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

Twistor Theory and Differential Equations

This is an elementary and self--contained review of twistor theory as a geometric tool for solving non-linear differential equations. Solutions to soliton equations like KdV, Tzitzeica, integrable chiral model, BPS monopole or Sine-Gordon arise from holomorphic vector bundles over T\CP^1. A different framework is provided for the dispersionless analogues of soliton equations, like dispersionless KP or $SU(\infty)$ Toda system in 2+1 dimensions. Their solutions correspond to deformations of (parts of) T\CP^1, and ultimately to Einstein--Weyl curved geometries generalising the flat Minkowski space. A number of exercises is included and the necessary facts about vector bundles over the Riemann sphere are summarised in the Appendix.Comment: 23 Pages, 9 Figure

Bulk Entanglement Gravity without a Boundary: Towards Finding Einstein's Equation in Hilbert Space

We consider the emergence from quantum entanglement of spacetime geometry in a bulk region. For certain classes of quantum states in an appropriately factorized Hilbert space, a spatial geometry can be defined by associating areas along codimension-one surfaces with the entanglement entropy between either side. We show how Radon transforms can be used to convert this data into a spatial metric. Under a particular set of assumptions, the time evolution of such a state traces out a four-dimensional spacetime geometry, and we argue using a modified version of Jacobson's "entanglement equilibrium" that the geometry should obey Einstein's equation in the weak-field limit. We also discuss how entanglement equilibrium is related to a generalization of the Ryu-Takayanagi formula in more general settings, and how quantum error correction can help specify the emergence map between the full quantum-gravity Hilbert space and the semiclassical limit of quantum fields propagating on a classical spacetime.Comment: 29 pages, 2 figure

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