Advancement of Optical Methods in Experimental Mechanics, Volume 3

River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Advancement of Optical Methods in Experimental Mechanics, Volume 3 Sanichiro Yoshida Luciano Lamberti Cesar Sciammarella Proceedings of the 2016 Annual Conference on Experimental and Applied Mechanics River Publishers

Conference Proceedings of the Society for Experimental Mechanics Series Series Editor Kristin B. Zimmerman, Ph.D. Society for Experimental Mechanics Bethel, CT, USA

River Publishers Sanichiro Yoshida • Luciano Lamberti • Cesar Sciammarella Editors Advancement of Optical Methods in Experimental Mechanics, Volume 3 Proceedings of the 2016 Annual Conference on Experimental and Applied Mechanics

Published, sold and distributed by: River Publishers Broagervej 10 9260 Gistrup Denmark www.riverpublishers.com ISBN 978-87-7004-937-5 (eBook) Conference Proceedings of the Society for Experimental Mechanics An imprint of River Publishers © The Society for Experimental Mechanics, Inc. 2017 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, or reproduction in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Preface Advancement of Optical Methods in Experimental Mechanics represents one of ten volumes of technical papers presented at the SEM 2016 Annual Conference & Exposition on Experimental and Applied Mechanics organized by the Society for Experimental Mechanics and held in Orlando, FL, on June 6–9, 2016. The complete Proceedings also includes volumes on: Dynamic Behavior of Materials; Challenges in Mechanics of Time-Dependent Materials; Experimental and Applied Mechanics; Micro- and Nanomechanics; Mechanics of Biological Systems and Materials; Mechanics of Composite & Multifunctional Materials; Fracture, Fatigue, Failure and Damage Evolution; Residual Stress, Thermomechanics & Infrared Imaging, Hybrid Techniques and Inverse Problems; and Joining Technologies for Composites and Dissimilar Materials. Each collection presents early findings from experimental and computational investigations on an important area within Experimental Mechanics, Optical Methods being one of these areas. With the advancement in imaging instrumentation, lighting resources, computational power and data storage, optical methods have gained wide applications across the experimental mechanics society during the past decades. These methods have been applied for measurements over a wide range of spatial domain and temporal resolution. Optical methods have utilized a full range of wavelengths from X-ray to visible lights and infrared. They have been developed not only to make two-dimensional and three-dimensional deformation measurements on the surface but also to make volumetric measurements throughout the interior of a material body. Hammond, LA Sanichiro Yoshida Bari, Italy Luciano Lamberti Chicago, IL Cesar A. Sciammarella v

Contents 1 A General Mathematical Model to Retrieve Displacement Information from Fringe Patterns . . . . . . . 1 C.A. Sciammarella and L. Lamberti 2 Full-Field High-Strain Evaluation from Wrapped ESPI Data Using Phasors .................... 25 Juuso Heikkinen and Gary S. Schajer 3 Dynamic Deformation with Static Load................................................ 35 S. Yoshida, H. Ono, T. Sasaki, and M. Usui 4 Full-Field Digital Holographic Vibrometry for Characterization of High-Speed MEMS........... 41 Payam Razavi, Cosme Furlong, and James D. Trolinger 5 Surface Orientation Measurement Using Sampling Moire Method............................ 49 Motoharu Fujigaki, Daiki Tomita, and Yorinobu Murata 6 DD-DIC: A Parallel Finite Element Based Digital Image Correlation Solver.................... 55 Jean-Charles Passieux, Robin Bouclier, and Jean-Noe¨l Pe´rie´ 7 ANewIn Situ Planar Biaxial Far-Field High Energy Diffraction Microscopy Experiment .......... 61 G.M. Hommer, J.S. Park, P.C. Collins, A.L. Pilchak, and A.P. Stebner 8 Thermal Strain Measurement Using Digital Image Correlation with Systematic Error Elimination.................................................... 71 Manabu Murata, Shuichi Arikawa, Satoru Yoneyama, Yasuhisa Fujimoto, and Yohei Omoto 9 Investigating the Tensile Response of Materials at High Temperature Using DIC................ 77 Guillermo Valeri, Behrad Koohbor, Addis Kidane, Michael A. Sutton, and Hubert Schreier 10 Hybrid Stereocorrelation for 3D Thermomechanical Field Measurements ...................... 83 A. Charbal, J.-E. Dufour, F. Hild, S. Roux, M. Poncelet, and L. Vincent 11 Experimental Characterization of the Mechanical Properties of 3D Printed ABS and Polycarbonate Parts ............................................. 89 Jason Cantrell, Sean Rohde, David Damiani, Rishi Gurnani, Luke DiSandro, Josh Anton, Andie Young, Alex Jerez, Douglas Steinbach, Calvin Kroese, and Peter Ifju 12 Experimental Determination of Transfer Length in Pre-stressed Concrete Using 3D-DIC.......... 107 Sreehari Rajan, Michael A. Sutton, Ning Li, Dimtris Rizos, Juan Caicedo, Sally Bartelmo, and Albert Lasprilla 13 Hybrid Infrared Image Correlation Technique to Deformation Measurement of Composites . . . . . . . . 115 Terry Yuan-Fang Chen and Ren-Shaung Lu 14 DIC Anisotropic Denoising Based on Uncertainty......................................... 121 Manuel Grewer and Bernhard Wieneke vii

