Biomedical Engineering Reference
In-Depth Information
9.4.8.5
Consequences of Finite Support
All what we said so far about expansion and reduction holds for infinite signals.
To adapt the method for finite signals, we considered the following requirements:
the expansion must be exact in the continuous sense, the projection identity
must hold, reduction followed by expansion must conserve the length of the
signal, and as much information as feasible should be conserved. These require-
ments are useful to guarantee the best possible use of the coarse-grid results at
the fine-grid level and are absolutely indispensable for multigrid minimization.
Traditionally, one represents the signal with exactly one coefficient per sam-
ple and assumes that the signal outside the region of interest follows some
known pattern, such as periodicity, or mirror-on-boundary conditions. We take
the mirror-on-boundary condition as an example, but the same kind of prob-
lems appear for other boundary conditions, too. First, the signal is forced to
be symmetric and thus flat at boundaries. Second, the boundary conditions for
both expansion and reduction are only conserved for odd number of samples,
otherwise the mirror position needs to change. Third, varying the length of the
signal by one does not change the length of the reduced version which makes it
impossible to recover the original length by expansion.
The centered pyramids [108] conserve the mirror position for even-length sig-
nals. Unfortunately, the expansion is no longer exact. Moreover, the constraint
of the size of the image being a power of two, together with the integer step size
h
, seems to be too restrictive.
Because of these considerations, we decided to dissociate the number of
B-spline coefficients from the length of the interest region. Initially, we extend
the signal by
(
n
−
1)
/
2
samples at each extremity which allows us to represent
any spline of degree
n
without constraints. We never move the boundaries of
our signal when expanding, although the number of B-splines might vary. In this
way, expansion is always exact while it adds extra knots at each end. Reducing
expanded signal recovers exactly the original. When reducing other signals, we
need to extend them to be able to use our efficient filtering technique. For this,
we choose to use the mirror-on-boundary conditions.
9.4.8.6
Image Size Change
The only trick when expanding and reducing the images is to adapt the de-
formation function accordingly. This is easily accomplished by multiplying the
