1. INTRODUCTIONSurface profile measurement by noncontact optical methodshas been extensively applied to automated optical inspection,defect detection of surface shape, and solid modeling. In sucha context, active methods based on the projection of struc-tured light are very popular and commercially successful. Atypical fringe projection profilometry (FPP) system consistsof a projection unit, an image acquisition module, and a pro-cessing module [1,2]. The implementation, as illustrated inFig. 1, involves (i) projecting a set of structured patterns(e.g., sinusoidal fringes) onto the observed object surface,(ii) recording the fringe images that are phase modulated bythe object’s height distribution, (iii) resolving the phase mod-ulation using a fringe analysis technique and a proper phaseunwrapping (PU) algorithm to acquire continuous phase dis-tribution, and (iv) calibrating the system for mapping the con-tinuous phase distribution to real-world 3D coordinates.
Common to most applications of FPP is the fact that theinteresting physical information, e.g., shape, deformation, anddisplacement, is related to the absolute phase ϕ. However,only phase ψ of modulo-2π, the so-called wrapped phase, canbe measured limited to the interval ½−π; πÞ. Thus, in order torecover the continuous phase distribution, a 2D PU algorithmhas to be used. Formally, PU is defined as given the wrappedphase ψ ∈ ½−π; πÞ, find the absolute phase ϕ ∈ ½−∞;∞Þ, andthe relationship between these two kinds of phases is asfollows:
ψ ¼ WðϕÞ ¼ ϕ − 2π� ϕ2π
�; ð1Þ
whereW is the wrapping operator and h·i rounds its argumentto the closest integer. In fact, PU is mathematically an ill-posed problem if no further constraint is taken into account,
because W is surjection rather than bijection. Thus, a con-straint is given that the absolute value of the gradient oftwo arbitrary adjacent absolute phases is less than π [3]. How-ever, PU is still a problem due to the existence of residues,which are the potential source of phase-error propagation [4].Thus, the problem for PU is to minimize the impact of thoseresidue-induced false jumps on the estimated unwrappedphase image. There are two ways to realize this purpose:algorithm driven and residue filtering driven. The existingalgorithm-driven PUs for 3D reconstruction can be classifiedinto four major categories: (i) path-following [5,6], (ii) mini-mum-norm [7–9], (iii) model-based [10], and (iv) Bayesian-based [11,12]. The advantage of these algorithm-drivenapproaches is that they minimize the number of phase jumpslarger than π between neighboring points of the estimated un-wrapped phases through searching optimal integration pathsor correcting the phase gradients to retrieve the most reliablesolution. The disadvantage is the computational burden of theoptimization procedure. For example, the computational com-plexity of the minimum spanning tree method is OðN log2 NÞ[13], the computational complexity of the branch-cut methodis OðN2Þ, and the computational complexity of the network-flow method reaches OðN3Þ.
In comparison, the motivation of the residue-filtering-driven approach is to identify residues and eliminate themin order to provide an irrotational field that can be integratedalong any path. This makes 3D reconstruction faster with si-milar accuracy. Capanni et al. [14] proposed an adaptive med-ian filter based on some histogram properties of the filteringwindow to identify noisy pixels in a wrapped phase image.However, this method introduces a smoothing impact onnoise-free phases, and noisy phases cannot be located in areaswith relatively dense noise. Lee et al. [15] presented a sigma
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filter as an adaptive extension of the averaging filter. Itaverages only selected phase values within the k1 × k2 neigh-borhood along the directions of phase fringes. This filter, how-ever, results in an undesired phase smoothing in residue-freeareas. Nico [16] was confined to the description of configura-tions of residues, and designed a special filter. This method isbased on the observation of the noisy phases inducing adja-cent phase inconsistencies. A combinatorial analysis, how-ever, shows that the filter designed is very complicated,owing to various configurations of the adjacent residues. For-naro et al. [17] addressed a maximum-likelihood PU based onthe phase difference, and the noise is postprocessed in theunwrapped phase image using a median filter. Recently,Qian et al. [18] proposed a windowed filter in the Fourier do-main, and the filtered amplitude is used as a quality map.Hitherto, no new method can remove the residues thoroughlywithin the minimum operating time.
