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c) 
Figure 1: Reconstruction of a pyramid from derivative images. 
(a) Reconstruction from noise-free data, using Wiener filtering with c — 107? and p = —2. 
(b) Reconstruction from noisy data, using Wiener filtering with c— 1 and p — —2. 
(c) Reconstruction from noisy data, using Wiener filtering with c — 107? and p = —2. 
(d) Reconstruction from noisy data, using regularization with A;=0.2, 45—0. 
of the noise characteristics. In fact, this type of regularization just results in an overall scaling by 1/(1+ Ay) as 
can be seen in Eq. 8. Ignored noise can, however, be compensated by a stabilizer of the form fZ, + 212, + Foy 
by using A; = c¢ as it was discussed in Sec. 2. 
4.3 Reconstruction from real images 
In a third experiment, the surface reconstruction was tested on real images of a dummy which were taken with 
a Sony 77CE video camera and a Sorac frame-store. The non-linear response curve of the camera and frame- 
store system was corrected by a lookup table (Bichsel, 1993). Three images were taken with light positions 
(-0.86m, 0.80m, 2.66m), (0.87m, 0.80m, 2.66m), and (0.0m, -0.92m, 2.66m), where light positions are measured 
in a camera centered coordinate system. Local surface normals were estimated with a least squares fit to a 
simplified Lambertian reflectance law. The normals were then converted to local surface derivatives p(z, y) and 
q(z, y) by using p(z, y) 2 —ni(z, y)/na(z, y) and a(z, y) = —n2(z, y)/ns(z, y). 
The original images are shown in Fig. 2(a-c). The surface was reconstructed using Wiener filtering with 
SNR parameters c = 1 and p — —2. This corresponds to regularization with A;=0 and \,=1. Using an SNR 
exponent p — —2 implies that the worst expected type of singularities in the desired depth map are edges in the 
derivatives and that the noise is white. The recovered height map is shown in Fig. 2(d). A shaded frontal view 
is shown in Fig. 2(e) and a shaded tilted view is shown in Fig. 2(f) where the tilt angle is 45 degrees. These 
figures show a good reconstruction quality for real images. 
5 CONCLUSIONS 
In this paper, a fast solution to photometric stereo was presented. The computational expense of this solution 
is dominated by four 1D DCTs and two 1D DSTs and is therefore on the order of L, Ly(l0g(Lz) + log(Ly)) 
where L, and L, are the image dimensions. 
A comparison between Wiener filtering and regularization led to the conclusion that the two are equivalent 
for a particular, reasonable SNR model and a particular choice of the regularization parameters. The Wiener 
filtering approach forces the user to make an explicit statistical model of signal and noise and returns the 
optimum filter for the specific model. The regularization approach, on the other hand, only makes implicit 
assumptions about statistical properties of signal and noise. The user is commonly unaware of these assumptions 
IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop “From Pixels to Sequences”, Zurich, March 22-24 1995