Homogeneous Coordinates (Projective Space) Let be a point in Euclidean space Change to homogeneous coordinates: Defined up to scale: Can go back to non-homogeneous representation as follows:

3-D Transformations: Translation Ordinarily, a translation between points is expressed as a vector addition Homogeneous coordinates allow it to be written as a matrix multiplication:

3-D Rotations: Euler Angles Can decompose rotation of about arbi-trary 3-D axis into rotations about the coordinate axes (“yaw-roll-pitch”) , where: the ordering is just a convention, and there are other representations (Clockwise when looking toward the origin)

3-D Transformations: Rotation A rotation of a point about an arbitrary axis normally expressed as a multiplication by the rotation matrix is written with homogeneous coordinates as follows:

3-D Transformations: Change of Coordinates Any rigid transformation can be written as a combined rotation and translation:

Pinhole Camera Model Image plane Optical axis Principal point Focal length Camera center Camera point Image point

Pinhole Perspective Projection Letting the camera coordinates of the projected point be leads by similar triangles to: this is why more distant objects appear smaller, of course

Projection Matrix Using homogeneous coordinates, we can describe perspective projection with a linear equation: (by the rule for converting between homogeneous and regular coordinates)

Example 1: 2D Translation Q: How can we represent translation as a 3x3 matrix? A: Using the rightmost column:

Translation Example of translation Homogeneous Coordinates tx = 2 ty = 1

Basic 2D Transformations Basic 2D transformations as 3x3 matrices Translate Scale Rotate Shear

Affine Transformations Affine transformations are combinations of … Linear transformations, and Translations Properties of affine transformations: Origin does not necessarily map to origin Lines map to lines Parallel lines remain parallel Ratios are preserved Closed under composition Models change of basis

Projective Transformations Affine transformations, and Projective warps Properties of projective transformations: Origin does not necessarily map to origin Lines map to lines Parallel lines do not necessarily remain parallel Ratios are not preserved Closed under composition Models change of basis

Matrix Composition Transformations can be combined by matrix multiplication p’ = T(tx,ty) R(Q) S(sx,sy) p

Homography (Projective Transformation) Definition: Projective transformation or Recall (set f=1) 8DOF

Homography Estimation Given N pairs of corresponding coordinates (u,v) <> (u’,v’), how to estimate eight-parameter transform H?

Computing the Homography 8 degrees of freedom in , so 4 pairs of 2-D points are sufficient to determine it Other combinations of points and lines also work 3 collinear points in either image are a degenerate configuration preventing a unique solution Direct Linear Transformation (DLT) algorithm: Least-squares method for estimating

Huang’s LS Method A

Direct Linear Transformation (DLT) Method Since vectors are homogeneous, are parallel, so Let be row j of , be stacked ‘s Expanding and rearranging cross product above, we obtain , where x’_i * first row + y’_i * second row = third row the w’s are the homogeneous coordinate; i used lambda in my definition

DLT Homography Estimation: Solve System Only 2 linearly independent equations in each , so leave out 3rd to make it 2 x 9 Stack every to get 2n x 9 Solve by computing singular value decomposition (SVD) ; is last column of Solution is improved by normalizing image coordinates before applying DLT x’_i * first row + y’_i * second row = third row

Applying Homographies to Remove Perspective Distortion this could be helpful with trying to recognize a building from different points of view—convert to a common one first. this sort of thing is also done to convert the ground plane in perspective to a bird’s eye view from Hartley & Zisserman 4 point correspondences suffice for the planar building facade

Homographies for Mosaicing from Hartley & Zisserman