15 An Applications-Oriented Measurement System Analysis of 3D Digital Image Correlation. . . . . . . . . 127 Jordan E. Kelleher and Paul J. Gloeckner 16 Preliminary Study on Determination Pointing-Knowledge of Camera-Pair Used for 3D-DIC. . . . . . . . 135 Chi-Hung Hwang, Wei-Chung Wang, and Shou Hsueh Wang 17 Analysis of Dynamic Bending Using DIC and Virtual Fields Method.......................... 143 Behrad Koohbor, Addis Kidane, Michael A. Sutton, and Xing Zhao 18 Elimination of Periodical Error for Bi-directional Displacement in Digital Image Correlation Method.................................................. 151 Shuichi Arikawa, Manabu Murata, Satoru Yoneyama, Yasuhisa Fujimoto, and Yohei Omoto 19 The Cluster Approach Applied to Multi-Camera 3D DIC System............................. 157 Thorsten Siebert, Karsten Splitthof, and Marek Lomnitz 20 Self-adaptive Isogeometric Global Digital Image Correlation and Digital Height Correlation. . . . . . . 165 J.P.M. Hoefnagels, S.M. Kleinendorst, A.P. Ruybalid, C.V. Verhoosel, and M.G.D. Geers 21 Ultrasonic Test for High Rate Material Property Imaging.................................. 173 F. Pierron and R. Seghir 22 The Virtual Fields Method to Rubbers Under Medium Strain Rates .......................... 177 Sung-ho Yoon and Clive R. Siviour 23 Inertial Impact Tests on Polymers for Inverse Parameter Identification........................ 187 F. Davis, F. Pierron, and Clive R. Siviour 24 Full-Field Identification Methods: Comparison of FEM Updating and Integrated DIC............ 191 A.P. Ruybalid, J.P.M. Hoefnagels, O. van der Sluis, and M.G.D. Geers 25 Finite Element Stereo Digital Image Correlation Measurement for Plate Model .................. 199 Jean-Emmanuel Pierre´, Jean-Charles Passieux, and Jean-Noe¨l Pe´rie´ 26 Measurement of Orthogonal Surface Gradients and Reconstruction of Surface Topography from Digital Gradient Sensing Method.............................. 203 Chengyun Miao and Hareesh V. Tippur 27 Opportunities for Inverse Analysis in Dynamic Tensile Testing.............................. 207 Steven Mates and Fadi Abu-Farha 28 Determination of the Dynamic Strain Hardening Parameters from Acceleration Fields ............ 213 J.-H. Kim, D.-H. Leem, F. Barlat, and F. Pierron 29 Image-Based Inertial Impact Tests on an Aluminum Alloy.................................. 219 S. Dreuilhe, F. Davis, Clive R. Siviour, and F. Pierron 30 Inverse Material Characterization from 360-Deg DIC Measurements on Steel Samples ............ 225 L. Cortese, K. Genovese, F. Nalli, and M. Rossi 31 Identification of Plastic Behaviour and Formability Limits of Aluminium Alloys at High Temperature.............................................. 233 G. Chiappini, L.M. Mattucci, M. El Mehtedi, and M. Sasso 32 Accurate Strain Distribution Measurement Based on the Sampling Moire´ Method............... 243 S. Ri, Y. Fukami, Q. Wang, and S. Ogihara 33 Full-Field Measurements of Principal Strains and Orientations Using Moire´ Fringes .............. 251 Q. Wang, S. Ri, Y. Takashita, and S. Ogihara 34 A Self-Recalibrated 3D Vision System for Accurate 3D Tracking in Hypersonic Wind Tunnel . . . . . . 261 Ran Chen, Meng Liu, Kai Zhong, Zhongwei Li, and Yusheng Shi viii Contents

35 Evaluating Stress Triaxiality and Fracture Strain of Steel Sheet Using Stereovision............... 271 D. Kanazawa, Satoru Yoneyama, K. Ushijima, J. Naito, and S. Chinzei 36 Shadowgraph Optical Technique for Measuring the Shock Hugoniot from Standard Electric Detonators .................................................... 279 Vilem Petr, Erika Nieczkoski, and Eduardo Lozano 37 Assessment of Fringe Pattern Normalisation for Twelve Fringe Photoelasticity.................. 295 Phani Madhavi Ch, Vivek Ramakrishnan, and Ramesh Krishnamurthi 38 Novel Scanning Scheme for White Light Photoelasticity.................................... 301 Vivek Ramakrishnan and Ramesh Krishnamurthi 39 Investigation of Non-equibiaxial Thin Film Stress by Using Stoney Formula.................... 307 Wei-Chung Wang, Po-Yu Chen, and Yen-Ting Wu 40 ESPI Analysis of Thermo-Mechanical Behavior of Electronic Components ..................... 321 C. Casavola, G. Pappalettera, and C. Pappalettere 41 Shear Banding Observed in Real-Time with a Laser Speckle Method.......................... 327 Pasi Karppinen, Antti Forsstr€om, Kimmo Mustonen, and Sven Bossuyt 42 Numerical and Experimental Eigenmode Analysis of Low Porosity Auxetic Structures ............ 335 L. Francesconi, M. Taylor, K. Bertoldi, and A. Baldi Contents ix

Contributors Fadi Abu-Farha Clemson University—International Center for Automotive Research, Greenville, SC, USA JoshAnton Mechanical and Aerospace Engineering Department, University of Florida, Gainesville, FL, USA Shuichi Arikawa Department of Mechanical Engineering Informatics, Meiji University, Kawasaki, Kanagawa, Japan A. Baldi Dipartimento di Ingegneria Meccanica, Chimica e dei Materiali, Universita` degli Studi di Cagliari, Cagliari, Italy F. Barlat GIFT, POSTECH, Pohang, Gyeongbuk, South Korea Sally Bartelmo Department of Civil and Environmental Engineering, University of South Carolina, Columbia, SC, USA K. Bertoldi School of Engineering and Applied Science, Harvard University, Cambridge, MA, USA Sven Bossuyt Department of Mechanical Engineering, Aalto University, Aalto, Finland Robin Bouclier Institut de Mathe´matiques de Toulouse, CNRS UMR 5219, INSA Toulouse, Universite´ de Toulouse, Toulouse, France Juan Caicedo Department of Civil and Environmental Engineering, University of South Carolina, Columbia, SC, USA Jason Cantrell Mechanical and Aerospace Engineering Department, University of Florida, Gainesville, FL, USA C. Casavola Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Bari, Italy A. Charbal LMT, ENS Cachan/CNRS/University of Paris-Saclay, Cachan, France CEA, DEN-SRMA, Universite´ of Paris-Saclay, Gif sur Yvette, France Terry Yuan-Fang Chen Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan, ROC RanChen State Key Laboratory of Material Processing and Die and Mould Technology, Huazhong University of Science and Technology, Wuhan, China Po-YuChen Department of Power Mechanical Engineering, National Tsing Hua University, Hsinchu, Taiwan, Republic of China G. Chiappini Dipartimento di Ingegneria Industriale e Scienze Matematiche, Universita` Politecnica delle Marche, Ancona, Italy S. Chinzei Kobe Steel, LTD, Kobe, Hyogo, Japan Phani Madhavi Ch Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai, India P.C. Collins Materials Science and Engineering, Iowa State University, Ames, IA, USA L. Cortese Faculty of Science and Technology—Free University of Bozen, Bolzano, Italy David Damiani Bartram Trail High School, Saint Johns, FL, USA F. Davis Engineering Materials Group, Faculty of Engineering and the Environment, University of Southampton, Southampton, UK xi

