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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 MSU CSE 803 Fall 2008 Stockman

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**Four Coordinate frames**

W: world, C,D: cameras, M: object model Need to relate all to each other. MSU CSE 803 Fall 2008 Stockman

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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 MSU CSE 803 Fall 2008 Stockman

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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 MSU CSE 803 Fall 2008 Stockman

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Stereo configuration 2 corresponding image points enable the intersection of 2 rays in W MSU CSE 803 Fall 2008 Stockman

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Stereo computation MSU CSE 803 Fall 2008 Stockman

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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) MSU CSE 803 Fall 2008 Stockman

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**Replace camera with projector**

Can calibrate a projector to W easily. Correspondence now means identifying marks. MSU CSE 803 Fall 2008 Stockman

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**Advantages/disadvantages of structured light**

MSU CSE 803 Fall 2008 Stockman

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**Grid projected on objects**

All grid intersects are integral MSU CSE 803 Fall 2008 Stockman

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**Computing surface normals**

Surface normals have been computed and then added to the image (augmented reality) MSU CSE 803 Fall 2008 Stockman

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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 MSU CSE 803 Fall 2008 Stockman

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**Translation of 3D point P**

Point in frame 1 Point in frame 2 Point in 3D parameters MSU CSE 803 Fall 2008 Stockman

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Scaling 3D point P MSU CSE 803 Fall 2008 Stockman

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**Rotation of P about the X-axis**

MSU CSE 803 Fall 2008 Stockman

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**Rotate P about the Y-axis**

MSU CSE 803 Fall 2008 Stockman

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**Rotate P about the Z-axis**

Looks same as 2D rotation omitting row, col 4 MSU CSE 803 Fall 2008 Stockman

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**Arbitrary rotation has orthonormal rows and columns**

MSU CSE 803 Fall 2008 Stockman

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**Example: camera relative to world**

MSU CSE 803 Fall 2008 Stockman

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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 MSU CSE 803 Fall 2008 Stockman

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**Transformation “calculus”: notation accounts for transforms**

Destination frame W Denotes transformation W T M T transforms points from model frame to world frame. (Notation from John Craig, 1986) Origin frame M MSU CSE 803 Fall 2008 Stockman

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**Apply transformations to points**

Point in model coordinates W W M P = T P M Point in world coordinates Transformation from model to world coordinates (instance transformation) MSU CSE 803 Fall 2008 Stockman

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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) MSU CSE 803 Fall 2008 Stockman

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**Composing transformations**

Projection parms. Parameters: rotation and translation cancel A A C T (w) T = T(p) W W C Two transformations are composed to get one transformation that maps points from the world frame to the frame A MSU CSE 803 Fall 2008 Stockman

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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 MSU CSE 803 Fall 2008 Stockman

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**Viewing model points M What’s in front of the camera?**

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Math for the steps Camera C maps 3D points in world W to 2D pixels in image I MSU CSE 803 Fall 2008 Stockman

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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 MSU CSE 803 Fall 2008 Stockman

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**Perspective projection: camera origin in the real image plane**

MSU CSE 803 Fall 2008 Stockman

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**Rigid transformation for change of coordinate frame**

3D coordinate frame of camera MSU CSE 803 Fall 2008 Stockman

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**Relate camera frame to world frame**

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**Change scene units to pixels**

To get into XV or GIMP image coordinates! This is a 2D to 2D transformation. MSU CSE 803 Fall 2008 Stockman

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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 MSU CSE 803 Fall 2008 Stockman

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The camera model, or matrix, is 3 x 4 and maps a homogeneous point in the world to a homogeneous pixel in the image. The ‘1’ is used to model translation MSU CSE 803 Fall 2008 Stockman

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