Point Spread Function-Based Approaches to Three-Dimensional Scene Reconstruction - 3D Computer Vision: Efficient Methods and Applications - page 184

( 1989 ) proposes to move the examined object towards the camera in small depth
intervals z known at high precision. If a point on the surface appears best fo-
cused in image n , the corresponding distance to the camera must be z 0 , and the
index n directly determines the distance between the scene point and the known
initial depth.
At this point, Nayar ( 1989 ) discusses the quantification of image blur in order
to determine the sharpest image, and suggests the 'sum-modified Laplacian' (SML)
according to

+

u + N v + N

∂ 2 I
∂x 2

∂ 2 I
∂y 2

F uv = (4.22)

x = u − N y = v − N

with I as the pixel grey value, where a summand is set to zero when it is smaller
than a predefined threshold. The sum is computed over an image window of size
( 2 N

1 ) pixels centred around the pixel at position (u, v) . The SML
focus measure is compared to the grey value variance, the sum of the Laplace-
filtered image pixels, and the Tenengrad measure. Nayar ( 1989 ) shows that the
SML measure for most regarded material samples yields the steepest and most ac-
curately located maximum. The focus measurements are interpolated by fitting a
Gaussian function to the maximum SML value and the two neighbouring measure-
ments in order to obtain depth measurements which are more accurate than the
interval z .
Xiong and Shafer ( 1993 ) suggest an efficient search method for determining the
maximum of the focus measure based on a Fibonacci search, corresponding to the
acquisition of measurements in increasingly smaller depth intervals. It is pointed out
by Xiong and Shafer ( 1993 ) that this approach is optimal with respect to the required
number of measurements as long as the depth-dependent focus measure displays a
unimodal behaviour. In the presence of noise, however, the focus measure is not
strictly unimodal but may display a large number of local maxima. Hence, Xiong
and Shafer ( 1993 ) propose to terminate the Fibonacci search for a search interval
width below a given threshold and adapt a Gaussian interpolation function to the
measurements acquired in this interval.
An advantage of the depth from focus method is that it yields precise depth val-
ues. On the other hand, it requires the acquisition of a considerable number of im-
ages as well as a highly controlled environment, including accurate knowledge of
the intrinsic camera parameters and the relative displacement between the object
and the camera. Applications of the depth from focus method include the industrial
quality inspection of small parts (Schaper, 2002 ) and the detection of obstacles in
the context of mobile robot guidance (Nourbakhsh et al., 1997 ).

+ 1 ) × ( 2 N +