Front Propagation, Level Set Methods - Computer Science 0 | CS T101, Study notes of Computer Science

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Medical Image Analysis

CS 593 / 791

Computer Science and Electrical Engineering Dept. West Virginia University

17th February 2006

Outline

(^1) Geometry of Implicit Curves

(^2) Level Set Methods

(^3) Introduction

(^4) Front Propagation

Evolving the embedding function by

∂ψ ∂t

logo

Smoothing the curve

Evolving the embedding function by ∂ψ ∂t

= −κ(x, y)

κ < 0 where contour is locally convex κ > 0 where contour is locally concave

Surface Representation

Surfaces are represented as the zero level set of an embedding function. This function is evolved, implicitly evolving the embedded curve (Eulerian approach) The previous (Lagrangian) approach was to track points on the interface.

Problems with ”snakes”

The Lagrangian approach does not handle Splitting / merging boundaries (topological change) Self-intersection Sharp corners of other discontinuities

Approach to segmentation

The curve will evolve with an inflation force, to reach protrusions in shape a curvature based speed, to keep the boundary smooth an image based speed, to stop the curve at image boundaries

Embed the curve, γ(t) , into ψ

Initialize ψ to be the signed distance to γ(t = 0 ).

ψ(x, y, t = 0 ) = ±d where d is the distance from (x, y) to γ(t = 0 ). d < 0 inside γ d > 0 outside γ

Curvature-Based Evolution

Decompose γ′(t) into components tangent and normal to γ(t).

0 = (∇ψ · (vN N(t) + vT T (t))) +

∂ψ ∂t = (∇ψ · vN N(t)) + ∂ψ ∂t since ∇ψ is perpendicular to the tangent to γ(t).

Substituting the level set definition for the normal to the embedded curve

0 = (∇ψ · (vN ∇ψ ||∇ψ||

∂ψ ∂t

Curvature-Based Evolution

We can rewrite this result

0 = (∇ψ · (vN

∇ψ ||∇ψ||

∂ψ ∂t

0 = vN (∇ψ · ∇ψ ||∇ψ||

∂ψ ∂t

0 = vN ||∇ψ||^2 ||∇ψ||

∂ψ ∂t 0 = vN ||∇ψ|| + ∂ψ ∂t

Stopping Term

For F (κ) = k(F 0 + F 1 ) We can use

k(x, y) =

1 + ||∇(Gσ ∗ I(x, y))||

as a stopping term since k(x, y) → 0 near edges in I(x, y).

Discretizing the evolution equation

The explicit discretization

ψ it,+j 1 − ψti,j ∆t = −ki,j ||∇ψt^ || − ki,j (κti,j ||∇ψt^ ||)

The advection term can lead to singularities, discretize using upwind finite differences The curvature term can be discretized using central differences.

In three dimensions

Level set is the surface ψ(x, y, z) = 0. Compute gradients in 3 dimensions.

Next Time

”Shape modeling with front propagation: A level-set approach.” More details of the level set implementation.