Implicit Geometry Overview • Imppctlicit Geom etry – Implicit functions • Points • Curves • Surfaces • Geometry toolbox • Calculus toolbox – Signed distance functions • Distance fifunctions • Signed distance functions • Geometry and calculus tool box Introduction • Tracking boundaries is an important problem – Image Processing – Machine Vision – Biometrics – Human Computer Interaction – Entertainment – Computational Physics – Computational Fluid Dynamics Introduction • Image Processing – Video, Image and Volume Segmentation Introduction • Machine Vision • Biometrics Introduction • Human Computer Interaction • Entertainment Introduction • Physical Simulations like Water Introduction • The Problem – How is it possible to efficiently and accurately represent the boundary? – How do you track the boundary ahead in time? Explicit Representation • Marker/String Methods • A standard approach to modeling a moving boundary ((/front/contour /interface )is to discretize it into a set of marker particles whose positions are known at any time. • To reconstruct the front the particles are linked by line segments in 2D and triangles in 3D. Explicit Representation • Marker/String Method – The idea is to advance the particles in the direction of the arrows, recalculate the arrows, advance the particles, and repeat – More particles should mean greater accuracy Explicit Representation • Marker/String Method Issues – The markers may potentially cross over one another and require untangggling – The boundary may split or merge Explicit Representation • Marker/String Method Issues – The front may expand or contract requiring the addition or removal of markers Implicit Vs. Explicit • Tracking the interface through points (markers) on its surface is a Lagrangian formulation. • Capturing an interface through the evolution of the implicit surface is a Eulerian formulation. Explicit Functions • Points – Explicitly divide a 1‐d region Ω into two sub‐regions with an interface at x = ‐1 and x = 1 Outside Ω+ Inside Ω‐ Outside Ω+ ‐2 ‐1012 Explicit Representation Interface 1,1 , 1 1, 1,1 Implicit Functions • Points – Implicitly divide a 1‐d region Ω into two sub‐regions with an interface at x = ‐1 and x = 1 Outside Ω+ Inside Ω‐ Outside Ω+ ‐2 ‐1012 Implicit Representation x x2 1 :x 0 Interface :x 0 :x 0 Implicit Functions • Points – An implicit representation provides a simple, numerical method to determine the interior, exterior and interface of a region Ω. – In the previous case the interior is always negative and the exterior is always positive. – The set of points where φ(x)=0isknownasthe zero level set or zero isocontour and will always give the interface of the region. Implicit Functions • in Rn , subdomains are n‐dimensional, while the interface has dimension n − 1. • We say that the interface has codimension one. Implicit Functions • Curves – In 2 spatial dimensions a closed curve separates Ω into two sub‐ regions. Outside Ω+ φ > 0 Inside Ω‐ φ < 0 Interface x x2 y 2 1 0 Implicit Functions Implicit Functions • Surfaces – In 3+ spatial dimensions higher dimensional surfaces are used to represent the zero level set of φ. Outside Ω+ φ > 0 Inside Ω‐ φ < 0 Interface x x2 y 2 z 2 1 0 Geometric Representation • For complicatedsurfacesin3Dtheexplicit representation can be quite difficult to discretize. One needs to choose a number of points on the two‐dimensional surface and record their connectivity. • Connectivity can changefordyy/namic/deformable surfaces, i.e., surfaces that are moving around. • One of the nicest properties of implicit surfaces is that connectivity does not need to be determined. Geometric Representations • Continuous representation – Explicit representation – Implicit representation • Discrete representation – Triangle meshes – Points/Particles Triangle Mesh Explicit vs Implicit Representation Explicit: • Pros: easier display • Cons: maintain connectivity, difficult to handle topology change. Implicit: • Pros: No connectivity, easier to handle topology change. • Cons: need extra step to display. Discrete Representations of Implicit Functions • Complicated 2D curves and 3D surfaces do not always have analytical descriptions. • The implicit function φ need to be stored with a discretization, i.e. in the implicit representation, we will know the values of the implicit function φ at only a finite number of data points and need to use interpolation to find the values of φ else where. Discrete Representations of Implicit Functions • Thelocationoftheinterface, the φ(x) =0zeroiso‐ contour need to be interpolated from the known values of φ at the data points. • This can be done using contour plotting alhlgorithms such as Marching Cubes algorithm. • The set of data points where the implicit function φ is defined is called a grid. • Uniform Cartesian grids are mostly used. • Other grids include unstructured, adaptive, curvilinear, etc. Iso‐contour extraction φ(x,y))0=0 φ(x,y)>0 φ(xyx,y)<0 Iso‐contour extraction