CV: 3D sensing and calibration Coordinate system changes; perspective transformation; Stereo and structured light MSU CSE 803 Fall 2008 Stockman

roadmap using multiple cameras using structured light projector 3D transformations general perspective transformation justification of 3x4 camera model MSU CSE 803 Fall 2008 Stockman

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

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

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

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

Stereo computation MSU CSE 803 Fall 2008 Stockman

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

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

Advantages/disadvantages of structured light MSU CSE 803 Fall 2008 Stockman

Grid projected on objects All grid intersects are integral MSU CSE 803 Fall 2008 Stockman

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

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

Translation of 3D point P Point in frame 1 Point in frame 2 Point in 3D parameters MSU CSE 803 Fall 2008 Stockman

Scaling 3D point P MSU CSE 803 Fall 2008 Stockman

Rotation of P about the X-axis MSU CSE 803 Fall 2008 Stockman

Rotate P about the Y-axis MSU CSE 803 Fall 2008 Stockman

Rotate P about the Z-axis Looks same as 2D rotation omitting row, col 4 MSU CSE 803 Fall 2008 Stockman

Arbitrary rotation has orthonormal rows and columns MSU CSE 803 Fall 2008 Stockman

Example: camera relative to world MSU CSE 803 Fall 2008 Stockman

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

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

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

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

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

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

Viewing model points M What’s in front of the camera? MSU CSE 803 Fall 2008 Stockman

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

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

Perspective projection: camera origin in the real image plane MSU CSE 803 Fall 2008 Stockman

Rigid transformation for change of coordinate frame 3D coordinate frame of camera MSU CSE 803 Fall 2008 Stockman

Relate camera frame to world frame MSU CSE 803 Fall 2008 Stockman

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

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

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