1 Chapter 2: Geometric Camera Models Objective: Formulate the geometrical relationships between image and scene measurements Scene: a 3-D function, g(x,y,z) Image: a 2-D function, f(x,y)

2 Contents: (i) Homogeneous coordinates (ii) Geometric transformations (iii) Intrinsic and extrinsic camera parameters (iv) Affine projection models 2.1. Elements of Analytical Euclidean Geometry Coordinate Systems ○ Right-handed coordinate system

3 coordinates of point P : position vector of point P O: origin; : basis vectors

4 ◎ Homogeneous Coordinates Advantages: (a) Some nonlinear systems can be transformed into linear ones (b) Equations written in terms of homogeneous coordinates become more compact. (c) A transformation, comprising rotation, translation, scaling, and perspective projection, can be written in a single matrix

5 ○ Point: ○ Plane equation: or where

6 ○ Sphere equation: where

7 ○ Quadric surface equation: where

Coordinate System Changes and Rigid Transformations Two subjects: (a) Coordinate system changes (b) Rigid transformations Consider two coordinate systems, A and B

9 ○ Coordinate System Changes Position vectors: Coordinate transformation:(?)

10 。 Translation vector : the vector translates the origin of coordinate system A to that of system B ○ Rigid Transformations

11 。 Rotation matrix : the 3 by 3 matrix rotates coordinate system A to coincide with system B

12 The 1 st column of is formed by projecting onto The columns of form frame A described in terms of frame B is formed by projecting onto The rows of form frame B described in terms of frame A The 1 st row of

13 ＊ Properties: (a) (b) : unitary matrix (c) (d) orthonormal matrices ○ Rigid Transformation: A rigid transformation preserves: (1) the distance between two points (2) the angle between two vectors

14 be their corresponding points in frame B, i.e., Proof: Let be two points in frame A Then, (1) Distance preservation

15 (2) Angle preservation (Assignment)

16 ○ Matrices can be multiplied in blocks ○ ○ In homogeneous coordinates: -- (2.7) where then

Camera Parameters 。 Intrinsic parameters -- Relate the actual camera coordinate system to the idealized camera coordinate system (1) the focal length of the lens f (2) the size and shape of the pixels (3) the position of the principal point (4) the angle between the two image axes Idealized cameraActual camera

18 。 Extrinsic parameters -- Relate the idealized camera coordinate system to a real world coordinate system (1) translation and (2) rotation parameters 。 Camera calibration -- estimates the intrinsic and extrinsic parameters of a camera Idealized camera coordinate system Real world Coordinate system

Intrinsic parameters Start with ideal perspective projection equations : scale : skew : shift parameters

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24 ○ The relationship between the physical image frame and the normalized one

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31 (Only if) – If, A: nonsingular A can always be factorized into Q: orthonormal matrix L : right upper triangular matrix

32 Compared with L, K : right upper triangular matrices Q, : orthonomal matrices : vectors W is a perspective projection matrix

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Affine Projection Models Orthographic Projection Models -- Objects are far from the camera Parallel Projection Models -- Objects are far and lie off the optical axis of the camera

36 Paraperspective Projection Models -- Objects lie near the optical axis Weak Perspective Projection Models -- Objects lie on the optical axis and their reliefs are ignored

37 Consider object reliefs in weak perspective projection

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