Luke DiSandro Mechanical and Aerospace Engineering Department, University of Florida, Gainesville, FL, USA S. Dreuilhe Engineering Materials Group, Faculty of Engineering and the Environment, University of Southampton, Southampton, UK J.-E. Dufour LMT, ENS Cachan/CNRS/University of Paris Saclay, Cachan, France Antti Forsstr€om Department of Mechanical Engineering, Aalto University, Aalto, Finland L. Francesconi Dipartimento di Ingegneria Meccanica, Chimica e dei Materiali, Universita` degli Studi di Cagliari, Cagliari, Italy Motoharu Fujigaki Graduate School of Engineering, University of Fukui, Fukui, Japan Yasuhisa Fujimoto Mitsubishi Electric Corporation Advanced Technology Research and Development Center, Amagasaki, Hyogo, Japan Y. Fukami National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Japan Department of Mechanical Engineering, Tokyo University of Science, Chiba, Japan Cosme Furlong Center for Holographic Studies and Laser Micro-mechaTronics (CHSLT), Worcester, MA, USA Mechanical Engineering Department, Worcester Polytechnic Institute, Worcester, MA, USA M.G.D. Geers Department of Mechanical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands K. Genovese School of Engineering, University of Basilicata, Potenza, Italy Paul J. Gloeckner Cummins Technical Center, Cummins Inc., Columbus, IN, USA Manuel Grewer Lavision GmbH, G€ottingen, Germany Rishi Gurnani College of Engineering, University of California at Berkeley, Berkeley, CA, USA Juuso Heikkinen Department of Mechanical Engineering, University of British Columbia, Vancouver, Canada F. Hild LMT, ENS Cachan/CNRS/University of Paris-Saclay, Cachan, France J.P.M. Hoefnagels Department of Mechanical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands G.M. Hommer Mechanical Engineering Department, Colorado School of Mines, Golden, CO, USA Chi-Hung Hwang ITRC, NARLabs, Hsinchu, Taiwan, ROC Peter Ifju Mechanical and Aerospace Engineering Department, University of Florida, Gainesville, FL, USA Alex Jerez Mechanical and Aerospace Engineering Department, University of Florida, Gainesville, FL, USA D. Kanazawa Department of Mechanical Engineering, Aoyama Gakuin University, Sagamihara, Kanagawa, Japan Pasi Karppinen ProtoRhino, Helsinki, Finland Jordan E. Kelleher Cummins Technical Center, Cummins Inc., Columbus, IN, USA Addis Kidane Department of Mechanical Engineering, University of South Carolina, Columbia, SC, USA J.-H. Kim GIFT, POSTECH, Pohang, Gyeongbuk, South Korea S.M. Kleinendorst Department of Mechanical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands Behrad Koohbor Department of Mechanical Engineering, University of South Carolina, Columbia, SC, USA Ramesh Krishnamurthi Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai, India Calvin Kroese Mechanical and Aerospace Engineering Department, University of Florida, Gainesville, FL, USA xii Contributors

L. Lamberti Dipartimento Meccanica, Matematica e Management, Politecnico di Bari, Bari, Italy Albert Lasprilla Department of Civil and Environmental Engineering, University of South Carolina, Columbia, SC, USA D.-H. Leem GIFT, POSTECH, Pohang, Gyeongbuk, South Korea NingLi Department of Mechanical Engineering, University of South Carolina, Columbia, SC, USA Zhongwei Li State Key Laboratory of Material Processing and Die and Mould Technology, Huazhong University of Science and Technology, Wuhan, China MengLiu State Key Laboratory of Material Processing and Die and Mould Technology, Huazhong University of Science and Technology, Wuhan, China Marek Lomnitz Dantec Dynamics GmbH, Ulm, Germany Eduardo Lozano Colorado School of Mines, Golden, CO, USA Ren-Shaung Lu Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan, ROC Steven Mates National Institute of Standards and Technology, Gaithersburg, MD, USA L.M. Mattucci Dipartimento di Ingegneria Industriale e Scienze Matematiche, Universita` Politecnica delle Marche, Ancona, Italy M. El Mehtedi Dipartimento di Ingegneria Industriale e Scienze Matematiche, Universita` Politecnica delle Marche, Ancona, Italy Chengyun Miao Department of Mechanical Engineering, Auburn University, Auburn, AL, USA Manabu Murata Department of Mechanical Engineering, Aoyama Gakuin University, Sagamihara, Kanagawa, Japan Yorinobu Murata Faculty of Systems Engineering, Wakayama University, Wakayama, Japan M.Murata Department of Mechanical Engineering, Aoyama Gakuin University, Sagamihara-shi, Kanagawa, Japan Kimmo Mustonen ProtoRhino, Helsinki, Finland Department of Physics, University of Vienna, Vienna, Austria J. Naito Kobe Steel, LTD, Kobe, Hyogo, Japan F. Nalli Faculty of Science and Technology—Free University of Bozen, Bolzano, Italy Erika Nieczkoski Colorado School of Mines, Golden, CO, USA S. Ogihara Department of Mechanical Engineering, Tokyo University of Science, Chiba, Japan Yohei Omoto Mitsubishi Electric Corporation Advanced Technology Research and Development Center, Amagasaki, Hyogo, Japan H. Ono Department of Chemistry and Physics, Southeastern Louisiana University, Hammond, LA, USA Department of Engineering, Niigata University, Niigata-shi, Niigata, Japan G. Pappalettera Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Bari, Italy C. Pappalettera Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Bari, Italy J.S. Park Materials Physics and Engineering X-ray Science Division, Advanced Photon Source, Argonne National Laboratory, Lemont, IL, USA Jean-Charles Passieux Universite´ de Toulouse, Institut Cle´ment Ader (INSA, ISAE, Mines Albi, UPS), CNRS UMR 5312, Toulouse, France Jean-Noe¨l Pe´rie´ Universite´ de Toulouse, Institut Cle´ment Ader (INSA, ISAE, Mines Albi, UPS), CNRS UMR 5312, Toulouse, France VilemPetr Colorado School of Mines, Golden, CO, USA Contributors xiii

