Integrated Frameworks for Three-Dimensional Scene Reconstruction - 3D Computer Vision: Efficient Methods and Applications - page 205

shading image. In general, the resulting depths of the ridges casting the shadows
are not identical any more—they are extracted from the reconstructed surface
profile z (m)

˜

. This yields a new profile z m + 1 (u, v) for the surface patch between

uv

the shadow lines.
4. The iteration index m is incremented: m 1.

5. Steps 2, 3, and 4 are repeated until the criterion

← m +

(z (m) z (m − 1 )

1 / 2
u,v <Θ z is

) 2

−

uv uv

0 . 01 pixel is used.
This iterative algorithm mutually adjusts in a self-consistent manner the depth pro-
files of the floor and the ridges that cast the shadows. It allows one to determine not
only the surface gradients p uv in the direction of the incident light, which can be
achieved by shape from shading alone, but to estimate the surface gradients q uv in
the perpendicular direction as well. It is especially suited for surface reconstruction
under coplanar light sources. Furthermore, the algorithm does not require any spe-
cial form of the shape from shading algorithm or the reflectance function used; thus
it can be extended in a straightforward manner to more than two shadows and to
shape from shading algorithms based on multiple light sources.

fulfilled. Again, the threshold value Θ z =

5.2.4 Experimental Evaluation Based on Synthetic Data

To demonstrate the accuracy of the surface reconstruction techniques presented in
the previous section, they are applied to synthetic data for which the ground truth is
known. A Lambertian reflectance function is assumed. In all examples of Fig. 5.14
we made use of the smooth surface constraint ( 3.20 ). In Figs. 5.14 a-c, the true
surface profile is shown to the left along with the synthetically generated images
used for reconstruction, the reconstructed profile in the middle, and the deviation
(z − z true ) between reconstructed and true depth to the right, respectively. The direc-
tions of illumination for the shading images are indicated by arrows. Figure 5.14 a
illustrates the result of the method described in Sect. 5.2.2 based on one shading
and one shadow image. Consequently, the surface gradients in the vertical image
direction are slightly under-estimated. In Fig. 5.14 b, two shading images and one
shadow image are employed, and a uniform surface albedo is used. This yields a
very accurate reconstruction result. The same illumination setting is assumed in
Fig. 5.14 c, but the albedo is strongly non-uniform. Here, the algorithm described in
Sect. 3.3.2 based on the ratio of the images is employed, yielding a less accurate but
still reasonable reconstruction result. The computed albedo map is shown next to the
reconstructed surface profile. Figure 5.14 d illustrates the performance of the algo-
rithm proposed in Sect. 5.2.3 on a synthetically generated object (left). In contrast to
traditional shape from shading, the surface gradients perpendicular to the direction
of incident light are revealed (middle). The single-image error term ( 3.19 ) was then
replaced by the ratio-based error term ( 3.50 ) for a reconstruction of the same syn-
thetic object but now with a non-uniform albedo (right). Consequently, two shading
images are used in combination with the shadow information. As a result, a similar
surface profile is obtained along with the surface albedo. Refer to Table 5.2 for a
detailed comparison between ground truth and reconstruction results in terms of the
root-mean-square error (RMSE).