A practical comparison between Zhang’s and Tsai’scalibration approaches

Wei LiDepartment of Civil &

Environmental EngineeringThe University of Auckland

Auckland, New Zealand

Trevor GeeDepartment of Computer

ScienceThe University of Auckland

Auckland, New Zealand

Heide FriedrichDepartment of Civil &

Environmental EngineeringThe University of Auckland

Auckland, New Zealand

Patrice DelmasDepartment of Computer

ScienceThe University of Auckland

Auckland, New Zealand

ABSTRACTWith the rise of affordable processing power and off-the-shelf apparatus supporting 3D imaging, there is a growingneed for reliable and fast calibration tools, enabling timelyaccurate data gathering. When confronted with a choice ofcamera calibration tools, Zhang’s and Tsai’s are not onlythe most cited, but also the most widely available solutions.Zhang’s calibration is often chosen by default, based on theassumption that it is more accurate. However, it typicallyinvolves extensive manual data gathering when comparedto the Tsai approach. Here, we demonstrate that there isno significant accuracy gain between Tsai’s or Zhang’s ap-proach in terms of stereo matching, given the variety of read-ily available 3D devices tested. Further to this, the trade-offbetween measurement accuracy compared to setup and dataacquisition time is decisively in favour of Tsai. This paperalso covers a new algorithm for the extraction of points fromimages of checkboards attached to calibration objects.

Categories and Subject DescriptorsH.2.0 [Image]: Calibration; D.2.2 [C++]: Implementation

General TermsCalibration

KeywordsACM proceedings, Tsai calibration, Zhang Calibration

1. INTRODUCTIONDigital cameras are cheap, versatile devices that are ex-tremely prevalent in the world today. From a computingPermission to make digital or hard copies of all or part of this work forpersonal or classroom use is granted without fee provided that copies are notmade or distributed for profit or commercial advantage and that copies bearthis notice and the full citation on the first page. Copyrights for componentsof this work owned by others than ACM must be honored. Abstracting withcredit is permitted. To copy otherwise, or republish, to post on servers or toredistribute to lists, requires prior specific permission and/or a fee. Requestpermissions from Permissions@acm.org.IVCNZ ’14, November 19 - 21 2014, Hamilton, New ZealandCopyright 2014 ACM 978-1-4503-3184-5/14/11$15.00.http://dx.doi.org/10.1145/2683405.2683443

perspective, digital cameras have huge potential as an in-put device, since they can capture large amounts of datain a quick and easy way. Calibration is often fundamen-tal to this process as it describes the relationship betweencaptured images and the scenes that they depict. The out-put of the calibration process is typically a set of intrinsicparameters describing the properties of the digital camera(such as focal length) and a set of extrinsic parameters de-scribing the pose of the camera (rotations and translationsfor example). In turn, these parameters allow a computersystem to understand the exact positioning of the camerawithin the scene. Once the position of the camera is known,a camera model (such as the pinhole camera model) can beused to perform tasks such as computing realistic real worldscene dimensions for robotic interaction; or understandingthe orientation of multiple images for 3D reconstruction.

One of the more popular calibration techniques is Zhang’scalibration technique [13] introduced in 1998. Zhang’s cal-ibration uses a pinhole camera model and a set of planarimages (recommended to be more than 3, but Zhang’s papershows that at least 11 are required to get sub-pixel perfor-mance) to derive a calibration result. Zhang’s technique isrespected for its flexibility and robustness in several stud-ies [2, 9]. Zhang’s calibration is also a basis for the calibra-tion module within the popular OpenCV SDK [1]1.

Another popular, but older calibration technique, was in-troduced by Roger Tsai in 1987, commonly known as Tsaicalibration [12]. Tsai’s technique is based on only a singleimage depicting a coplanar calibration object that typicallyhas at least 100 points, which are mapped between the im-age and real world coordinates. Tsai’s calibration is alsowell respected, however some authors argue [11] that Tsai’stechnique is less accurate and less robust than Zhang’s tech-nique.

