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Level Sets Method2011

# Level Sets Method2011 - We will focus on 5 Deriving the...

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11/15/2011 1 Level Sets Method Introduction Images Curves as isolevels of an image, image as a surface

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11/15/2011 2 Parametrized Curves p N T Arc Length, Unit Tengent, and Curvature Arc length: Unit tangent vector (verify: ) Curvature tensor: So 𝑑𝑇 𝑑𝑠 and 𝑁 ? are parallel and (for some 𝜅(?) , the curvature) ? = 0 ? = ?
11/15/2011 3 Two Important Formulas Verify and Curves as Iso-Level of a Function u Differentiate (*) again: Consider (*) (**)

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11/15/2011 4 Formulas for Curvature of Level Curves From | T’(s) |=1, we have Thus We can re-write the formula as Boundary Detection Functional We will apply calculus of variation to some image analysis problems. First let us consider the problem of detecting the boundary of an object in an image. Recall that boundary points are where the intensity changed most. This means, the magnitude of the gradient of the intensity is very large on the boundary points.

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Unformatted text preview: We will focus on 11/15/2011 5 Deriving the Equations • Let ° = ° (?, ?) denote the curve (in ? ) that evolve with time (an artificial parameter ? ). To minimize ± 2 , we set To minimize ± 1 , we set Similarly, Example OF Mean Curvature Motion 11/15/2011 6 From Curve to Level Sets Method Curve evolution equation °±(?, ?) °? = ²³´ ± 0, ? = ± µ (?) In level set notation: So Plug in the curve evolution equation ´ = the unit normal of ±, Curve Evolution in Level Set Method • Inward (always on your left-side) normal of u=0 : • So, we can write ´ = − ∇? ∇? 11/15/2011 7 Boundary and Initial Conditions Hamilton-Jacobi Equation 11/15/2011 8 11/15/2011 9 11/15/2011 10 11/15/2011 11 11/15/2011 12 11/15/2011 13 11/15/2011 14 Homework • Verify all the cirlcled formulas....
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Level Sets Method2011 - We will focus on 5 Deriving the...

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