Spatial and temporal carrier fringe pattern demodulation using the one-dimensional continuous wavelet transform: Recent progress, challenges, and suggested developments Munther A. Gdeisat a,Ã , Abdulbasit Abid b , David R. Burton a , Michael J. Lalor a , Francis Lilley a , Chris Moore c , Mohammed Qudeisat a a The General Engineering Research Institute (GERI), Liverpool John Moores University, James Parsons Building Room 114, Byrom Street, Liverpool L3 3AF, United Kingdom b Umm Al-Qura University, College of Computer and Information Systems, Computer Engineering Department, P.O. Box 715 Makkah 21955, Saudi Arabia c North Western Medical Physics Department, Christie Hospital, Wilmslow Road, Withington, Manchester M20 4BX, United Kingdom article info Available online 7 August 2009 Keywords: Continuous wavelet transform Fringe pattern analysis Phase retrieval abstract This paper presents a thorough discussion on the application of the one-dimensional continuous wavelet transform (1D-CWT) in order to retrieve phase information in temporally and spatially tilted fringe patterns and highlights recent progress and challenges. The paper also suggests some possible future developments for this method. The advantages and drawbacks of the one-dimensional continuous wavelet transform technique are discussed here and in this context are compared to the widely used methods of Fourier fringe analysis, phase stepping and the windowed Fourier transform. A description is given of the manner in which the CWT phase gradient and phase estimation methods may be used to extract the phase of fringe patterns, and these two methods are compared and contrasted. Five different ridge extraction algorithms are explained and the performance of three of these is evaluated. To alleviate the distortions that may occur at the image borders and at regions close to holes in fringe patterns, two methods are described and evaluated for extending the image edges and for filling in holes within fringe patterns. A novel mother wavelet is presented which has been designed to improve the ability of the continuous wavelet transform to analyse fringe patterns that contain sudden phase variations. The sampling and structural conditions that are required to obtain ‘correct’ phase are also discussed. & 2009 Elsevier Ltd. All rights reserved. 1. Introduction Fringe pattern analysis using digital computers has recently seen significant interest due to its widespread application in science, medicine and industry [1,2]. Many methods for retrieving phase information from fringe patterns have been researched in considerable depth and some have now reached a mature state. For example, Fourier transform profilometry (FTP) [3], phase stepping (PS) or phase shifting [4] and more recently windowed Fourier transform (WFT) profilometry [5] and wavelet transform profilometry (WTP) [6,7], have all been the topics of active research. Wavelet transform profilometry has many potential advan- tages over the Fourier transform and phase stepping methods. Initially, this paper will summarise the problems that are inherent to Fourier transform profilometry and phase stepping. The paper will then move on to discuss the application of wavelet transform methods to fringe pattern analysis in order to resolve these issues [7–20]. The WTP technique utilises either the one-dimensional continuous wavelet transform (1D-CWT) [7], or the two-dimen- sional continuous wavelet transform (2D-CWT) [6], in order to extract the phase of a fringe pattern. In both methods, the phase extraction process can be carried out by using either phase estimation approaches [6,7,9,13,16,20] or via phase gradient (frequency estimation) approaches [6,11,17,18,21,22]. The majority of the WTP methods proposed in the literature belong to one of these four generic types. In many cases it may be found that the 1D-CWT method in conjunction with the phase estimation approach is often the most suitable method for many practical applications [6]. All the WTP results in this paper have been produced using the 1D-CWT method. In conjunction with a previous paper by the authors [6], it is hoped that this paper will act as a thorough and rigorous review of the wavelet transform profilometry method. The reader is strongly advised to read the paper presented in [6] prior to reading this paper. ARTICLE IN PRESS Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/optlaseng Optics and Lasers in Engineering 0143-8166/$- see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2009.07.009 Ã Corresponding author. E-mail addresses: m.a.gdeisat@ljmu.ac.uk (M.A. Gdeisat), abid@uqu.edu.sa (A. Abid), d.r.burton@ljmu.ac.uk (D.R. Burton), m.j.lalor@ljmu.ac.uk (M.J. Lalor), f.lilley@ljmu.ac.uk (F. Lilley), Chris.Moore@physics.cr.man.ac.uk (C. Moore). Optics and Lasers in Engineering 47 (2009) 1348–1361