An algorithm to increase the residues of wrapped-phase in spatial domain

Transcription

1 An algorithm to increase the residues of rapped-phase in spatial domain Guangliang Du a, Minmin Wang a, Canlin Zhou a*,shuchun Si a, Hui Li a, Zhenkun Lei b,yanjie Li c a School of Phsics, Shandong Universit, Jinan 251, China b Department of Engineering Mechanics, Dalian Universit of Technolog, Dalian 11624, China c School of Civil Engineering and Architecture, Universit of Jinan, Jinan, 2522, China * Corresponding author: Tel: ; address: canlinzhou@sdueducn Abstract In phase unrapping, the locations and densities of residues are indicative of the severit of the unrapping problem The residues are used to detect and evade inconsistent phase areas Gdeisat et al proposed an algorithm to increase the number of residues in a rapped-phase map to improve the results of phase unrapping But this method ill take much time to make the Fourier transform, inverse Fourier transform, select and shift the spectral components, and there is no theoretical analsis on h the frequenc shift can increase the number of residues In vie of the above problems, e proposed an algorithm to increase the number of residues in a rapped-phase map, hich onl uses a simple multipl operation in spatial domain to realize frequenc shift b taking advantage of the frequenc shift propert of Fourier transform Besides that, e discuss the relationship beteen the number of residues and frequenc shift Finall, the experimental evaluation is conducted to prove the validit of the proposed method Experimental results demonstrated that the proposed method can speed up more than 5% Keords phase unrapping; residues; Goldstein branch-cut; Fourier transform 1 Introduction Phase-based fringe projection technique is an important method in three-dimensional (3D) shape measurement It has been extensivel investigated and idel used in man fields for its simple device and higher accurac [1-4] Since the phase obtained b phase demodulation method is rapped in (, ), phase unrapping become one of the critical steps of 3D shape measurement [5-7] Most phase-unrapping algorithms can be classified into to categories: temporal phase unrapping and spatial phase unrapping Temporal phase unrapping method requires to or more rapped phase maps [8-11], hile the

2 spatial phase unrapping method requires onl one rapped phase map to complete unrapping Among them, path independent phase unrapping algorithm is an active kind of algorithms Goldstein et al [12] proposed the famous branch-cut method in 1988, hich has been idel used Because this algorithm is ver dependent on the placement of the branch cuts, there ill be a lot of errors in the unrapping results if the noise is too much, so man people then proposed some improved methods Flnn etal [13] proposed the qualit guided path folloing method in 1996 Xu [14] proposed a region-groing phase unrapping algorithm Fornaro [15] et al proposed a multi-channel phase unrapping algorithm based on phase differences in 25 Hu proposed a color fringe projection method [16] Zhang proposed a stair phase coding technolog [17] Chen [18] proposed a method making use of the three primar color channels associated ith digital projectors Goldstein [19] proposed a smart temporal unrapping that temporall unraps the phase data such that small motion beteen frames is accounted for and phase data are unrapped consistentl beteen frames Liu [2] proposed a phase retrieval method using a composite fringe ith multi-frequenc Souza [21] proposed a to-dimensional phase unrapping algorithm based on the theor of residues hich is essential in the development of branch-cut algorithms These methods make great progress in phase unrapping Recentl, Gdeisat et al proposed a method to increase the number of residues in to-dimensional phase-rapped images that contain discontinuities [22], hich can be used to improve the performance of path independent phase unrapping algorithm such as branch-cut algorithm Hoever, according to our on experience ith the method, Gdeisat s method has the folloing disadvantages: (1) In the literature, there is no theoretical analsis on h the frequenc shift can increase the number of residues (2) This method needs Fourier transform, inverse Fourier transform, select and shift the spectral components, these procedures increase the calculation complexit and the processing time consuming as ell Here, in order to improve the calculation efficienc as ell as simplif its procedures, e present an algorithm to increase the number of residues of rapped-phase in spatial domain, hich takes advantage of the frequenc shift propert of to-dimensional(2d) Fourier transform [23] This method is a good solution to the problems of Gdeisat s method The capabilit of the presented method is demonstrated b both theoretical analsis and experiments The paper is organized as follos Section 2 introduces the principle of the sstem Section 3 presents the experimental results Section 4 summarizes this paper 2 Theor 21 Gdeisat s method

