Biomedical Engineering Reference
In-Depth Information
Applying Dirichlet boundary conditions, it gives the system of linear equations
with a matrix, the off diagonal elements of which are symmetric and negative.
Diagonal elements are positive and dominate the sum of absolute values of the
nondiagonal elements in every row. Thus, the matrix of the system is symmetric
and diagonally dominant M-matrix which imply that it always has a unique solu-
tion. The M-matrix property gives us the minimum-maximum principle, which
can be seen by the following simple trick. We may temporarily rewrite (11.23)
in the equivalent form
τ
b n 1
u p +
a n 1
pq ( u p u q ) = u n 1
(11.26)
p
p
q C p
and let max( u 1 ,..., u n M ) be achieved in the node p . Then the second term
on the left-hand side is non-negative and thus max( u 1 ,..., u n M ) = u p u n 1
p
max( u n 1 ,..., u n M ). In the same way we can prove the relation for minimum
and together we have
u n 1
p
u p max
p
u p max
p
u n 1
p
min
p
min
p
1 n N ,
(11.27)
,
which by recursion imply the desired stability estimate (11.24).
So far, we have said nothing about evaluation of g T included in coefficients
(11.22). Since image is piecewise constant on pixels, we may replace the con-
volution by the weighted average to get I 0
σ
: = G σ I 0
(see e.g. [37]) and then
relate discrete values of I 0
σ
to pixel centers. Then, as above, we may construct its
piecewise linear representation on triangulation and in such way we get constant
value of I 0
σ
on every triangle T T h . Another possibility is to solve numerically
a linear heat equation for time t corresponding to variance σ with initial datum
given by I 0 (see e.g. [3]). The convolution represents a preliminary smoothing
of the data. It is also a theoretical tool to have bounded gradients and thus a
strictly positive weighting coefficient g 0 . In practice, the evaluation of gradients
on discrete grid (e.g., on triangulation described above) always gives bounded
values. So, working on discrete grid, one can also avoid the convolution, espe-
cially if preliminary denoising is not needed or not desirable. Then it is possible
to work directly with gradients of piecewise linear representation of I 0
in the
evaluation of g T .
Our co-volume scheme in this paper is designed for the specific mesh (see
Fig. 11.18) given by the rectangular pixel structure of 2D image. For sim-
plicity of implementation and for the reader's convenience, we will write the
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