In this paper, we analyze the configurations of noise-induced residues, introduce clustering into the problem of re-sidue filtering, design a set of directional windows to involvemore similar phases, and propose a feasible filter to removeresidues without the need of searching optimal integrationpaths or correcting the phase gradients. The remainder ofthe paper is organized as follows. Section 2 briefly introducesthe principle of the FPP system. Section 3 analyzes the char-acteristics of residues. Section 4 addresses our motivationsand the proposed filter. Section 5 demonstrates the validityof our approach in the profile measurement system with arbi-trarily arranged devices. Section 6 presents the conclusion.
2. PRINCIPLE OF FRINGE PROJECTIONPROFILOMETRYIn the FPP system, a set of structured fringe patterns is pro-jected on the object surface. The captured images are phasemodulated by the object height distribution. Thus, the task isto analyze the deformed fringes to determine the profile of themeasured object. The main steps consist of fringe analysis,PU, and system calibration.
A. Fringe AnalysisIn the fringe projection technique, an object shape is evalu-ated through a phase distribution extracted from the capturedfringes. In a general FPP system, the illumination is divergentand a nonlinear carrier will be introduced. Figure 2 shows the
optical geometry of the FPP system. When a sinusoidal fringeis projected onto a 3D diffusing object, its deformed image canbe expressed as
where ðx; yÞ is the coordinate on the reference plane, Aðx; yÞis the reflected intensity of the object surface, Bðx; yÞ is thefringe contrast, φ denotes the phase-shifting increment,ψðx; yÞ ∈ ½−π; πÞ, and nðx; yÞ ∈ Z. The task is to obtainψðx; yÞ from captured fringe images. According to the princi-ple of phase-shifting technique, when N measurements I1,I2; � � � ; IN of I are made with a phase increment 2π
N, we can
acquire
ψði; jÞ ¼ arctan
PNk¼1 Ikði; jÞ sin
�2kπN
�P
Nk¼1 Ikði; jÞ cos
�2kπN
� ; ð3Þ
where ði; jÞ is the pixel coordinate on the imaging plane. It isnotable that although the fringe spacing is varied in the x di-rection, the phase-shifting increment is constant for any pointðx; yÞ on the object.
B. Phase UnwrappingIn practice, the presence of shadows, low modulations, fringediscontinuities, and noise makes PU difficult. In the next
Fig. 1. (Color online) Work flow in FPP: (a) projection of sinusoidal fringes, (b) image acquisition, (c) fringe analysis and PU, and (d) phase-to-height conversion.
Fig. 2. Optical geometry for fringe analysis: the reference plane oxy,the projection plane opxpyp, and the imaging plane ocxcyc are arbitra-rily arranged; Q represents an arbitrary point on the object; A and B
indicate the original fringe point projected at Q and the imaging pointof Q respectively; Op and Oc denote the lens centers of the projectorand the camera, respectively.
Jiang et al. Vol. 28, No. 2 / February 2011 / J. Opt. Soc. Am. A 215
sections, we will analyze how to eliminate the impact of noise-induced residues on PU to improve the running speed of thesystem. Given that all the residues are removed, PU can beoperated in the wrapped phase image using Eqs. (4) and(5) to acquire a continuous phase image:
where ϕði; jÞ is the unwrapped phase, Δ1ϕði; jÞ, Δ2ϕði; jÞ arethe discrete partial derivatives of neighboring unwrappedphases, Δ1ψði; jÞ, Δ2ψði; jÞ are the discrete partial derivativesof neighboring wrapped phases, and n1ði; jÞ, n2ði; jÞ ∈ Z areselected to make ψ1ði; jÞ, ψ2ði; jÞ ∈ ½−π; πÞ, respectively.