3 In 3-D shape measurement, surface relief, shado and phase noise ma cause local errors in the unrapping phase, Goldstein called the local errors as residues [12] In order to detect and calculate the residue, a 2 2 indo is defined in a rapped phase diagram, and then the difference beteen adjacent pixels(phase gradient value) is calculated b the folloing formula in a fixed direction hich can be either clockise or counterclockise Then the phase gradient value is accumulated 1 W ( ( i, j 1) ( i, j)) 2 W ( ( i 1, j 1) ( i, j 1)) 3 W ( ( i 1, j) ( i 1, j 1)) 4 W ( ( i, j) ( i 1, j)) 4 ( i) q i1 2 here W is the rapping phase operator, ( i, j )is the rapped phase at (i,j) Mark the upper left corner of the 2 2 indo as the center point When q=, the center point is not a residue; hen q>, it is a positive residue; on the contrar, it is a negative residue In the original rapped phase map, move the indo, e can detect and locate the residues in the hole to-dimensional phase map The locations and densities of residues are indicative of the severit of the phase unrapping problem and graphicall indicate here problems in the unrapping process are likel to occur The residues can be used to place branch cuts that are used as barriers to prohibit the phase-unrapping path from passing through them Gdeisat[22] thinks that the residues are the indicators of poor qualit regions of the phase, and increase their number is helpful to detect and avoid the inconsistent phase areas So, the proposed a method that uses the Fourier transform to increase the number of residues in the original rapped phase map The details for Gdeisat s method can be found in [22],The main stages of Gdeisat s algorithm are described as follos: (1)convert the original rapped-phase map into the complex arra ( (2) make the Fourier transform to ( to obtain ( u, v) (3) select and move the spectrum to obtain ( u- u, v- v ) c (1) c (4) make the inverse Fourier transform to obtain cs x u u v v 1 (, ) F [ ( -, - )] (5) generate the ne rapped phase map from ( (6)calculate the ne residues from the ne rapped phase map (7)construct qualit map or branch cut or mask from the ne residues (8)unrap the original rapped phase b the unrapping algorithm The method in ref [22] can improve considerabl the performance of the Goldstein algorithm b using ne residues as mask Experiments ill be described in the third section 22 Our method cs

4 In the literature 22, the preprocessing method to increase the number of residues can improve the accurac of the qualit guided or branch-cut phase-unrapping algorithm, but it also has some disadvantages, that is, (1) there is no discussion on h the frequenc shift can increase the residues (2) this method needs tice Fourier transform, and it needs to search and select the spectral components, hich ill occup large amount of processing time Based on our analsis of the ref [22], e put forard the corresponding solutions The basic idea of our method is that the complex procedures such as Fourier transformation, frequenc selection and inverse transformation are not required, hich are replaced b a simple multipl operation in spatial domain In this paper, e onl use the four-step phase shift algorithm to obtain the rapped phase Hoever, the method proposed in this paper also can be used to process rapped phase maps that have been extracted using other algorithms such as three-step phase shift algorithm Assuming that the four phase-shifted fringe images is as follos, I ( cos[2f x 2f ( ] 1 I 2( cos[2f xx 2f ( ] 2 I ( cos[2f x 2f ( ] 3 x x 3 I 4( cos[2f xx 2f ( ] 2 (2) Where f x and f are the spatial carrier frequenc along the X axis and the Y axis respectivel, is the modulation index The rapped phase can be extracted using the ell-knon Eq(3) belo b the four-step phase shift algorithm ( tan 1 I I 4 3 I I 2 1 W (2f x 2f ( ) x (3) Where 1 tan is the four quadrant arctangent operator, and ( is the rapped phase hich is rapped in (, ) The folloing three equations convert the rapped phase map into the complex arra ( c R( cos[ ( ] I( sin[ ( ] ( R( ji( c (4) Where j is equal to 1 In Gdeisat s method, first, make the Fourier transform to ( c as shon in Eq(5) ( u, v) [ ( ] (5) c c

