Motivation: Level Sets. Input Data Noisy. Easy Case Use Marching Cubes. Intensity Varies. Non-uniform Exposure. Roger Crawfis

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1 Level Set Motivation: Roger Crawfi Slide collected from: Fan Ding, Charle Dyer, Donald Tanguay and Roger Crawfi 4/24/2003 R. Crawfi, Ohio State Univ. 109 Eay Cae Ue Marching Cube Input Data Noiy 4/24/2003 R. Crawfi, Ohio State Univ /24/2003 R. Crawfi, Ohio State Univ. 111 Non-uniform Expoure Intenity Varie 4/24/2003 R. Crawfi, Ohio State Univ /24/2003 R. Crawfi, Ohio State Univ. 113

2 Multiple problem Applying Marching Cube (threholding) 4/24/2003 R. Crawfi, Ohio State Univ /24/2003 R. Crawfi, Ohio State Univ. 115 Applying A Threhold Applying A Threhold 4/24/2003 R. Crawfi, Ohio State Univ /24/2003 R. Crawfi, Ohio State Univ. 117 Four (contour) Level What To Do? Uer Intervention!!! We ee it!! It right there!!! Well, the edge get pretty fuzzy. Two tep proce: 1. Draw an initial curve (or urface) within the deired region. 2. Expand that interface outward toward the edge of our deired region. 4/24/2003 R. Crawfi, Ohio State Univ /24/2003 R. Crawfi, Ohio State Univ. 119

3 Interface An interface (or front) i a boundary between two region: inide and outide. In 2-D, an interface i a imple cloed curve: Propagating Interface How doe an interface evolve over time? At a pecific moment, the peed function F (L, G, I) decribe the motion of the interface in the normal direction. 4/24/2003 R. Crawfi, Ohio State Univ /24/2003 R. Crawfi, Ohio State Univ. 121 Propagating Interface Motion Under Curvature Speed F(L,G,I) i a function of 3 type of propertie: Local depend on local geometric information (e.g., curvature and normal direction) Global depend on the hape and poition of the front (e.g., integral along the front, heat diffuion) Independent do not depend on the hape of the front (e.g., an underlying fluid velocity that paively tranport the front) Example: Motion by curvature. Each piece move perpendicular to the curve with peed proportional to the local curvature. mall negative motion large poitive motion 4/24/2003 R. Crawfi, Ohio State Univ /24/2003 R. Crawfi, Ohio State Univ. 123 Motion Under Curvature Motion Under Curvature Curvature κ i the invere of the radiu r of the oculating circle. 4/24/2003 R. Crawfi, Ohio State Univ /24/2003 R. Crawfi, Ohio State Univ. 125

4 Functional Repreentation Eulerian framework: define fixed coordinate ytem on the world. For every world point x, there i (at mot) one value y = f t (x). Falling now example: Functional Repreentation However, many imple hape are multivalued; they are not function regardle of the orientation of the coordinate ytem. y t= 0 x = f ( ) = f ( ) y t= 1 x 4/24/2003 R. Crawfi, Ohio State Univ /24/2003 R. Crawfi, Ohio State Univ. 127 Parametric Repreentation Parametric Repreentation Spatially parameterize the curve x by o that at time t the curve i x t (), where 0 S and the curve i cloed: x t (0) = x t (S). Point on initial curve. Gradient (wrt time) i the peed in normal direction. Normal i perpendicular to curve, a i curvature. 4/24/2003 R. Crawfi, Ohio State Univ. 128 For motion under curvature, peed F depend only on local curvature κ the equation of motion i thu: yx x y where curvature i κ t ( ) = 2 2 3/ 2 ( x + y ) x and the normal i nt ( ) = x xt ( ) = F( κ t ( )) nt ( ), t = ( x 4/24/2003 R. Crawfi, Ohio State Univ. 129 T T [ x y ] [ y x ] 2 + y 2 1/ 2 ) = ( x 2 + y 2 1/ 2 ) Particle Method In order to compute, dicretize the parameterization into moving particle which recontruct the front. Known under a variety of name: marker particle technique, tring method, nodal method. t = time tep = parameterization tep S = # meh particle n n ( x, y ) = location of point i at time n t i i Problem Statement Generally, given: An initial front Equation that govern it evolution How do we imulate the front evolution? Called an initial value problem Given the initial poition Solve for a poition at a future time 4/24/2003 R. Crawfi, Ohio State Univ /24/2003 R. Crawfi, Ohio State Univ. 131

5 More formally: Given ome initial front Г: More formally: (2) And a function F that pecifie the velocity of the front in the normal direction: 4/24/2003 R. Crawfi, Ohio State Univ /24/2003 R. Crawfi, Ohio State Univ. 133 More formally: (3) Solve for Г at ome future time Level et method are ued to track an interface Water/air interface, for example What Wrong with the Obviou Solution? Why i a level et method neceary? There eem to be a more intuitive way to olve thi problem 4/24/2003 R. Crawfi, Ohio State Univ /24/2003 R. Crawfi, Ohio State Univ. 135 Marker/String Method Why not jut connect ome control point (in 3D, triangulate): Marker/String Method (2) And run the imulation on the point? 4/24/2003 R. Crawfi, Ohio State Univ /24/2003 R. Crawfi, Ohio State Univ. 137

