Camera matrix について

Words near each other

Dictionary Lists

mini英和辞書

mini和英辞書

翻訳と辞書　辞書検索 [ 開発暫定版 ]

スポンサードリンク

Camera matrix ：ウィキペディア英語版

Camera matrix

In computer vision a camera matrix or (camera) projection matrix is a

3 \times 4

matrix which describes the mapping of a pinhole camera from 3D points in the world to 2D points in an image.
Let

\mathbf

be a representation of a 3D point in homogeneous coordinates (a 4-dimensional vector), and let

\mathbf

be a representation of the image of this point in the pinhole camera (a 3-dimensional vector). Then the following relation holds
:

\mathbf \sim \mathbf \, \mathbf

where

\mathbf

is the camera matrix and the

\, \sim

sign implies that the left and right hand sides are equal up to a non-zero scalar multiplication.
Since the camera matrix

\mathbf

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.
== Derivation ==
The mapping from the coordinates of a 3D point P to the 2D image coordinates of the point's projection onto the image plane, according to the pinhole camera model is given by
:

\begin y_1 \\ y_2 \end = \frac \begin x_1 \\ x_2 \end

where

(x_1, x_2, x_3)

are the 3D coordinates of P relative to a camera centered coordinate system,

(y_1, y_2)

are the resulting image coordinates, and ''f'' is the camera's focal length for which we assume ''f'' > 0. Furthermore, we also assume that ''x₃ > 0''.
To derive the camera matrix this expression is rewritten in terms of homogeneous coordinates. Instead of the 2D vector

(y_1,y_2)

we consider the projective element (a 3D vector)

\mathbf = (y_1,y_2,1)

and instead of equality we consider equality up to scaling by a non-zero number, denoted

\, \sim

. First, we write the homogeneous image coordinates as expressions in the usual 3D coordinates.
:

\begin y_1 \\ y_2 \\ 1 \end = \frac \begin x_1 \\ x_2 \\ \frac \end \sim \begin x_1 \\ x_2 \\ \frac \end

Finally, also the 3D coordinates are expressed in a homogeneous representation

\mathbf

and this is how the camera matrix appears:
:

\begin y_1 \\ y_2 \\ 1 \end \sim \begin 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \frac & 0 \end \, \begin x_1 \\ x_2 \\ x_3 \\ 1 \end

\mathbf \sim \mathbf \, \mathbf

where

\mathbf

is the camera matrix, which here is given by
:

\mathbf = \begin 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \frac & 0 \end

,
and the corresponding camera matrix now becomes
:

\mathbf = \begin 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \frac & 0 \end \sim \begin f & 0 & 0 & 0 \\ 0 & f & 0 & 0 \\ 0 & 0 & 1 & 0 \end

The last step is a consequence of

\mathbf

itself being a projective element.
The camera matrix derived here may appear trivial in the sense that it contains very few non-zero elements. This depends to a large extent on the particular coordinate systems which have been chosen for the 3D and 2D points. In practice, however, other forms of camera matrices are common, as will be shown below.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
■ウィキペディアで「Camera matrix」の詳細全文を読む

スポンサードリンク

翻訳と辞書 : 翻訳のためのインターネットリソース