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2 Advances in Mechanical Engineering

Sphere in dierent positions

Light plane

Projector

Reference board

!e plane through sphere centers

Figure 1: 8e principle of the calibration.

board as the known radius of each sphere. By moving thestandard board into several positions, the height matrix,which gives the relationship between matching diGerencesand height, can be deduced based on successive geometricalcalculation. According to the model of calibration and theobtained height matrix, the structured light sensor of 3Dmeasurement can be calibrated. Inspired by this method, acalibrationmethod for line structured light vision sensor onlyusing a white sphere and a black reference board is presented(see Figure 1).

By moving the sphere into several positions on the Mxedreference board, we can get a plane through the centersof spheres in diGerent positions. 8e center of each sphereunder the camera coordinate system can be gained from itscorresponding sphere projection on the image plane. 8enthe plane through the sphere centers (a virtual plane in orderto express the reference board easily as illustrated in Figure 1)can be obtained by planar Mtting easily. As the distance fromthe sphere center to the reference board is a known constantand the normal vector of the plane through sphere centershas been deduced, the representation of the reference boardunder camera coordinate system can be solved.8en a groupof collinear feature points are obtained from the light stripeon the reference board.

Moving the target randomly into more than two diGerentpositions, we can get enough feature points of the structured-light plane easily. In this method, all feature points projectedon the reference board can be used to Mt the structured-lightplane; meanwhile, the sphere projection is not concernedwith the sphere orientation, which can make the calibrationmore accurate and robust.

2. Measurement Model of LSLVS

8e location relationship between the camera in LSLVS andthe structured-light plane projector remains unchangeablein the process of calibration and measurement. So thestructured-light plane can be expressed as a Mxed function,

x

yX

Y

O

Z

Projector Cameracoordinate

system

Imagecoordinate

system

Lightplane

o

P

p

Imageplane

Figure 2: 8e measurement model of the LSLVS.

which is deMned as (1), under the camera coordinate system.Consider

[, , , ] [, , , 1] = 0, (1)where , , , and are the parameters of the struc-tured-light planes expression.

8e measurement model of LSLVS is illustrated inFigure 2. - is the Camera Coordinate System (CCS),while - is the Image Coordinate System (ICS). Under theCCS, the center of projection of the camera is at the origin andthe optical axis points in the positive direction. A spatialpoint is projected onto the plane with = 0, referred toas the image plane under the CCS, where 0 is the eGectivefocal length (EFL). Suppose the point = (, , 1) is theprojection of = (, , ) on the image plane. Under theundistorted model of the camera, namely the ideal pinholeimaging model, , and the center of projection arecollinear.8e fact can be expressed by the following equation:

[[1]]= [[0 0 0 00 0 0 00 0 1 0

]][[[[

1]]]]. (2)

Practically, the radial distortion and the tangential distor-tion of the lens are inevitable. When considering the radialdistortion, we have the following equations:

= (1 + 12 + 24) = (1 + 12 + 24) ,

(3)

where 2 = 2 + 2, (, ) is the distorted image coordinate,(, ) is the idealized one, and 1, 2 are the radial distortioncoe[cients of the lens.

3. Calibration of the LSLVS

In our method, the calibration can be executed by thefollowing three key points: (1) work out the coordinates ofthe sphere centers in diGerent positions under the CCS,