Advanced Computer GraphicsComputer Animation Implicit Surfaces Spring 2002 Professor Brogan Many slides from Brian Wyvill’s online materials at U. Calgary
Papers for Tuesday • Spacetime Constraints, Witkin and Kass • Siggraph • Deep-water Animation and Rendering • Gamasutra.com, Sept 26, 2001
Implicit Surfaces • Surfaces defined by points that satisfy • f(P) = 0 Implicit Function • Example, a circle • Parametric • x=r cos(a) • y=r sin(a) • Implicit • x2 + y2 + r2 = 0
Implicit Surface Modeling • Useful for modeling natural and smooth/organic synthetic phenomena • Living forms, liquids, clouds • Each primitive is represented by a skeletal element which contributes in defining a scalar field • Every point in space is assigned a scalar value equal to shortest distance to a skeletal element
Implicit Surface Modeling • Simplest skeletal element is a point • A (distance) contour of that point defines the (surface of the) model • Ex: two points approach and their contours blend
Combining Skeletal Primitives
Blending Skeletal Elements • Define a simple surface as: • A central point, C • A radius of influence, R • A density function, f() • A threshold value, T • All points, P, for which dist (P, C) < R • Implicit surface = f (dist(P, C)) – T = 0
Blending Skeletal Elements • Example: metaballs • f(dist) = • Surface drawn where f(dist) – T = 0 Surface drawnat this radius Value of T
Blending Skeletal Elements • Example: two metaballs • Surface drawn where: • f(dist1) + f(dist2) – T = 0 Do we drawsurface here? r1 r2 r2 r1
Blending Skeletal Elements • Yes http://www.lifl.fr/~triquet/implicit/video/blend.mpg
Blending Skeletal Elements • In general, surface defined by multiple surface elements is:
Blending Lines • Lines can bulge when their ends meet • Usually not the desired effect • Tinkering with density function and line separation can fix
Versions of Density Functions
Modeling a Dinosaur (Wyvill)
Modeling Blood http://www.lifl.fr/~triquet/implicit/video/blood.mpg
Solving Implicit Functions • Root finding • Given a function f, we wish to find the set of x values (1D points) that satisfy f(x)=0 • From calculus, the Intermediate Value Theorem states: “as x varies from a to b, the continuous function f takes on every value between f(a) and f(b)”
Solving Implicit Functions • If f(a) and f(b) have opposite signs, the root is said to be bracketed in the interval [a, b]
Bracketing the roots • Uniformly subdividespace • Evaluate f( ) for eachboundary • Transition from + to –defines bracket
Bisection search • Iteratively subdivide to find exact zero
2-D Implicit Surfaces • Surface defined by f(X) = 0; X = (x, y) • Uniformly subdivide • Evaluate f( ) at all points • Bracket roots
2-D Implicit Surfaces • Find intersection point along all grid lines with +/- values at endpoints • Edge defined by crossing points
2-D: Adaptive Subdivision • Recursively subdivide cells containing a surface crossing down to a threshold size • Calculate crossing values and connect dots to form polygon edge
2-D: Continuation Methods • Overlay fine-grain uniform subdivision • Find start (seed) point that lies on surface • Find cell containing seed point • “Grow” the set of cells across surface • Evaluate adjacent cells to find next surface crossing cells
Potential Problem #1 : Bounding the Domain Space • May not completely contain the object • May miss disconnected components • Will result in “clipping”
Potential Problem #2 : Discretization Error • Too large uniform cell size: • May not be able to converge (entirely miss the surface) • Too large adaptive threshold cell size: • Misses small, completely-containedfeatures • Coarse resolution model • Incorrect topology: • ambiguous cells • connects or breaks components • Too small cells sizes are inefficient and more susceptible to numerical error
Potential Problem #3 : Disconnected Components • Uniform cell subdivision may find them, but inefficiently • Continuation methods may miss them • must have a seed pt on each component • Related to discretization error
Potential Problem #4 : Ambiguous Surface-Crossing Cells • Many cell vertex polarity configurations are ambiguous • Possible to polygonize in different arrangements • May even result in disjoint polygons • Solutions: • Detect and recursively subdivide • Choose a consistent convention (e.g. always join positive pts)
Moving on to 3-D • Conceptually the same • Discretize space and find crossing points on 12 edges of cube • This can be tricky to connect the crossing points into a polygon correctly
3-D: Finding cell polygon face vertices Algorithmic method: • Begin with any edge-surface intersection point (1) • Proceed to negative corner (white open pts) and then clockwise about the cube face (w.r.t outside) until another intersection point is found (2) • Repeat for each subsequent face (34, 45, 51)
3-D: Tetrahedral Decomposition • Tetrahedra reduce ambiguity and produce correct meshes, but many more polygons result • Diagonal edges cut across cube faces • Must make sure adjacent cubes have aligned tetrahedron edges to preserve topological correctness
3-D Tetrahedral Decomposition • Five Six
3-D Implicit Surface Evaluation • Particle Methods • Turk, Hoppe, Szeliski, Witkin, Heckbert (cool paper) • Constrain particles to exist on isosurface • Allow particles to split and join • Particle seeks to balance space between neighbors • A constrained optimization problem
Collision Detection • Easy to detect colliding blobbies • Evaluate sample points from one object in the implicit equation of another • Deforming the blobbies is more complex • Define collision boundary, F1(p) = F2(p) • Create addition term, G(p) that is added to density function, F(p) to prevent passing through collision boundary • Add terms to G(p) so blobbies preserve volume as they are retarded by collision boundary
Spacetime Constraints • Find min of f(x) = y • Find zeros (roots) of derivative function, f’(x) • Newton Raphson Method does this • Find min of f(X) = Y; X and Y vectors • Same method, but find where Jacobian is equal to zero
Spacetime Constraints • Null space of a function • The inputs of a function for which the function evaluates to zero • Finite differences • Approximate derivatives (inverse of Euler integration) • Solving for all forces in time simultaneously