While it is easy to dismiss Tsai’s calibration technique on ac-curacy alone, it should be noted that it takes much less time

1OpenCV can be found at http://opencv.org

166

to perform Tsai’s calibration, than the significantly morelabour intensive Zhang approach, based on quantity of im-ages alone. Thus the focus of this paper is to investigate howmuch the accuracy of Tsai’s approach deviates from that ofZhang’s approach and whether the extra accuracy associatedwith Zhang’s approach is worth all that extra work.

In the next section we introduce work that others have donewith respect to the Zhang and Tsai calibration techniques.Then, in section 3, we cover the details of our Zhang andTsai calibration pipelines implementation, as well as the gen-eral methodology used for our experiments. Following that,we represent our results in section in section 4. Finally, weconclude this article with section 5 that details our analysisof the results that we acquired from our experiments.

2. LITERATURE REVIEWTsai’s calibration method is a two-step algorithm, in whichthe first step solves the linear aspects of the calibration prob-lem, while the second step addresses the non-linear aspectsdescribed by a simple distortion model. Tsai’s calibration isoften associated with a coplanar calibration object, of whichonly a single image is required for calibration.

Zhang’s approach involves capturing multiple images of thesame planar surface in different poses around the cameras’field of view (FOV). Zhang’s algorithm places more empha-sis on the non-linear aspect of the calibration problem andtherefore is based on a more complex model in comparisonto Tsai’s.

Over the years, there have been several studies conducted inorder to establish the best camera calibration approaches.One prominent study was done by Salvi et al [10] whichlooked at calibration algorithms developed during the period1982–1998. Salvi used re-projection error measurements inboth distorted and undistorted images to evaluate the qual-ity of the calibration algorithms. Tsai’s calibration approachwas identified as one of the top calibration algorithms of thisera. However, unfortunately, Zhang’s calibration techniquewas too new to be featured in this work.

Feng et al. [2] analysed both algorithms from both a mathe-matical standpoint and then experimentally using laser beamsin order to establish the quality of each algorithm. In theirconclusion, they note that one disadvantage of Tsai’s cali-bration is that it does not determine the true optical cen-tre from calibration. Experimentally, Tsai performs well,however overall, they conclude that Zhang is more robust,complete and accurate.

Ricalfe-Viala et al. [9] argue the case between Zhang andTsai in order to motivate their choice of Zhang for experi-mentation. In their argument they claim that Tsai’s calibra-tion has a weaker distortion model, a difficult to constructcalibration object and a high sensitivity to measurement er-rors. While Zhang’s calibration object is much more easy toconstruct, has more flexibility due to the fact that the cali-bration object and camera can be moved during the calibra-tion process, and the ability to choose how many calibrationimages to acquire dependant on accuracy needs. Much oftheir argument, with regards to accuracy and robustness, isbased on work by Sun et al. [11].

Sun et al. [11] used synthetic images to test the calibrationdifferences between Zhang and Tsai. They then followed uptheir results with tests against real world images. They usedreprojection error to measure accuracy. Their conclusionwas that Zhang’s approach is far superior to that of Tsai;however they described the process of measuring points inTsai’s calibration as laborious, while they had clearly au-tomated the corner extraction for their Zhang calibrationpipeline. Thus, perhaps, had they automated their Tsaicorner extraction too, the results would have been closer.

3. METHODOLOGYIn order compare the behaviour and efficiency of using Zhang’sand Tsai’s calibration in terms of stereo applications, we as-sembled a typical stereo pipeline. The next set of subsectionsdescribe each component of that pipeline, and how each partwas crafted to test our hypothesis.

3.1 CamerasWhen considering which camera to use for our calibration,it was decided that our experiments should not be biased to-wards a single camera type. Thus we decided to perform ourexperiments against a variety of cameras that were availableto us: GoPro HERO2, GoPro HERO3, Fuji W3 and Basler.