φ(x,y))0=0 φ(x,y)>0 φ(xyx,y)<0 Iso‐contour extraction φ(x,y))0=0 φ(x,y)>0 φ(xyx,y)<0 Iso‐contour extraction φ(x,y))0=0 φ(x,y)>0 φ(xyx,y)<0 Marching Squares φ(x,y))0=0 φ(x,y)>0 φ(xyx,y)<0 Four unique cases (after considering symmetry) Marching Squares Principle of Occam’s razor: If there are multiple possible explanations of a phenomenon that are consistent with the data, choose the simplest one. Marching Squares Linear interpolation f(x,y)=0 f(xyx,y)>0 f(x,y)<0 fi, j+1 > 0fi+1, j+1 > 0 f < 0 i, j fi+1, j > 0 Marching Squares fi, j = a < 0 f = b > 0 fi+x, j=0 i+1, j Marching Squares fi, j = a < 0 f = b > 0 fi+x, j=0 i+1, j x / h = (‐a) / (b‐a) x = ah / (a‐b) Marching Squares Marching Squares Marching Squares‐ambiguity More information are needed to resolve ambiguity Iso‐contour extraction φ(x,y))0=0 φ(x,y)>0 φ(xyx,y)<0 Marching Cubes Algorithm Lorensen & Cline: Marching Cubes: A high resolution 3D surface construction algorithm. In SIGGRAPH 1987. Geometry Toolbox • Inside/outside • Boolean operation/CSG • The gradient of the iliiimplicit fifunction • The normal of the implicit function • Mean curvature of the implicit function • Numerical approximations Inside/outside function Determine whether a point is inside/outside the interface • Easier for Implicit function, just look at the sign of x : IidInside : :x 0 Outside : :x 0 Boundary : :x 0 • Difficult for explicit representation: Inside/outside function Difficul t for eepctxplicit represe ntat io n: • A standard procedure is to cast a ray from the point in question to some far‐off place that is known to be outside the interface. • Then if the ray intersects the interface an even number of times, the point is outside the interface. • Otherw ise, therayitintersec ts the itinter face an odd number of times, and the point is inside the interface. Boolean Operations of Implicit Functions • φ(x) = min (φ1(x), φ2(x)) is the union of the interior regions of φ1(x) and φ2(x). • φ(x) = max (φ1(x), φ2(x)) is the intersection of the interior regions of φ1(x) and φ2(x). • φ(x) = ‐φ1(x) is the complement of φ1(x) . • φ(x) = max (φ1(x), ‐φ2(x)) represents the subtraction of the interior regions of φ1(x) by the interior regions of φ2(x). Constructive Solid Geometry (()CSG) Gradient/Normal of Implicit Function Mean Curvature of the Interface Calculus Toolbox • Characteristic function • Heaviside function • Delta fifunction • Volume integral • Surface integral • Numerical approximation Signed Distance Functions • Signed distance functions are a subset of implicit function defined to be positive on the exterior, negative on the interior with | (x) 1| Distance Functions • A distance function is defined as: dx min x xI where xI • so: dx | x xc |, where xc is the closest point on the boundary to x dx 0, where x Distance function Signed Distance Functions • A signed distance function adds the appropriate signing of the interior vs. exterior. d(x) x (x) d(x) 0 x d(x) x • Signed distance function share all the properties of implicit function with new ppproperties such as: | (x) 1| • Given a point x , the closest point xc on the interface is xc x (x)N Signed Distance Functions • 1D Example: – Implicit function x x2 1 – Signed distance function x | x | 1 • 2D Example: – Implicit function x, y x2 y 2 1 – Signed distance function x, y x2 y 2 1 • 3D Example: – Implicit function x, y, z x2 y 2 z 2 1 – Signed distance function x, y, z x2 y 2 z 2 1 Signed Distance Functions Boolean Operations of Signed Distance Functions • φ(x) = min (φ1(x), φ2(x)) is the union of the interior regions of φ1(x) and φ2(x). • φ(x) = max (φ1(x), φ2(x)) is the intersection of the interior regions of φ1(x) and φ2(x). • φ(x) = ‐φ1(x) is the complement of φ1(x) . • φ(x) = max (φ1(x), ‐φ2(x)) represents the subtraction of the interior regions of φ1(x) by the interior regions of φ2(x). Geometry and Calculus Toolboxes of the Signed Distance Functions Since | (x) 1| • The Delta function is: (x) ((x)) • The Surface integral is: f (x) ((x))d(x) Geometry and Calculus Toolboxes of the Signed Distance Functions Since | (x) 1| • The normal of the signed distance function is: N • Mean curvature of the the signed distance function is: k Where is the Laplacian of xx yy zz Signed Distance Functions • In two spatial dimensions the signed distance function for a circle with a radius r and a center (x0, y0) is given as: 2 2 x x x0 y y0 r Overview • Implicit Geometry • Level Set Methods Overview • Level set methods – Motion in an externally generated velocity field • Convection • Upwind differencing • Hamilton‐Jacobi ENO//,WENO, TVD Runge‐Kutta – Motion involves mean curvature • Equations of motion • Numerical discretization • Convection‐diffusion equations Overview – Hamilton‐Jacobi Equation • Connection with Conservation Laws • Numerical discretization – Motion in the Normal Direction • The Basic Equation • Numerical discretization • Adding a Curvature‐Dependent Term • Adding an External Velocity Field References • S.