6 Ocean Wave Think of an air/water interface with two wave racing toward each other: Ocean Wave (2) What happen to the control point when the wave collide? 4/24/2003 R. Crawfi, Ohio State Univ /24/2003 R. Crawfi, Ohio State Univ. 139 Shock Event known a a hock Below formation called a wallowtail Shock (2) How to fix up the control point? Fixing wallowtail known a de-looping Very difficult Some method exit in 2D No robut 3D method o far 4/24/2003 R. Crawfi, Ohio State Univ /24/2003 R. Crawfi, Ohio State Univ. 141 Changing Topology Example: two fire merge into a ingle fire. Changing Topology In particle method: Difficult (and expenive) to detect and change the particle chain Much more difficult a dimenionality increae 4/24/2003 R. Crawfi, Ohio State Univ. 142 Buoy! 4/24/2003 R. Crawfi, Ohio State Univ. 143

7 Difficultie With Particle Method Intability Local ingularitie Management of particle: remove, reditribute, connect Level Set Formulation Recat problem with one additional dimenion the ditance from the interface. And then ue Marching Cube to extract the urface. z t= = φ 0( x, y) 4/24/2003 R. Crawfi, Ohio State Univ /24/2003 R. Crawfi, Ohio State Univ. 145 Level Set Method Contour evolution method due to J. Sethian and S. Oher, Difficultie with nake-type method Hard to keep track of contour if it elf-interect during it evolution Hard to deal with change in topology The level et approach: Define problem in 1 higher dimenion Define level et function z = φ(x,y,t=0) where the (x,y) plane contain the contour, and z = igned Euclidean ditance tranform value (negative mean inide cloed contour, poitive mean outide contour) 4/24/2003 R. Crawfi, Ohio State Univ /24/2003 R. Crawfi, Ohio State Univ. 147 How to Move the Contour? Move the level et function, φ(x,y,t), o that it rie, fall, expand, etc. Contour = cro ection at z = 0 Level Set Surface The zero level et (in blue) at one point in time i a lice of the level et urface (in red) 4/24/2003 R. Crawfi, Ohio State Univ /24/2003 R. Crawfi, Ohio State Univ. 149

8 Level Set Surface Level Set Surface Later in time the level et urface (red) ha moved and the new zero level et (blue) define the new contour 4/24/2003 R. Crawfi, Ohio State Univ /24/2003 R. Crawfi, Ohio State Univ. 151 Level Set Formulation How to Move the Level Set Surface? The interface alway lie at the zeroth level et of the function Φ, i.e., the interface i defined by the implicit equation Φ t (x, y) = 0. 4/24/2003 R. Crawfi, Ohio State Univ Define a velocity field, F, that pecifie how contour point move in time Baed on application-pecific phyic uch a time, poition, normal, curvature, image gradient magnitude 2. Build an initial value for the level et function, φ(x,y,t=0), baed on the initial contour poition 3. Adjut φ over time; current contour defined by φ(x(t), y(t), t) = 0 Φ + F Φ = 0 t 2 2 Φ Φ Φ F + + t x y = 0 4/24/2003 R. Crawfi, Ohio State Univ Speed Function F(K) = F + F 1(k) = 0 ( 1 εk) ( x,y) ( 1 εk) F(K) = k I Example: Shape Simplification F = 1 0.1κ where κ i the curvature at each contour point k k I I 1 = 1 + G I = e - Gσ I σ ( x,y) ( x,y) 4/24/2003 R. Crawfi, Ohio State Univ /24/2003 R. Crawfi, Ohio State Univ. 155

9 Example: Segmentation Digital Subtraction Angiogram F baed on image gradient and contour curvature Example (cont.) Initial contour pecified manually 4/24/2003 R. Crawfi, Ohio State Univ /24/2003 R. Crawfi, Ohio State Univ. 157 Reult egmentation uing Fat marching Reult vein egmentation No level et tuning 4/24/2003 R. Crawfi, Ohio State Univ. 158 No level et tuning With level et tuning 4/24/2003 R. Crawfi, Ohio State Univ. 159 Reult vein egmentation continued Reult egmentation uing Fat marching Original Our reult Sethian reult (Fat marching + (Level et only) Level et tuning) 4/24/2003 R. Crawfi, Ohio State Univ. 160 No level et tuning 4/24/2003 R. Crawfi, Ohio State Univ. 161

10 Reult brain egmentation continued Reult brain image egmentation No level et tuning With level et tuning 4/24/2003 R. Crawfi, Ohio State Univ. 162 # of iteration = 9000 # of iteration = Fat marching only, no level et tuning 4/24/2003 R. Crawfi, Ohio State Univ. 163