The GoPro HERO2 & GoPro HERO3 cameras have highbarrel distortion and wide angle lenses. In order to adjustfor this, calibration targets are placed 50 cm in front of theGoPro cameras. As our aim is to capture stereo images,we also made use of the tandem housings designed for thesecameras. The tandem housing also comes with synchroni-sation cables that are extremely important for capturing si-multaneous images. For the capture of the required Zhangcalibration images, the automatic image capture mode wasset to 2 seconds.

The Fuji W3 stereo camera is a consumer level camera withtwo standard lenses. The W3 has significantly less lens dis-tortion than the GoPro cameras. The Fuji W3 camera alsohas an automatic capturing mode that was used for the cap-ture of Zhang calibration images.

The Basler cameras are industrial cameras that are attachedto the computer via a connection to a Gigbit network card.

Details of the cameras are summarized in table 3.1.

Table 1: Camera Details (note that the distortionvalues come from our own calibration)

Camera Size Baseline Distortion(pixels) (mm) (mm−2)

Hero 2 3840 x 2880 35 High Barrelκ1=0.038

Hero 3 4000 x 3000 35 High Barrelκ1=0.033

Fuji W3 2048 x 1526 75 Low Barrelκ1=0.004

Basler 2046 x 1086 40 Medium Barrelκ1=0.01

3.2 Calibration Object

167

Figure 1: Off-the-shelf cameras and their exampleimages below. From left to right: GoPro HERO2,GoPro HERO3, Fuji W3, Basler.

Figure 2: Tsai calibration object (left), Zhang cali-bration object (right)

We now turn our attention to the calibration object. In thecase of Zhang’s calibration, the calibration object is typicallya single planar surface, whereby the points of interest on theobject are often the corners of a grid. For Tsai’s calibration,the calibration object is similar to that used for Zhang’sapproach, except that it has two coplanar surfaces.

In our study, we used a glass tile to construct a calibrationfor Zhang’s calibration. Glass was specifically chosen forits smooth surface type and its relative lack of flexibility.An A3 checker board was printed using a laser printer withaccuracy of 20 um, and specifically glued to the calibrationobject so as to avoid unwanted bulges on the attached sheetof paper, which could lead to inaccuracy.

For Tsai’s calibration, the University of Auckland’s Depart-ment of Civil and Environmental Engineering specificallymanufactured a smooth inflexible perspex calibration objectfeaturing two perfectly coplanar surfaces. We decided to at-tach our checkerboard to the inside of the two surfaces ratherthan the outside, to facilitate the use of automatic point ex-traction. The rationale was that from the perspective pro-jection view of a camera, having the calibration object onthe inside makes the calibration object appear ”butterfly”shaped and larger (since the edges move towards the cam-era and not away from the camera). It is admitted that itis more difficult to construct a good quality Tsai calibrationobject than it is to construct a good quality Zhang calibra-tion object. However, the construction process only needsto be done once.

3.3 Point ExtractionThe automatic extraction of corner points from calibrationobjects is crucial to both accuracy and calibration time.Our initial strategy was to perform corner extraction usingOpenCV’s findChessBoardCorners function. Unfortunately,it was our experience, that this function was not reliable,with an average of 40% of the calibration images deemedtoo difficult by the algorithm. While OpenCV’s algorithm

has better performance for Zhang’s calibration images, itsfailures were fairly consistent across all image types. Wedecided to develop our own point extraction application.

Our strategy was to initially start with a basic template for acorner, which is a 2 x 2 grid, with alternate black and whitecells. This template can be easily constructed in OpenCV.Next, we perform block matching throughout the image inorder to find likely locations where corners could be found.Typically, this block matching process yields clusters of highprobability points around the location of each corner of thecalibration object. Our strategy was then to identify these”blobs” using OpenCV’s blob detector and to find the cen-troid of these blobs to yield actual corner points. Naturally,our template only captures half the corners in the calibra-tion object, so it needs to be inverted and then the invertedtemplate is used to find the remaining corners.

