Riesz Transforms for the Isotropic Estimation of the Local Phase of Moir´ e Interferograms Thomas B¨ ulow , Dieter Pallek , and Gerald Sommer Christian–Albrechts–Universit¨ at zu Kiel Institute of Computer Science, Cognitive Systems tbl,gs @ks.informatik.uni-kiel.de DLR (German Aerospace Center) G¨ ottingen Institute of Fluid Mechanics Dieter.Pallek@dlr.de Abstract. The estimation of the local phase and local amplitude of 1-D signals can be realized by the construction of the analytic signal. This includes the evalu- ation of the signal’s Hilbert transform, which performs a phase shift. In the past, different definitions of the analytic signal of multidimensional signals have been proposed, all of which are based on different combinations of partial and total Hilbert transforms. None of these approaches is isotropic. We propose the use of Riesz transforms which are known to mathematicians as appropriate gener- alizations of the Hilbert transform to -D. This approach allows the isotropic estimation of the intrinsically 1-D local image phase. Applications to Moir´ e in- terferograms are shown. 1 Introduction The estimation of the local phase and the local amplitude is an important step in many signal and image processing tasks. A second crucial task in image processing is the estimation of the local orientation. Usually these two tasks are treated separately. The methods used for the phase and amplitude estimation are based on the evaluation of the analytic signal of the input signal 1 which involves the calculation of the signal’s Hilbert transform. In practical problems either the analytic signal itself is evaluated, or the analytic signal of a band-pass filtered version of the input signal is considered. The latter can be constructed by the application of quadrature filters or, approximately, by using Gabor filters. 2-D Gabor filters are now widely used in image processing. These filters are ori- entation selective and allow the estimation of the local phase provided that the local orientation is known or has been estimated in a previous processing step. Actually, 2- D Gabor filters, like their 1-D correspondents, rely on an analytic signal, the partial analytic signal, which is defined as the line-wise evaluation of the 1-D analytic signal wrt. a predefined orientation. This presents, besides others, one possible extension of the analytic signal to 2-D. Unfortunately, none of these extensions allows the estimation of a smooth phase map of images containing arbitrary orientations. 1 We assume all signals to be real-valued.