Jean-Emmanuel Pierre´ Universite´ de Toulouse, Institut Cle´ment Ader (INSA, ISAE, Mines Albi, UPS), CNRS UMR 5312, Toulouse, France F. Pierron Engineering Materials Group, Faculty of Engineering and the Environment, University of Southampton, Southampton, UK A.L. Pilchak Air Force Research Laboratory, Wright-Patterson AFB, Ohio, OH, USA M. Poncelet LMT, ENS Cachan/CNRS/University of Paris Saclay, Cachan, France Sreehari Rajan Department of Mechanical Engineering, University of South Carolina, Columbia, SC, USA Vivek Ramakrishnan Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai, India Payam Razavi Center for Holographic Studies and Laser Micro-mechaTronics (CHSLT), Worcester, MA, USA Mechanical Engineering Department, Worcester Polytechnic Institute, Worcester, MA, USA S. Ri Research Institute for Measurement and Analytical Instrumentation, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Japan Dimtris Rizos Department of Civil and Environmental Engineering, University of South Carolina, Columbia, SC, USA SeanRohde Mechanical and Aerospace Engineering Department, University of Florida, Gainesville, FL, USA M. Rossi Universita` Politecnica delle Marche, Ancona, Italy S. Roux LMT, ENS Cachan/CNRS/University of Paris-Saclay, Cachan, France A.P. Ruybalid Department of Mechanical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands T. Sasaki Department of Engineering, Niigata University, Niigata-shi, Niigata, Japan M. Sasso Dipartimento di Ingegneria Industriale e Scienze Matematiche, Universita` Politecnica delle Marche, Ancona, Italy Gary S. Schajer Department of Mechanical Engineering, University of British Columbia, Vancouver, Canada Hubert Schreier Correlated Solutions Inc., Irmo, SC, USA C.A. Sciammarella Department of Mechanical, Materials and Aerospace Engineering, Illinois Institute of Technology, Chicago, IL, USA R. Seghir Engineering and the Environment, University of Southampton, Southampton, UK Yusheng Shi State Key Laboratory of Material Processing and Die and Mould Technology, Huazhong University of Science and Technology, Wuhan, China Thorsten Siebert Dantec Dynamics GmbH, Ulm, Germany Clive R. Siviour Department of Engineering Science, University of Oxford, Oxford, UK O. van der Sluis Department of Mechanical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands Karsten Splitthof Dantec Dynamics GmbH, Ulm, Germany A.P. Stebner Mechanical Engineering Department, Colorado School of Mines, Golden, CO, USA Douglas Steinbach Mechanical and Aerospace Engineering Department, University of Florida, Gainesville, FL, USA Michael A. Sutton Department of Mechanical Engineering, University of South Carolina, Columbia, SC, USA Y. Takashita Department of Mechanical Engineering, Tokyo University of Science, Chiba, Japan M. Taylor Department of Mechanical Engineering, Santa Clara University, Santa Clara, CA, USA Hareesh V. Tippur Department of Mechanical Engineering, Auburn University, Auburn, AL, USA xiv Contributors

Daiki Tomita Graduate School of Systems Engineering, Wakayama University, Wakayama, Japan James D. Trolinger MetroLaser, Inc., Laguna Hills, CA, USA K. Ushijima Department of Mechanical Engineering, Tokyo University of Science, Tokyo, Japan M. Usui Department of Engineering, Niigata University, Niigata-shi, Niigata, Japan Guillermo Valeri Department of Mechanical Engineering, University of South Carolina, Columbia, SC, USA C.V. Verhoosel Department of Mechanical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands L. Vincent CEA, DEN-SRMA, Universite´ of Paris-Saclay, Gif sur Yvette, France Shou Hsueh Wang Department of Power Mechanical Engineering, National Tsing Hua University, Hsinchu, Taiwan, ROC Q. Wang Research Institute for Measurement and Analytical Instrumentation, National Institute of Advanced Industrial Science and Technology, Tsukuba, Ibaraki, Japan Wei-Chung Wang Department of Power Mechanical Engineering, National Tsing Hua University, Hsinchu, Taiwan, ROC Bernhard Wieneke Lavision GmbH, G€ottingen, Germany Yen-Ting Wu Department of Power Mechanical Engineering, National Tsing Hua University, Hsinchu, Taiwan, Republic of China Satoru Yoneyama Department of Mechanical Engineering, Aoyama Gakuin University, Sagamihara-shi, Kanagawa, Japan Sung-ho Yoon Department of Engineering Science, University of Oxford, Oxford, UK S. Yoshida Department of Chemistry and Physics, Southeastern Louisiana University, Hammond, LA, USA Andie Young Mechanical and Aerospace Engineering Department, University of Florida, Gainesville, FL, USA XingZhao Department of Mechanical Engineering, University of South Carolina, Columbia, SC, USA KaiZhong State Key Laboratory of Material Processing and Die and Mould Technology, Huazhong University of Science and Technology, Wuhan, China Contributors xv

