Point Spread Function-Based Approaches to Three-Dimensional Scene Reconstruction - 3D Computer Vision: Efficient Methods and Applications - page 172

Fig. 4.4 Two intensity
profiles, extracted orthogonal
to a slightly defocused ( solid
curve ) and a strongly
defocused ( dashed curve )
object boundary, respectively

gradient, and σ as the PSF radius in pixel units (cf. Sect. 1.4.8 ). The error function
erf (t) =

√ π 0 e − s 2 ds is the step response following from the Gaussian PSF. The
edge is steep, i.e. well focused, for σ →

2

0, while the amount of defocus increases

for increasing σ .
Subbarao and Wei ( 1992 ) introduce a computationally efficient method to de-
termine the width of the PSF based on the evaluation of one-dimensional intensity
profiles rather than two-dimensional image regions. The one-dimensional profiles
are obtained by summing the pixel rows. The fast Fourier transform algorithm used
to compute the amplitude spectrum assumes that the image continues periodically at
its borders, which may introduce spurious high spatial frequency components when
the grey values at the borders are not equal. To circumvent this problem, Subbarao
and Wei ( 1992 ) perform a pixel-wise multiplication of the grey values of the image
with a two-dimensional Gaussian weighting function such that the pixel grey values
decrease from the centre of the image towards its borders.
Another computationally efficient scheme to determine the PSF difference be-
tween two images, which is based on the inverse S-transform, is introduced by Sub-
barao and Surya ( 1994 ), involving local spatial filtering operations rather than a
two-dimensional Fourier transform. Their approach allows PSFs of arbitrary, non-
Gaussian shape by introducing a generalisation of the parameter σ as the square
root of the second central moment of the arbitrarily shaped PSF.
Watanabe et al. ( 1996 ) present a real-time system for estimating depth from de-
focus which relies on a specially designed telecentric lens which generates two dif-
ferently focused images using a prism element. A light pattern is projected into the
scene being analysed such that depth values can also be obtained for uniform sur-
faces. The 'pillbox' function, which corresponds to a blur circle of uniform intensity,
is used as a PSF instead of a Gaussian function.
Chaudhuri and Rajagopalan ( 1999 ) introduce the 'block shift-variant blur
model', which takes into account the mutual influences of the PSF widths of
neighbouring image regions. Furthermore, a variational approach which enforces a
smooth behaviour of the PSF width is introduced. Based on a maximum likelihood
approach, a criterion for the optimal relative blurring of the two regarded images
is inferred. These frameworks are extended by Chaudhuri and Rajagopalan ( 1999 )
to a depth estimation based on several images of the scene acquired at different