Reconstruction and Geometric
Algorithms
Romain Vergne
Prashant Goswami
2013 - 2014
PLAN
• Introduction
• Parameterized / Ordered data
– Curve Interpolation
– Surface Interpolation
– Curve Approximation
– Surface Approximation
Least Squares
___ : y = ax + b
Least Squares
Line : y = ax + b
Least Squares
___ : y = ax + b
Least Squares
Gaussian
___ : y = ax + b
Weighted Least Squares
Gaussian
Parabola : y = ax2 + bx + c
Weighted Least Squares
Gaussian
Parabola : y = ax2 + bx + c
Weighted Least Squares
Gaussian
Cubic : y = ax3 + bx2 + cx + d
Weighted Least Squares
Gaussian
Cubic : y = ax3 + bx2 + cx + d
Principal Component Analysis
On two random variables x and y
Principal Component Analysis
On two random variables x and y
Principal Component Analysis
On an image
Total Least Squares
Scattered Points
Total Least Squares
y= ax + b
Total Least Squares
ax + by + c = 0
Total Least Squares
Comparison
Moving Least Squares
Moving Least Squares
w = box function
Moving Least Squares
w = triangle function
Moving Least Squares
w = Gaussian function
Moving Least Squares
Neighborhood influence, σ = 1
Moving Least Squares
Neighborhood influence, σ = 2
Moving Least Squares
Neighborhood influence, σ = 3
Moving Least Squares
Neighborhood influence, σ = 5
Moving Least Squares
Neighborhood influence, σ = 10
PLAN
• Introduction
• Parameterized / Ordered data
• Reconstruction of given data
– Explicit Methods
– Implicit Methods
Convex Hull
Incremental
Convex Hull
Incremental
Convex Hull
Incremental
Convex Hull
Incremental
Convex Hull
Incremental
Convex Hull
Incremental
Convex Hull
Incremental
Convex Hull
Incremental
Convex Hull
Incremental
Convex Hull
Incremental
Convex Hull
Incremental
Convex Hull
Incremental
Convex Hull
Incremental
Convex Hull
Quick Hull
Convex Hull
Quick Hull
Convex Hull
Quick Hull
Convex Hull
Quick Hull
Convex Hull
Quick Hull
Convex Hull
Quick Hull
Convex Hull
Quick Hull
Convex Hull
Quick Hull
Convex Hull
Quick Hull
Convex Hull
Quick Hull
Convex Hull
Quick Hull
Convex Hull
Quick Hull
Convex Hull
Quick Hull
Convex Hull
Quick Hull
Convex Hull
Quick Hull
Delaunay Triangulation
Properties
Delaunay Triangulation
Properties
Voronoi Diagram Ѵ of E
Delaunay Triangulation
Properties
Delaunay Triangulation Ϯ of E
Delaunay Triangulation
Properties
Delaunay Triangulation Ϯ of E
Delaunay Triangulation
Properties
Property of empty circumscribed circles
Delaunay Triangulation
Properties
Property of empty circumscribed circles
Delaunay Triangulation
Properties
Property of empty circumscribed circles
Delaunay Triangulation
Properties
Property of empty circumscribed circles
Delaunay Triangulation
Properties
Property of empty circumscribed circles
Delaunay Triangulation
Properties
Property of empty circumscribed circles
Delaunay Triangulation
Properties
Property of empty circumscribed circles
Delaunay Triangulation
Properties
Property of empty circumscribed circles
Delaunay Triangulation
Properties
Property of empty circumscribed circles
Delaunay Triangulation
Properties
Property of empty circumscribed circles
Delaunay Triangulation
Properties
Center of circumscribed circles = Vertices of Voronoi diagram
Delaunay Triangulation
Properties
Center of circumscribed circles = Vertices of Voronoi diagram
Delaunay Triangulation
Properties
Example of non-Delaunay triangulation
Delaunay Triangulation
Properties
Example of non-Delaunay triangulation
Delaunay Triangulation
Incremental Algorithm
We consider an initial triangulation and you
want to add P
– find the triangle containing P
– destroy the triangles that pose problem
– construct new triangles
– add new triangles to the initial triangulation
Delaunay Triangulation
Algorithm Flip
– find the triangle containing P
– destroy the triangles that pose problem
– construct new triangles
– add new triangles to the initial triangulation
Delaunay Triangulation
Algorithm Flip
Delaunay Triangulation
Construction by Convex Hull
Points Pi( xi, yi )
Delaunay Triangulation
Construction by Convex Hull
Points Pi projected( xi, yi , Pi.Pi)
Delaunay Triangulation
Construction by Convex Hull
Convex hull of Pi
Delaunay Triangulation
Construction by Convex Hull
Lower Convex hull of Pi
Delaunay Triangulation
Construction by Convex Hull
Projection of the point on plane z = 0
Delaunay Triangulation
Construction by Convex Hull
Ellipse (Intersection of a plane with the convex hull)
Delaunay Triangulation
Construction by Convex Hull
Projected ellipse = circle circumscribing the triangle
Delaunay Triangulation
Construction by Convex Hull
Delaunay Triangulation in dim d = Convex hull in dimension d+1
Algorithm Crust
2D
Algorithm Crust
2D
Delaunay Triangulation T of P and S = center of circumscribed circles
Algorithm Crust
2D
Delaunay Triangulation T of P and S = center of circumscribed circles
Algorithm Crust
2D
Delaunay Triangulation T’ of P  S
Algorithm Crust
2D
Crust on P = Edges in T’ connecting points in P
Algorithm Crust
2D
Crust on noisy data
Algorithm Crust
2D
Crust on noisy data
Algorithm Crust
3D
Bibliography
• Cours de Nicolas Szafran 2011/2012
(http://www-ljk.imag.fr/membres/Nicolas.Szafran/)
• Scattered Data Interpolation and
Approximation for Computer Graphics
(Siggraph Asia course 2010)
• Implicit surface reconstruction from point
clouds (Johan Huysmans’s thesis)