Perspective Projection
Orthographic and Affine Projections
Perspective Projection
Orthographic and Affine Projections
Perspective Projection
Figure 6, below, illustrates an ideal pinhole at O, at a fixed distance in front of
an image plane. Assuming that the pinhole is made within an enclosure, only light travelling through the pinhole may reach the image plane.
As light travels in straight lines, each image point corresponds to a particular directional ray passing through the pinhole. This is the familiar
perspective projection. the optical axis lies along the line OZ, which is perpendicular to the XY plane.
To compute the position of an image point, P', in the 2D plane, which arises from a point, P, in 3D space, we can define vectors,
Here
where
and so,
In component form, and by similar triangles, this can be written as,
Of course, a pinhole camera has in theory an inifinitesimally small hole, and therefore gathers no light, which makes it slightly impractical for a computer vision system.
Click here to see figure 6: Pinhole perspective and orthographic projection
An ideal lens produces the perspective projection (with several types of distortion), but also gathers light. As in the pinhole case, the ray through the centre of the lens illustrated in Figure 6, above, is undeflected, but the other rays are focussed to the same image point as the central ray. An ideal lens has the disadvantage that it only focusses light from points at a specific distance, -z, given by the lens equation,
Other points are blurred to some extent. The range of acceptably focussed points is a function of the lens focal length
and aperture, and is commonly referred to as the depth of field.
The field of view (FOV) of an imaging system is the angle of the cone of directions encompassed by the scene being imaged and can be envisaged by connecting the edges of the image plane to the centre of projection. Typically a normal lens has a FOV of 40-50 degrees, and a wide angle lens a FOV much greater than this.
A telephoto lens has a restricted FOV of less than 40 degrees, and tends to approximate to orthographic projection.
The orthographic projection of Figure 6 can be modelled by rays which travel parallel to the optic axis, as opposed to rays passing
through the origin. For example, if we form the image of a plane at distance
where
where
Comments to: Sarah Price at ICBL.
and 
connecting the origin, O, to P and P' respectively, where
is the distance from the image plane to the pinhole, and
and 
are the coordinates of the point P' in the image plane.
Clearly, 
and 
are collinear and differ only by a (negative) scale factor. Assuming the ray from P to P' makes an angle, 
, with the optical axis, the z-axis in this case, then the length of 
is given by
is the unit vector along the optical axis.
Similarly
Orthographic and Affine Projections
from the
camera, then we can define m, the lateral magnification, as the ratio of the distance between two points in the image to two points in the image plane. Thus,
is the distance of the plane from the optical origin
and x and y, and 
and 
,
refer to the separation of two points in x and y in the world and image planes respectively. In this case, the projection equations read simply,
and 
is the average value of -z.
When the lateral magnification is 1 (or constant), no matter the distance of the object from the lens, this is true orthographic projection.
If the magnification varies as the object to lens distance varies ( similar to perspective, but retaining the parallel projection), then this is termed
affine projection.
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(Last update: 4th July, 1996)