5 here [] is the 2D Fourier transform operator, and the terms u and v are the vertical and horizontal frequencies respectivel The 2D Fourier transform of the rapped phase c ( u, v) is shifted aa from the origin using the indices u and v (the frequenc shift should not exceed a quarter of the image size [22]), then do the inverse Fourier transform 1 [ ( u -u, v - v )] (6) cs ( c here -1 [] is the inverse 2D Fourier transform operator, u, v is moving distance We can obtain the phase map as follos I{ ( }, ) tan 1 cs x W (2 ( f u) x 2 ( f v ) ( )) R{ ( } x s ( cs (7) here I{} represents the imaginar part, and R{} represents the real part of the complex arra ( cs From Eq(7), it is not difficult to understand that h the frequenc shift can increase the number of residues in the rapped phase map Shifting the spectrum in the frequenc domain equals to increasing the spatial carrier frequenc of projected fringe, thus increase the number of phase raps in the phase map Therefore, the sensitivit of projected fringe pattern is enhanced and the number of residues become more B analzing the implementation process in the ref [22], e can see Gdeisat s method requires tice Fourier transform, once spectrum search and selection (If e select spectrum ith an interactive approach in the frequenc domain, it is not suitable for the automatic implementation of the algorithm If e use the automatic method to search for peak in the frequenc domain, it is necessar to design a band-pass filter indo after locating the peak position, the process is relativel cumbersome), and finall it need to move the spectrum Therefore, in the actual operation, the process is relativel complex To solve this problem, We simplif and speed up the implementation process b the frequenc shift propert of 2D Fourier transform As shon in Ref [23], the frequenc shift propert of 2D Fourier transform can be ritten as: F( u -u, v - v ( j2 ( uox / mv / n)) ) [ f ( e ] (8) here F( u, v) is the Fourier transform of f (, m, n are the length of f ( along the X axis and the Y axis respectivel B Eq(8), e can directl obtain ( cs b multipling e ( j2 ( uox / mv / n)) b the result ( of Eq(4) c c cs ( u -u, v - v 1 ( [ ) [ ( e c c ( u -u, v - v ( j2 ( u x / mv / n)) o )] ( e c ] ( j2 ( u x / mv / n)) o (9)

6 B Eq(9), e can realize frequenc shift in spatial domain b the frequenc shift propert of 2D Fourier transform It is obvious that our method does not require Fourier transformation, spectrum selection, spectrum shift and inverse Fourier transformation Thus, the proposed method can greatl save the computing time The phase ( s can be obtained directl b calculating the phase angle of the complex number ( cs The main stages of our algorithm are summarized as follos: (1) convert the original rapped-phase map into the complex arra ( c (2) multipl ( c b e ( j2 ( uox / mv / n)) to obtain ( cs (3) generate the ne rapped phase map from ( (4) calculate the ne residues from the ne rapped phase map (5) construct qualit map or branch cut or mask from the ne residues (6) unrap the original rapped phase b the unrapping algorithm The folloing experiment is used to verif the proposed algorithm 3 Experiments In this section, for evaluating the real performance of our method, e test our method on a series of experiments Belo, e ill describe these experiments and practical suggestions for the above procedure We develop a fringe projection measurement sstem, hich consists of a DLP projector (Optoma EX762) driven b a computer and a CCD camera ( DH-SV41FM) Fig 1 shos the schematic of fringe-projection profilometr sstem, here P is the projection center of the projector, C is the camera imaging center, and D is an arbitrar point on the tested object The captured image is 768 pixels ide b 576 pixels high The surface measurement softare is programmed b MATLAB ith I GHz The tested object is a hand Fig 2 shos the captured image, and the rapped phase obtained b four-step phase shift algorithm is shon in Fig 3 Fig 4 shos the location of residues in the rapped-phase map b [12], there are a total of 47 residues Fig5 shos the unrapped phase b branch-cut phase-unrapping method [12], e can see that some of the results are incorrect Fig6 is the rapped phase obtained b Gdeisat s method and the proposed method, that is sa, the frequenc of 2D Fourier transform spectrum is shifted using values u=5,v=5 in frequenc and spatial domain respectivel The results processed b both method are same Compared ith Fig6 and Fig3, e can see, as the spectrum moves, the carrier frequenc becomes larger, so the number of phase raps in Fig6 are increased significantl Fig 7 shos the location of residues in Fig6 b [12], there are a total of 77 residues The number of residues are increased The rapped phase map shon in Fig3 is unrapped again b branch-cut cs

