Page 139 - Introduction to Autonomous Mobile Robots

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1 1 1 Chapter 4
--- = --- + --- (4.19)
f d e
If the image plane is located at distance from the lens, then for the specific object
e
voxel depicted, all light will be focused at a single point on the image plane and the object
e
voxel will be focused. However, when the image plane is not at , as is depicted in figure
4.19, then the light from the object voxel will be cast on the image plane as a blur circle.
To a first approximation, the light is homogeneously distributed throughout this blur circle,
R
and the radius of the circle can be characterized according to the equation

Lδ
R = ------ (4.20)
2e
δ
L is the diameter of the lens or aperture and is the displacement of the image plan
from the focal point.
Given these formulas, several basic optical effects are clear. For example, if the aperture
or lens is reduced to a point, as in a pinhole camera, then the radius of the blur circle
approaches zero. This is consistent with the fact that decreasing the iris aperture opening
causes the depth of field to increase until all objects are in focus. Of course, the disadvan-
tage of doing so is that we are allowing less light to form the image on the image plane and
so this is practical only in bright circumstances.
The second property that can be deduced from these optics equations relates to the sen-
sitivity of blurring as a function of the distance from the lens to the object. Suppose the
image plane is at a fixed distance 1.2 from a lens with diameter L = 0.2 and focal length
R
f = 0.5 . We can see from equation (4.20) that the size of the blur circle changes pro-
δ
portionally with the image plane displacement . If the object is at distance d = 1 , then
δ
from equation (4.19) we can compute e = 1 and therefore = 0.2. Increase the object dis-
δ
tance to d = 2 and as a result = 0.533. Using equation (4.20) in each case we can com-
pute R = 0.02 and R = 0.08 respectively. This demonstrates high sensitivity for
defocusing when the object is close to the lens.
In contrast, suppose the object is at d = 10 . In this case we computee = 0.526 . But if
the object is again moved one unit, to d = 11 , then we compute e = 0.524 . The resulting
blur circles are R = 0.117 and R = 0.129 , far less than the quadrupling in when the
R
obstacle is one-tenth the distance from the lens. This analysis demonstrates the fundamental
limitation of depth from focus techniques: they lose sensitivity as objects move farther
away (given a fixed focal length). Interestingly, this limitation will turn out to apply to vir-
tually all visual ranging techniques, including depth from stereo and depth from motion.
Nevertheless, camera optics can be customized for the depth range of the intended appli-
f
cation. For example, a zoom lens with a very large focal length will enable range resolu-