Biomedical Engineering Reference
In-Depth Information
2. Remove the smallest (top) node of the tree.
When the top node of the tree is removed, the child of the top node,
whose value for
φ
is smallest, is chosen to be the new top node. This
process of promoting the smallest child up the tree is then propagated
down until a node with less than two children is detected. This process
preserves the property of the tree that parent nodes always have a smaller
value of
φ
than the children.
3. Add a new node to the tree.
When a grid point is moved from the set
D
to
T
, it is also added to
the tree. Since the initial estimate for
φ
at this point is likely to be larger
than any of those already in the tree, it is best to add the node to an outer
branch. For purposes of efficiency, care should be taken to keep the tree as
balanced as possible, hence the new node should be added to the sparsest
part of the tree. Once the node is appended, an up-sweep is performed to
ensure proper placement.
4. Change the key value of an element in the tree.
When a grid point value is changed, it may require the tree to be re-
sorted. If the value of the node is increased, then a down-sweep is done,
and if the value is decreased, an up-sweep is done.
Initialization of the Fast Marching Method
The best form of initialization is where the exact solution is assigned to all
the points in the original set
A
. These are all the nodes which are immedi-
ately adjacent to the initial interface. Most often, the exact solution is not
known, and the initial values for the set
A
must be approximated from the initial
data.
The method for initializing the set
A
given in [105, 106] is only first-order
accurate, and can be prone to errors which will propagate through the remainder
of the calculation. It was shown in [22] that a more accurate method is available,
which can drive higher order fast marching method solutions.
The underpinning of this higher degree of accuracy around the initial front
is the use of a bicubic interpolation function
p
which is a second-order ac-
curate local representation of a level set function
φ
, i.e.
p
(
x
)
≈
φ
(
x
). The in-
terpolation function
p
(
x
) can serve many purposes, including second-order
