Correction of a Minor Error in the Implementation Software and the Original Paper of Tsai's Calibration

Computer Science Department of The University of Auckland CITR at Tamaki Campus (http://www.citr.auckland.ac.nz)

CITR-TR-89 May 2001

Correction of a Minor Error in the Implementation Software and the Original Paper of Tsai's Calibration

Method

Yongbao Zhang ¹

Abstract

This note reports about a minor error in the original paper on Tsai's calibration method and in the corresponding implementation software available for free downloads on the Web. Tsai's camera calibration method allows a determination of eleven camera calibration parameters.

There is an incorrectness in the formula for calculating the X –

component of the translation vector T. This report corrects this minor problem.

1 Center for Image Technology and Robotics Tamaki Campus, The University of Auckland, Auckland, New Zealand

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are reserved by the author(s).

in the Implementation Software and

the Original Paper of Tsai's Calibration Method

YongbaoZhang

CenterforImageTechnology andRob otics

TamakiCampus,TheUniversity ofAuckland,Auckland,NewZealand

Abstract. This note rep orts ab out a minor error in the original pa-

p er on Tsai's calibration metho d and in the corresp onding implemen-

tationsoftware available forfreedownloads on theWeb.Tsai'scamera

calibration metho d allowsadetermination ofeleven cameracalibration

parameters.Thereisan incorrectness intheformulaforcalculating the

X-comp onentofthetranslationvectorT.Thisrep ortcorrectsthisminor

problem.

1 Introduction

Camera calibration is an imp ortant task in computer vision, see [3]. Tsai's

metho disthemostwidelyusedcalibrationtechnique. Ithasb eenprop osed by

RogerY.Tsai in[1]and [4].Thereisacorresp ondingimplementationsoftware

which can b e downloadedfromthe website [2].For b oth of the originalpap er

and theimplementationsoftware, theformulaofcalculatingthe X-comp onent

of thetranslationvector T is erroneous. Inthis littlenote, we willdiscuss this

minorerrorandhowtocorrect it.

2 Error in the Original Paper

We address line 2 in the right columnof page 333 in[1],i.e. the equation for

calculatingtheX-comp onentT

x

ofthetranslationvector.ItisclearthattheX

comp onentT

x

ofthetranslationvectorshouldb e calibratedaftercalibrationof

thehorizontalscalefactorS

x

, b ecausethefourth transformationparametera

4

isdened asfollows:

a

4

= T

x S

x

T

y

; (1)

and,obviously,theX-comp onentT

x

ofthetranslationvectorshouldb eobtained

by

T

x

= a

4 T

y

S

(2)

T

x

=a

4 T

y

: (3)

Theoriginalpap ermissesthehorizontalscalefactorS

x .

3 Error in the Implementation Software

The implementation software of Tsai'scalibration metho dcan b e downloaded

fromthewebsite[2].Ithasb eenwrittenbyRegWillsonin1995.Thissoftwareis

freewareandiswidelyusedinthecomputerversioneldandtherob oticseld.

Theminorincorrectness rep ortedab ove is also existent inthis software.In

function ncccompute TxandTy () of lecal main:c, the X-comp onent T

x of

translation vector should not b e calculated here. It should b e calculatedafter

thecalibrationofthehorizontalscalefactorS

x

andtheformulaofT

x

should b e

Equ.2ratherthanEqu.3

Inordertocorrectthiserror,weremovethecommandofcalculatingT

x from

thefunction

ncccompute TxandTy ()

andputit intothefunction

ncccompute sx()

oflecal main:c.Wecalculate T

x

aftercalculationofS

x .

Sensors of CCD camerasare mostly square nowadays.Therefore, the hori-

zontalscalefactorS

x

isnearly1formostofallcameras,andthereconstruction

error causedbytherep orted implementationerror isveryminorconsequently.

References

1. R.Y.Tsai:AversatileCameraCalibration TechniqueforHigh-Accuracy3DMa-

chineVisionMetrologyUsingO-the-ShelfTVCamerasandLenses.IEEEJour-

nalofRoboticsandAutomation,Vol.RA-3,No.4,August1987, pages323-344.

2. R.Willson :TsaiCameraCalibrationSoftware(1995),http://almond.srv.cs.cmu

.edu/afs/cs/usr/rgw/www/TsaiCode.html(lastvisited:22 nd

May,2001)

3. R. Klette, K.Schl uns, A.Koschan: Computer Vision: Three-dimensional Data

fromImages.Springer,Singap ore(1998).

4. R.Y.Tsai:AneÆcientandaccuratecameracalibration techniquefor3Dmachine

vision. Pro c. Internat. Conf. Computer Visionand Pattern Recognition(1986)