Chapter 1 A General Mathematical Model to Retrieve Displacement Information from Fringe Patterns C.A. Sciammarella and L. Lamberti Abstract The extraction of the displacement field and its derivatives from fringe patterns entails the following steps: (1) information inscription; (2) data recovery; (3) data processing; (4) data analysis. Phase information is a powerful representation of the information contained in a signal. In a previous work, the above mentioned steps were formulated and discussed for a 1D signal, indicating that the extension to 2-D was a non trivial process. Proceeding along the same line of thought when one moves from the one dimension to two dimensions it is necessary to consider a 3D abstract space to generate the additional dimension that can handle the analysis of 2D signals and simultaneously extend the Hilbert transform to 2D. In this study the basic theory developed in the preceding reference is further elaborated to produce a version of the monogenic function yielding the necessary answers to the previously described processes. The monogenic signal, a 3D vector in a Cartesian complex space, is graphically represented by a Poincare sphere which provides a generalization of the Hilbert transform to a 2D version of what is called the generalized Hilbert transform or Riesz transform. These theoretical derivations are supported by the actual application of the theory and corresponding algorithms to 2D fringe patterns and by comparing the obtained results with known results. Keywords 2D signals • Displacement and strain determination • Generalized Hilbert (Riesz) transform • Poincare sphere 1.1 Introduction In [1], the present authors developed a one dimensional mathematical model of fringe patterns analysis based on the general Theory of Signal Analysis. This paper now deals with a generalization of the one dimension model derivations to 2-D. The extension to a higher dimension requires the review of some basic concepts of image signal analysis. To simplify the derivations we will consider the signal analysis on plane surfaces. The extension to general surfaces in the space requires further developments that cannot be covered on a single paper. The information to be decoded is recorded as level of gray in a 2D sensor through a device composed of optical and electronic circuits commanded by software, a measure of the light intensity of the imaged field. At this point the details of the process of data generation will set aside and the paper will concentrate in the process of information extraction. The recorded levels of gray must be converted into data that provide displacement fields and the displacement derivatives in the case of deformed bodies or geometrical parameters and their derivatives. In [1], it is shown that data conversion in one dimension requires the description of gray levels in terms of 2D complex functions (analytical functions) that lead to the introduction of the concept of phasor: Isp ) xð Þ¼Isp xð Þe 2πjϕ xð Þ ð1:1Þ The symbol)indicates a vector in the complex plane. A phasor in the complex plane is characterized by two separate pieces of information: amplitude related to the light intensity at the considered point and a phase representing the optical path followed by the recorded wave front from a selected reference point where the phase is assumed to be zero. The classical definition of phase in optics is, C.A. Sciammarella Department of Mechanical, Materials and Aerospace Engineering, Illinois Institute of Technology, 10 SW 32nd St., Chicago, IL 60616, USA L. Lamberti (*) Dipartimento Meccanica, Matematica e Management, Politecnico di Bari, Viale Japigia 182, Bari 70126, Italy e-mail: luciano.lamberti@poliba.it #The Society for Experimental Mechanics, Inc. 2017 S. Yoshida et al. (eds.), Advancement of Optical Methods in Experimental Mechanics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-41600-7_1 1

ϕ xð Þ¼ 2πδ xð Þ p ð 1:2Þ where δ(x) is the optical path and p is pitch of the sinusoidal function, unit of measure utilized to evaluate a path length and convert distances into an angle. The optical path length of the light arriving at an image is given by, δ xð Þ¼ð S0 0 n xð Þdx ð1:3Þ where n(x) is the index of refraction of the medium along the path followed by the light from a certain reference point to another point following a trajectory. A question than can arise is: Why to begin with the review of the phase concept? The answer to this question is found in [2]: the phase concept is a fundamental tool to develop a consistent theory of image analysis. There is another important aspect to the concept of phase, the definition of local phase implicit in Eq. (1.1) and the more general concept of global phase expressed by Eq. (1.3). The phase concept is associated with the notion of vector. When one introduces the definition of local phase or phase at a point for a 1D signal, one introduces an additional dimension to the mathematical model required to associate one dimensional functions with the phase concept. This additional dimension corresponds not to the actual space but to the complex plane. It is a fundamental concept in the Gabor’s analytic signal theory [3], basic starting point of many developments in Signal Analysis and in Optics. A complementary development to the analytic signal theory in one dimension is the Hilbert transform [4] that converts cosines into sines and is a unitary transform that changes the phase of the signal of π/2, leaving the signal amplitude unchanged. The Hilbert transform takes the original signal, a level of gray or intensity in some scale, and associates the gray level with an analytical function: Isp xð Þ¼Ip xð ÞþIq xð Þj ð1:4Þ where the symbol j is the imaginary versor, Ip(x) is the recorded signal (in-phase signal) and Iq(x) is the in-quadrature signal that provides the phase, ϕ xð Þ¼arctg Iq xð Þ Ip xð Þ ð 1:5Þ and Isp ) xð Þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffifffii I2 p þI 2 q q ð1:6Þ where the double bar symbol indicates the modulus of the vector in the complex plane. A fundamental property of the Hilbert transform is to provide a definition of local phase concept that is ancillary to the definition provided by Eq. (1.3) but applies to a single point of the gray level continuum of a one-dimensional signal provided that the gray levels are smooth functions with smooth derivatives in R2 that symbolizes the 2D continuum. The preceding conditions are ideal conditions that are not satisfied by actual signals. Recorded signals are inherently stochastic, hence in actual applications it is necessary to apply to the recorded gray levels smoothing procedures to approximate with certain error the theoretical ideal continuum signal. One should keep in mind these two separated aspects of the local phase definition, the theory behind this definition that is a consequence of the continuum theory and the procedures needed to implement applications of the mathematical model to actual experimental signals. In the literature of analysis of actual optical signals there is a very extensive treatment of the subject of separating stochastic and deterministic information. In this section and following sections the emphasis is on the continuum model, the stochastic aspect will be introduced later on in the paper. This is a very important simplification for the subject matter of the paper, fringe pattern information retrieval. Later on we will indicate the impact of the assumption of continuity in the handling of actual stochastic signals. The aim of the current paper is to extend the derivations presented in the framework of a one dimensional model continuum model, [1], to a two dimensional continuum case. 2 C.A. Sciammarella and L. Lamberti

1.2 Two Dimensional Sinusoidal Functions The next step in this process is to define the properties of two dimensional sinusoidal functions, a generalization of one dimensional sinusoidal functions utilized in the one dimensional continuum [1]. Figure 1.1a shows a two dimensional sinusoidal signal, it has an amplitude and a period p as is the case in one dimension but has an additional degree of freedom, the local orientation. Figure 1.1b illustrates the 2D sinusoid as a signal in 2-D. The yellow line shows a line of equal intensity (phase); the normal n provides the orientation of the signal, angle θ, and the vector r identifies a point of phase ϕ in the uniform field of the 2D sinusoidal signal. As shown by Eq. (1.2), the phase is computed with respect to a selected point O (center of coordinates) and is evaluated—Eq. (1.3)—as an angle that provides the number of cycles of the unit of measure p, a rational number n. The red line corresponds to points of equal number of cycles, since as the orientation of the vector r changes, it also changes the projected pitch p that is the unit measure to convert distances into angles. Comparing a 2D signal with a 1D sinusoidal signal, as mentioned before, there is an additional degree of freedom, the angle θ(see Fig. 1.1b). The considerations that follow are very important because the information that we want to retrieve is connected with a model, the continuum mechanics of solids that has its basis on the differential geometry approach to the continuum deformation with specific requirements for the signal and its successive derivatives. What this last sentence means is: specific requirements are imposed on the signal and its derivatives. The information that we want to obtain is a tensorial field that requires in the case of orthogonal Cartesian Coordinates a family of two orthogonal carrier fringes illustrated in Fig. 1.2. The vertical fringes and the horizontal fringes are represented in the frequency plane of the Fourier Transform (FT) by power spectrum dots whose coordinates are for example of the form (fx, fy) ¼(10, 0) for the x-axis, and the coordinates of the point in the negative frequencies are (fx, fy) ¼( 10, 0), that is a reflection with respect to the vertical axis. In analogous fashion, for the horizontal fringes we have (fx, fy) ¼(0, 10) and for the negative frequency (fx, fy) ¼(0, 10). In Fig. 1.2, the system of coordinates is selected as a left-handed system according with the usual practice in image analysis literature as opposed to Fig. 1.1 where a right-handed reference system is used. Fig. 1.1 (a) 2D sinusoidal signal; (b) additional parameter θ to define a two dimensional signal Fig. 1.2 Representation of a 2D cosinusoidal even signal when θ ¼0 and θ ¼π/2 and the corresponding representation in the frequency plane represented by a sensor with square pixels 1 A General Mathematical Model to Retrieve Displacement Information from Fringe Patterns 3