While we found the above strategy to be reasonable accu-rate, we found that on more difficult images there was some-times a small amount of false positives and false negatives.Thus we implemented a quick manual interface to fix errors.The interface functions as follows: If a mouse click is de-tected and a detected point is in a close vicinity, then thepoint is removed; otherwise the local vicinity is searched tofind the best location to place a point (using the same strat-egy as before) and a new point is added to the best locationfound.

The next problem concerns the order that points are ex-tracted. Order matters since each point is associated with aworld coordinate and randomly extracting points from theimage makes associating that point with the world coordi-nate far more complex.

Our solution begins with the assumption that all the pointsare the vertices in a grid of known dimensions and that allcorner detection problems have been removed. Our firststep is to find the corners of the grid. To do this, we find theconvex hull and then find the top two mutually exclusive setsof points that are furthest apart. Once the corners are found,we can construct four lines outlining the grid (we avoid thediagonals by only connecting adjacent corner points). Thisoutline can then be used to detect the orientation of thegrid, which is important since Tsai calibration objects arecoplanar.

Our orientation detection takes advantage of the fact that aborder around a projection of two coplanar grids normallyhas one pair of edges that are straight while the other pairis not. Thus orientation is detected by determining whichlines are closest to the set of detected points. In practice,we found that using parallel pairs of edges with the sameorientation was more reliable than considering each edge in-dividually. From the best pair of edges we can then select theedge that best represents a column or row. After that, wecan systematically extract columns or rows (of know size) bysystematically removing clusters (of column or row size) thatare closest to the best edge and sorting each cluster basedon their distance from the top left corner of the grid. Wecontinue this process until all the points have been placedinto a matrix. Once in a matrix, points can be extracted inany desired order.

168

Figure 3: Point Extraction for different calibrationobjects.

Figure 3 is a screen-shot of the featured point extraction ap-plication. While we have used OpenCV’s drawChessBoard-Corners function to render the points and the order of ex-traction, the actual logic behind that extraction is that ofthe algorithm described above.

3.4 Rectification without CalibrationRectification is the process of warping images, so that theyappear to be captured in a rig consistent with canonicalepipolar geometry. Canonical epipolar geometry assumesthe sensor planes of all cameras involved to lie in the sameplane with each row on a sensor plane of any camera alignedwith the same row on every other camera’s sensor plane.This configuration reduces the search space for correspond-ing pixels (the stereo matching problem) to a single dimen-sion, making the problem much more tractable.

Normally, rectification is performed using the intrinsic andextrinsic data that a calibration algorithm has determined.However, without calibration data, one can still use theproperties of the canonical epipolar geometry to find a set ofwarps that would yield a rectified pair. An example of such acalibration approach was proposed by Hartley [5]. A typicalstrategy begins with finding a sparse set of known matchingpoints in the images to be rectified. Next, a system of equa-tions can be formulated to derive a special matrix known asthe Fundamental matrix. This matrix defines the relation-ship between a point in one image and the matching lineof projection (epipolar line) in the other. The Fundamentalmatrix thus contains all the information that is required tocalculate a set of transformations to produce rectified im-ages.

Unfortunately, rectification is an ill-posed problem wherethere is not a single unique solution. Thus, while rectifi-cation without calibration finds a valid solution to the rec-tification problem, this solution often lacks the constraintsof projective geometry that only calibration data can pro-vide. In light of this, rectification without calibration is aperfect baseline for establishing the lower bound of perfor-mance that calibrated rectification should outperform.

The implementation of rectification without calibration usedin this paper is the one found in OpenCV, which is basedon [5].

