Fast quality-guided flood-fill phase unwrapping algorithm for three-dimensional fringe pattern profilometry

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1 Univesity of Wollongong Reseach Online Faculty of Infomatics - Papes (Achive) Faculty of Engineeing and Infomation Sciences 2010 Fast quality-guided flood-fill phase unwapping algoithm fo thee-dimensional finge patten pofilomety Ke Chen Univesity of Wollongong Jiangtao Xi Univesity of Wollongong, jiangtao@uow.edu.au Yanguang Yu Univesity of Wollongong, yanguang@uow.edu.au Joe F. Chichao Univesity of Wollongong, chichao@uow.edu.au Publication Details K. Chen, J. Xi, Y. Yu & J. F. Chichao, "Fast quality-guided flood-fill phase unwapping algoithm fo thee-dimensional finge patten pofilomety," in Optical Metology and Inspection fo Industial Applications, 2010, pp Reseach Online is the open access institutional epositoy fo the Univesity of Wollongong. Fo futhe infomation contact the UOW Libay: eseach-pubs@uow.edu.au

2 Fast quality-guided flood-fill phase unwapping algoithm fo theedimensional finge patten pofilomety Abstact A fast quality-guided flood-fill phase unwapping algoithm is poposed fo eal-time 3D Finge Patten Pofilomety (FPP) system. The poposed appoach consists of thee steps. Fist, based on the phase maps acquied by phase shift pofilomety (PSP) techniques, a quality map is geneated accoding to the phase vaiance adjacent pixels on the wapped phase map. Accoding to the quality map, the phase map is divided into seveal pats which ae categoised as eithe apid phase changing aeas o smooth phase changing aeas. Then quality-guided flood-fill phase unwapping algoithm is applied to apid phase changing aeas and nonguided path-following algoithm is used in the smooth phase changing aea. The poposed appoach is much faste than the conventional non-guided path-following algoithm, and it is moe obust than the non-guided path-following algoithm. Expeiments ae caied out to veify the pefomance. Disciplines Physical Sciences and Mathematics Publication Details K. Chen, J. Xi, Y. Yu & J. F. Chichao, "Fast quality-guided flood-fill phase unwapping algoithm fo theedimensional finge patten pofilomety," in Optical Metology and Inspection fo Industial Applications, 2010, pp This confeence pape is available at Reseach Online:

