Camera Models A camera is a mapping between
Camera Models A camera is a mapping between the 3 D world and a 2 D image The principal camera of interest is central projection
Central Projection • Cameras modeling Central projection are specialization of the general projective camera. It is examined using the tools of projective geometry. • Specialized models fall into two classes: • (1) camera with a finite centre. (2) cameras with centre at infinity Affine camera is an important example
Finite cameras • The basic pin hole model(most specialized)
Pinhole camera • A point in space with coordinates • X =(x, y, z)T is mapped to a point on the image plane. • • (x, y, z)T ____ ( f x/z , f y/z)T (5. 1) The centre of projection is called the camera centre
Principal axis and principal point. • The line from the camera centre perpendicular to the image plane is called the principal axis and point where it intersects the image plane is called the principal point. • The plane parallel to the image plane and passing through the camera centre is called the principal plane
Central projection using homogenous coordinates • If the world and image points are represented by homogenous vectors, then • P = diag( f , f, 1) [I ! 0]
Image and camera coordinate systems
Principal point offset • The expression (5. 1) assumes that the origin of the coordinates in the image plane is the principal point. In practice, it may not be as follows. • (x, y, z)T ____ ( f x/z + px , f y/z + py )T • Where ( px, py) T are the coordinates of the principal point.
Principal point offset 2 • Now writing
Principal point offset 3 • Then (5. 3) has the concise form • x = K [ I ! 0] xcam (5. 5) • (x, y, z, 1) as Xcam as the camera is assumed to be located at the origin of a Euclidean coordinate system with the principal axis of the camera pointing straight down the z-axis. K is called the camera calibration matrix.
The Euclidean transformation between the world and camera coordinate frame
Camera rotation and translation • represents the coordinates of the camera centre in the world coordinate frame. is an inhomogeneous 3 -vector in world coordinate frame • R is 3 x 3 rotation axis •
Camera Rotation and translation • • Putting (5. 5) and (5. 6) together leads to •
Camera Rotation and translation 2 • The parameter contained in K are called internal camera parameters • The parameter of R and which relate the orientation and position to a world frame are called the external parameters. A more convenient form of the camera matrix is • • P = K [ R ! t] (5. 8)
CCD Cameras • The CCD camera may have rectangular pixels, where unit distances in x and y directions are mx and my, then • x 0 = mx px , and y 0 = my py • ax = f mx , and ay = f my
Finite projective camera • S is the skew parameter • A camera is called a finite projective camera. It has 11 degree of freedom, same as a 3 x 4 matrix defined up to an arbitrary scale
General projective Camera
The projective camera A general projective camera P maps world point X to image points x according to x = PX
Camera centre
Camera centre 2
Column vectors • The columns of P are pi , i = 1, 2, 3, 4 • Then p 1 , p 2 , p 3 are the vanishing points of the world coordinate x , y, z axes respectively. • For example: x axis ahs direction D =(1, 0, 0, 0), which is imaged at p 1 = PD • The column p 4 is the image of the world origin
The three image points defined by the columns pi, i= 1, 2, 3 of the projection matrix are the vanishing points of the directions of the world axes
Row vectors
Principal plane
Axis planes
Summary of the properties of a projective camera P=[M ! p 4]
Summary of the properties of a projective camera 2
Principle point
Principle axis
Principle axis 2
Principle axis 3
Two of the three planes defined by the rows of the projection matrix
Action of a projective camera on points
Back projection of points to rays
Back projection of points to rays 2
Back projection of points to rays 3
Depth of points
Depth of points 2
Linear optics
Decomposition of the camera matrix
Finding camera orientation and internal parameters
Finding camera orientation and internal parameters 2
Example 5. 2 The camera matrix
Euclidean vs Projective space
Euclidean and affine interpretation
Cameras at infinity
Affine cameras
Increasing focal length from left to right
Affine camera 2
Affine camera 3
Affine camera 4
Focal length increases as the object distance between the camera increases
The image remains the same size, but perspective effects diminish
Perspective vs weak perspective projection
Orthographic projection
Orthographic projection 2
Scaled orthographic projection
Weak perspective projection
Affine camera
A general affine camera
A general affine camera 2
More properties of the affine camera
General cameras at infinity
Pushbroom camera
Pushbroom camera 2
Pushbroom camera 3
Pushbroom camera – mapping of line
Line camera
Line camera 2
Acquisition geometry of a pushbroom camera
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