C. System CalibrationIn Fig. 2, the unwrapped phase ϕB at point B must equate tothe phases at points Q and A, respectively, which is the es-sence of calibration for an arbitrarily arranged FPP system.Du and Wang [19] derived a mathematical description of theout-of-plane height distribution. The analytic equation is asfollows:
where h or zQ is the out-of-plane height of point Q on the ob-ject; ϕB is the unwrapped phase of point B on the imagingplane; ðiB; jBÞ is the pixel coordinate of B; and c1–c5 andd0–d5 are constants produced by the system geometric param-eters, including the lens focuses of the camera and projector,the relative positions among the CCD image plane, projectionplane, and reference plane.
To calculate the out-of-plane height h, the unknown coeffi-cients must be determined first. A proper way is to use the non-linear least-squares algorithm, e.g., the Levenberg–Marquardtmethod. The least-squares error can be expressed as
E ¼Xmk¼1
�1þ c1ϕk þ ðc2 þ c3ϕkÞik þ ðc4 þ c5ϕkÞjk
do þ d1ϕk þ ðd2 þ d3ϕkÞik þ ðd4 þ d5ϕkÞjk− z
gk
�2;
ð7Þwhere zg denotes the heights of the gauge blocks andm is thenumber of points on the gauge blocks used in the calculation. Alargerm generally yields a higher accuracy. Two gauge objectswith uniform but different heights are required for calibratingthe system, because using a single gauge block with a uniformheight would produce indeterminate solutions for nonlinearequations.
3. CHARACTERISTICS OF NOISE-INDUCEDRESIDUESA. Localization of ResiduesWe assume that the projected fringes are sampled sufficiently,in that, Nyquist sampling theorem is satisfied, and no abruptshape change occurs on the object. A general method utilizesthe irrotational property to check phase inconsistency [8], anda residue is located by evaluating the sum of the phase gradi-
ents counterclockwise (or clockwise) around each set of fouradjacent points:
where Δði; jÞ ∈ f−1; 0;þ1g, ψ1ði; jÞ and ψ2ði; jÞ are explainedin Eq. (5). For a given 2 × 2 pixel neighborhood, if Δði; jÞ isdifferent from zero, the 2 × 2 area is called a residue in thispaper. If the summation takes the value þ1, it is referred toas positive residue (positive polarity); if the summation takesthe value −1, it is referred to as negative residue (negative po-larity). Figure 3 shows a phase image and its correspondingresidue image.
B. Configurations of ResiduesThe noise-induced residue is caused by the random fluctua-tion of phases due to noise in the fringe images. Of the fourphase gradients in Eq. (8), only three are independent. For agiven 2 × 2 pixel neighborhood, it can be shown that the sum-mation defined in Eq. (8) can be different from zero only if anodd number, one or three, of its four phase gradients round toa nonzero value. Some triads cannot occur, and only triadssatisfying the following rules are observed: (i) two consecu-tive phase gradients greater than π in modulus must be ofan opposite sign and (ii) two nonconsecutive phase gradientsgreater than π in modulus must share the same sign, and thethird one must have the opposite sign. As Jiang and Cheng [20]depicted, most of the residues have only one phase gradientgreater than π in modulus. From a quantitative viewpoint, theadjacent residues are the majority (more than 70% [16]). Inthis paper, we classify the noise-induced residues into fourconfigurations: (i) a couple of adjacent opposite-sign residuesin the form of left–right or up–down, (ii) a couple of adjacentopposite-sign residues in the form of diagonal, (iii) a couple ofdisjoint opposite-sign residues, and (iv) more than two resi-dues joined together. Figure 4 depicts the four categoriesof residues. Besides, Karout et al. [21] indicate that a singleresidue only occurs close to the borders of the phaseimage—the simple fact of their border location causing theiropposite-sign residue to lie outside the field of measurement.
4. CLUSTERING-DRIVEN RESIDUE FILTERA. Motivations
• The operating speed of the PU algorithm determines therunning speed of a FPP system. Although the algorithm-driven
Fig. 3. (a) 256 × 256 phase image and (b) corresponding residueimage of (a); the white and black flags denote positive and negativeresidues, respectively.
