3D Computer Vision II. Reminder Camera Models
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1 3D Computer Vision II Reminder Camera Models Nassir Navab" based on a course given at UNC by Marc Pollefeys & the book Multiple View Geometry by Hartley & Zisserman" October 27, 2009"
2 Outline Camera Models" Geometric Parameters of a Finite Camera" Projective Camera Model" Camera Center, Principal Plane, Axis Plane, Principal Point, Principal Ray" Decomposition of Camera Matrix" Cameras at Infinity" Other Camera Models (Pushbroom and Line Cameras)" 2"
3 Pinhole Camera Model" Mapping between 3D world and 2D image" Central projection" Models are described in matrices with particular properties" 3"
4 Homogeneous Coordinates" 4"
5 Central Projection" 5"
6 Principal Point Offset" principal point (perpendicular intersection point of principal axis and image plane) where are the coordinates of the principal point 6"
7 Principal Point Offset" where is called camera calibration matrix 7"
8 Camera Rotation and Translation" Inhomogeneous coordinates where represents the point in world coordinates represents the same point in camera coordinates represents the coordinates of the camera origin in the world coordinate frame 8"
9 Camera Rotation and Translation" Homogeneous coordinates projection to image plane from camera coordinates projection to image plane from world coordinates 9"
10 Extrinsic and Intrinsic Parameters" where projection matrix of a general pinhole camera with 9 DOF intrinsic camera parameters with 3 DOF extrinsic camera parameters with each 3 DOF (camera orientation, position in world coordinates) 10"
11 Camera Rotation and Translation" No explicit camera center where from 11"
12 CCD Cameras Non-Square Pixels" number of pixels per unit distance 4 DOF 10 DOF 12"
13 Skew Parameter" skew parameter 5 DOF finite projective camera with 11 DOF 13"
14 Finite Projective Camera Summary" projection matrix 11 DOF (5+3+3) non-singular 14"
15 Finite Projective Camera Decomposition of P" non-singular 3x3 matrix (8 DOF) decompose projection matrix P in K,R,C QR matrix decomposition 15"
16 Finite Projective Camera Summary" where Camera matrices P are identical with the set of homogeneous 3x4 matrices for which the left 3x3 sub-matrix is non-singular" If rank(p) = 3, but rank(m) < 3, then camera at infinity" if rank(p) < 3 the matrix mapping will be a line or a point and not a plane (not a 2D image) 16"
17 Outline Camera Models" Geometric Parameters of a Finite Camera" Projective Camera Model" Camera Center, Principal Plane, Axis Plane, Principal Point, Principal Ray" Decomposition of Camera Matrix" Cameras at Infinity" Other Camera Models (Pushbroom and Line Cameras)" 17"
18 Camera Anatomy" Camera center" Column vectors" Principal plane" Axis plane" Principal point" Principal ray" 18"
19 Camera Center" P has a 1D null-space we will prove that the 4-vector C is the camera center points on a line through A and C since All 3D points on the line are mapped on the same 2D image point, and thus the line is a ray through the camera center Finite cameras: Infinite cameras: 19"
20 Column Vectors" Column vectors are the image points which project the axis directions (X,Y,Z) and the origin Example for the image of the y-axis is the image of the world origin 20"
21 Row Vectors" Represent geometrically particular world planes. row vectors column vectors 21"
22 Row Vectors of the Projection Matrix" p 1 is defined by the camera center and the line x=0 on the image. p 2 is defined by the camera center and the line y=0 on the image. Example p 2 respectively for p 1 22"
23 Principal Plane" Plane through camera center and parallel to the image plane. if X is on the principle plane points X are imaged on the line at infinity especially 23"
24 Principal Point" The line through camera center and perpendicular to principal plane is the principal axis. The intersection of the principal axis with the image plane is the principal point. principal point normal direction to the principal plane where and third row of M
25 Principal Axis Vector" Ambiguity that principal axis points towards the front of the camera (positive direction) towards the front of the camera direction unaffected by scaling since 25"
26 Forward Projection" Maps a point in space on the image plane Vanishing points Only M affects the projection of vanishing points 26"
27 Back-Projection to Rays" Points on the reconstructed ray camera center C (pseudo-inverse) Ray is the line formed by those two points intersection of the ray with the plane at infinity 27"
28 Depth of Points" (PC=0) If, w can be interpreted as the dot product of the ray CX with the principal ray direction then m 3 is a unit vector pointing in positive axis direction Suppose. Then 28"
29 Depth of Points: Examples" 29"