7 phase-unrapping method [11,22], but this time these residues shon in Fig7 used as mask that is provided to guide the branch-cut phase-unrapping method [12,22] The resulting unrapped phase map is shon in Fig8 Comparing both unrapped phase in Fig8 and Fig5 demonstrated that the increase the number of residues can improve the accurac of branch-cut phase-unrapping method In order to compare the time beteen Gdeisat s method and the proposed method, e conduct a comparative experiment Table 1 lists the comparisons of time consuming beteen these to methods In the comparisons, all processed fringe patterns have the pixels sizes of 768 x 576, the computational platform is a personal laptop ith Intel Core i5-457 CPU at 32GHz and a 4GB RAM We use MATLAB 214a on the same computer to process the same fringe pattern method Gdeisat s method 146s The proposed method 712s Table1 Comparisons of time consuming of the to methods As can be seen from table 1, the difference of time consuming beteen Gdeisat s method and the proposed method is obvious, the speed of the proposed algorithm is improved b about 5% In the proposed method, the calculation procedures such as Fourier transformation, frequenc selection and inverse transformation are not required, therefore, it can save large amount of processing time 4 Conclusion Time consuming In this paper, e propose an algorithm to increase the number of residues of rapped-phase in spatial domain hich is an extension of Gdeisat s method Our method overcomes the main disadvantages that Gdeisat s method encounters The proposed method eliminates Fourier transformation, inverse Fourier transformation and frequenc selection in Gdeisat s method, and achieves the frequenc shift b a simple multipl operation in spatial domain So the proposed method can save large amount of processing time Experimental results demonstrated that the proposed method can speed up more than 5% Acknoledgment This ork as supported b the National Natural Science Foundation of China (Grant nos and ) The support is gratefull acknoledged References 1ZK Lei, C Wang,CL Zhou, Multi-frequenc inverse-phase fringe projection profilometr for nonlinear phase error compensation,opt Lasers Eng; 215, 66 : Qudeisat M, Gdeisat M, Burton D, et al A simple method for phase raps elimination or reduction in spatial fringe patterns Opt Commun; 211, 284(21): TC Liu, CL Zhou, SC Si, et al, Improved differential 3D shape retrieval, Opt Lasers Eng; 215, 73: Du GL,Zhang CR,Zhou CL,etal, Iterative to-step temporal phase-unrapping applied to high sensitivit three-dimensional profilometr, Opt Lasers Eng; 216,79:22-28

8 5 Zhou C, Liu T, Si S, et al An improved stair phase encoding method for absolute phase retrieval Opt Lasers Eng; 215, 66: M Rivera, FJ Hernandez-Lopez and A Gonzalez, Phase unrapping b accumulation of residual maps Opt Lasers Eng; 215, 64: Jeught S V, Sijbers J, Dirckx J J J Fast Fourier-Based Phase Unrapping on the Graphics Processing Unit in Real-Time Imaging Applications J Imaging; 215, 1(1): HO Saldner and JM Huntle, Temporal phase unrapping: application to surface profiling of discontinuous object, Appl Opt; 1997, 36 (13): Li Lulu, Si Xianu, Dou Yunfu, et al, Error analsis and algorithm design of temporal phase unrapping, J Sichuan Univ (Nat Sci Ed); 212, 49 (1): P Cao, J Xi, Y Yu, et al, 3D shape measurement based on projection of triangular patterns of to selected frequencies, Opt Express; 214, 22 (23): Y Fu, Wang Wei and Xiao Huirong, Three-dimensional profile measurement based on modified temporal phase unrapping algorithm, Optik; 214, 124 (6): R M Goldstein, H A Zebker, L Werner Satellite Radar Interferometr: To-Dimensional Phase Unrapping Radio Sci; 1988; 23(4): Flnn T J Consistent 2-D phase unrapping guided b a qualit map,geoscience and Remote Sensing Smposium; 1996 IGARSS'96'Remote Sensing for a Sustainable Future', International IEEE; : Xu W, Cumming I A region-groing algorithm for InSAR phase unrapping, IEEE Trans Geosci Remote Sens; 1999, 37(1): Fornaro G, Pauciullo A Sansosti E Phase difference-based multichannel phase unrapping IEEE Trans Image Process; 25,14(7): L Hu, F Da, L Wang, A Novel Color Fringe Projection Method for 3D Measurement of Colorful Objects, Acta Opt Sin; 212, 32(2): Y Wang, S Zhang, Novel phase-coding method for absolute phase retrieval, Opt Lett; 212,37(11) : K Chen, J Xi, Y Yu,et al, Three-dimensional measurement of object surfaces ith complex shape and color distribution based on projection of color fringe pattern, Appl Opt; 213,52(3) : Goldstein, G,K Creath Quantitative phase microscop: automated background leveling techniques and smart temporal phase unrapping Appl Opt; 215, 54(16): T Liu, C Zhou, Y Liu, et al, Deflectometr for phase retrieval using a composite fringeappl Opt; 214,94: de Souza J, Oliveira M, dos Santos P Branch-cut algorithm for optical phase unrapping Opt Lett; 215, 4(15): Gdeisat M A, Burton D R, Lille F, et al Aiding phase unrapping b increasing the number of residues in to-dimensional rapped-phase distributions Appl Opt; 215, 54(34): RN Braceell The Fourier Transform and its applications Stanford Universit 25

9 Fig1Optical path of phase measuring profilometr Fig 2the first pattern of the sinusoidal fringe Fig 3 the initial rapped phase

10 Fig 4 the location of residues in the rapped-phase map Fig5 the unrapped phase b branch-cut phase-unrapping method Fig6 the ne rapped phase obtained b the proposed method(u=5,v=5)

11 Fig7 the location of residues in Fig6 Fig8 the unrapped phase b branch-cut phase-unrapping method