216 J. Opt. Soc. Am. A / Vol. 28, No. 2 / February 2011 Jiang et al.
PUs can minimize the adverse impact of residues, it is verytime consuming to search integration paths or correct thephase gradients. Thus, the performance of the system mightbe reduced drastically. From the viewpoint of rapidity, thepath-independent PU based on residue filtering is superiorto the algorithm-driven one. Consequently, the main task re-duces to filtering the residues effectively.
• Because residues are the potential source of phase-errorpropagation in the process of PU, we only have to filter thephases within a residue area rather than all the phases inthe wrapped phase image. In an actual FPP system, residuesare the minority relative to correct phases, and it is reasonableto regard these noisy phases inducing residues as outliers oranomalies locally. From the viewpoint of anomaly detection,the correct phases belong to large and dense clusters, whilethe noisy phases (or anomalies) belong to small or sparse clus-ters. Thus, we formulate the problem of residue filtering as asimple problem of 1D data clustering, as illustrated in Fig. 5.
• For steep slope areas, the fringe is tightly packed.Squared windows, such as the one for the boxcar filter, willcontainmore dissimilar phases. Thiswill destroy the continuityof fringes and make the PU incorrect. Thus, we want to designdirectional windows to make the wrapped phases within thefiltering window havemore similar values. Thismakes filteringmore effective and preserves the details of the fringes.
B. Identification of the Filtering WindowTo filter the noise-induced residues, the local wrapped phasesin the operating window have to be unwrapped, because thesediscontinuous phases are improper to measure their similar-ity, and we are not able to identify the local fringe orientationas Lee et al. [15] depicted. We should choose an effective PUto unwrap the pixels within the window to continuous phasesreliably.
We use 16 directional windows, as shown in Fig. 6. The win-dow size is set to 9 × 9 empirically, and the windows enclosethe four categories of residues as shown in Fig. 4. Only thewhite pixels in windows are included for computation. Thewindow to be selected should be approximately parallel tothe local fringe. In the implementation, these windows areconvolved with the phase image. The center of the adjacentresidue area serves as the center of the windows. We take ad-vantage of local statistics to analyze the similarities of localunwrapped phases, and corresponding variances are com-puted in all 16 windows individually using Eqs. (9) and (10):
v ¼ 1N
XNi¼1
ðxi − �xÞ2; ð9Þ
�x ¼ 1N
XNi¼1
xi; ð10Þ
where N is the number of the local unwrapped phases, xi isthe ith phase, v and �x are the variance and mean of the N
Fig. 4. Configurations of residues: (a) a couple of adjacent opposite-sign residues in the form of up–down, (b) a couple of adjacentopposite-sign residues in the form of diagonal, (c) a couple of disjointopposite-sign residues, and (d) two couples of adjacent opposite-sign residues joined together. gð·Þ denotes one pixel in the adjacentresidue area.
Fig. 5. This is a clustering example. A local window contains a pairof adjacent residues in the form of the diagonal in a wrapped phaseimage. The seven phases in this pair of residues are shown by a biggerand bold style. The seven phases can be classified into three groupseasily based on the knowledge of clustering. In fact, −0:0008 is thenoisy phase inducing this pair of residues.
Fig. 6. Sixteen directional windows for the filtering of noisy phases.Only the white pixels are included in the computation.
Jiang et al. Vol. 28, No. 2 / February 2011 / J. Opt. Soc. Am. A 217
phases. Note that the phases, coincided with the white pixelsin windows, do not include the pixels in the adjacent residuearea. The window, with the minimum variance, is selected asthe filtering window.
C. Unsupervised ClusteringBecause we are discussing a clustering problem, the k-meansand the single-linkage (SL) algorithms are selected as candi-dates for their excellent performance. For the problem of 1Ddata clustering, the computational complexity of the SL algo-rithm is Oðn2Þ with the n samples, and the complexity of thek-means algorithm is OðncTÞ with c clusters and T iterations[22]. Because the number of clusters of an arbitrary residuearea cannot be definite, SL, with unsupervised attribute, is se-lected as the basis of the proposed noise-residue-filteringalgorithm rather than k-means clustering.
