Phase-unwrapping algorithm for noisy phase-map processing J. A. Quiroga and E. Bernabeu Automated fringe-pattern processing is important in a great number of industrial applications, such as optical data testing and quality control. One of the main problems that arises with these processes is the automated phase unwrapping of the phase map associated with the fringe pattern. Usually the phase map presents problems such as noise, and low-modulation areas. A new phase-unwrapping algorithm with high noise immunity is presented. The algorithm is easily implemented and can process arbitrary shapes. The main features of this algorithm are the use of a queue for the processing of arbitrary shapes and a selection criterion that determines which pixels are going to be processed. Key words: Phase unwrapping, automated interferometry, fringe-pattern processing. 1. Introduction The most usual techniques for the calculation of the phase P(x,y) associated with a fringe pattern imply the shifting of a reference phase through known increments (phase-sampling interferometry) or by addition of a substantial tilt to the wave front that produces carrier fringes and then by a further Fou- rier transformation of the fringe pattern.' In all these cases the phase is calculated by means of an inverse trigonometric function (arctan or arccos). All these functions return only principal values, i.e., values in the [- r, r]interval, generating a discontinu- ous phase map wrapped into a [-Fr, rr] interval. For sampled functions the phase jumps tend to r as the sampling frequency reaches the Nyquist frequency, and they tend to 27ras the sampling frequency rises compared with the Nyquist frequency. Further- more, for noise-free fringe patterns it is enough to seek jumps greater than r and to correct them by addition or subtraction of a 2r offset until the difference between collateral pixels is less than r. These clean fringe patterns are not the usual case; usually they are affected by several error sources, which one must keep in mind to design a phase unwrapping algorithm. The authors are with the Departamento de Optica, Universidad Complutense de Madrid, Facultad de Ciencias Fsicas, Ciudad Universitaria s/n, Madrid 28040, Spain. Received 13 August 1993; revised manuscript received 19 April 1994. 0003-6935/94/296725-07$06.00/0. © 1994 Optical Society of America. The error sources that most frequently arise in a fringe pattern are as follows: (a) Noise, electronic or speckle. The electronic noise is produced during the acquisition of the image. Speckle is due to the reflection of a coherent light beam in rough surfaces. (b) Low-modulation points that are due to areas of low visibility. The low-modulation points appear as fluctuations in the phase module 2rr, which might introduce errors in the phase-unwrapping process. (c) Abrupt phase changes that are due to object discontinuities. This type of error can lead to logical inconsistencies in the processed phase map; i.e., the phase difference between two points is path depen- dent. (d) Violation of the sampling theorem. The fringe pattern must be sampled correctly for the recovering of all the information from the phase module 2 r. Thus there must be at least two sampling points per fringe. This imposes a maximum spatial variation of the wave front of r rad per sampling point on the detector. In phase-sampling interferometry the sam- pling condition is more restrictive; at least three sampling points per fringe are needed.' The algorithm presented in this paper can handle in an automatic way the first two problems listed above [(a) and (b)]. The fourth problem, (d), can be overcome with very little change in the algorithm with the technique described in Ref. 2. To solve the third problem, (c), with this technique, one needs to have a priori knowledge of the object or to use the techniques described in Refs. 3 and 4. 10 October 1994 / Vol. 33, No. 29 / APPLIED OPTICS 6725