AlignPlus 2D Hand-Eye 
Calibration Concepts 
5/12/2014 
Version 1.0 
 
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VisionPro 
5481712, 5495537, 5548326, 5583954, 5602937, 5640200, 5751853, 5768443, 5825913, 5850466, 5872870, 5901241, 5943441, 5978080, 5978521, 5987172, 
6005978, 6039254, 6064388, 6075881, 6137893, 6141033, 6167150, 6215915, 6240208, 6324299, 6381366, 6381375, 6411734, 6421458, 6459820, 6490375, 
6516092, 6563324, 6658145, 6687402, 6690842, 6697535, 6718074, 6748110, 6771808, 6804416, 6836567, 6850646, 6856698, 6920241, 6959112, 6963338, 
6973207, 6975764, 6985625, 6993177, 6993192, 7006712, 7016539, 7043081, 7058225, 7065262, 7088862, 7164796, 7190834, 7242801, 7251366, 7313761, 
EP0713593, JP3522280, JP3927239 
DataMan 
5742037, 5943441, 6215915, 6236769, 6282328, 6381375, 6408109, 6457032, 6690842, 6941026, 7175090, 7181066, 7412106, 7427028, 7549582, 7604174, 
7614563, 7617984, US-2005-0087601-A1, US-2006-0131418-A1, US-2006-0131419-A1, US-2006-0133757-A1, US-2007-0090193-A1, US-2007-0091332-A1, 
US-2007-0152064-A1, US-2007-0170259-A1, US-2008-0004822-A1, US-2008-0011855-A1, US-2008-0142604-A1, US-2008-0143838-A1, US-2008-0158365-A1, 
US-2009-0090781-A1, US-2009-0108073, US-2009-0121027-A1, US-2009-0166424-A1, US-2009-0294541-A1, WO06065619A1, EP1687752 
CVL 
5495537, 5548326, 5583954, 5602937, 5640200, 5717785, 5751853, 5768443, 5825483, 5825913, 5850466, 5859923, 5872870, 5901241, 5943441, 5949905, 
5978080, 5987172, 5995648, 6002793, 6005978, 6064388, 6067379, 6075881, 6137893, 6141033, 6157732, 6167150, 6215915, 6240208, 6240218, 6324299, 
6381366, 6381375, 6408109, 6411734, 6421458, 6457032, 6459820, 6490375, 6516092, 6563324, 6658145, 6687402, 6690842, 6718074, 6748110, 6751361, 
6771808, 6798925, 6804416, 6836567, 6850646, 6856698, 6920241, 6959112, 6975764, 6985625, 6993177, 6993192, 7006712, 7016539, 7043081, 7058225, 
7065262, 7088862, 7164796, 7190834, 7242801, 7251366, EP0713593, JP3522280,  JP3927239 
VGR 
5495537, 5602937, 5640200, 5768443, 5825483, 5850466, 5859923, 5949905, 5978080, 5995648, 6002793, 6005978, 6075881, 6137893, 6141033, 6157732, 
6167150, 6215915, 6324299, 6381375, 6408109, 6411734, 6421458, 6457032, 6459820, 6490375, 6516092, 6563324, 6658145, 6690842, 6748110, 6751361, 
6771808, 6804416, 6836567, 6850646, 6856698, 6959112, 6975764, 6985625, 6993192, 7006712, 7016539, 7043081, 7058225, 7065262, 7088862, 7164796, 
7190834, 7242801, 7251366 
OMNIVIEW 
AlignPlus 2D Hand-Eye Calibration Concepts 
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Legal Notices 
6215915, 6381375, 6408109, 6421458, 6457032, 6459820, 6594623, 6804416, 6959112, 7383536 
CVL Vision Library 
5495537, 5548326, 5583954, 5602937, 5640200, 5717785, 5751853, 5768443, 5825483, 5825913, 5850466, 5859923, 5872870, 5901241, 5943441, 5949905, 
5978080, 5987172, 5995648, 6002793, 6005978, 6064388, 6067379, 6075881, 6137893, 6141033, 6157732, 6167150, 6215915, 6240208, 6240218, 6324299, 
6381366, 6381375, 6408109, 6411734, 6421458, 6457032, 6459820, 6490375, 6516092, 6563324, 6658145, 6687402, 6690842, 6718074, 6748110, 6751361, 
6771808, 6798925, 6804416, 6836567, 6850646, 6856698, 6920241, 6959112, 6975764, 6985625, 6993177, 6993192, 7006712, 7016539, 7043081, 7058225, 
7065262, 7088862, 7164796, 7190834, 7242801, 7251366, EP0713593, JP3522280,  JP3927239 
SMD 4 
5995648, 5850466, 6751361, 6690842, 6563324, 6490375, 5949905, 5978080, 6137893, 6167150, 6075881, 6748110, 5859923, 6411734, 6324299, 6516092, 
7190834, 6658145, 6836567, 6850646, 6975764, 6985625, 6993192, 7006712, 7043081, 7058225, 7065262, 7088862, 7164796, 7251366, 6856698, 6002793, 
6005978, 6771808, 6804416, 7016539, 6959112, 5602937, 7242801, 5640200, 5495537, 5768443, 5825483, 6421458, 6459820, 6215915, 6381375, 6457032, 
6157732, 6408109, 6141033, 6026176, 6442291, 6151406, 6396942, 6614926, 5371690, 5845007, 5943441, 6963338, 5805722, 5909504, 5933523, 5964844, 
5974169, 5987172, 6078700, 6252986, 6278796, 6307210, 6408429, 6424734, 6526165, 6571006, 6639624, 6681039, 6748104, 6813377, 6853751, 6898333, 
6950548, 6993177, 7139421, 5757956 
BGA II and BGA III 
5495537, 5602937, 5640200, 5768443, 5801966, 5825483, 5850466, 5859923, 5949905, 5978080, 5995648, 6002793, 6005978, 6026176, 6055328, 6075881, 
6115042, 6118893, 6130959, 6137893, 6141009, 6141033, 6151406, 6157732, 6167150, 6215915, 6289117, 6324299, 6353676, 6381375, 6396942, 6408109, 
6411734, 6421458, 6442291, 6457032, 6459820, 6490375, 6516092, 6563324, 6577775, 6614926, 6658145, 6690842, 6748110, 6751361, 6771808, 6804416, 
6836567, 6850646, 6856698, 6959112, 6975764, 6985625, 6993192, 7006712, 7016539, 7043081, 7058225, 7065262, 7088862, 7164796, 7190834, 7242801, 
7251366 
Wire Bonder 
5495537, 5532739, 5581632, 5602937, 5640199, 5640200, 5642158, 5676302, 5754679, 5757956, 5768443, 5825483, 5835622, 5850466, 5859923, 5861909, 
5949905, 5978080, 5991436, 5995648, 6002793, 6005978, 6035066, 6061467, 6075881, 6137893, 6141033, 6157732, 6167150, 6215915, 6289492, 6324299, 
6381375, 6408109, 6411734, 6421458, 6457032, 6459820, 6490375, 6516092, 6563324, 6658145, 6690842, 6748110, 6751361, 6771808, 6804416, 6836567, 
6850646, 6856698, 6959112, 6975764, 6985625, 6993192, 7006712, 7016539, 7043081, 7058225, 7065262, 7088862, 7164796, 7171036, 7190834, 7242801, 
7251366 
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Contents 
