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function based on the uniqueness and smoothness assumptions. Cooperative meth-
ods have shown to give strong results in various publications. However, limitations
include the higher computational effort compared to local methods and the depen-
dence on a good initialization. Furthermore, depth boundaries tend to be blurred,
since rectangular support regions are employed. Zhang and Kambhamettu [17] try
to overcome these problems by using image segmentation to estimate an appropriate
support window and ensure that the initial matches are correct.
Global approaches
Global approaches formulate stereo matching in terms of a global energy function,
which consists of two terms and takes typically the following form:
E
(
d
)=
E
data
(
d
)+
λ
E
smooth
(
d
)
.
(8)
The first term measures the distance between corresponding pixels, while the sec-
ond one enforces the smoothness of the disparity field and
is a positive constant
weighting the two terms. Several different energy minimization algorithms have
been proposed to solve Eq. (8). The most common approach is dynamic program-
ming, which uses the ordering and smoothness constraints to optimize correspon-
dences in each scanline. The matching costs of all points on a scanline describe the
disparity search space. Finding the correct disparities is akin to finding the minimum
cost path through this space. The most significant limitation of dynamic program-
ming is its inability to enforce smoothness in both horizontal and vertical directions.
The work of [15] proposes a way to cope with this problem while maintaining the
dynamic programming framework. Recently, powerful algorithms have been devel-
oped based on graph cuts [18, 12] and belief propagation [19] for minimizing the
full 2-D global energy function. The idea is to cast the stereo matching problem as a
pixel labelling problem to find the minimum cut through a certain graph. Variational
approaches have also been very effective for minimizing Eq. (8) via an iterative
scheme derived from the associated Euler-Lagrange differential equations [20, 21].
However, these techniques often are computationally demanding and they require
a careful study for the discretization of the associated partial differential equations.
Besides, they require the determination of the Lagrange parameter
λ
which may be
a difficult task. The latter problem becomes even more involved when a sum of reg-
ularization terms has to be considered to address multiple constraints, which may
arise in the problem. The work in [22] attempts to overcome these difficulties by for-
mulating the stereo matching problem in a set theoretic framework. Each available
constraint is then represented by a convex set in the solution space and the intersec-
tion of these sets constitutes the set of admissible solutions. An appropriate convex
quadratic objective function is finally minimized on the feasibility set using an effi-
cient block-iterative algorithm which offers great flexibility in the incorporation of
a wide range of complex constraints.
λ
