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Unformatted text preview: Project 2 Level Sets Method in Curve/Surface Evolution Background: Level sets method uses implicit equations (instead of parametric equations) to
represent curves/ surfaces (by tracking the curves or surfaces via the 0level sets of the Level Set
Function (LSF)). In mathematical notations, (using 2D closed curve as an example) c: (x,y) = E(q),a S q S b,5(a) = 3(1))
TT
ii c: {(x,y) (1506,37) = 0} 1U 2U 30 40 50 50 3'0 SI] 90 100 The graph of The 0Ievel set of From a single curve/ surface to a family of curves/ surfaces: When we add an additional time
variable, t, we can write ct: (mm) = 3(t,q),a S q S 19,501): 5(b),t 2 0
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it Ct: {Cm/)I ¢(t,x,y) = 0},t 2 0. In our lectures, we derived the curve evolution equation from the parametric form 65 — = F 1V
at
to the level set evolution equation (for the LSF)
ad) — F IV l
at _ ¢ ' (Make sure you understand the notations — what are the independent variables, the gradient
operator applied to what, etc.) More generally, the following equation can be (and has been) used:
(2—9:) = g lngIdiv + Vg  W!) + ag V§b
where a is a constant and g is an edge indicator ﬁanction (which could be deﬁned by
1
g 2 m where Go is a Gaussian kernel with standard deviation 0' and I = I (x, y) denotes an image
intensity ﬁanction deﬁned on a domain 9.) Numerical implementation: We went through some issues for the numerical implementation of
the level set method (like using matrices to store the discrete form of the LSFs, initialization of
the LSF using the signed distance ﬁmction, when to use central difference and when and where
to use the upwind scheme, the need for reinitialization, etc.) In this project, you are asked to
experiment with the implementation of level set method provided at http://barissumengen.com/level set methods/index.html#tutorial Requirement on your report and presentation: Please download the Matlab code, read the mfiles,
and run (and modify if necessary) the test.m function. Your report for this project is a summary
of your experimental results (for example, state how you changed the testm file, experiment
with your own images if you prefer) on at least two types of examples (the above web pages
describe 5 such examples — follow that style for your description of your experiments). In your
presentation, I will ask questions about your understanding of the level set method and the
related implementation issues (so read the mfiles like curvature.m and evolve_kappa.m, and see
how spatial discretization is done — note that ENO and WENO used in many mfiles are just
more advanced/accurate methods to discretize derivatives). So, even though you are not required
to put any theory part in your report, you need to understand how the method works and be able
to interpret your results. Due date: December 1, 2011 ...
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 Fall '11
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