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focus settings. Furthermore, a simulated annealing-based approach for estimating
the PSF width based on a focused and a blurred image is developed, and a method
for a coupled estimation of the PSF width and the pixel grey values of the focused
image is proposed. The mathematical frameworks established by Chaudhuri and
Rajagopalan ( 1999 ) are especially useful in the presence of a strongly non-uniform
PSF or considerable depth discontinuities.
In order to increase the depth accuracy at the borders of objects in the scene,
where the depth map is discontinuous, McCloskey et al. ( 2007 ) suggest to estimate
the amount of image blur based on elliptical regions oriented orthogonal to the di-
rection of the maximum variation of the depth.
Namboodiri et al. ( 2008 ) propose a regularisation approach to the estimation of
dense depth maps that also applies for scene parts in which no surface structure is
present. Based on the framework outlined by Chaudhuri and Rajagopalan ( 1999 ), a
Markov random field method is developed which allows one to determine the max-
imum a posteriori solution for the depth map using graph cut-based optimisation.
Section 4.2.2 describes to which extent relatively small depth differences across
surfaces can be recovered with the depth from defocus method under realistic cir-
cumstances. Section 4.2.3 introduces a framework for the determination of absolute
depth values across broad ranges, introducing an appropriate empirical calibration
approach.
4.2.2 Determination of Small Depth Differences
For clarity, in this section the radius Σ in frequency space of the Gaussian modu-
lation transfer function (MTF) of the optical system, which results from the Fourier
transform of the Gaussian PSF, is utilised as a measure for the image blur. The pre-
sentation in this section is adopted from d'Angelo and Wöhler ( 2005c , 2008 ). The
observed image blur decreases with increasing values of Σ , and it is Σ
1 , such
→∞
=
that a perfectly sharp image is characterised by Σ
0.
The calibration procedure for estimating depth from defocus then involves the
determination of the lens-specific characteristic curve Σ(z
, corresponding to σ
z 0 ) . For this purpose
two pixel-synchronous images of a rough, uniformly textured plane surface con-
sisting of forged iron are acquired, inclined by 45 with respect to the optical axis.
The image part in which the intensity displays maximum standard deviation (i.e.
most pronounced high spatial frequencies) is sharp and thus situated at distance z 0 .
A given difference in pixel coordinates with respect to that image location directly
yields the corresponding depth offset (z z 0 ) . The first image I 1 is taken with a
small aperture, i.e. f/ 8, resulting in virtually absent image blur, while the second
image I 2 is taken with a large aperture, i.e. f/ 2, resulting in a perceivable image
blur that depends on the depth offset (z
z 0 ) .
The images are partitioned into quadratic windows, for each of which the av-
erage depth offset (z
z 0 ) is known. After Tukey windowing, the MTF radius Σ
in frequency space is computed based on the ratio
I 1 u v ) of the
amplitude spectra of the corresponding windows of the second and the first image,
I 2 u v )/
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