Optics and Lasers in Engineering 46 (2008) 446–455 Three-dimensional Fourier Fringe Analysis Hussein S. Abdul-Rahman à , Munther A. Gdeisat, David R. Burton, Michael J. Lalor, Francis Lilley, Abdulbasit Abid General Engineering Research Institute (GERI), Liverpool John Moores University, James Parsons Building Room 114, Byrom Street, Liverpool L3 3AF, UK Received 26 November 2007; received in revised form 14 January 2008; accepted 14 January 2008 Available online 21 February 2008 Abstract Over the years two-dimensional Fourier Fringe Analysis (2D-FFA) has demonstrated both its capability and its relative robustness in analysing fringe patterns within a short time-frame from static objects. Nowadays, there is an increasing demand to measure dynamic objects. Today 2D-FFA is seen as a fast and flexible method of processing fringe patterns for dynamic objects. But it is still inherently a 2D approach, i.e. it deals with three-dimensional data (video sequences) on an individual 2D frame-by-frame basis. In this paper, a novel three-dimensional Fourier Fringe Analysis (3D-FFA) algorithm is proposed to demodulate fringe pattern sequences taken from dynamic objects. This technique processes the stack of fringe patterns as a single 3D volume, not as a set of individual 2D frames that are each processed in isolation. The proposed algorithm has been evaluated on both computer simulated and real dynamic objects. Results show that the proposed technique is able to demodulate fringe pattern volumes successfully. r 2008 Elsevier Ltd. All rights reserved. Keywords: Fringe analysis; Phase measurements; 3D measurements 1. Introduction Many techniques have been proposed for the analysis of fringe patterns such as: Fourier Fringe Analysis (FFA) [1–10], three-dimensional windowed Fourier Fringe Ana- lysis [11], phase stepping [12–16], direct phase detection [17], wavelet transform fringe analysis [18–22], phase locked loop methods [23,24] and many other techniques. These methods all vary in accuracy, the number of frames required and processing time. The aim of any fringe pattern analysis algorithm is to obtain the phase information that is encoded into the fringe pattern. Fourier Transform Fringe Analysis (FFA) is a very popular technique that is well known to researchers working in the field of non-contact measurement. The history of the FFA algorithm began in 1982 when Takeda et al. [1] first suggested the use of the one-dimensional Fourier transform to analyse fringe patterns. The use of 1D signal processing techniques to analyse what are basically two-dimensional images is probably this the main reason that this first approach did not find its way into practical applications. The obvious development of Takeda’s algorithm was to extend it to two dimensions in order to make it more suitable for the analysis of fringe patterns. Many research- ers worked to improve Takeda’s 1D-FFA algorithm, extending it into two dimensions and successfully adapting it for specific applications. These include: Bone et al. [3], Burton and Lalor [5], Li et al. [4] and Gorecki [25]. The two-dimensional Fourier Fringe Analysis (2D-FFA) technique has been used for the measurement of both static objects and also for the measurement of dynamic objects whose surface shape changes with time [26]. But the 2D-FFA technique is not very suitable for the measure- ment of dynamic objects. In the case of static objects, a single fringe pattern is analysed in order to measure the object’s 3D surface. On the other hand, measuring the surface of a dynamic object requires the recording of a ARTICLE IN PRESS www.elsevier.com/locate/optlaseng 0143-8166/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2008.01.004 à Corresponding author. E-mail addresses: hussein_nemer@yahoo.com (H.S. Abdul-Rahman), m.a.gdeisat@ljmu.ac.uk (M.A. Gdeisat), 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).