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Reconstruction from Point Cloud(GATE-540)

Dr.Çağatay ÜNDEĞER

InstructorMiddle East Technical University, GameTechnologies

&

General ManagerSimBT 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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Outline

• Reconstruction from point clouds

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Goals• Develop 3D Analysis Algorithms:

– Reconstruction– Segmentation– Feature Detection– Labeling– Matching – Classification– Retrielval– Recognition– Clustering

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Goal of Surface Reconstruction

• Have a set of unorganized points• Reconstruct a surface model that best

approximates the real surface

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Data Sources

• Surfaces from range data• Surfaces from contours (slices of images)• Interactive surface sketching

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Terminology

• A surface: compact, orientable two dimentional monifold

• A simplicial surface: A piecewise linear surface with triangular faces

• X = {x1, ..., xn} : sampled data points on or near an unknown surface M

• M = {y1, ..., yn} : real points on unknown surface M that maps X

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Terminology

• p-dense : ??

• ei or δ : maximum error of data source

xi = yi ± ei

• 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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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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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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Tangent Plane

• Nbhd(xi) : k-neighborhood of xi

• Tangent plane center of xi (Oi) : centroid of Nbhd(xi)

• Tangent plane normal of xi (Ni) : determined using principle component analysis of Nbhd(xi)

sampled point xi

k-neighborhood of xi

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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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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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Consistent Tangent Plane Orientation

• Model the problem as a graph optimization• Each Oi will have a corresponding Vi in graph

• Connect Vi and Vj is Oi and Oj are “sufficiently” close.

• Cost on edges encodes the degree to which Ni and Nj are consistently oriented.

• Maximize the total cost on the graph.– NP-hard– Use an approximation algorithm.

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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 oi is in the k-neighborhood of oj.

– Result is called Reimannian Graph.

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Reimannian Graph

• A connected graph

Reimannian GraphEMST

over tangent planes

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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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A Good Propagation Order

• Favor propagation from oi to oj if unoriented planes are nearly parallel.

• Assign cost as 1 – |Ni*Nj|

– A cost is small if parallel

• A fovorable propagation order:– Travers mimimum spanning tree (MST)

dot product

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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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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 oi

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Computing Distance Function

z = p – ((p-oi)*Ni)*Ni

If d(z,X) < (p+δ) then // graph is p-dense

f(p) = (p-oi)*Ni

Else

f(p) = undefined

• Defined ones create a zero set (estimate for M)

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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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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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Collapse Edges

• Contain triangles with arbitrary poor ascpect ratio.

• Collapse edges in post processing

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Collapse Edges

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Sample Results