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GATE-540 1 Reconstruction from Point Cloud (GATE-540) Dr.Çağatay ÜNDEĞER Instructor Middle East Technical University, GameTechnologies & General Manager SimBT Inc. e-mail : cagatay@undeger.com Game Technologies Program – Middle East Technical University – Spring 2010 Reference: Hugues et al, Surface Reconstruction from Unorganized Points

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GATE-540 2 Outline Reconstruction from point clouds

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GATE-540 3 Goals Develop 3D Analysis Algorithms: –Reconstruction –Segmentation –Feature Detection –Labeling –Matching –Classification –Retrielval –Recognition –Clustering

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GATE-540 4 Goal of Surface Reconstruction Have a set of unorganized points Reconstruct a surface model that best approximates the real surface

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GATE-540 5 Data Sources Surfaces from range data Surfaces from contours (slices of images) Interactive surface sketching

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GATE-540 6 Terminology A surface: compact, orientable two dimentional monifold A simplicial surface: A piecewise linear surface with triangular faces X = {x 1,..., x n } : sampled data points on or near an unknown surface M M = {y 1,..., y n } : real points on unknown surface M that maps X

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GATE-540 7 Terminology p-dense : ?? e i or δ : maximum error of data source x i = y i ± e i Features of M that are small compared to δ will not be recoverable. It is not possible to recover features of M in regions where insufficient sampling has occured.

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GATE-540 8 Problem Statement & Algorithm Goal: –To determine a surface N that approximates an unknown surface M An algorithm proposed by Hugues et al, 1992. Consists of two stages: 1) Estimate signed geometric distance to the unknown surface M 2) Estimate unknown surface M using a contouring algorithm

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GATE-540 9 Define a Signed Distance Function Associate an oriented plane (tangent plane) with each of the data points. Tangent plane is a local linear approximation to the surface. Used to define signed distance function to surface. N sampled point tangent plane signed distance estimated surface point

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GATE-540 10 Tangent Plane Nbhd(x i ) : k-neighborhood of x i Tangent plane center of x i (O i ) : centroid of Nbhd(x i ) Tangent plane normal of x i (N i ) : determined using principle component analysis of Nbhd(x i ) sampled point x i k-neighborhood of x i

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GATE-540 11 Principle Component Analysis Involves a mathematical procedure that transforms a number of possibly correlated variables into a smaller number of uncorrelated variables called principal components. Normal of tangent plane might be found in opposite direction ??

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GATE-540 12 Consistent Tangent Plane Orientation If two neigbors are consistently oriented, –Their tangent planes should be facing almost the same direction. –Otherwise one of them should be flipped.

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GATE-540 13 Consistent Tangent Plane Orientation Model the problem as a graph optimization Each O i will have a corresponding V i in graph Connect V i and V j is O i and O j are “sufficiently” close. Cost on edges encodes the degree to which N i and N j are consistently oriented. Maximize the total cost on the graph. –NP-hard –Use an approximation algorithm.

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GATE-540 14 Euclidian Minimum Spanning Three (EMST) Surface is assumed to be a single connected component, –The graph should be connected. A simple connected graph for a set of points that tends to connect neighbors is EMST. EMST over tangent planes is not sufficiently dense! –Enrich it by adding an edge (i,j) if o i is in the k-neighborhood of o j. –Result is called Reimannian Graph.

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GATE-540 15 Reimannian Graph A connected graph Reimannian Graph EMST over tangent planes

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GATE-540 16 Simple Algorithm Arbitrarily choose an orientation for some plane Propogate the orientation to neigbors in Reimannian Graph. Order of propagation is important!

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GATE-540 17 A Good Propagation Order Favor propagation from o i to o j if unoriented planes are nearly parallel. Assign cost as 1 – |N i *N j | –A cost is small if parallel A fovorable propagation order: –Travers mimimum spanning tree (MST) dot product

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GATE-540 18 Assigning Orientations Assign +z orientation to point in graph that has largest z coordinate. Travers the tree in depth first order. Oriented tangent planes

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GATE-540 19 Computing Distance Function Signed distance f(p) f(p) of a point p to unknown surface M: –distance between p and closest point z Є M –multiplied by ±1 depending on the side of the surface p lies in –z is unknown, thus use closest o i

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GATE-540 20 Computing Distance Function z = p – ((p-o i )*N i )*N i If d(z,X) < (p+δ) then// graph is p-dense f(p) = (p-o i )*Ni Else f(p) = undefined Defined ones create a zero set (estimate for M)

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GATE-540 21 Contour Tracing Contour tracing is to extract iso-surface from a scalar function. A variation of matching cubes is used Cube sizes should be less than p+δ –But larger increases the speed and reduces the number of triangle facets created

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GATE-540 22 Contour Tracing Visit the cubes only intersect the zero set. No intersection if the signed distance is unefined in any vertex within a cube.

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GATE-540 23 Collapse Edges Contain triangles with arbitrary poor ascpect ratio. Collapse edges in post processing

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GATE-540 24 Collapse Edges

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GATE-540 25 Sample Results

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