D. Proposed Filtering Algorithm1. Locate the residues in the wrapped phase image. Resi-
dues are tested using Eq. (8). An accompanying flag matrix,the same size as the corresponding wrapped phase image,is introduced to mark the locations of the residues. The2 × 2 pixel neighborhood forming a residue is marked “1”(white), and the other pixels are marked “0” (black).
2. Identify the filtering window. The window size is set to9 × 9 empirically. We choose Costantini’s approach [8] to un-wrap the pixels locally within the windows to continuousphases. The sixteen directional windows in Fig. 6 convolvewith a residue area, and the corresponding variances arecomputed individually. The window, with the minimumvariance, is selected as the filtering window, as described inSubsection 4.B.
3. Locate the noisy phases and remove them. Thewrapped correspondences of the local unwrapped phasesin the filtering window serve as the input of the unsupervisedclassifier. The SL algorithm is used to classify the selectedwrapped phases fψ1;ψ2; � � �g in the filtering window into dif-ferent groups fc1; c2; � � �g. The phases within the minimal clus-ter cmin [23] are regarded as the noisy phases based on theassumption of an unsupervised anomaly detection. The as-sumption is that normal data instances belong to large anddense clusters, while anomalies either belong to small orsparse clusters [24]. Thus, the noisy phases are replaced bythe median phase within the maximal cluster cmax [25].
4. Judge whether there is still any residue. If any residueoccurs, go back to step 1 until all the residues are removed.
After all the residues have been removed, Δ1ϕði; jÞ andΔ2ϕði; jÞ in Eq. (5) can be estimated by ψ1ði; jÞ and ψ2ði; jÞreliably. Then we can unwrap the filtered wrapped phase im-age using Eq. (4) to acquire consistent unwrapped phaseimage.
5. EXPERIMENTSIn this section, the filter is used as a preprocessing step of PUto remove noise-induced residues. The filtering performanceis evaluated by applying the filter to synthetic and real phasesignals. We demonstrate the capability of the proposed filterin three aspects:
• processing tightly packed fringes with dense noise-induced residues effectively,
• removing noisy phases inducing residues withoutreducing the detailed information, and
• substituting the residue-filtering-driven PU for thealgorithm-driven one reliably.
We test our algorithm on simulated and real phase images.The simulated data allow a quantitative validation of the meth-od. We compare the proposed filter with the complex averagefilter [26] and the local histogram-based filter [14] from theviewpoints of residue reduction and fringe preservation, whilewe use real data to compare the path-independent PU based onthe proposed filter with the algorithm-driven methods [6–8]from the viewpoint of rapidity. The accuracy is also demon-strated after calibrating the FPP system in a less controlled si-tuation. Besides, the reasonwhywe choose these comparative
Fig. 7. (Color online) (a) Corresponding phase image of a set of four256 × 256 fringe images contaminated by random noise, (b) the re-wrapped phase fringe based on the proposed filter, (c) the rewrappedphase reconstruction using complex average filter with area A
blurred, and (d) the rewrapped phase reconstruction using local his-togram based filter with areas B and C blurred.
Fig. 8. Schematic diagram of our FPP system. There are fourmodules: fringe projection, image acquisition, fringe analysis, andhigh-precision motion.
218 J. Opt. Soc. Am. A / Vol. 28, No. 2 / February 2011 Jiang et al.
algorithms is that they are representative and widely used inpractice.
We simulate four 256 × 256 phase-shifted fringes as the ob-ject to be constructed. The fringe images are contaminated bydense random noise, and the corresponding phase image isshown in Fig. 7(a). In the proposed algorithm, the residuesare filtered iteratively, because new residues may occur whenprevious residues are removed. This means that a wrappedphase, not inducing a residue, is not always a noise-freeone. Figure 7(b) is the rewrapped phase reconstruction usingthe proposed filter. The fringes are clear and unbroken, indi-cating that the proposed filter removes the noise-inducedresidues thoroughly and preserves the fringe boundarieseffectively.