3.5 A Tsai ImplementationThe implementation of Tsai presented in this paper is ac++ implementation written by the authors. The imple-

mentation follows the paper of Roger Tsai [12]. In terms ofthe non-linear aspect of Tsai’s algorithm, we noticed thatmany implementations (such as that by Fernandez et al. [3])use Levenberg-Marquardt [6] as an iterative optimisationscheme. Since the Levenberg-Marquardt algorithm is es-sentially a hybrid of Gauss-Newton and Gradient Descentalgorithms, we decided to formulate our optimisation as asimplified Levenberg-Marquardt by simultaneously perform-ing both Gauss-Newton and Gradient descent and selectingthe best result on each iteration. Our implementation wastested against synthetic data with known distortion valuesand it was found that accuracy was adequate for our pur-poses.

3.6 A Zhang ImplementationThe implementation of Zhang’s algorithm presented in thispaper is based on the free ANSI C version provided by Fer-nandez et al. [3].

3.7 UndistortThe output of calibration is a set of intrinsic and extrin-sic parameters that describe the image acquisition process.Additionally, both the Tsai and Zhang algorithms also yielda model of the distortion. Typically, this distortion modelshould be used to remove distortion prior to rectification.

In order achieve this, we wrote a simple C++ applicationthat acquires initial images and a distortion model as inputs,and yields undistorted images as output. In order to calcu-late the undistorted image, our algorithm’s main strategy isto loop through all the pixels of the undistorted image, andfor each pixel, calculate the distorted location of that pixel.Bilinear interpolation is then used to resolve the RGB valueat the required image location.

xd = xu(1+k1r2+k2r

4+k5r6)+2k3xuyu+k4(r2+2x2u) (1)

yd = yu(1+k1r2+k2r

4+k5r6)+k3(r2+2y2u)+2k4xuyu (2)

The above two equations are the equations for the Zhangdistortion model. Clearly, derivation of the distorted lo-cation, given the undistorted location, is just a matter ofsubstitution.

xu = xd(1 + k1r2), yu = yd(1 + k1r

2) (3)

However equations 3 are from Tsai’s distortion model. Theseequations are more problematic since they are written fromthe perspective of the undistorted coordinate. In order to re-solve this, we square the equations 3 and add them togetherto get equation 4.

k21rsd3 + 2k21r

sd2 + rsd − rsu = 0 (4)

Where rsu, rsd are the undistorted and distorted squared ra-dial distances from the image’s principal point.

This is a cubic equation and Cardano’s formula can be usedto solve for rsd. Once rsd is known, the ratio of x and ycoordinates to rsu can be used to find the distorted versions.

169

Figure 4 shows the result of our undistortion algorithm, us-ing a the distortion model of Tsai.

Figure 4: Distorted and undistorted images

3.8 Calibrated RectificationAfter distortion has been removed, the next logical step isto perform calibrated rectification. The rectification imple-mented in this paper is a c++ application written by theauthors and based on the algorithm proposed by Fusielloet al. [4]. The main idea behind this algorithm is to calcu-late a new Camera matrix (Qu) and then find the transformthat will convert the old Camera matrix Qo (found throughcalibration) to the new one as per equation 5.

T = QnQ−1o (5)

3.9 Experimental ConditionsAs the main purpose of our experimentation was to providea fair and balanced comparison between the Zhang and Tsaiapproaches, it was important to conduct our experiments ina controlled environment. During these sessions we used ex-actly the same cameras for all image acquisitions, the samepoint extraction algorithm and the same rectification algo-rithms. Experiments were performed in laboratory condi-tions with careful control of lighting courtesy of a HL-1000Quartz bulb lighting system.

Timing was another important aspect of our experimentalprocess. As it was our aim to reflect the performance ofan experienced researcher in the lab, timing was only per-formed, once the individuals performing calibration were fa-miliar with the calibration approach.

