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245 
FAST PHOTOMETRIC STEREO: REGULARIZATION VS. WIENER 
FILTERING 
Martin Bichsel 
University of Zurich 
Department of Computer Science 
MultiMedia Laboratory 
Winterthurerstrasse 190 
CH-8057 Zurich 
Switzerland 
Tel: +41-1-257 43 58, Fax: +41-1-363 00 35, E-mail: mbichsel@ifi.unizh.ch 
KEY WORDS: Photometric stereo, Wiener filtering, regularization 
ABSTRACT: Photometric stereo produces estimates of the local surface derivatives which must be integrated 
to a consistent surface. Traditionally, this integration was addressed as a regularization problem which was solved 
iteratively on a discrete grid in the spatial domain. In this paper, a Wiener filtering approach is presented and 
compared to the regularization approach. For Wiener filtering and regularization, a fast algorithm is derived 
in the Fourier domain, avoiding time consuming iterative procedures. This results in fast algorithms with a 
computational cost that is dominated by the Fast Fourier Transform. 
1 INTRODUCTION 
Photometric stereo (Woodham, 1980), (Ikeuchi, 1981) is an attractive method for shape reconstruction since it 
measures gradient data directly, handles albedo changes, does not require expensive hardware, and since the 
required image acquisition can be very fast (a few frames of the video camera) (Saji, 1992). 
In photometric stereo, images of a fixed object are taken under various illumination conditions. Assuming 
a known reflectance map for each illumination condition, each image imposes a constraint on the normal 
and reflectance coefficient at every image point. For the entire class of m-lobed reflectance maps (including. 
Lambertian surfaces, Phong model, and Torrance-Sparrow Model) three lights are sufficient to yield a unique 
surface normal at each image point as long as no self-shadowing occurs (Tagare, 1991). The measured images 
can be combined to local surface normals or, equivalently, to x- and y-derivatives of the depth map in a highly 
efficient way by using lookup tables (Nakamura, 1991). Therefore, this part of the problem will not be discussed 
in the present paper. 
Due to noise, unfortunately, the estimated x- and y-derivatives of the depth map are, in general, not directly 
integrable into a depth map because the integral over a closed path may be non-zero. Traditionally, this problem 
was solved by applying regularization theory. In this paper, Wiener filtering is presented as an alternative to 
the regularization approach. The connection between the two approaches is discussed and it is shown that the 
traditional regularization model corresponds to Wiener filtering with a particular noise model. 
Both, Wiener filtering and regularization lead to a matrix equation if either method is applied to data points 
on a discrete grid. Traditionally, this equation was solved iteratively, requiring 100-1000 iterations (Nakamura, 
1991). In this paper it is shown that both approaches have a very efficient solution in the Fourier domain. 
2 FROM DERIVATIVES TO DEPTH MAPS 
Let us assume that photometric stereo has given us the derivatives p(z,y) — fz(z,y) and q(z, y) = fy(z,y) 
where p and q are the x- and y-slopes of the surface height f(x, y) where a small height corresponds to a large 
distance to the camera. The measured images can be converted into the derivative images in a highly efficient 
way by using lookup tables (Nakamura, 1991). Therefore, this conversion is not discussed in the present paper. 
In practice, the derivative data is known on a discrete and regular grid of size L, by L,. After discretization 
we end up with the matrix equations 
p = D.f +n, (1) 
q = D,f + n, (2) 
IAPRS, Vol. 30, Part 5SW1, ISPRS Intercommission Workshop “From Pixels to Sequences”, Zurich, March 22-24 1995