extends all real valued signals of dimension n to a (n + 1) - dimensional vector valued monogenic signal by means of the generalized first order Hilbert transform (Riesz transform). In this technical report we present the 2D analytic signal as a novel generalization of the 2D monogenic signal which now extends the original 2D signal to a multivector valued signal in conformal space by means of higher order Hilbert transforms and by means of a hybrid matrix geometric algebra representation. The 2D analytic signal can be interpreted in conformal space which delivers a descriptive geometric interpretation of 2D signals. One of the main results of this work is, that all 2D signals exist per se in a 3D projective subspace of the conformal space and can be analyzed by means of geometric algebra. In case of 2D image signals the 2D analytic signal enables now the rotational invariant analysis of lines, edges, corners and junctions

Algebraic Representation and Geometric Interpretation of Hilbert Transformed Signals

This thesis covers a fundamental problem of local phase based signal processing: the isotropic generalization of the classical one dimensional analytic signal (D. Gabor) to higher dimensional signal domains. The classical analytic signal extends a real valued one dimensional signal to a complex valued signal by means of the classical 1D Hilbert transform. This signal extension enables the complete analysis of local phase and local amplitude information for each frequency component in the sense of Fourier analysis. In case of two dimensional signal domains, e.g. for images, additional geometric information is required to characterize higher dimensional signals locally. The local geometric information is called orientation, which consists of the main orientation and apex angle for two superimposed one dimensional signals. The problem of two dimensional signal analysis is the fact that in general those signals could consist of an unlimited number of superimposed one dimensional signals with individual orientations. Local phase, amplitude and additional orientation information can be extracted by the monogenic signal (M. Felsberg and G. Sommer) which is always restricted to the subclass of intrinsically one dimensional signals, i.e. the class of signals which only make use of one degree of freedom within the embedding signal domain. In case of 2D images the monogenic signal enables the rotationally invariant analysis of lines and edges. In contrast to the 1D analytic signal the monogenic signal extends all real valued signals of dimension n to a (n+1) - dimensional vector valued monogenic signal by means of the generalized first order Hilbert transform, which is also known as the Riesz transform. The analytic signal and the monogenic signal show that a direct relation between analytical signals and their algebraic representation exists. This fact has motivated the work and the results of this thesis, namely the extension of the 2D monogenic signal to more general 2D analytic signals, their algebraic representation, and their most geometric embedding. In case of more general 2D signals the geometric algebra will be shown to be a natural representation, and the conformal space as the geometric embedding for the signal interpretation. In this thesis we present 2D analytic signals as generalizations of the 2D monogenic signal which now extend the original 2D signal to a multi-vector valued signal in homogeneous conformal space by means of higher order Hilbert transforms, and by means of a so called hybrid matrix geometric algebra representation. The 2D analytic signal and the more general multi-vector signal will be interpreted in conformal space which delivers a descriptive geometric interpretation and algebraic embedding of signals. In case of 2D image signals the 2D analytic signal and the multi-vector signal enable the rotationally invariant analysis of lines, edges, corners and junctions in one unified framework. Furthermore, additional local curvature can be determined by first order generalized Hilbert transforms without the need of derivatives. This so called conformal monogenic signal can be defined for any signal domain

Curvature Detection by Integral Transforms

In various fields of image analysis, determining the precise geometry of occurrent edges, e.g. the contour of an object, is a crucial task. Especially the curvature of an edge is of great practical relevance. In this thesis, we develop different methods to detect a variety of edge features, among them the curvature. We first examine the properties of the parabolic Radon transform and show that it can be used to detect the edge curvature, as the smoothness of the parabolic Radon transform changes when the parabola is tangential to an edge and also, when additionally the curvature of the parabola coincides with the edge curvature. By subsequently introducing a parabolic Fourier transform and establishing a precise relation between the smoothness of a certain class of functions and the decay of the Fourier transform, we show that the smoothness result for the parabolic Radon transform can be translated into a change of the decay rate of the parabolic Fourier transform. Furthermore, we introduce an extension of the continuous shearlet transform which additionally utilizes shears of higher order. This extension, called the Taylorlet transform, allows for a detection of the position and orientation, as well as the curvature and other higher order geometric information of edges. We introduce novel vanishing moment conditions which enable a more robust detection of the geometric edge features and examine two different constructions for Taylorlets. Lastly, we translate the results of the Taylorlet transform in R^2 into R^3 and thereby allow for the analysis of the geometry of object surfaces

Tensor based generalization of monogenic wavelets for coherent multiscale local phase analysis of color images

4 pagesInternational audienceWe propose a new color extension of the monogenic wavelet transform. Monogenic wavelets give a coherent representation of scalar images through a local phase concept and an underlying orientation analysis. We here define a color extension of the monogenic framework. The underlying local geometric analysis and phase concept are generalized by using the color structure tensor. Resulting transform appears to be a clear improvement of our previous work - that is the only proposition of color monogenic wavelets to our knowledge. This is shift and rotation invariant and efficiently represents multiscale color lines and edges