Project 2 - Project 2 Level Sets Method in Curve/Surface...

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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 0-level 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 0-Ievel 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 TT 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 re-initialization, 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 m-files, 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 m-files like curvature.m and evolve_kappa.m, and see how spatial discretization is done — note that ENO and WENO used in many m-files 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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This note was uploaded on 01/16/2012 for the course MAP 4371 taught by Professor Xli during the Fall '11 term at University of Central Florida.

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Project 2 - Project 2 Level Sets Method in Curve/Surface...

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