Figure 7(c) illustrates the rewrapped phase fringe based onthe complex average filter. Obviously, area A is blurred, indi-cating that there is residue-induced erroneous unwrapping.Using a local histogram-based filter, there are similar phenom-ena at B and C, as shown in Fig. 7(d). Each of the three dis-cussed algorithms applies a filtering window; however, theresults are quite different. The fundamental reason is thatthe selected window used in the proposed algorithm variesadaptively according to the local orientation of the fringe.The phases within this window are classified into correctphase groups and noisy phase groups accurately. The noisyphases are removed according to the assumption of unsuper-
vised anomaly detection. On the contrary, although the com-plex average filter is feasible to filter random noise in fringeimages, the selection of window size may impact the preser-vation of the fringe edges and the accuracy of the PU, parti-cularly for the case of narrow spacing fringes. On the otherhand, if the noise is dense, the shape of the local histogrammay be distorted and a clear peak may not be distinguished,so that it becomes difficult to decide whether there is a phasejump crossing the window, and the rate of false detection in-creases consequently. Besides, the selection of threshold σ(see [14]) may impact the filtering effect significantly. If σis too small, the window will smooth the noise-free phases,and if σ is too big, the noisy phases inducing residues cannotbe identified. Thus, the criterion to determine optimal σ mustbe constructed beforehand.
We also verify the rapidity of path-independent PU basedon the proposed filter using real phase signals. The surfaceof a container is selected as the object, because the corre-sponding measurement is relatively complex in natural envir-onment. This can verify our filter convictively. Our FPPsystem is shown in Fig. 8. Four phase-shifted sinusoidal fringeimages are projected on the surface of a container, and thecorresponding deformed fringes are captured by the camerasuccessively. All the captured images share the same size
Fig. 9. Actual size of the observed surface of a container.
Fig. 10. Process of 3D reconstruction: (a) one fringe image of a partial surface of the container, (b) the wrapped phase image of (a), (c) the residueimage corresponding to (b), and (d) the reconstruction result using PU based on the proposed filter.
Jiang et al. Vol. 28, No. 2 / February 2011 / J. Opt. Soc. Am. A 219
1024 × 768, and the corresponding measured area is450mm × 320mm. The actual size of the observed surfaceis shown in Fig. 9. Figure 10 illustrates the process of 3D re-construction of a partial surface of the container. The numberof residues reduces from 589 to 0 within three iterations. ThePU result using our filter is shown in Fig. 10(d). Obviously, noerror propagation occurs, and the time consumed is only 6:5 s.Although the algorithm-driven methods can acquire the sameresult, the operation for PU is time consuming. From the view-point of rapidity, the sequence in ascending order is PUsbased on minimum-cost-flow, branch-cut, least-squares, andSL, as shown in Table 1. Table 2 lists the computational com-plexity of the major procedures for separate approaches. Theproblem of minimum cost flow is equivalent to the problem oflinear programming, and this procedure is the most time con-suming. For the branch-cut algorithm, flood filling is used toprevent the integration path from crossing the branch cuts.The least-squares-driven PU is a computationally efficientalgorithm. However, the least-squares procedure tends tospread the errors rather than containing them within a limitedset of points. In comparison with the above three methods, themost time-consuming procedure is the locating residues for anSL-driven PU. In Table 2, N is the pixel number of an image; Tis the iteration number of residue filtering, T ≪ N ; nr is thenumber of residues in each iteration, nr ≪ N ; and n is thepoint number within the filtering window, n ≪ N .
