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http://en.wikipedia.org/wiki/Pinhole_camera
Cameras using small apertures, and the human eye in bright light both act like a pinhole camera. The smaller the hole, the sharper the image, but the dimmer the projected image. Optimally, the size of the aperture, should be 1/100 or less of the distance between it and the screen.

http://en.wikipedia.org/wiki/Projection_matrix
a projection is a linear transformation P from a vector space to itself such that P2 = P. It leaves its image unchanged. Though abstract, this definition of "projection" formalizes and generalizes the idea of graphical projection.

http://en.wikipedia.org/wiki/Orthographic_projection
Increasing the focal length and distance of the camera to infinity in a perspective projection results in an orthographic projection. It is a form of parallel projection, where the view direction is orthogonal to the projection plane.

http://en.wikipedia.org/wiki/Homogeneous_coordinate
The homogeneous coordinates of a point of projective space of dimension n are usually written as (x : y : z : ... : w), a row vector of length n + 1, other than (0 : 0 : 0 : ... : 0). Two sets of coordinates that are proportional denote the same point of projective space: for any non-zero scalar c from the underlying field K, (cx : cy : cz : ... : cw) denotes the same point. Therefore this system of coordinates can be explained as follows: if the projective space is constructed from a vector space V of dimension n + 1, introduce coordinates in V by choosing a basis, and use these in P(V), the equivalence classes of proportional non-zero vectors in V.

http://en.wikipedia.org/wiki/Affine_transformation
In geometry, an affine transformation or affine map or an affinity (from the Latin, affinis, "connected with") between two vector spaces (strictly speaking, two affine spaces) consists of a linear transformation followed by a translation:
              x \mapsto A x+ b
In the finite-dimensional case each affine transformation is given by a matrix A and a vector b, satisfying certain properties described below.
Physically, an affine transformation is one that preserves

1. Collinearity between points, i.e., three points which lie on a line continue to be collinear after the transformation
2. Ratios of distances along a line, i.e., for distinct colinear points p1, p2, p3, the ratio | p2 − p1 | / | p3 − p2 | is preserved

In general, an affine transform is composed of zero or more linear transformations (rotation, scaling or shear) and a translation (or "shift"). Several linear transformations can be combined into a single matrix, thus the general formula given above is still applicable.

http://en.wikipedia.org/wiki/Projective_space
The idea of a projective space relates to perspective, more precisely to the way an eye or a camera projects a 3D scene to a 2D image. All points which lie on a projection line (i.e. a "line-of-sight"), intersecting with the focal point of the camera, are projected onto a common image point. In this case the vector space is R3 with the camera focal point at the origin and the projective space corresponds to the image points.
( For example, in the standard geometry for the plane two lines always intersect at a point except when the lines are parallel. In a projective representation of lines and points, however, such an intersection point exists even for parallel lines, and it can be computed in the same way as other intersection points. )

http://en.wikipedia.org/wiki/Projective_geometry
Projective geometry is the most general and least restrictive in the hierarchy of fundamental geometries, i.e. Euclidean - metric (similarity) - affine - projective. It is an intrinsically non-metrical geometry, whose facts are independent of any metric structure. Under the projective transformations, the incidence structure and the cross-ratio are preserved. It is a non-Euclidean geometry. In particular, it formalizes one of the central principles of perspective art: that parallel lines meet at infinity and therefore are to be drawn that way. In essence, a projective geometry may be thought of as an extension of Euclidean geometry in which the "direction" of each line is subsumed within the line as an extra "point", and in which a "horizon" of directions corresponding to coplanar lines is regarded as a "line". Thus, two parallel lines will meet on a horizon line in virtue of their possessing the same direction.

http://en.wikipedia.org/wiki/Camera_resectioning
When a camera is used, light from the environment is focused on an image plane and captured. This process reduces the dimensions of the data taken in by the camera from three to two (light from a 3D scene is stored on a 2D image). Each pixel on the image plane therefore corresponds to a shaft of light from the original scene. Camera resectioning determines which incoming light is associated with each pixel on the resulting image.

http://en.wikipedia.org/wiki/Camera_matrix
Since the camera matrix  is involved in the mapping between elements of two projective spaces, it too can be regarded as a projective element. This means that it has only 11 degrees of freedom since any multiplication by a non-zero scalar results in an equivalent camera matrix.

http://en.wikipedia.org/wiki/Euclidean_space
An n-dimensional space with notions of distance and angle that obey the Euclidean relationships is called an n-dimensional Euclidean space.
An essential property of a Euclidean space is its flatness. Other spaces exist in geometry that are not Euclidean.

http://en.wikipedia.org/wiki/Euclidean_transformation
The Euclidean group E(n) is a subgroup of the affine group for n dimensions, and in such a way as to respect the semidirect product structure of both groups.
1. by a pair (A, b), with A an n×n orthogonal matrix, and b a real column vector of size n; or
2. by a single square matrix of size n + 1, as explained for the affine group.
 

http://en.wikipedia.org/wiki/Orthogonal_matrix
Q^T Q = Q Q^T = I . \,\!


http://en.wikipedia.org/wiki/Singular_value_decomposition

M = U\Sigma V^*, \,\!

where U is an m-by-m unitary matrix over K, the matrix Σ is m-by-n diagonal matrix with nonnegative numbers on the diagonal, and V* denotes the conjugate transpose of V, an n-by-n unitary matrix over K. Such a factorization is called a singular-value decomposition of M.

  • The matrix V thus contains a set of orthonormal "input" or "analysing" basis vector directions for M
  • The matrix U contains a set of orthonormal "output" basis vector directions for M
  • The matrix Σ contains the singular values, which can be thought of as scalar "gain controls" by which each corresponding input is multiplied to give a corresponding output.

A common convention is to order the values Σi,i in non-increasing fashion. In this case, the diagonal matrix Σ is uniquely determined by M (though the matrices U and V are not).


http://en.wikipedia.org/wiki/Total_least_squares
total least squares = errors in variables = rigorous least squares = orthogonal regression

http://en.wikipedia.org/wiki/RQ_decomposition#Relation_to_RQ_decomposition


posted by maetel