Biomedical Engineering Reference
In-Depth Information
improve the convergence and accuracy of the numerical solutions derived
in the “single scale” methods. The multigrid method appears to be one of the
most successful multiscale methods for solving the SFS problems. The multi-
grid method associated with vision problems was studied in the early 1960s
by Fedorenko and Bakhvalov [3] and later in the work of many others, for
example, [1, 56, 57]. For readers without previous experiences in the multi-
grid method, excellent tutorials can be found in the topics [8] and [44, 62]. A
classic book on multigrid method is [23]. Some other novel multiscale meth-
ods are developed recently, for example, the panel-clustering method, wavelet-
based matrix compression, and the hierarchical matrix decomposition
technique.
5.3 Numerical Algorithms and Their
Implementations
It is understood that the problem of shape from shading is fundamentally a very
difficult mathematical problem. Nevertheless, some methods were developed
while attempting to solve the problem numerically for some particular cases.
Most of the methods work for a particular type of images.
SFS techniques can be roughly divided into four groups:
(1)
Minimization approach
. The method is to reconstruct the shape of the
object by minimizing the well-designed energy function over the entire
image. The idea is similar to that in deformable models used in image seg-
mentation which segment the object by minimizing certain predefined en-
ergy functions. The constraint functions can be the brightness constraint,
the gradient constraint, the smooth constraint, etc. (See Section 5.2.3.2).
(2)
Propagation approach
. The method starts from some initial curve, which
is defined as special points such as the brightest or the darkest points in
the image, and propagates the shape information across the entire image.
(See 5.2.2).
(3)
Local approach
. The method reconstructs the shape by assuming local
surface type. The surface is derived by matching derivatives of measured
intensity to those assumed spherical surface.
