Class4

# Class4 - HW-QEM Simplification based of Quadrics(distance...

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CS101.3 2002 1 HW -QEM Simplification based of Quadrics (distance to original planes of the surface).

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CS101.3 2002 2 Set of planes For a vertex v with an associated set of planes, define the distance to be the sum of squared distance to all the planes 2 2 ) ( ) ( ) ( i i T i i i plane d v n v D v E + = = v
CS101.3 2002 3 Quadric error metric This means that for a given plane The fundamental quadric Q is: ) ) ( 2 ) ( ( ) ( 2 2 d v dn v nn v v D T T T + + = c v b Av v v Q T T + + = 2 ) ( 0 ) ( = + d v n T ) , , ( 2 d dn nn Q T = (Rewritten distance eqn)

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CS101.3 2002 4 What have we done Now with a quadric Q = (A, b,c) And the equation: We can compute distances to planes in a nice fashion c v b Av v v Q T T + + = 2 ) (
CS101.3 2002 5 Summary Select candidate edges Allocate a quadric for each vertex v i For each face f i = (j, k, l) compute the quadric Q i n Add the fundamental quadric to the vertex quadrics Q j , Q k , Q l (weighted)

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CS101.3 2002 6 Summary Continued For each edge(v i , v j ) n Apply consistency check n Compute Q i +Q j n Select target position n Place pair in priority queue on cost of v ) ( v Q
CS101.3 2002 7 Summary Continued Repeat until desire approximation n Remove top edge from PrioQ n Perform contraction n Set n For each affected edge (v i , v k ) compute target position and cost update PrioQ v j i i Q Q Q + =

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CS101.3 2002 8 Question How do you represent the world in the computer? What are the properties we want of these data structures?
CS101.3 2002 9 Surface reconstruction How do we acquire 3D Models? n Artists n Scanning data n Laser scanners n Medical scanners n Simulations

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CS101.3 2002 10 Created data Artists create models in the computer using some nice higher order smooth surface representation (nurbs, etc.)
CS101.3 2002 11 Acquired data Scanners & Simulations n Points (laser scanners, other 3d photography type techniques) n Volumes

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CS101.3 2002 12 Laser Scanners How they work – 3d photography Sweep a 1D or 2D sensor over the object – combine measurement.
CS101.3 2002 13 What to do with Points Surface reconstruction from points n Reconstruction of piece-wise linear reconstruction n Cleanup &Simplification n Perhaps fit a higher order smooth surface

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CS101.3 2002 14 Reconstruct Piece-wise Try to construct a nice triangulation at first… Alpha-hulls & crusts n Using Voronoi regions and delaunay triangulation to construct a surface
CS101.3 2002 15 Properties Delaunay triangulation n empty circumcircle for every face n every edge possesses disk not containing any other point closest point outside shrink by ε

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CS101.3 2002 16 Voronoi Diagram Dual of Delaunay n regions which are closest to a given site n Good for talking about neighbors…
CS101.3 2002

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Class4 - HW-QEM Simplification based of Quadrics(distance...

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