Page 143 - Introduction to Autonomous Mobile Robots

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1
2
2
--------- if ( x – x ) + ( y – y ) ) ≤ R 2 Chapter 4
(
hx y x y R,( , , , ) = 2 g f g f (4.23)
,
g g f f xy πR
2
2
0 if (( x – x ) + ( y – y ) ) > R 2
g
f
g
f
Intuitively, point contributes to the image pixel x y,( g g ) only when the blur circle of
P
point contains the point x y,( g g ) . Now we can write the general formula that computes
P
the value of each pixel in the image, f x y,( ) , as a function of the point spread function and
the focused image:
,
,,,
gx y,( ) = ∑ hx y xyR )fxy) (4.24)
,
(
(
,
g g g g xy
,
xy
g
R
This equation relates the depth of scene points via to the observed image . Solving
R
for would provide us with the depth map. However, this function has another unknown,
and that is , the focused image. Therefore, one image alone is insufficient to solve the
f
depth recovery problem, assuming we do not know how the fully focused image would
look.
Given two images of the same scene, taken with varying camera geometry, in theory it
g
f
will be possible to solve for as well as because stays constant. There are a number
R
of algorithms for implementing such a solution accurately and quickly. The classic
approach is known as inverse filtering because it attempts to directly solve for R , then
extract depth information from this solution. One special case of the inverse filtering solu-
tion has been demonstrated with a real sensor. Suppose that the incoming light is split and
sent to two cameras, one with a large aperture and the other with a pinhole aperture [121].
The pinhole aperture results in a fully focused image, directly providing the value of . f
With this approach, there remains a single equation with a single unknown, and so the solu-
tion is straightforward. Pentland [121] has demonstrated such a sensor, with several meters
of range and better than 97% accuracy. Note, however, that the pinhole aperture necessi-
tates a large amount of incoming light, and that furthermore the actual image intensities
must be normalized so that the pinhole and large-diameter images have equivalent total
radiosity. More recent depth from defocus methods use statistical techniques and charac-
terization of the problem as a set of linear equations [64]. These matrix-based methods have
recently achieved significant improvements in accuracy over all previous work.
In summary, the basic advantage of the depth from defocus method is its extremely fast
speed. The equations above do not require search algorithms to find the solution, as would
the correlation problem faced by depth from stereo methods. Perhaps more importantly, the
depth from defocus methods also need not capture the scene at different perspectives, and
are therefore unaffected by occlusions and the disappearance of objects in a second view.