If we return to the concept of phase defined in Eq. (1.3) and compute the phase of a point defined by the vectorr ¼xi þyj in the direction of the normal n (Fig. 1.1b) it follows ϕ r; θ ð Þ¼ 2π rk k p ð 1:7Þ where the double bar indicate the modulus of the vector in agreement with Eq. (1.3). To understand the developments that follow, it is necessary to come back to the concept of local phase that can be introduced [1] via the Hilbert transform. The concept of local phase is a fundamental step in the whole process described in this paper and will be dealt with later on in the paper since it involves the transition between the ideal continuum and the actual recorded stochastic signals. The extension of local phase to 2D sinusoidal signals includes an additional degree of freedom, the angle θ indicated in Fig. 1.1b. The concept of phase requires a 2D vectorial field since it is associated with a vectorial function. The information captured by a sensor is given by levels of gray, a scalar quantity. This scalar function in the case of a 1D signal is connected to a 2D scalar potential in the complex plane that will be calledV • ; the symbol • indicates that the scalar potential is associated with a given point of coordinate x in the one dimensional continuumℒ1, that has a certain reference zero point from where the coordinate x is computed. In Eq. (1.4), the complex notation of [4] is utilized to represent an analytical function, for a more general approach in view to the extension to 3-D. A complex plane defined by the versors i)and j)is introduced, thus avoiding the utilization of quaternions that are the extension of the complex notation beyond 2-D. Returning to the complex plane required to introduce the concept of local phase, the gradient of the scalar potential is given by grad V • ¼G2 ¼ ∂V • ∂xc i) þ ∂V • ∂yc j) ð1:8Þ where: i)and j)are the versors in the complex plane (introducing different symbols fromi andj that represent the versors in the physical space); xc andyc are the coordinates in the complex plane; the subscript “2” indicates 2D gradient vector in the complex space. The sinusoidal signal is represented by gray levels defined by a function of the form, Ve • ¼Ip cos 2π p x þϕ0 ð1:9Þ The upper script “e” expresses the fact that the selected function is a cosine, an even function. It is possible to see that the local phase depends on the selection of the phase at the reference point. Computing the dot product of the ∇operator with the vector G2, the divergence of the field is obtained as: ∇•G2 ¼ ∂2 V • ∂xc 2 þ ∂2 V • ∂yc 2 ð 1:10Þ Calling V • xc ¼ ∂V • ∂xc and Vyc • ¼∂V • ∂yc , and computing the vector product “ ” of the ∇operator with the G2 vector, it follows: ∇ G2 ¼ ∂V • yc ∂xc ∂V • xc ∂yc !k) ð1:11Þ Since the field is a scalar field, the divergence of the field is zero and the rotor is also zero. Two equations can be derived: ∂2 V • ∂xc 2 þ ∂2 V • ∂yc 2 ¼ 0 ð1:12Þ 4 C.A. Sciammarella and L. Lamberti

∂V • yc ∂xc ∂V • xc ∂yc ¼ 0 ð1:13Þ These equations mean that the potential function in the complex plane must satisfy the Cauchy-Riemann equations, ∂V • xc ∂xc ¼ ∂V • yc ∂yc ð 1:14Þ ∂V • xc ∂yc ¼ ∂V • yc ∂xc ð 1:15Þ Equation (1.12) implies that the gray level potential V • to define a local phase must be a solution of Laplace’s equation in the complex plane. The solutions of the Laplace’s equation are part of the theory of potentials; these solutions are known to be harmonic functions. The field is conservative and the vectorial field is the gradient of a potential scalar field. The meaning of Eqs. (1.14) and (1.15) is that, in order to define a local phase, successive derivatives of gray levels must satisfy the above conditions. Furthermore, considering the full complex field, these equations are the conditions for V • ρc ð Þ dρc, where ρc ¼xc i ) þyc j ) is a given direction in the complex plane, to be an exact differential or, in other words, that is a potential such that the integral of the field is independent of the pathway followed. This conclusion leads to the complex function, z xð Þ¼Ve • xð Þþ j )Vo • xð Þ ð1:16Þ where Vo • xð Þ represents the odd component of the signal. Through Eq. (1.16) one gets the connection between the Hilbert transform, holomorphic functions and the levels of gray as a potential function leading to the definition of a local phase. For example, if Ve • is of the form given by Eq. (1.8), through the Hilbert transform we will obtain, Vo • xð Þ¼Iq sin 2π p x þϕ0 ð1:17Þ Each one of these derivatives can be computed from the information recorded in the image sensor. For each point of the ℒ1 domain, one can plot the gray levels as V • xð Þ. From Eq. (1.8) ∂V • ∂xc ∂V xð Þ • ∂x and ∂V xc ð Þ • ∂yc ∂V xð Þ • ∂y can be obtained and, finally, complementary derivatives are obtained from Eqs. (1.13) and (1.14). In summary, to represent the deformation of a continuous field the derivatives must satisfy the above relationships for a one dimensional signal. However, recorded signals will be contaminated by different signals that we designate as noise. Whatever processes that are applied to the signal to remove noise they must get successive derivatives satisfying the above conditions. All previous developments correspond to gray levels in one dimension. To introduce the definition of local phase for the 2D sinusoidal signal shown in Fig. 1.1 it is necessary to resort to a 3D complex space. Figure 1.1 showed a 2D cosinusoidal function which has the same parameters as a 1D signal but also additional parameter, the orientation θ. The normal nto the fringe trajectory shown in Fig. 1.1 provides the orientation of the signal at a given point of the physical space and the angleθ defines the orientation of the segment of curve with respect to a selected reference system. Some notations that will be useful in the developments that follow are now introduced. The unit normal to fringes in a point of a sinusoidal signal (Fig. 1.1b) is, n ¼x cos θi þy sin θj ð1:18Þ The above relationship is converted into cycles per unit length by multiplying Eq. (1.18) by 2π/p. Introducing the concept of wave vector for the sinusoidal signal, it can be written: k ¼ 2π p cos θi þ 2π p sin θj ¼kxi þkyj ð1:19Þ The wave vector is an alternative way to define the orientation of a segment of a sinusoidal signal in 2-D and relates it to the projections of the trajectory into the reference axis x–y. From Eq. (1.19), the local value of θ is given by 1 A General Mathematical Model to Retrieve Displacement Information from Fringe Patterns 5

