3-D Euclidean Space - Vectors Rigid Body Motion and A “free” vector is defined by a pair Image Formation of points : Jana Kosecka http://cs.gmu.edu/~kosecka/cs682.html Coordinates of the vector : CS682, Jana Kosecka CS682, Jana Kosecka Rotation Matrices in 3D 3D Rotation of Points – Euler angles Rotation around the coordinate axes, counter-clockwise: 1 0 0 •3 by 3 matrices R (α) = 0 cosα − sinα • 9 parameters – only three degrees of freedom x • Representations – either three Euler angles 0 sinα cosα • or axis and angle represntation P’ cos β 0 sin β Y’ γ R (β ) = 0 1 0 y − β β y P sin 0 cos x cosγ − sinγ 0 X’ • Properties of rotation matrices (constraints between the R (γ ) = sinγ cosγ 0 z elements) z 0 0 1 CS682, Jana Kosecka CS682, Jana Kosecka 1 Rotation Matrices in 3D Canonical Coordinates for Rotation •3 by 3 matrices Property of R • 9 parameters – only three degrees of freedom • Representations – either three Euler angles Taking derivative • or axis and angle representation Skew symmetric matrix property By algebra • Properties of rotation matrices (constraints between the elements) By solution to ODE Columns are orthonormal CS682, Jana Kosecka CS682, Jana Kosecka 3D Rotation (axis & angle) Rotation Matrices Solution to the ODE Given How to compute angle and axis with or CS682, Jana Kosecka CS682, Jana Kosecka 2 Rigid Body Motion – Homogeneous 3D Translation of Points Coordinates Translate by a vector 3-D coordinates are related by: Homogeneous coordinates: P’ t Y’ x’ x P Homogeneous coordinates are related by: z’ y z CS682, Jana Kosecka CS682, Jana Kosecka Rigid Body Motion – Homogeneous Properties of Rigid Body Motions Coordinates Rigid body motion composition 3-D coordinates are related by: Homogeneous coordinates: Rigid body motion inverse Homogeneous coordinates are related by: Rigid body motion acting on vectors CS682, Jana Kosecka Vectors are only affected by rotationCS682, Jana – 4Koseckath homogeneous coordinate is zero 3 Rigid Body Transformation Rigid Body Motion • Camera is moving • Notion of a twist • Relationship between velocities Coordinates are related by: Camera pose is specified by: CS682, Jana Kosecka CS682, Jana Kosecka Image Formation – Perspective Projection Pinhole Camera Model “The Scholar of Athens,” Raphael, 1518 Pinhole Frontal pinhole CS682, Jana Kosecka CS682, Jana Kosecka 4 More on homogeneous coordinates Pinhole Camera Model In homogenous coordinates – these represent the 2-D coordinates Same point in 3D Homogeneous coordinates The first coordinates can be obtained from the second by division by W What if W is zero ? Special point – point at infinity – more later In homogeneous coordinates – there is a difference between point and vector CS682, Jana Kosecka CS682, Jana Kosecka Image Coordinates Calibration Matrix and Camera Model Pinhole camera Pixel coordinates metric coordinates Linear transformation Calibration matrix pixel (intrinsic parameters) coordinates Projection matrix Camera model CS682, Jana Kosecka CS682, Jana Kosecka 5 Calibration Matrix and Camera Model Radial Distortion Pinhole camera Pixel coordinates Nonlinear transformation along the radial direction More compactly Transformation between camera coordinate Systems and world coordinate system Distortion correction: make lines straight CS682, Jana Kosecka CS682, Jana Kosecka Image of a line – homogeneous Image of a point representation Homogeneous coordinates of a 3-D point Homogeneous representation of a 3-D line Homogeneous coordinates of its 2-D image Projection of a 3-D point to an image plane Homogeneous representation of its 2-D image Projection of a 3-D line to an image plane CS682, Jana Kosecka CS682, Jana Kosecka 6 Image of a line – 2D representations Visual Illusions Representation of a 3-D line Projection of a line - line in the image plane Special cases – parallel to the image plane, perpendicular When λ -> ∞ - vanishing points In art – 1-point perspective, 2-point perspective, 3-point perspective CS682, Jana Kosecka CS682, Jana Kosecka Vanishing points Ames Room Illusions Different sets of parallel lines in a plane intersect at vanishing points, vanishing points form a horizon line CS682, Jana Kosecka CS682, Jana Kosecka 7 More Illusions Which of the two monsters is bigger ? CS682, Jana Kosecka 8.