HOME

TheInfoList



OR:

The pinhole camera model describes the mathematical relationship between the
coordinate In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
s of a point in three-dimensional space and its projection onto the image plane of an ''ideal''
pinhole camera A pinhole camera is a simple camera without a lens but with a tiny aperture (the so-called '' pinhole'')—effectively a light-proof box with a small hole in one side. Light from a scene passes through the aperture and projects an inverted image ...
, where the camera aperture is described as a point and no lenses are used to focus light. The model does not include, for example, geometric distortions or blurring of unfocused objects caused by lenses and finite sized apertures. It also does not take into account that most practical cameras have only discrete image coordinates. This means that the pinhole camera model can only be used as a first order approximation of the mapping from a
3D scene This is a glossary of terms relating to computer graphics. For more general computer hardware terms, see glossary of computer hardware terms. 0–9 A B ...
to a 2D
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
. Its validity depends on the quality of the camera and, in general, decreases from the center of the image to the edges as lens distortion effects increase. Some of the effects that the pinhole camera model does not take into account can be compensated, for example by applying suitable coordinate transformations on the image coordinates; other effects are sufficiently small to be neglected if a high quality camera is used. This means that the pinhole camera model often can be used as a reasonable description of how a camera depicts a 3D scene, for example in
computer vision Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the human ...
and computer graphics.


Geometry

The geometry related to the mapping of a pinhole camera is illustrated in the figure. The figure contains the following basic objects: * A 3D orthogonal coordinate system with its origin at O. This is also where the ''camera aperture'' is located. The three axes of the coordinate system are referred to as X1, X2, X3. Axis X3 is pointing in the viewing direction of the camera and is referred to as the '' optical axis'', ''principal axis'', or ''principal ray''. The plane which is spanned by axes X1 and X2 is the front side of the camera, or ''principal plane''. * An image plane, where the 3D world is projected through the aperture of the camera. The image plane is parallel to axes X1 and X2 and is located at distance f from the origin O in the negative direction of the X3 axis, where ''f'' is the focal length of the pinhole camera. A practical implementation of a pinhole camera implies that the image plane is located such that it intersects the X3 axis at coordinate ''-f'' where ''f > 0''. * A point R at the intersection of the optical axis and the image plane. This point is referred to as the ''principal point'' or ''image center''. * A point P somewhere in the world at coordinate (x_1, x_2, x_3) relative to the axes X1, X2, and X3. * The ''projection line'' of point P into the camera. This is the green line which passes through point P and the point O. * The projection of point P onto the image plane, denoted Q. This point is given by the intersection of the projection line (green) and the image plane. In any practical situation we can assume that x_3 > 0 which means that the intersection point is well defined. * There is also a 2D coordinate system in the image plane, with origin at R and with axes Y1 and Y2 which are parallel to X1 and X2, respectively. The coordinates of point Q relative to this coordinate system is (y_1, y_2) . The ''pinhole'' aperture of the camera, through which all projection lines must pass, is assumed to be infinitely small, a point. In the literature this point in 3D space is referred to as the ''optical (or lens or camera) center''.


Formulation

Next we want to understand how the coordinates (y_1, y_2) of point Q depend on the coordinates (x_1, x_2, x_3) of point P. This can be done with the help of the following figure which shows the same scene as the previous figure but now from above, looking down in the negative direction of the X2 axis. In this figure we see two
similar triangles In Euclidean geometry, two objects are similar if they have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly wi ...
, both having parts of the projection line (green) as their
hypotenuse In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equ ...
s. The catheti of the left triangle are -y_1 and ''f'' and the catheti of the right triangle are x_1 and x_3 . Since the two triangles are similar it follows that : \frac = \frac or y_1 = -\frac A similar investigation, looking in the negative direction of the X1 axis gives : \frac = \frac or y_2 = -\frac This can be summarized as : \begin y_1 \\ y_2 \end = -\frac \begin x_1 \\ x_2 \end which is an expression that describes the relation between the 3D coordinates (x_1,x_2,x_3) of point P and its image coordinates (y_1,y_2) given by point Q in the image plane.


Rotated image and the virtual image plane

The mapping from 3D to 2D coordinates described by a pinhole camera is a perspective projection followed by a 180° rotation in the image plane. This corresponds to how a real pinhole camera operates; the resulting image is rotated 180° and the relative size of projected objects depends on their distance to the focal point and the overall size of the image depends on the distance ''f'' between the image plane and the focal point. In order to produce an unrotated image, which is what we expect from a camera, there are two possibilities: * Rotate the coordinate system in the image plane 180° (in either direction). This is the way any practical implementation of a pinhole camera would solve the problem; for a photographic camera we rotate the image before looking at it, and for a digital camera we read out the pixels in such an order that it becomes rotated. * Place the image plane so that it intersects the X3 axis at ''f'' instead of at ''-f'' and rework the previous calculations. This would generate a ''virtual (or front) image plane'' which cannot be implemented in practice, but provides a theoretical camera which may be simpler to analyse than the real one. In both cases, the resulting mapping from 3D coordinates to 2D image coordinates is given by the expression above, but without the negation, thus : \begin y_1 \\ y_2 \end = \frac \begin x_1 \\ x_2 \end


In homogeneous coordinates

The mapping from 3D coordinates of points in space to 2D image coordinates can also be represented in homogeneous coordinates. 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 3 \times 4
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 homo ...
and the \, \sim means equality between elements of projective spaces. This implies that the left and right hand sides are equal up to a non-zero scalar multiplication. A consequence of this relation is that also \mathbf can be seen as an element of a projective space; two camera matrices are equivalent if they are equal up to a scalar multiplication. This description of the pinhole camera mapping, as a linear transformation \mathbf instead of as a fraction of two linear expressions, makes it possible to simplify many derivations of relations between 3D and 2D coordinates.


See also

*
Camera resectioning Camera resectioning is the process of estimating the parameters of a pinhole camera model approximating the camera that produced a given photograph or video; it determines which incoming light ray is associated with each pixel on the resulting im ...
*
Collinearity equation The collinearity equations are a set of two equations, used in photogrammetry and computer stereo vision, to relate coordinates in a sensor plane (in two dimensions) to object coordinates (in three dimensions). The equations originate from the ...
* Entrance pupil, the equivalent location of the pinhole in relation to object space in a real camera. * Exit pupil, the equivalent location of the pinhole in relation to the image plane in a real camera. * Ibn al-Haytham *
Pinhole camera A pinhole camera is a simple camera without a lens but with a tiny aperture (the so-called '' pinhole'')—effectively a light-proof box with a small hole in one side. Light from a scene passes through the aperture and projects an inverted image ...
, the practical implementation of the mathematical model described in this article. * Rectilinear lens


References


Bibliography

* * * * *{{cite book, author=Gang Xu and Zhengyou Zhang, title=Epipolar geometry in Stereo, Motion and Object Recognition, publisher=Kluwer Academic Publishers, year=1996, isbn=0-7923-4199-6, url = https://books.google.com/books?id=DnFaUidM-B0C&pg=PA7&dq=pinhole+intitle:%22Epipolar+geometry%22 Geometry in computer vision Cameras