reconstruction and geometric...

Reconstruction and Geometric Algorithms

Romain Vergne

2014 - 2015

InterpolationFunction Parameters

Linear Shepard

Shepard Polynomial

Linear Shepard

Shepard Polynomial

Hermite Spline

Tensor Product

LinearHermite

Laplace/Poisson InterpolationFunctions f( x, y ) = z

Original

3D Version

Subdivision Curves

Chaikin 1 Chaikin 2

Catmull-Clark (approx) 4 points (interp)

Subdivision Surfaces

Data Loop

Catmull-Clark

Subdivision Surfaces

Triangulated Surface Subdivision Butterfly

Triangulated Surface Butterfly modified

Least Squares

y = ax + b y = ax2 + bx + c

y = ax3 + bx2 + cx + d

Total Least Squares

ax + by + c = 0

PCA LS vs. TLS

Moving Least Squares

Convex Hull

Incremental

Quick Hull

Delaunay Triangulation

Relationship with the convex hull

• Introduction

• Parameterized / Ordered data

• Reconstruction of unorganized data

– Explicit methods

– Implicit methods

Implicit Methods

Explicit vs. Implicit

Data in the reference frame Oxy

Line fitting using least square

Coordinate system Bxy attached to the line

Approximating Y = f( X ) in the coordinate system Bxy

Reorientation of Normals

Choice of neighbors of each point

Determination of reference

Obtaining non-oriented normals

Objective: To re-orient normals

Construction of the proximity graph G of the set of points Pi

Edge e = ( i, j ) : d( Pi, pj ) ≤ r and i ≠ j

Valuation of the graph v(e) (value of the edge e(i, j))v(e) = 1 - |ni nj|

Computation of A : Minimum spanning tree of G

Before re-orientation

Traverse the tree and re-orient

Points + Normals. Iso-values f( x ) = 0

Corresponding Surface

Effect of the selection of neighborhood

Curve ReconstructionScattered Data

Points as input Pi = ( xi, yi )

Curve ReconstructionAlgorithm

Triangulation (as per Delaunay)

Triangulation (as per Delaunay)And associated values zi

Triangular mesh with plane Pi = ( xi, yi )And associated values zi

How to determine the isovalue curve (piecewise linear)

(A) : Binary marking peaks Pi

Partition the heights in 2 sets

Curve ReconstructionAlgorithmTraversal of triangles and construction of isovalue curve. For

each triangle [S1, S2, S3]

Curve ReconstructionAlgorithmTraversal of triangles and construction of isovalue curve. For

each triangle [S1, S2, S3]

Curve ReconstructionScattered DataTraversal of triangles and construction of isovalue curve. For

each triangle [S1, S2, S3], 8 possible markings

Curve ReconstructionScattered DataTraversal of triangles and construction of isovalue curve. For

each triangle [S1, S2, S3], 8 possible markings

Curve ReconstructionAlgorithmTraversal of triangles and construction of isovalue curve

Same sign

Curve ReconstructionAlgorithmTraversal of triangles and construction of isovalue curve.

Same sign : no intersection with the plane z = v

Different signs

Different signs: intersection with the plane z = v

Curve ReconstructionAlgorithm(B) Traversal of triangles and construction of isovalue curve.

Different signs: intersection with the plane z = v

Objective : To find the segment AB

Intersection of the plane with edge PQ

For each triangle, calculate 0 or 1 segment

Computed data = isovalue curve

Curve ReconstructionAlgorithm(C) : Strategy for path of triangles

A. In no particular order

How to obtain the isovalue curve ?

B. Follow up curve

(0) Initial triangulation

B. Follow up curve

(1) Marking non-intersection triangles

B. Follow up curve

(1) Marking non-intersection triangles

B. Follow up curve

(2) To : open components (external stopped)

B. Follow up curve

(2) b : open components (successive neighborhoods)

B. Follow up curve

(3) a : closed components (finding departure)

B. Follow up curve

(3) b : closed components (successive neighborhoods)

B. Follow up curve

Marching Squares

Same principle with square !

Marching Squares

Same principle with square !https://www.youtube.com/watch?v=uChG0Ziy74k

With Normals

Points + Normals

With Normals

Embedding in a regular grid

With Normals

Strategies : Closest point / average distances (MLS) ….

With Normals

Signed distance < 0

With Normals

Signed distance = 0

With Normals

Signed distance > 0

With Normals

http://www.labri.fr/perso/guenneba/videos/APSS_sig07.avi

Same approach but using spheres instead of planes

Isovalue Surfaces

Marching Cubes

15 families for 256 situations

Marching Cubes

15 families for 256 situations

https://www.youtube.com/watch?v=LfttaAepYJ8https://www.youtube.com/watch?v=LC0IbOZ0jkw

Marching TetrahedronsDecomposition of a cube into 6 tetrahedrons

Marching Tetrahedrons

[CDEG]

Decomposition of a cube into 6 tetrahedrons

[CDEG], [DFGH],

[CDEG], [DFGH], [DEFG]

[CDEG], [DFGH], [DEFG], [BCDE]

[CDEG], [DFGH], [DEFG], [BCDE], [BDEF],

[CDEG], [DFGH], [DEFG], [BCDE], [BDEF], [ABCE]

[CDEG], [DFGH], [DEFG], [BCDE], [BDEF], [ABCE] Non-minimal decomposition

But with good neighboring connection

For a tetrahedron

Marching TetrahedronsFor a tetrahedron : 16 configurations

No intersection with the iso-value surface

Vertices with same sign

Vertex A has different sign than B, C and D

Intersection of 3 edges [AB], [AC], [AD] with the iso-value

Triangle [IJK] obtained

Vertices A, C have different sign than B, D

Intersection of 3 edges [AB], [AD], [CB], [CD] with the iso-value

2 triangles [IJK], [JKL]

Intersection of 3 edges [AB], [AD], [CB], [CD] with the iso-value

2 triangles [IJK], [JKL] OR 2 triangles [IJL], [IKL]

Marching TetrahedronsSmooth

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)

• The Method of Least Squares (Steven J. Miller)

• Course of Gael Guennebad (moving least squares) http ://www.labri.fr/perso/guenneba/

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