3-D D Euclidean Space - Vectors

Transcription

1 3-D D Euclidean Space - Vectors Rigid Body Motion and Image Formation A free vector is defined by a pair of points : Jana Kosecka Coordinates of the vector : 3D Rotation of Points Euler angles Rotation around the coordinate axes, counter-clockwise: clockwise: z Y y P X γ P x R x( α) = 0 cosα sinα 0 sinα cosα cos β 0 sin β R = y ( β ) sin β 0 cos β cosγ sinγ 0 R ( γ ) = z sinγ cosγ Rotation Matrices in 3D 3 by 3 matrices 9 parameters only three degrees of freedom Representations either three Euler angles or axis and angle represntation Properties of rotation matrices (constraints between the elements) 1

2 Rotation Matrices in 3D 3 by 3 matrices 9 parameters only three degrees of freedom Representations either three Euler angles or axis and angle representation Canonical Coordinates for Rotation Property of R Taking derivative Skew symmetric matrix property Properties of rotation matrices (constraints between the elements) By algebra By solution to ODE Columns are orthonormal 3D Rotation (axis & angle) Rotation Matrices Solution to the ODE Given How to compute angle and axis with or 2

3 3D Translation of Points Translate by a vector Rigid Body Motion Homogeneous Coordinates 3-D coordinates are related by: Homogeneous coordinates: P Y z z t x P x Homogeneous coordinates are related by: y Rigid Body Motion Homogeneous Coordinates Properties of Rigid Body Motions 3-D coordinates are related by: Rigid body motion composition Homogeneous coordinates: Rigid body motion inverse Homogeneous coordinates are related by: Rigid body motion acting on vectors th homogeneous coordinate is zero CS682, Jana Vectors are only affected by rotation 4Kosecka 3

4 Rigid Body Transformation Rigid Body Motion Camera is moving Notion of a twist Coordinates are related by: Relationship between velocities Camera pose is specified by: Image Formation Perspective Projection Pinhole Camera Model The Scholar of Athens, Raphael, 1518 Pinhole Frontal pinhole 4

5 More on homogeneous coordinates In homogenous coordinates these represent the Same point in 3D Pinhole Camera Model 2-D coordinates 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 Image Coordinates Calibration Matrix and Camera Model Pinhole camera Pixel coordinates metric coordinates Linear transformation Calibration matrix (intrinsic parameters) pixel coordinates Projection matrix Camera model 5

6 Calibration Matrix and Camera Model Pinhole camera Pixel coordinates Radial Distortion Nonlinear transformation along the radial direction More compactly Transformation between camera coordinate Systems and world coordinate system Distortion correction: make lines straight Image of a point Image of a line homogeneous 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 6

7 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 Vanishing points Ames Room Illusions Different sets of parallel lines in a plane intersect at vanishing points, vanishing points form a horizon line 7

8 More Illusions Which of the two monsters is bigger? 8