3 Fast Quality-guided flood-fill phase unwapping algoithm fo thee-dimensional Finge Patten Pofilomety Ke Chen, Jiangtao Xi, Yanguang Yu, Joe F. Chichao School of Electical, Compute & Telecommunications Engineeing, Univesity of Wollongong, Austalia; ABSTRACT A fast quality-guided flood-fill phase unwapping algoithm is poposed fo eal-time 3D Finge Patten Pofilomety (FPP) system. The poposed appoach consists of thee steps. Fist, based on the phase maps acquied by phase shift pofilomety (PSP) techniques, a quality map is geneated accoding to the phase vaiance adjacent pixels on the wapped phase map. Accoding to the quality map, the phase map is divided into seveal pats which ae categoised as eithe apid phase changing aeas o smooth phase changing aeas. Then quality-guided flood-fill phase unwapping algoithm is applied to apid phase changing aeas and non-guided path-following algoithm is used in the smooth phase changing aea. The poposed appoach is much faste than the conventional non-guided path-following algoithm, and it is moe obust than the non-guided path-following algoithm. Expeiments ae caied out to veify the pefomance. Keywods: Phase unwapping, Finge Patten pofilomety (FPP), Thee-dimension measuement 1. INTRODUCTION Finge Patten Pofilomety (FPP) has been an active aea of eseach as an effective technology fo non-contact 3D shape measuement. A typical FPP system consists of a digital pojecto, a CCD camea and a compute. The coe idea of FPP system is that, the digital pojecto geneates a goup of image pattens and pojects them onto the object suface to be measued [2]. Due to the vaiance of the suface shape, the image pattens ae distoted which ae acquied by the CCD camea. The distoted images cay the infomation of the 3D suface shape and ae analysed by the compute, yielding the suface shape of the object [1][2][3]. A class of popula appoaches fo FPP ae based on phase detection (PD), whee images pattens ae sinusoidal o peiodic. With PD, the defomed pojected finge pattens can be modelled as the esult of phase modulation of the oiginal finge pattens. Both finge pattens can be descibed by phase maps, and the 3D shape can be eteaved based on analysis of the phase maps. Theefoe, most existing PD based appoaches consist of the following steps (1) acquie the finge pattens (2) extact the phase maps, and (3) econstuct the 3D shapes. A majo poblem associated with the second step is that the extacted phase maps ae wapped into the ange fom π toπ. In othe wods, in contast to the tue phaseφ, the extacted phase φ becomes φ = φ± 2mπ, whee m is an intege making π < φ π. Obviously, when φ vaies on the phase map acoss the points ± π, ± 3 π, ± 5 π,... etc, the wapped φ exhibits jumps o dops of 2π, causing discontinuity in the wapped phase maps. In ode to obtain the 3D shapes, the tue phase maps ae equied and thus must be ecoveed fom the wapped phase maps. Such a pocess is called phase unwapping which is a citical step in the PD based FPP appoaches. Although phase unwapping is athe simple in its pinciple, its implementation in pactice is a challenging task. In the cases that the tue phase is continuous, slow vaying and not affected by noise, phase unwapping can be caied out by adding o subtacting 2π to the esult of unwapped phase map when the wapped one eaches π oπ. Howeve, when the acquied wapped maps ae coupted by unwanted noise o distubance, o the tue phase maps have discontinuities lage than 2 π, the task becomes vey difficult as wong decision may be made egading the addition o subtaction of 2π. In 3D shape measuement with PD based FPP, the late situations ae athe common as the FPP systems ae inevitably influenced by vaious noise o distubance, and fo objects with a complex and discontinues suface shapes, the tue phase map ae discontinuous in natue. Consequently, phase unwapping unde these situations has been an active aea of eseach duing the past two decades. Optical Metology and Inspection fo Industial Applications, edited by Kevin Hading, Peisen S. Huang, Tou Yoshizawa, Poc. of SPIE Vol. 7855, 78550X 2010 SPIE CCC code: X/10/$18 doi: / Poc. of SPIE Vol X-1 Downloaded Fom: on 04/29/2013 Tems of Use:

4 Seveal phase unwapping algoithms have been poposed to ecove the tue phase map. These algoithms can be classified as thee categoies: (1) global algoithms, (2) egion gowing algoithms, and (3) path-following algoithms [2]. The global algoithms utilize a global function to fomulate the unwapping algoithm [8]. Algoithms in this class ae obust but computationally intensive, that is they can accuately unwap complex phase maps but take too much time. The egion gowing algoithms divide phase map into seveal egions. Each egion is unwapped individually and then meged with its adjoin egions to fom a whole unwapped phase map [6]. In contast to global algoithms, egion gowing algoithms ae less computational intensive but also less obust. The path-following algoithms cay out phase unwapping along a continuous path on the phase map. Compaed with global and egion gowing algoithms, path-following algoithms ae less computational complex. Thee ae thee classes of path following algoithms: (1) fixed path-following methods [8], (2) banch-cut methods [5], and (3) qualityguided methods [1]. The simplest methods among these thee classes ae fixed path-following method. The fixed pathfollowing methods undetake phase unwapping along a fixed path, and thus ae vey fast, but they ae not able to ecove the tue phase if thee is a jump o dop of phase moe than 2π between two adjacent pixels (phase jump) o phase noise. In contast to fixed path-following methods, banch-cut methods ae moe obust. With the banch-cut methods, phase jumps in a phase map ae detected and maked by cuves, efeed to the cuts. The unwapping path is pohibited to go ove the cuts and hence wong phase unwapping decisions can be limited. With quality-guided methods, the unwapping path is detemined based on the pixels quality [1], and thus the methods ae moe obust than othe pathfollowing algoithms. Two path-following phase unwapping algoithms ae commonly used in FPP systems, which ae the non-guided pathfollowing which belongs to fixed path-following methods and the quality-guided flood-fill which belongs to quality guided methods [1][2][3]. The non-guided path-following is simple and fast. Howeve, this appoach is not suitable fo complex phase maps. In contast to non-guided path-following, quality-guided flood-fill method is suitable fo complex phase maps, but it suffes fom the poblem of long time to complete a phase unwapping task [2][3][7][8], thus making itself not suitable fo eal-time measuement. Consequently, it is necessay to find a new method with impoved pefomance in tems of both obustness, i.e. unwapping accuacy, and speed. In this pape, we popose a new unwapping appoach called fast quality-guided flood-fill. The idea of the poposed appoach is to combine non-guided path-following method and quality-guided flood-fill method togethe. In the poposed appoach, the wapped phase map is divided into two types of aeas accoding to a quality map, including the apid phase changing aeas and smooth phase changing aeas. Then quality-guided flood-fill method is applied to the apid phase changing aea and non-guided path-following method to the smooth phase changing aea. Expeiments show that both speed and obustness can be ensued with the poposed appoach. The following of this pape is oganised as follows. In Section 2, existing appoaches ae intoduced. In Section 3, poposed appoach is shown. In Section 4, expeiments esults ae demonstated. Finally, the conclusions ae dawn. 2. EXISTING APPROACHES 2.1 Quality maps As mentioned above, the challenges associated with the phase unwapping lie in the complexity of the unwapped phase map. When the wapped phase maps ae smooth and slow vaying, it is quite easy to ecove the tue phase by nonguided path-following appoach. Howeve, fo the wapped phase with shap changes o discontinuities, the tasks ae vey difficult. Based on such a scenaio people intoduced a metic called quality map to evaluate unwapped phase maps in tems of the difficulty associated with thei unwapping. Let us use s( xy, ) to denote the light intensity of a finge patten, whee x and y ae pixel index numbes. The coesponding wapped phase map will be φ ( x,, and accodingly the quality map will take Qxy, (, ) with its element detemined accoding to a quality paamete of each pixel on the wapped phase map. The quality paamete at a cetain pixel can be measued by phase vaiance between adjoin pixels [2][3][8]. The phase vaiance between adjoin pixels at pixel ( x, can be calculated by [3][8]: Poc. of SPIE Vol X-2 Downloaded Fom: on 04/29/2013 Tems of Use:

5 [ φ ( x, φ ( x, y 1) + φ ( x, φ ( x, y + 1) + φ ( x, φ ( x 1, + φ ( x, φ ( x 1, ) ] 1 Δφ ( x, = + y (1) 4 whee, φ ( x, is the wapped phase at pixel ( x,, and Δφ( x, [0,2π ]. The quality paamete at the pixel ( x, can be detemined by the following: Δφ( x, Q( x, = 1 (2) 2π Obviously, 0 Qxy (, ) 1, and the smalle the phase vaiance Δ φ( x,, the lage the quality paamete Qxy (, ). By calculating the quality paametes fo all the pixels on the phase map using Equation (1) and (2), we ae able to obtain Qxy (, ) fo all the pixels. If we use an image of the same size as the phase map to epesent Qxy (, ) with the bightest (white) fo 0 and the dakest fo 1, the esulting image is called quality map. 2.2 Non-guided path-following algoithm In the family of path-following appoaches, the simplest one is non-guided path-following method poposed by Schafe and Oppenheim [9]. The path of unwapping is simply vetical o hoizontal fom one bode of the phase map to the othe. Let us assume that the unwapping diection is hoizontal and the wapped phase map size is M N. The unwapped phase φ of pixel x, y ) can be calculated as follows: ( m n φ ( xm 1, yn) +Δ φ( xm, yn) + 2 π if Δφ( xm, yn) π φ( xm, yn) = φ( xm 1, yn) +Δφ( xm, yn) if π <Δ φ( xm, yn) < π φ ( xm 1, yn) +Δφ( xm, yn) 2 π if Δφ( xm, yn) π whee Δ φ( xm, yn) = φ( xm, yn) φ( xm 1, yn), m [1, M], n [0, N]. φ ( x, y ) is the wapped phase of pixel ( x, y ). m n m n As shown in Equation (3), the unwapping pocess will stat at pixel ( x y 0 0) and unwap all the pixels in a same ow. Then, the pocess goes to pixel ( x y 0 1) and unwap all the pixels of that ow. The pocess will continue until all ows ae unwapped. 2.3 Quality-guided flood-fill algoithm As descibed above, non-guided path-following appoaches cay out phase unwapping on ow-by-ow o column-bycolumn path basis on the phase map. In case thee is a phase jump o noise in the wapped phase map, the methods fail to ecove the tue phase and these eos will popagate to the following pocessed pixels. In ode to solve the poblem, quality-guided flood-fill algoithm is poposed [2][3][8]. The details of quality-guided flood-fill algoithm ae shown below. In ode to clealy descibe the pocess, let us assume that we have a cuso with its position indicating the pixel on which phase unwapping is cuently caied out. Step 1. Fistly, find out the pixel with the highest quality paamete on the quality map Q ( x,. We assume that phase value of this point on the wapped phase map is the same as the tue phase (o we ae able to unwap this point to yield the tue phase value). Put the cuso to this point and mak the point as unwapped, and then stat the unwapping pocess as follows. Step 2. Check the quality paametes of the fou pixels suounding the cuso, that is, the pixels to the left and the ight, the ones above and below. Find out the pixel with the highest quality paamete and move the cuso to the pixel and then unwap the phase value. Step 3. Let φ (n) indicates the wapped phase of cuent cuso point, φ ( n 1) and φ( n 1) indicate the wapped phase and unwapped phase of pevious point espectively. The unwapped phase of cuent cuso point φ(n) can be detemined as following: (3) Poc. of SPIE Vol X-3 Downloaded Fom: on 04/29/2013 Tems of Use:

6 φ ( n 1) +Δ φ( n) + 2 π if Δφ( n) π φ( n) = φ( n 1) +Δφ( n) if π <Δ φ( n) < π φ ( n 1) +Δφ( n) 2 π if Δφ( n) π whee Δ φ ( n) = φ ( n) φ ( n 1). Upon completion of the above, the pixel is maked as unwapped. Step 4. Check the suounding pixels aound unwapped pixels and select the pixel with the highest quality paamete and move the cuso to the pixel, then epeat Step 3 until all pixels have been unwapped. The advantage of quality-guided flood-fill is that phase unwapping is always caied out following the paths of high quality pixels, that is, with elatively small vaiance fom pixel to the next. As phase unwapping eos ae moe likely occu at pixels with lowe quality, such a quality-guided appoach will be able to impove the obustness of phase unwapping. (4) 3. PROPOSED APPROACH Ou poposed appoach combines non-guided path-following and quality-guided flood-fill togethe. The details of the appoach ae shown below: Fistly, the quality map is geneated accoding to quality paametes as descibed in Section 2.1. The pixels on the quality map ae classified into two categoies by means of a theshold associated with the quality paamete Q th ( x,. The pixels with quality paamete lage than the theshold ae categoized as the high quality pixels, and the emaining as the low quality pixels. The theshold Q th ( x, should be neithe too big no too small, othewise eithe phase unwapping obustness o speed will be degaded. Secondly, we mak the quality map by ectangles in such a way that only the high quality pixels ae inside these ectangles. As the esult, the quality map will be divided into diffeent type of aeas. The aeas inside the ectangles ae chaacteized by smooth phase vaiation and hence efeed to as high quality aeas, and the pats outside the ectangles ae called low quality aeas due to apid phase vaiation. Thidly, afte apid and smooth phase changing aeas ae identified, a stat point will be selected. The stat point can be any high quality pixel on the bounday of the quality map. Then, the ode fo phase unwapping pocess will be detemined by following citeia to keep unwapping path continuous: (1) The aeas close to the stat point will be unwapped ealie than the aeas with lage distance to stat the point. (2) If thee ae seveal aeas have a simila distance to the stat point, the high quality aeas will be unwapped ealie than the low quality aeas. Then, the phase unwapping pocess will be commenced by following this ode. The high quality aeas and low quality aeas ae unwapped using the non-guided path following and quality-guided flood-fill algoithms descibed above espectively. Finally, the esults of all the aeas ae meged togethe to yield a tue phase map. 4. EXPERIMENTS In ode to demonstate the poposed appoach and compae it with existing phase unwapping algoithms, we applied them to images acquied by a 3D shape measuement system in ou laboatoy. In the expeiment system, the sinusoidal finge pattens ae geneated by a HITACHI CP-X260 digital pojecto, and a Duncan Tech MS3100-RGB 3CCD digital camea is utilized to captue the images. The digital camea is placed on top of the pojecto with a distance of 350 mm between thei lenses. The distance between the efeence plane and camea lens is 1295 mm. The thee-step phaseshifting pofilomety (3-PSP) has been used to acquie phase infomation and calculate suface height of object. In the fist expeiment, we used a milk bottle as the taget and Figue 1(a) shows the camea captue of measued object. Figue 1(b) (c) (d) shows the thee-step PSP finge images. The esolution of image is pixels. The phase shift between each step is 2 π. The wapped phase map obtained using the 3-PSP is shown in Figue 1(e). Then we convet 3 Poc. of SPIE Vol X-4 Downloaded Fom: on 04/29/2013 Tems of Use:

7 the phase map into the quality map using Equation (1) and (2) shown in Figue 2 (a). Both consideing the unwapping obustness and speed, a theshold of quality paamete at Q th ( x, =0.20 is applied to categoize high and low quality aeas. The stat point ( x, y 0 0) is at uppe ight cone of the quality map. In Figue 2(b). The aeas maked by alphabets ae high quality aeas and the aeas maked by numbes ae low quality aeas. Accoding to the citeia of odeing, the ode of those aeas fo phase unwapping pocess is A, B, C, D, E, F, 1, G, H, I, 2. (a) (b) (c) (d) (e) Figue 1. (a) The camea captue of measued object. (b) (c) (d) Thee-step PSP finge images with stating phases at 0, 2 π 4, π espectively. (e) Wapped phase map. 3 3 Poc. of SPIE Vol X-5 Downloaded Fom: on 04/29/2013 Tems of Use:

8 (a) (b) Figue 2. (a) The quality map fo conventional quality-guided method. (b) The quality map fo poposed method. Fo compaison pupose, we econstucted the shape of measued object using the phase maps ecoveed by non-guided path-following algoithm, the poposed algoithm and quality-guided flood-fill algoithm espectively, and the esults ae shown in Figues 3 (a) (b) and (b). We can see that, in tems of smoothness of the econstucted shape which eflects the phase unwapping obustness, the poposed appoach is close to quality-guided flood-fill algoithm and much bette than the non-guided path-following algoithm. (a) (b) (c) Figue 3. Thee-dimensional econstuction esults using (a) Non-guided path-following algoithm, (b) Poposed algoithm, (c) Quality-guided flood-fill algoithm. We also caied out expeiments on a plaste palm model. Figue 4(a) is the camea captue of measued object. Figue 4(b) shows the quality map. Figue 4(c) shows the smooth and apid phase changing aeas in the quality map. We choose the stat point to be the uppe ight cone of the quality map. The econstucted shapes fo this set of expeiments ae shown in Figue 5. Poc. of SPIE Vol X-6 Downloaded Fom: on 04/29/2013 Tems of Use:

9 Based on the expeiments we can say that the poposed appoach is able to achieve obustness pefomance close to that of quality-guided flood-fill algoithm, but with much less computations and hence can be much faste. In ode to compae the computational buden, let us conside the numbe of phase unwapping opeations in Equation (3) (4) and the numbe of compaison opeations. Fo a phase map of the size M N, the non-guided flood-fill algoithm equies MN phase unwapping opeations only, but the quality-guided flood-fill algoithm equies the same numbe of unwapping opeations and additional CM (, N) compaisons, given by the following: N M N + MN M N 2 C( M, N) M N + M N MN + 2 (5) Fom Equation (5), we can see that the computational buden associated with non-guided path-following algoithm is much lowe than that of quality-guided flood-fill algoithm. When the poposed appoach is applied to the two expeiments, the whole phase map is divided into 9 and 11 high quality aeas with the total size of and espectively, and 2 low quality aeas with the total size of and espectively. With egad to expeiments esults, the poposed appoach is able to significant educe the computation. Table 1 give compaison of the time consumption fo each algoithm. (a) (b) (c) Figue 4. (a) Camea captue of a plaste palm model. (b) The quality map fo conventional quality-guided method. (c) The quality map fo poposed phase unwapping method. Poc. of SPIE Vol X-7 Downloaded Fom: on 04/29/2013 Tems of Use:

10 (a) (b) (c) Figue 5. (a) Thee-dimensional econstuction esults using (a) Non-guided path-following algoithm, (b) Poposed algoithm (c) Quality-guided flood-fill algoithm. Table.1. Compaison of Phase Unwapping Time of Diffeent Algoithms Non-guided Quality-guided Poposed path-following flood-fill method Unwapping time fo bottle s s s Unwapping time fo palm s s s The computation time was acquied by a Deltacom desktop (Intel Q9400@2.66GHz, 4 G memoy; the image esolution is pixels 5. CONCLUSIONS We pesented a fast quality-guided flood-fill phase unwapping algoithm fo thee-dimensional FPP system based on the combination of two conventional phase unwapping algoithms. The quality map was geneated accoding to phase vaiance between adjacent pixels of the wapped phase map. This quality map was divided into two types of aeas, namely smooth and apid phase changing aeas. A non-guided path-following algoithm was applied to the smooth phase changing aeas and a quality-guided flood-fill phase unwapping algoithm was applied in the apid phase changing aeas. As demonstated by theoetical analysis and expeiment esults, the poposed appoach is chaacteized by much less computational buden but is able to achieve the same level of 3D measuement obustness in compaison with conventional quality-guided flood-fill algoithm. Poc. of SPIE Vol X-8 Downloaded Fom: on 04/29/2013 Tems of Use:

11 REFERENCES [1] Zhang, S., Li, X. and Yau, S., Multilevel quality-guided phase unwapping algoithm fo eal-time theedimensional shape econstuction, Applied Optics, 46, (2007). [2] Zappa, E. and Busca, G., Compaison of eight unwapping algoithms applied to Fouie-tansfom pofilomety, Optics and Lases in Engineeing, 46, (2008). [3] Su, X. and Chen, W., Reliability-guided phase unwapping algoithm: a eview, Optics and Lases in Engineeing, 42, (2004). [4] He, X., Kang, X., Tay, C., Quan, C. and Shang, H., Poposed algoithm fo phase unwapping, Applied Optics, 41, (2002). [5] Salfity, M. F., Ruiz, P.D., Huntley, J. M., Gaves, M. J., Cusack, R. and Beauegad, D.A., Banch cut suface placement fo unwapping of unde sampled thee-dimensional phase data: application to magnetic esonance imaging ateial flow mapping, Applied Optics, 34, (1995). [6] Baldi, A., Phase unwapping by egion gowing, Applied Optics, 42, (2003). [7] Heaez, M., Boticaio, J. G., Lalo, M. J. and Buton, D. R., Agglomeative clusteing-based appoach fo two-dimensional phase unwapping, Applied Optics, 44, (2005). [8] Ghiglia, D. C. and Pitt, M. D., Two-Dimensional Phase Unwapping-Theoy, Algoithms, and Softwae, John Wiley & Sons, INC. New Yok, (1998). [9] Schafe, R. W., and Oppenheim, A. V., Digital Signal Pocessing, Pentice-Hall, Englewood Cliffs, New Jesey, (1975). Poc. of SPIE Vol X-9 Downloaded Fom: on 04/29/2013 Tems of Use: