1 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Numerical geometry of shapes...

1Numerical geometry of non-rigid shapes Lecture II – Numerical Tools

Numerical geometryof shapes

Lecture II – Numerical Tools

non-rigid

Alex Bronstein


Shape matching blueprints

Isometric embedding

A. Elad & R. Kimmel, 2003


Shape matching blueprints

Compute canonical formsEXTRINSIC SIMILARITY OF CANONICAL FORMS

INTRINSIC SIMILARITY

= INTRINSIC SIMILARITY



Numerical ingredients

Shape discretization

Metric discretization

How to compute geodesic distances?

Discrete embedding problem

How to compute the canonical forms?

Can we do better?

EMBEDDING

EMBEDDING

MATCHING


Discrete Geodesics andShortest Path ProblemsDiscrete Geodesics andShortest Path Problems


Shapes as graphs

Cloud of points Edges Undirected graph=+


Discrete geodesic problem

Local length function

Path length

Length metric in graph


Dijkstra’s algorithm

INPUT: source point

Initialize and for the rest of the graph;

Initialize queue of unprocessed vertices .

While

Find vertex with smallest value of ,

For each unprocessed adjacent vertex ,

Remove from .

OUTPUT: distance map .E.W. Dijkstra, 1959


Troubles with the metric

Inconsistent metric approximation!


Metrication error

Graph induces

inconsistent metric

SOLUTION 1

Change the graph

Both sampling & connectivity

Sampling theorems guaranteeconsistency for some conditions

SOLUTION 2

Change the algorithm

Stick to same sampling

Discrete surface rather than

graph

New shortest path algorithm

SOLUTION 1

Discretized shapeDiscrete metric

SOLUTION 2

Discretized metric


Forest fire

Fermat’s Principle (of Least Action):

Fire chooses the quickest path to travel.

Pierre de Fermat (1601-1665)

Fermat’s Principle (of Least Action):

Fire chooses the shortest path to travel.


Eikonal equation

Source

Equidistant contour

Steepest distance

growth direction

Eikonal equation


Fast marching methods (FMM)

A family of numerical methods for

solving eikonal equation

Simulates wavefront propagation

from a source set

A continuous variant of Dijkstra’s

algorithm

Consistently discretized metric

J.N. Tsitsiklis, 1995; J. Sethian, 1996, R. Kimmel & J. Sethian, 1998; A. Spira & R. Kimmel, 2004


Fast marching

Dijkstra’s update

Vertex updated from

adjacent vertex

Distance computed

from

Path restricted to graph edges

Fast marching update

Vertex updated from

triangle

Distance computed

from and

Path can pass on mesh faces


Fast marching update step

Update from triangle

Compute from

and

Model wave front propagating from

planar source

unit propagation direction

source offset

Front hits at time

Hits at time

When does the front arrive to ?

Planar source


Fast marching update step

is given by the point-to-plane distance

Solve for parameters and using the point-to-plane distance

…after some algebra

where



Uses of fast marching

Geodesic distances

Minimal geodesics

Voronoi tessellation &

sampling

Offset curves


Marching even faster

Heap-based update

Unknown grid visiting

order

Inefficient use of cache

Inherently sequential

Raster scan update

Regular access to

memory

Can be parallelized

Suitable only for regular

grid


Raster scan fast marching

Parametric surface

Parametrization domain sampled on Cartesian grid

Four alternating scans


Raster scan fast marching

4 scans=1 iteration 2 iterations 3 iterations

4 iterations 5 iterations 6 iterations

Several iterations requiredfor non-Euclidean geometries


Marching even faster

Heap-based update

Irregular use of memory

Sequential

Any grid

Single pass,

Raster scan update

Regular access to

memory

Can be parallelized

Only regular grids

Data-dependent

complexity


O. Weber, Y. Devir, A. Bronstein, M. Bronstein & R. Kimmel, 2008

Parallellization

Rotate by 450

On NVIDIA GPU 50msec per distance map on 10M vertices

200M distances per second!


Embedding ProblemsEmbedding Problems


Isometric embedding


Mapmaker’s problem


Mapmaker’s problem

A sphere has non-zero curvature,

therefore, it is not isometric to the plane

(a consequence of Theorema egregium)

Karl Friedrich Gauss (1777-1855)


Minimum distortion embedding




Ingredients:

Discretized shape

Discretized metric

Euclidean embedding space

Embedding is a configuration of points in represented

as a matrix with the Euclidean metric

Embedding distortion (stress) function

e.g., quadratic

Numerical procedure to minimize



Multidimensional scaling (MDS)


Minimization of quadratic stress

Quadratic stress

where

Its gradient

Non-linear non-convex function of variables


Minimization of quadratic stess

Start with some

Repeat for

Steepest descent step

Until convergence

OUTPUT: canonical form

Can converge to local minimum

Minimum defined modulo congruence


Examples of canonical forms

Canonical forms

Near-isometric deformations of a shape


Examples of canonical forms

Embedding distortion limitsdiscriminative power!

J.N. Tsitsiklis, 1995; J. Sethian, 1996, R. Kimmel & J. Sethian, 1998; A. Spira & R. Kimmel, 2004


Non-Euclidean Embedding

Non-Euclidean Embedding


Euclidean embedding


Non-Euclidean embedding


Spherical embedding

Richer geometry than Euclidean (asymptotically Euclidean).

Minimum embedding distortion obtained for shape-dependent radius.


The ultimate embedding space

Embed one shape directly into the other

If shapes are isometric, embedding is distortionless

Otherwise, distortion is the measure of dissimilarity


Generalized embedding problem

A. Bronstein, M. Bronstein & R. Kimmel, 2006


Generalized multidimensional scaling

Representation

How to represent arbitrary points on ?

Metric approximation

How to compute distance terms between arbitrary

points on ?

Numerical procedure

How to minimize stress?



Representation

Triangle

index

Convex

combination

Barycentric

coordinates=+

Triangular mesh

How to represent an arbitrary point on ?



Metric approximation

How to approximate distance between arbitrary

points?

Precompute distances between all pairs of vertices on



Generalized stress

Fix all variables except for

Quadratic in

Convex in



Minimization algorithm

Initialize

Select corresponding to maximum gradient

Compute minimizer

If constraints are active

translate to adjacent triangle

Iterate until convergence…



GMDS in action

CANONICAL FORMS (MDS, 500 points)

MINIMUM DISTORTION EMBEDDING(GMDS, 50 points)



Summary

Discrete geodesic problem

Dijkstra – inconsistent discrete metric

Fast marching – consistentlydiscretized metric


MDS – embedding distortion is an enemy

Non-Euclidean embedding

Generalized embedding

GMDS – embed one surface into the other

Embedding distortion is the measure ofdissimilarity

Date post:	20-Jan-2016
Category:	Documents
View:	227 times
Download:	2 times

1 Numerical geometry of non-rigid shapes Lecture II – Numerical Tools Numerical geometry of shapes...

Documents