4. RESULTSIn this section we present the results of our experimentation.

4.1 Practical ComplexityDuring our experimentation process, it was necessary tokeep track of how long each calibration process took. Sincethe actual execution time of our algorithms was quite fast,we focused on the stages of the process that required themost manual interaction. These stages were identified asthe following: Capturing images, manually correcting thepoint detection, extracting images from the cameras andlaunching algorithms. It was found that extracting imagesand launching applications took roughly the same time, irre-spective of the calibration strategy used, thus it was decidedto exclude these from scrutiny as well.

The above Table 2 outlines the time taken (as per a mo-bile phone stop watch) for the most labour intensive (andthus the most time absorbing) activities in our experimentalpipeline.

Table 2: Average time taken for experiment on cho-sen manual activities

Activity Tsai (sec) Zhang (sec)The acquisition of images 15 305Correcting Point detection 60 997

Table 3: Average projective error results in pixels

GP2 GP3 W3 Basler

L R L R L R L R

Zh m 0.4 0.38 0.34 0.33 0.39 0.34 0.25 0.17

σ 0.26 0.35 0.22 0.3 0.23 0.2 0.15 0.09

Ts m 0.8 0.88 0.74 0.66 0.75 0.82 0.37 0.36

σ 0.55 0.55 0.67 0.47 0.47 0.42 0.22 0.21

Table 4: Rectification error results in pixels

GP2 GP W3 Basler

Zhang m 0.07 0.58 0.26 0.19

σ 0.06 0.64 0.19 0.09

Tsai m 0.36 0.17 0.31 0.42

σ 0.37 0.21 0.39 0.38

Uncalibrated m 0.52 0.52 0.47 0.43

σ 0.58 0.52 0.49 0.52

4.2 Re-projection ErrorThe intrinsic and extrinsic data that is produced by a cali-bration algorithm can be succinctly formulated as a specialprojection matrix, known as the camera matrix. The cam-era matrix models the camera’s operation of converting 3Dscene coordinates into image plane coordinates. The qualityof this model may be evaluated, given a set of actual 3Dworld coordinates, by simply comparing the variance be-tween actual image coordinates and those forecasted by themodel. In our case, we measure the discrepancy in terms ofEuclidean distance. Results are in Table 3.

4.3 Rectification ErrorThe goal of rectification is to align images in canonical epipo-lar geometry. This is typically evaluated by checking whethera set of known pixel locations in corresponding images havebeen resolved to the same row (or column, depending onthe orientation of the image). As the process of calibrationnormally relies on such a set of points (extracted from thecalibration object), this would be the obvious set of points totest against. The procedure is relatively simple: (i) Performrectification and retrieve the warp transformations that therectification algorithm outputs. (ii) Apply the warp trans-formations to the corresponding calibration points in the leftand right images. (iii) Calculate error for each point basedon the Euclidean distance differences in the y-direction. (iv)Calculate the mean error distance (as shown in Table 4).

4.4 Reconstruction errorAs a final step, we decided to use Minh Nguyen’s [8] web-based application to perform stereo matching and 3D recon-struction. Using these depth maps and the location of our

170

Table 5: Reconstruction error resultsGP2 GP W3 Basler

Zhang m 0.38 0.44 0.44 0.68

σ 0.33 0.38 0.34 0.62

Tsai m 0.8 0.51 1.06 0.87

σ 0.6 0.45 0.87 0.63

calibration points in those images, we were able to approxi-mate least-squares planar surfaces of our calibration objects.An error was calculated by finding the Euclidean distancedifference between the point on the expected plane and theactual location of the point. The value of this step is basedon the notion that good performance in rectification shouldresult in good performance with regards to stereo matching(see Table 5).

Figure 5: The stereo-matching pipeline. From top tobottom: Gopro Hero3, Fuji W3; From left to right:left, undistorted and depth map images using Tsaicalibrationa and the guided DP stereo matching [7].

