Biomedical Engineering Reference
In-Depth Information
target image can be found by trying all possible shifts in some small neighbor-
hoods [34, 56].
9.2.4.3
Dynamic Programming
For integer 1D problems,
dynamic programming
is applicable [57] with com-
plexity proportional to the number of decisions to be taken (size of the image)
and the number of possible outcomes (shifts). The stereo matching problem with
cameras with known relative geometry also falls into this category [58]. These
are nice examples of how the a priori knowledge about the image formation
process simplifies the registration problem by imposing very strong geometrical
constraints on the correspondence field.
9.2.4.4
Partial Differential Equations
A partial differential equation can describe an evolution in time of the deforma-
tion field converging to the desired solution [32, 52]. (Time-independent PDE's
derived directly by Euler-Lagrange formulas from the variational formulation are
seldom used.) The PDE parameters can vary with time, to enforce robustness
at first and relax the constraints toward the end, to allow for precise registra-
tion. Various numerical methods can be used to solve the PDEs, the princi-
pal ones being finite difference relaxation method [59], finite elements method
(FEM) [60, 61], hierarchical finite element bases [62], multigrid methods [63],
and wavelets [64, 65].
9.2.4.5
Multidimensional Optimization Methods
Many non-linear registration methods lead to a non-linear multidimensional op-
timization problem. Various optimization methods are used, depending on the
size and structure of the problem. The most popular choices include the Powell
method,
gradient descent
, conjugated gradients, and variations of the Newton
method, such as the
Marquardt-Levenberg
algorithm [59, 66].
Some minimization algorithms are described using different paradigms, such
as 'demons' [67], but usually can be explained in terms of a coordinate descent.
9.2.4.6
Multiresolution
Multiresolution [7,36,68] on the feature space (usually image size) helps to speed
up the process and to increase its robustness. It is based on approaching the