Introduction ..........................................................................................................................6 
How Hand-Eye Calibration is Performed ............................................................................... 11 
Train Time ................................................................................................................................... 11 
Run Time ..................................................................................................................................... 11 
Calibration Process ...................................................................................................................... 12 
Train-Time Calibration Process ................................................................................................ 12 
Run-Time Process Using Calibration Results ............................................................................ 16 
Hand-eye Calibration Coordinate Spaces .............................................................................. 18 
Raw2D ......................................................................................................................................... 18 
Home2D ...................................................................................................................................... 19 
Stage2D ....................................................................................................................................... 21 
Plate2D ........................................................................................................................................ 23 
Camera2D .................................................................................................................................... 25 
Handedness of Coordinate Spaces .............................................................................................. 26 
Transforms .......................................................................................................................... 27 
General Transforms .................................................................................................................... 27 
Home2DFromStage2D.............................................................................................................. 27 
UncorrectedHome2DFromStage2D ......................................................................................... 27 
Raw2DFromHome2D................................................................................................................ 27 
ConvertUncorrectedHome2DFromStage2DToHome2DFromStage2D() and 
ConvertHome2DFromStage2DToUncorrectedHome2DFromStage2D() Conversion Functions
 .................................................................................................................................................. 27 
Raw2DFromCamera2D ............................................................................................................. 27 
Stationary Camera Configuration transforms ............................................................................. 27 
Home2DFromStationaryCamera2D ......................................................................................... 27 
Stage2DFromMovingPlate2D ................................................................................................... 28 
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Contents 
Moving Camera Configuration transforms ................................................................................. 28 
Home2DFromStationaryPlate2D .............................................................................................. 28 
Stage2DFromMovingCamera2D ............................................................................................... 28 
Transforms Overview .................................................................................................................. 29 
Motion Stage Validation....................................................................................................... 31 
What AlignPlus Hand-Eye Calibration Does ........................................................................... 32 
Motion Capability Limitations ..................................................................................................... 32 
Fundamental Requirements and Residuals ................................................................................ 33 
Fixed Placement Pose of the Calibration Target and Cameras ................................................ 33 
Precision of Motion Stage ........................................................................................................ 33 
In-Plane Motion........................................................................................................................ 34 
Indications of Violations to Requirements ............................................................................... 34 
Residual Errors ......................................................................................................................... 34 
Multi-Camera Hand-Eye Calibration...................................................................................... 36 
Sample application .............................................................................................................. 37 
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Introduction 
 
