Image Processing Reference
In-Depth Information
Figure 4.2: Reconstruction results for a very flexible plastic sheet. In spite of the many creases, the overall
shape is correctly recovered up to small errors due to erroneous correspondences. © 2007 IEEE.
[ 2005 ], Sim and Hartley [ 2006 ] that expressed the minimization of the true reprojection error as a
Second Order Cone Programming (SOCP) problem Boyd and Vandenberghe [ 2004 ]. In its general
form, an SOCP can be written as
f T x
c i x
subject to
A i x
b i 2
d i , 1
where f is the vector that defines the objective function, A i is a matrix, b i and c i are vectors, and d i
is a scalar. Problems of this type are convex and, thus, have a unique minimum that can be found
very effectively using available packages such as SeDuMi Sturm [ 1999 ]. Furthermore, SOCP can
be used to formulate problems more general than linear programming, quadratic programming and
quadratically-constrained quadratic programming.
For the specific case of deformable surface reconstruction, minimizing the reprojection error
can be expressed as
γ, x
subject to ( P 1 u i P 3 ) h i ,( P 2 v i P 3 ) h i 2 γ P 3 ˜
q i , 1
i N c ,
q i = q i , 1 T
where P k contains the k th
line of the projection matrix, and
is the vector of ho-
mogeneous coordinates of the 3D point matching the i th
q i is obtained from the
vertex coordinates x and barycentric coordinates. γ is an additional slack variable that encodes the
maximum reprojection error for all feature points.
While a solution of the above problem minimizes the true reprojection error, it still is under-
constrained. This is why temporal consistency was introduced in Salzmann et al. [ 2007a ], to prevent
the orientation of mesh edges from varying excessively from one frame to the next, as illustrated in
Fig. 4.3 . This can be expressed as additional SOCP constraints of the form
feature point.
2 λl i,j ,
v t + 1
v j
v i
v t + 1
+ l i,j
v j
v i 2
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