θ kð Þ¼arctg ky ffiffiffiffiffiffiffiffiffiffiffiffiffifffii k2 x þk 2 y q ð1:20Þ At every given point of a cosinusoidal fringe field defined by the position vector r ¼xi þyj there is a phasor represented by Eq. (1.1) and the local orientation of the signal defined by the angle θ. The addition of one more parameter, the angle θ requires to extend the definition of the gray levels as a potential scalar functionVr • in a 2D space, wherer ¼ xi þyjis an upper script that indicates that the potential corresponds to a point inℛ2, the 2D continuum. Following the same steps applied in two dimensions and recalling that levels of gray are scalar quantities, gradVr • ¼G3 rð Þ¼ ∂Vr • ∂xc i) þ ∂Vr • ∂yc j) þ ∂Vr • ∂zc k) ð1:21Þ where the versors i), j)and k)indicate a Cartesian coordinates system in a 3D complex space, the subscript “3” indicates that one is dealing with a 3D vector in the complex space. The divergence of the field is determined as: ∇•G3 rð Þ¼ ∂2 Vr • ∂xc 2 þ ∂2 Vr • ∂yc 2 þ ∂2 Vr • ∂zc 2 ð 1:22Þ Calling Vr xc • ¼∂V r • ∂xc , Vr yc • ¼∂V r • ∂yc , Vr zc • ¼∂V r • ∂zc , and computing the vectorial product of the ∇operator with the vector G3(r), it follows: ∇ G3 rð Þ¼ ∂Vr zc • ∂yc ∂Vr yc • ∂zc 0 @ 1 A i) þ ∂Vr xc • ∂zc ∂Vr zc • ∂xc 0 @ 1 A j) þ ∂Vr yc • ∂xc ∂Vr xc • yc 0 @ 1 A k) ð1:23Þ Since we are dealing with a scalar potential, the divergence is zero. Hence, it can be written: ∂2 Vr • ∂xc 2 þ ∂2 Vr • ∂yc 2 þ ∂2 Vr • ∂zc 2 ¼ 0 ð1:24Þ Equation (1.24) indicates that the potential Vr • satisfies the Laplace’s equation in the complex 3D space. The meaning of this equation is the same as for two dimensions. The fact that the rotor is zero implies, ∂Vr xc • ∂yc ¼ ∂Vr yc • ∂zc ¼ ∂2 Vr • ∂yc∂zc ð 1:25Þ ∂Vr xc • ∂zc ¼ ∂Vr zc • ∂xc ¼ ∂2 V • ∂xc∂zc ð 1:26Þ ∂Vr yc • ∂xc ¼ ∂Vr xc • ∂yc ¼ ∂2 Vr • ∂xc∂yc ð 1:27Þ Equations (1.26)–(1.28) are the conditions for the existence of a scalar potential in the 3D complex space and are equivalent to the Cauchy-Riemann conditions in the two dimensional case. The above derivations indicate that the information contained in a 2D cosinusoidal fringe pattern is described mathematically by a conservative 3D vectorial field in the complex space. Similarly with the one dimensional case the derivatives that appear in the preceding developments can be computed in the 2D real space as recorded in the sensor. The difference with the one dimensional case is now that the information is in the form of a Monge’s type surface where the gray level is of the form, 6 C.A. Sciammarella and L. Lamberti

Vr • zc ¼F xc; yc ð Þ ð1:28Þ where F(xc, yc) indicates a 2D function. From Eq. (1.28), it is possible to get all the derivatives that appear in the preceding developments of the 3D complex field proceeding in a similar way to that utilized in the one dimensional case. Since we are dealing with Continuum Mechanics problems operating on tensorial entities, a family of orthogonal cosinusoidal signals must be defined in Cartesian coordinates as shown in Fig. 1.2. The orthogonal modulated fringe patterns project the displacement vectors in two orthogonal directions. These projections are not independent from each other since they are tied together by the compatibility conditions of the continuum and involve also the Eqs. (1.25)–(1.27) that both systems of fringes must satisfy at the same points of the image. Conclusions similar to the one dimensional case can be obtained: the successive derivatives of the gray levels must satisfy the above conditions to define a scalar potential. Hence, the passage from the actual signals to the continuum signals requires operations that must enforce the above conditions as close as it may be feasible. This conclusion is very important because the change of the orientation of the fringes is related to the curvature of the fringes, the larger is the local change of orientation the more important is the effect of the orientation on the derivatives of the gray levels function. 1.3 The Monogenic 2D Signal The extension of the one dimension approach of signal analysis to multiple dimensions has been the object of a large number of papers (see, for example, [5–8] and the references cited therein). This study will apply the complex Riesz transform presented in the preceding publications. In these four publications are introduced the required arguments to create a transform equivalent of the Hilbert transform in a multidimensional space. To achieve this purpose, the concept of monogenic function is introduced. The original derivation of the monogenic signal concept has its foundations on the algebra of quaternions that is connected to Lie algebra isomorphisms. In this paper, a variation of the original arguments is introduced. The derivations fit the mappings originally developed by Poincare and that for the particular field considered in this study, a 2D flat field, are graphically represented by a Poincare sphere [8] that it is utilized in the field of birefringent optics and in photoelasticity to define the different forms of polarization. The relationship of Poincare sphere and the concept of phase of the components of polarized light has been the object of several publications (see, for example, [9–11]). The connection between the preceding applications of the Poincare sphere and the phase concept and the current version introduced in this paper is a subject of a great deal of interest but is beyond the purpose of this paper. The motivation in the current version follows from the isomorphism pointed out in [12]. It has been shown that in order to introduce the concept of phase in one dimensional signals, it is necessary to resort to a 2D vectorial field, similarly for 2D signals it is necessary to introduce a vector field in the 3D complex space. The 3D phasor representing the gray levels in 2D has an amplitude that corresponds to the intensity of the signal, a phase that corresponds to the optical path information, and introduces a new variable that corresponds to the orientation of the 2D sinusoid in the physical plane defined by the normal n Eq. (1.18), a function of the angle θ defined in Fig. 1.1. Figure 1.3 illustrates the Poincare sphere notation. The vector amplitude is defined by the following components: (a) the components of the gray levels Ix, Iy associated with the versors i )and j), respectively; (b) to these two components it is added a third component Iq, corresponding to the versor k ). The complex amplitude vector in the complex space is given by, Isp ¼Ix i ) þIy j ) þIq k ) ð1:29Þ and corresponds to the radius of a Poincare sphere shown in Fig. 1.3a. This sphere represents the local phase and amplitude at a point in the 2D continuum of gray levels. In the coordinate plane i), j), sphere equator, Eq. (1.29) becomes: Ip ¼Ix i ) þIy j ) ð1:30Þ From Fig. 1.3, it follows: Ix ¼ Ip cos θ ð1:31Þ 1 A General Mathematical Model to Retrieve Displacement Information from Fringe Patterns 7