To confirm further that the estimated unwrapped phase im-age is almost correct,wewill calibrate the profilemeasurementsystem for mapping the unwrapped phases to real world 3Dcoordinates. In order to avoid an undetermined solution forthe problem of nonlinear least squares, two gauge blocks, withuniform top heights of 9.115 and 13:085mm, are employed.Figure 11 shows one projected fringe image on the two gaugeblocks. When the Levenberg–Marquard algorithm is used, we
should set a suitable initial value to prevent the solutionfrom converging to the local minimum. In Eq. (7), let thecontent within the brackets equal zero, for k ¼1; 2; � � � ;m, m > 11, and then we get
−ϕ1 � � � −ϕm
−i1 � � � −im−ϕ1i1 � � � −ϕmim−j1 � � � −jm
−ϕ1j1 � � � −ϕmjmzg
1 � � � zgm
ϕ1zg
1 � � � ϕmzgm
i1zg
1 � � � imzgm
ϕ1i1zg
1 � � � ϕmimzgm
j1zg
1 � � � jmzgm
ϕ1j1zg
1 � � � ϕmjmzgm
0BBBBBBBBBBBBBBBB@
1CCCCCCCCCCCCCCCCA
T c1c2c3c4c5d0d1d2d3d4d5
0BBBBBBBBBBBBBBBB@
1CCCCCCCCCCCCCCCCA
¼
11111111111
0BBBBBBBBBBBBBBBB@
1CCCCCCCCCCCCCCCCA
: ð11Þ
Equation (11) can be described as
Ax ¼ b: ð12Þ
From Eq. (12), we have
x ¼ ðATAÞ−1ATb: ð13Þ
Thus, we can estimate an initial x0 using Eq. (13). Substitutingx0 into the Levenberg–Marquardt algorithm, we obtain x¼ð−2:59×10−3;3:03×10−5;2:80×10−8;1:42×10−3;4:39×10−8;2:78× 10−3;4:58 × 10−6;2:23 × 10−7;−5:33 × 10−10;2:88×10−6;1:89 ×10−9ÞT . Finally, we retrieve the height map using Eq. (6). Theheight map is shown in Fig. 12. After repeated measurements,we list the accuracy analysis of the height measurement of the
Fig. 11. Projected fringe image on the two gauge blocks.
Table 2. Comparison of Computational Complexity between the Proposed Algorithm and Several Famous
Fig. 12. Height map of a partial surface of the measured container.
220 J. Opt. Soc. Am. A / Vol. 28, No. 2 / February 2011 Jiang et al.
top surface in Table 3: hs is the actual value, hmax is the max-imum measured value, hmin is the minimum measured value, �his the average measured value, σmax is the maximum deviation,and �σ is the average deviation. Consequently, the error ofrepeated measurements is less than 2%.
6. CONCLUSIONIn terms of the characteristics of noise-induced residues, a fastclustering-driven residue filter is proposed, and it is embeddedin the arbitrarily arranged FPP system. It uses a set of direc-tional windows to include more similar phases, and it classi-fies the noisy phases inducing residues and the other correctphases within the filtering window into separate clusters ac-curately. The noisy phases are removed based on the assump-tion of anomaly detection. The filter is designed to make themeasurement system substitute the residue-filtering-drivenPU for the algorithm-driven one. Experimental results demon-strated the filter’s effectiveness in residue reduction, fringepreservation, and time savings. After all the residues are re-moved, the error of repeated measurements is less than 2%.
ACKNOWLEDGMENTSThe authors would like to thank Dacheng Tao and Wei Bianfrom Nanyang Technological University for their constructivecomments, and Bruno Luong from FOGALE Nanotech for hishelp with compiling the program of “2D phase unwrapping al-gorithm based on network flow.”The work described in this paper is supported by the NationalNatural Science Foundation of China (NSFC) (grant60806050), the Knowledge Innovation Program of the ChineseAcademy of Sciences (CAS) (KGCX2-YW-156, KGCX2-YW-154), the Key Laboratory of Robotics and Intelligent Systemof Guangdong Province (2009A060800016), the ShenzhenTechnology Project (JC200903160416A), and the ShenzhenNanshan Research Project (2009016).
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Table 3. Analysis of the Measured Height Data
(Unit: mm)
hs hmax hmin�h σmax �σ
Top surface 34.20 35.92 32.90 34.82 1.92 0.62
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