30 Outline Camera Models" Geometric Parameters of a Finite Camera" Projective Camera Model" Camera Center, Principal Plane, Axis Plane, Principal Point, Principal Ray" Decomposition of Camera Matrix" Cameras at Infinity" Other Camera Models (Pushbroom and Line Cameras)" 30"
31 Camera Matrix Decomposition" Finding the camera center C numerically: find right null-space by SVD of P Algebraically: where 31"
32 Camera Matrix Decomposition" Finding the camera center C Any plane π going through C will be a linear combination of the three planes defined by the rows of P. Therefore: where 32"
33 Camera Matrix Decomposition" Finding the camera orientation and internal parameters Decompose using RQ decomposition =( Q R ) -1 = R -1-1 Q Ambiguity removed by enforcing positive diagonal entries 33"
34 When is Skew Non-zero?" arctan(1/s) 1 γ for CCD/CMOS, always s=0 Image from image, s 0 possible (non coinciding principal axis) resulting camera: where H is a 3x3 homography 34"
35 Euclidean vs. Projective Spaces" General projective interpretation Meaningful decomposition in K,R,t requires Euclidean image and space Camera center is still valid in projective space Principal plane requires affine image and space Principal ray requires affine image and Euclidean space 35"
36 Outline Camera Models" Geometric Parameters of a Finite Camera" Projective Camera Model" Camera Center, Principal Plane, Axis Plane, Principal Point, Principal Ray" Decomposition of Camera Matrix" Cameras at Infinity" Other Camera Models (Pushbroom and Line Cameras)" 36"
37 Cameras at Infinity" Cameras with their center lying at infinity M is singular Two types of cameras at infinity: Affine and non-affine cameras 37"
38 Affine Cameras" Definition: An affine camera is a camera with a camera matrix P in which the last row p 3T is of the form (0,0,0,1) T. Points at infinity are mapped to points at infinity 38"
39 Affine Cameras" 39"
40 Affine Cameras" distance of the world origin from the camera center in direction of the principal ray modifying p 34 corresponds to moving along principal ray 40"
41 Affine Cameras" Combine tracking back and zooming magnification factor k=d t /d 0 remains fixed 41"
42 Error in Employing Affine Cameras" point on plane parallel with principal plane and through origin, then general points (not on the parallel plane) with distance from the plane 42"
43 Error in Employing Affine Cameras"
44 Affine Imaging Conditions" Approximation should only cause small error 1. Δ much smaller than d 0 2. points close to principal ray (i.e. small field of view) 44"
45 Decomposition of P " absorb d 0 in K 2x2 alternatives, because 8dof (3+3+2), not more 45"
46 Summary of Parallel Projections" canonical representation calibration matrix principal point is not defined 46"
47 Hierarchy of Affine Cameras" dropping the z-coordinate orthographic projection (5dof) 47"
48 Hierarchy of Affine Cameras" scaled orthographic projection (6dof) 48"
49 Hierarchy of Affine Cameras" weak perspective projection (7dof) 49"
50 Hierarchy of Affine Cameras" Affine camera (8dof) full generality of an affine camera Affine camera is a projective camera with principal plane at infinity Affine camera maps parallel world lines to parallel image lines No center of projection, but direction of projection P A D=0 50"
51 General Camera at Infinity" M is singular, but last row not zero Camera center is on plane at infinity Principal plane is not plane at infinity Images of points at infinity are in general not mapped to infinity on the image plane
52 Outline Camera Models" Geometric Parameters of a Finite Camera" Projective Camera Model" Camera Center, Principal Plane, Axis Plane, Principal Point, Principal Ray" Decomposition of Camera Matrix" Cameras at Infinity" Other Camera Models (Pushbroom and Line Cameras)" 52"
53 Pushbroom Cameras" (11dof) Straight lines are not mapped to straight lines! (otherwise it would be a projective camera) 53"
54 Line Cameras" (5dof) Null-space PC=0 yields camera center Also decomposition 54"
55 Summary Camera Models" Photometric and radiometric properties of a camera" Geometric parameters of a finite camera" Projective cameras" Camera anatomy (camera center, principle plane, principle point, and principle axis)" Camera matrix decomposition (camera center, orientation, and intrinsic parameter" Cameras at infinity" Affine cameras" Non-affine cameras" Alternative models (pushbroom cameras, line cameras)" 55"
56 Literature on Camera Models" Chapter 6 in R. Hartley and A. Zisserman, Multiple View Geometry, 2 nd edition, Cambridge University Press, " Chapter 3 in O. Faugeras, Three-dimensional Computer Vision, MIT Press, 1993." Chapter 2 in E. Trucco and A. Verri, Introductory Techniques for 3-D Computer Vision, Prentice Hall, 1998." H. Gernsheim, The Origins of Photography, Thames and Hudson, 1982." A. Shashua. Geometry and Photometry in 3D Visual Recognition, Ph.D. Thesis, MIT, Nov AITR " 56"
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