5. CONCLUSIONSThe focus of this study was to evaluate the performance ofZhang’s and Tsai’s calibration in the context of multi-imagestereo. Typically, the multi-image stereo pipeline consists ofthe following steps: image acquisition, feature point extrac-tion, calibration, distortion removal, rectification and finallystereo matching. In this process, calibration is an importantcog in a pipeline of operations, where each step has an im-pact on the quality of the final result.

Whether calibration is based on Tsai or Zhang, results willbe inaccurate if point extraction is inaccurate. Thus it isessential that point extraction is automated as far as possi-ble in order to improve on accuracy. Here, we demonstratedthat the extraction of points can be, for the most part, au-tomated for both calibration approaches.

We then used standard implementations of the Zhang andTsai algorithms, followed by tried and trusted algorithmsfor rectification and stereo matching. We also developedour own application for the removal of radial distortion fromimages based on models outputted by calibration.

However, while our aim was to make our pipeline as accurateas possible, we need to bear in mind that imaging is basedon digital sampling, and thus the size of a pixel representsa lower bound to the achievable quality. With this in mind,

while the tables in our result section clearly show that Zhangis slightly ahead in accuracy, we pose the question, is thisgain in accuracy essential?

To test this theory we performed one final experiment: usingTsai’s calibration we attempted to acquire a disparity mapof an outdoor scene. The result can be seen in Figure 5. Wefeel that this result hints that a reasonable output depthmapcan be achieved using Tsai’s calibration. Coupled with thequicker turn-around times for Tsai’s calibration (see Table1), it surely makes Tsai a first choice contender when itcomes to the selection of a calibration algorithm for multi-image stereo.

6. REFERENCES[1] G. Bradski and A. Kaehler. Learning OpenCV:

Computer vision with the OpenCV library. O’ReillyMedia, Inc., 2008.

[2] X. Feng, M. Cao, H. Wang, and M. Collier. Thecomparison of camera calibration methods based onstructured-light measurement. In IEEE Congress onImage and Signal Processing, pages 155–160, 2008.

[3] M. Fernandez, L. Teixeira, and M. Gattass. Ansi Cimplementation of classical camera calibrationalgorithms: Tsai and Zhang, Sept 2014.

[4] A. Fusiello, E. Trucco, and A. Verri. A compactalgorithm for rectification of stereo pairs. MachineVision and Applications, 12(1):16–22, 2000.

[5] R. Hartley. Theory and practice of projectiverectification. International Journal of ComputerVision, 35(2):115–127, 1999.

[6] J. More. The Levenberg-Marquardt algorithm:implementation and theory. In Numerical analysis,pages 105–116. Springer, 1978.

[7] M. Nguyen, Y.-H. Chan, P. Delmas, andG. Gimel’farb. Symmetric dynamic programmingstereo using block matching guidance. IEEEIVCNZ’13 Proc., pages 88–93, 2013.

[8] M. Nguyen, G. Gimel’farb, and P. Delmas. Web-basedon-line computational stereo vision. IEEE IVCNZ’08Proc., pages 1–6, 2008.

[9] C. Ricolfe-Viala and A.-J. Sanchez-Salmeron.Improved camera calibration method based on atwo-dimensional template. In Pattern Recognition andImage Analysis, pages 420–427. Springer, 2007.

[10] J. Salvi, X. Armangue, and J. Batlle. A comparativereview of camera calibrating methods with accuracyevaluation. Pattern recognition, 35(7):1617–1635, 2002.

[11] W. Sun and J. Cooperstock. Requirements for cameracalibration: Must accuracy come with a high price? InIEEE 2005 Workshops on Application of ComputerVision, pages 356–361, 2005.

[12] R. Tsai. A versatile camera calibration technique forhigh-accuracy 3D machine vision metrology usingoff-the-shelf tv cameras and lenses. IEEE Journal ofRobotics and Automation, 3(4):323–344, 1987.

[13] Z. Zhang. A flexible new technique for cameracalibration. IEEE PAMI, 22(11):1330–1334, 2000.

171