Introduction 
AlignPlus hand-eye calibration allows you to calibrate images obtained with your cameras to the 
platform on which the object to be inspected is and whose position relative to your cameras 
moves. The images you obtain using your cameras may exhibit lens distortion and perspective 
distortion, your motion system input may be offset from the actual position it moves to and it 
may have systematic errors as well. AlignPlus hand-eye calibration allows you to view and 
inspect features of objects in an undistorted manner (with physically correct length units) and 
with placement in the native coordinate system of the motion system. That is, it shows you the 
object to be inspected in its real physical appearance and it tells you where that object is (in the 
native coordinate space). 
The following are determined regarding your motion system during calibration:  
•  Origin and axis directions of native coordinate space – the native coordinate space of 
the motion system is determined as a mapping from the image coordinate space for 
each camera  
•  Systematic errors along the axes of movement – systematic motion system errors are 
corrected 
 
Note: Even a configuration with a perfect motion system (that is, one that performs the 
movements as required within precision thresholds) and no image distortions requires hand-eye 
calibration because the mappings between each image coordinate space and the native 
coordinate space need to be determined. These mappings enable positioning image features in 
the native coordinate space. In other words, by establishing these mappings, correspondence is 
established between features found in images taken by the cameras to the physical coordinates 
of these features in the native coordinate space.  
To get basic feedback on your motion stage and identify problems before hand-eye calibration, 
you perform motion stage validation before hand-eye calibration. 
AlignPlus hand-eye calibration is a 2D calibration, it calibrates images in the plane of the 
platform on which the object to be inspected is. This tool provides no 3D information. 
There are two types of scenarios: either the platform moves with the object to be inspected and 
the cameras are stationary or the cameras move and the platform with the object to be 
inspected is stationary.  
 
 
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The following figure illustrates this with a single camera. 
Introduction 
 
 
The motion system performs planar movement: 
• 
• 
It can move in the X and Y directions,  
It can perform Theta rotation based on your inputs. 
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The following figures demonstrate the X and Y movements and the Theta rotation of the motion 
stage. 
Introduction 
 
 
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