Chapter 10 Extension of the Monogenic Phasor Method to Extract Displacements and Their Derivatives from 3-D Fringe Patterns C. A. Sciammarella and L. Lamberti Abstract Determination of displacement field and its derivatives from fringe patterns entails four steps: (1) information inscription; (2) data recovery; (3) data processing; (4) data analysis. Phase retrieval and processing are very important in fringe analysis. In [1], these steps were discussed for 1D signals, introducing a 2D abstract space as phase evaluation requires a vectorial function. The Hilbert transform allows to obtain signal in-quadrature defining local phase. A 3D abstract space must be generated to handle the analysis of 2D signals and simultaneously extend Hilbert transform to 2D. The theory of a monogenic function introduced in [2] is elaborated here: the 3D vector in a Cartesian complex space is graphically represented by a Poincare sphere. This provides a generalization of the Hilbert transform to a 2D version of the Riesz transform, a modified version of that described in [2]. Theoretical derivations are supported by actual application of theory and corresponding algorithms to 2D fringe patterns and by comparing obtained results with literature. Keywords 2D signals • Displacements and strains of 3D solid bodies • Poincare hyper-sphere • Heart tissue deformation 10.1 Introduction In preceding papers [1, 2], the process of extracting displacement information and its derivatives has been analyzed from the point of view of application of the Image Analysis Science basic framework. Fringe pattern analysis is a sub discipline of the above-mentioned science that is applied to a mathematical model of the kinematics of the continuum. Fringe pattern analysis is also applied in the metrology of bodies. Both fields of deformation analysis and metrology information gathering have many common aspects but the 3D metrology of objects requires the utilization of many additional resources of the Image Analysis Science than the analysis of deformations. This paper will be limited to the analysis of deformations of 3D solids. In [1, 2], a new approach to fringe patterns analysis was introduced. For historic reasons, methods of displacement information determination have been associated with developments of techniques that utilize different approaches to generate and decode signals containing displacement information. Apart from the pathways of different methods, the same basic science is behind the utilized approaches and the same basic rules and restrictions apply to all of them. 10.2 Determination of the Displacement Field in 3D The first step in generating displacement data in 3D is to have a carrier on the volume under observation. A carrier is a known signal that upon deformation of the analyzed body will be modified from a certain known configuration, called the reference configuration or base configuration, to another configuration, the deformed configuration. The displacement information may be extracted from a deterministic signal or from a random signal. The basic methodology is the same, the random signal introduces additional challenges and difficulties but does not modify the process of extracting the displacement C.A. Sciammarella ( ) Department of Mechanical, Materials and Aerospace Engineering, Illinois Institute of Technology, 10 SW 32nd St, 60616, Chicago, IL, USA e-mail: sciammarella@iit.edu L. Lamberti Dipartimento Meccanica, Matematica e Management, Politecnico di Bari, Viale Japigia 182, 70126, Bari, Italy © The Society for Experimental Mechanics, Inc. 2018 L. Lamberti et al. (eds.), Advancement of Optical Methods in Experimental Mechanics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-63028-1_10 63
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