MSU CSE 803 Fall 2008 Stockman1 CV: 3D sensing and calibration Coordinate system changes;...
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Transcript of MSU CSE 803 Fall 2008 Stockman1 CV: 3D sensing and calibration Coordinate system changes;...
MSU CSE 803 Fall 2008 Stockman
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CV: 3D sensing and calibration
Coordinate system changes; perspective transformation; Stereo and structured light
MSU CSE 803 Fall 2008 Stockman
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roadmap using multiple cameras using structured light projector 3D transformations general perspective
transformation justification of 3x4 camera model
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Four Coordinate frames
W: world,
C,D: cameras,
M: object model
Need to relate all to each other.
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Can we recognize?
Is there some object M
That can be placed in some location
That will create the two images that are observed?
Discover/compute what object and what pose
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Need to relate frames to compute
relate camera to world using rotations and translations
project world point into real image using projection
scale image point in real image plane to get pixel array coordinates
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Stereo configuration
2 corresponding image points enable the intersection of 2 rays in W
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Math for stereo computations need to calibrate both cameras to
W so that rays in x,y,z reference same space
need to have corresponding points find point of closest approach of
the two rays (rays are too far apart point
correspondence error or crude calibration)
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Replace camera with projector
Can calibrate a projector to W easily. Correspondence now means identifying marks.
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Computing surface normals
Surface normals have been computed and then added to the image (augmented reality)
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Relating coordinate frames need to relate camera frame to
world need to rotate, translate, and scale
coordinate systems need to project world points to the
image plane all the above are modeled using
4x4 matrices and 1x4 points in homogeneous coordinates
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Translation of 3D point PparametersPoint in 3D Point in
frame 1
Point in frame 2
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Rotate P about the Z-axis
Looks same as 2D rotation omitting row, col 4
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exercise verify that the 3 x 3 rotation matrix
is orthonormal by checking 6 dot products
invert the 3 x 3 rotation matrix invert the 4 x 4 matrix verify that the new matrix
transforms points correctly from C to W
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Transformation “calculus”: notation accounts for transforms
TW
M
Denotes transformation
Origin frame M
Destination frame W
T transforms points from model frame to world frame. (Notation from John Craig, 1986)
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Apply transformations to points
TM
WP
MP =W
Point in model coordinates
Point in world coordinates
Transformation from model to world coordinates (instance transformation)
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Matrix algebra enables composition
Let M and N be 4 x 4 matrices and let P be a 4 x 1 point
M ( N P ) = ( M N ) P we can transform P using N and then
transform that by M, or we can multiply matrices M and N and then apply that to point P
matrix multiplication is associative (but not commutative)
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Composing transformations
T (W
CT(
C
AT =W
A
Two transformations are composed to get one transformation that maps points from the world frame to the frame A
Parameters: rotation and translation
Projection parms.cancel
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Deriving form of the camera matrix
We have already described what the camera matrix does and what
form it has; we now go through the steps to justify it
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Math for the stepsCamera C maps 3D points in world W to 2D pixels in image I
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Perspective transformation: camera origin at the center of projection
This transformation uses same units in 3D as in the image plane
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Rigid transformation for change of coordinate frame
3D coordinate frame of camera
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Change scene units to pixels
To get into XV or GIMP image coordinates! This is a 2D to 2D transformation.
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Final result a 3x4 camera matrix maps 2D image pixels to 3D rays maps 3D rays to 2D image pixels obtain matrix via calibration
(easy) obtain matrix via reasoning (hard) do camera calibration exercise