Point Spread Function-Based Approaches to Three-Dimensional Scene Reconstruction - 3D Computer Vision: Efficient Methods and Applications - page 176

Fig. 4.8 Relation between
the offset b =| b − b 0 | of
the image plane and the
radius c of the circle of
confusion

Although this corresponds to an absolute depth error of only 0 . 3 %, due to the rela-
tively small depth extension of the scene of about 4 mm, the extracted set of three-
dimensional points contains reasonably reliable information only about the average
surface gradient, not about higher order surface properties. It is shown in Sects. 5.3
and 5.6 that in combination with triangulation-based and intensity-based methods,
such depth from defocus measurements may nevertheless contribute useful infor-
mation to the three-dimensional reconstruction of a surface or an object, as pointed
out by d'Angelo and Wöhler ( 2005c ).

4.2.3 Determination of Absolute Depth Across Broad Ranges

According to Kuhl et al. ( 2006 ) and Wöhler et al. ( 2009 ), in the geometric optics
approximation a point in a scene is transformed into a circle of confusion of radius
c

= b/( 2 κ) (4.11)

in the image plane, where b and κ is the f-stop number expressing

the focal length in terms of the aperture diameter (cf. Fig. 4.8 ). Empirically, it is
found that for small

=| b − b 0 |

the resulting amount σ of defocus can be modelled by a
zero-mean Gaussian, which is symmetric in

| b |

| b | :

exp

(b) 2
φ 2

1
φ 1

σ (b) = − + φ 3 . (4.12)

The radius c of the circle of confusion and the PSF radius σ are related to each other
in that σ is a monotonically increasing nonlinear function of c . Hence, the symmet-
ric behaviour of c(b) apparent from Fig. 4.8 implies a symmetric behaviour of
σ (b) .
Depending on the constructional properties of the lens, analytic forms dif-
ferent from ( 4.12 ) but also symmetric in b may better represent the ob-
served behaviour of the PSF. For example, Krauß ( 2006 ) determines the PSF ra-
dius σ based on the adaptation of a function of sigmoidal shape of the form
I(u) σ) according to ( 1.90 ), t o an inten-

sity profile extracted orthogonal to an object boundary. Setting σ

= a tanh [ ζ(u − u 0 ) ]+ b , where ζ = 1 /( 2 ˜

σ/ √ π/ 8 yields

=˜