Iy ¼ Ip sin θ ð1:32Þ Figure 1.3b shows that the following relationship applies: tanϕ¼ Iq Ip ð 1:33Þ There are two in-quadrature quantities Ip and Iq andaphase ϕthat defines a local phase for a signal of orientationθin the2D space. The above derived equations lead to the following relationships between the intensities, defining Isp as the modulus of the vector Isp. Ix ¼Isp cosϕcosθ ð1:34Þ Iy ¼Isp cosϕsinθ ð1:35Þ Iq ¼Isp sinϕ ð1:36Þ Finally, the monogenic signal can be represented by, Mr s ¼Isp cos ϕcos θ i ) þcos ϕsin θ j ) þsin ϕk ) ð1:37Þ The upper script indicates that it corresponds to a point r of the 2D continuum. The angle θdefines the longitude of the point under consideration referred to the i) k )plane in the complex space. The angle ϕis the latitude of the point with respect to the equatorial plane and provides the local phase associated with the actual signal. The above derived relationships from local gray levels at a given point of a 2D image provide local orientation and the local phase. If a sinusoidal signal is such that the normal n i, then θ ¼0 and the corresponding representation in the Poincare sphere is shown in Fig. 1.3b. It can be seen from Fig. 1.2, for the vertical fringes that measure horizontal displacements the angle of the normal is θ ¼0. This case is depicted by the Poincare sphere of Fig. 1.3b. An alternative space can be considered replacing the light intensities by the frequencies. Defining the energy of the vector Isp in the frequency space as I2 sp ¼I 2 x þI 2 y þI 2 q ð1:38Þ Fig. 1.3 (a) Poincare sphere of the levels of gray representing light intensities; (b) Poincare sphere of the levels of gray for θ ¼0 8 C.A. Sciammarella and L. Lamberti

and taking into consideration the FT energy theorem, the following equation holds true in the frequency space: f sp ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifffifi f 2 x þf 2 y þf 2 q q ð1:39Þ The Poincare sphere in the intensity space depicted in Fig. 1.3a can be transformed into the Poincare sphere of the frequency space by replacing intensities by the corresponding frequencies. All the derivations made for the 3D complex space that are related to the intensities are valid also for the frequencies. 1.4 The Riesz Transform All the quantities that define the Poincare sphere can be obtained directly by applying the generalized Hilbert transform or Riesz transform defined in [5–7]. The Riesz transform can be computed in the physical space or in the frequency space defined by the analytic function theory [4]. Equation (1.37) provides the monogenic vector corresponding to a given point of the ℛ2 gray levels continuum represented graphically by a Poincare sphere in a 3D complex space. The monogenic function vector Isp has three components Ix, Iy and Iq, and its position in space is defined by two angles, θ and ϕ, a total of five unknown quantities. These quantities are related by Eqs. (1.34)–(1.36). Since of these three equations only two are independent, only three quantities (i.e. Ix, Iy and Iq) must be determined while Ip is the level of gray captured by the sensor. The Riesz transform of gray levels of an image in the spatial domain associates with each point of the continuum two orthogonal convolution kernels (Chap. 4 of [7]): hx rð Þ¼ x 2π x2 þy2 ð Þ 3 2 ð 1:40Þ hy rð Þ¼ y 2π x2 þy2 ð Þ 3 2 ð 1:41Þ where r ¼xi þyj. These kernels yield: Ix rð Þ¼ x 2π x2 þy2 ð Þ 3 2 ∗∗Ip rð Þ ð1:42Þ Iy rð Þ¼ y 2π x2 þy2 ð Þ 3 2 ∗∗Ip rð Þ ð1:43Þ In the above equations, Ip(r) is the gray level at rwhile the ** symbol [13] denotes a 2D convolution in physical space. These kernels satisfy the following relationships with the corresponding quantities in the frequency space: hx rð Þ!Hf x fr ð Þ¼ f x ffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 2 x þf 2 y q ð1:44Þ hy rð Þ!Hf y fr ð Þ¼ f y ffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 2 x þf 2 y q ð1:45Þ where the symbols Hf x and Hf y indicate Hilbert transform in 2-D as it is called in [7]. The generic denomination multidimensional Hilbert transform is used in place of the Riesz transform and the frequency space corresponds to the analytic frequency space [4]. It should be noted that the operatorsHf x andHf y of Eqs. (1.44) and (1.45) define the cos θand sinθterms in the frequency plane, consistently with Eqs. (1.34) and (1.35). Then 1 A General Mathematical Model to Retrieve Displacement Information from Fringe Patterns 9

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