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DIRECT COMPUTATION OF DEPTH FROM SHADING FOR
PERSPECTIVE PROJECTION
Kousuke Wakabayashi, Norio Tagawa and Kan Okubo
Graduate School of System Design, Tokyo Metropolitan University, Hino-shi, Tokyo, Japan
Keywords:
Depth Recovery, Direct Computation, Shape from Shading, Perspective Projection.
Abstract:
We present a method for recovering shape from shading in which the surface depth is directly computed.
The already proposed method solving the same problem assumes that images are captured under the parallel
projection, and hence, it can be correctly used only for the relative thin objects compared with the distance from
the camera. If this method is formally extended for the perspective projection completely, the complicated
calculations for differential are required. This gives rise to unstable recovery. In this study, we examine an
extension of this method so as to treat the perspective projection approximately. In order to keep the simplicity
of the original method, we propose the simple approximation of the derivative of the surface with respect to
the image coordinate.
1 INTRODUCTION
Various algorithms for shape from shading have
been enthusiastically studied, but most of them com-
pute the surface orientation rather than surface depth
(Brooks and Horn, 1985), (Szeliski, 1991), (Zhang
et al., 1999). Computing surface orientation gives
rise to two fundamental difficulties. First, the recov-
ering problem is under-constrained, i.e. for the each
point in an image, there is one observation but two un-
known. To solve this problem, additional constraints,
such as smoothness of the orientations, are required
to obtain a unique solution. Secondly, arbitrary two
functions p(x,y) and q(x,y) on an image will not gen-
erally correspond to the orientations of some continu-
ous and differential surface.
Horn (Horn, 1990) developed a method which
considered solving for three functions simultane-
ously: a surface function Z(x, y) was recovered in ad-
dition to p(x, y) and q(x, y), which should represent
the surface orientation. In this paper, we use the cap-
ital letter (X,Y,Z) for a three-dimensional point and
the small case letter (x,y) for an image point. The
objective function in (Horn, 1990) includes a term
(Z
X
− p)
2
+ (Z
Y
− q)
2
which makes these three func-
tions to represent the same surface, but the actually
recovered surface Z never exactly corresponds to the
orientations (p,q).
Thereafter, Leclerc and Bobick (Leclerc and Bo-
bick, 1991) developed a direct method for recover-
ing shape from shading, which directly find a sur-
face Z(x,y) that minimizes the photometric error. In
this method, the surface orientation is represented ex-
plicitly as the derivative of Z(x, y), and the objective
function is minimized with respect to Z(x, y). By this
method, additional constraints to ensure integrability
of the surface orientation is not needed to be con-
sidered. However, this method assumes the parallel
projection for imaging, and hence applicability of it
is low. To recover shape collectively using various
schemes including shape from stereo (Lazaros et al.,
2008) and shape from motion (Simoncelli, 1999),
(Bruhn and Weicke, 2005), the perspective projection
has to be considered.
If this method is formally extended for the per-
spective projection, the objective function becomes
complicated, and hence, the computation becomes
unstable. To treat the perspective projection effec-
tively with keeping the simplicity of the original
method, we propose an approximation method for the
objective function, and confirm the intrinsic perfor-
mance of it numerically.
2 SHAPE FROM SHADING
Almost methods for shape from shading are based on
the image irradiance equation:
I(x, y) = R(n(x,y)), (1)
445
Wakabayashi K., Tagawa N. and Okubo K..
DIRECT COMPUTATION OF DEPTH FROM SHADING FOR PERSPECTIVE PROJECTION.
DOI: 10.5220/0003854804450448
In Proceedings of the International Conference on Computer Vision Theory and Applications (VISAPP-2012), pages 445-448
ISBN: 978-989-8565-03-7
Copyright
c
2012 SCITEPRESS (Science and Technology Publications, Lda.)