1 numerical geometry of non-rigid shapes shortest path problems numerical geometry of non-rigid...

1Numerical geometry of non-rigid shapes Shortest Path Problems

Numerical geometry of non-rigid shapes

Shortest path problems

Alexander Bronstein, Michael Bronstein, Ron Kimmel© 2007 All rights reserved


Shortest path problem

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Shortest paths in graphs

Object is sampled at vertices

Approximated as undirected graph

Edges represent adjacent samples

Length function measures local length that can be

approximated as

Path in graph between two vertices and given by

Path length – sum of lengths of all edges



Approximate the intrinsic metric as the shortest path length in

graph

Alternative formulation: given a source point , compute the

distance function

Bellman’s principle of optimality: if is the shortest path

between and , and is a point on it, then the sub-paths

and are the shortest paths between and ,

and and , respectively.



Consequence: there exists some such that

How to select such a point ?

Since has to be the shortest distance,

Problem of computing is reduced to smaller sub-problems

of computing

Dynamic programming


Dijkstra’s algorithm

Dynamic programming successive approximation procedure

Initially, set ; for the rest of the vertices, set

Mark all vertices as unprocessed

While there are still unprocessed vertices

Select with the smallest

Update all adjacent points

Mark as processed

Return the distance function


Dijkstra’s algorithm

Every vertex is processed only once

Update step propagates distance to adjacent vertices

If the previously computed distance is shorter, value is not updated

Typically, there are vertices in the graph

Extraction of the minimum distance can be done in

using a heap

Total algorithm complexity


Metrication error

“True” distance is

Distance measured in graph

Error persists, no matter how the sampling is refined


Metrication error

No flaw in Dijkstra’s algorithm

The graph induced the metric, inconsistent with the Euclidean

metric

Problem can be resolved by changing the sampling

So far the path was restricted to graph edges

Represent the surface as a triangle mesh

Allow the path to pass on mesh faces

Fast marching – a “continuous Dijkstra”

algorithm

Numerical solution of the eikonal equation


Fast marching

Dijkstra

Path restricted to graph

edges

Adjacent vertices are

updated

A vertex is updated from

another vertex

Fast marching

Path can pass on mesh

edges

Adjacent triangles are

updated

A vertex is updated from two

vertices within a triangle


Fast marching update step

Given a triangle with vertices , ,

Given distances and

We need to compute

Model planar source described by

Wavefront propagates from the source

Hits at time


Fast marching update step

From triangle geometry and ,

find the source parameters

Propagation direction: (1 DOF)

Offset: (1 DOF)

Solve

Quadratic equation with two solutions

corresponding to

Smallest solution is inconsistent

and is discarded

Compute


Consistency & monotonicity

Consistency condition:

Wavefront hits and before hitting

Monotonicity condition:

Increase of or increases


Geometric interpretation

Consistency condition: update direction must form acute angles with

normals to triangle edges

Monotonicity condition: update direction must come from within the

triangle

ConsistentMonotoneConsistent & Monotone


Monotonicity

Monotonicity: update must come from within the triangle

If is not within the triangle, force it to lie inside

The update direction will coincide with one of the edges

Dijkstra-type update


Consistency: acute triangle

When triangle is acute, both conditions hold for any direction


Consistency: obtuse triangle

When triangle is obtuse, some consistency condition is violated for

some directions


Fast marching on obtuse meshes

Inconsistent solution if the mesh contains obtuse triangles

Remeshing is costly

Solution: split obtuse triangles by adding virtual connections to

non-adjacent vertices

Done as a pre-processing step in


Fast marching algorithm

Geodesic distances

Minimal geodesics

Voronoi tessellation


Fast marching on parametric surfaces

Object is given as a parametric surface

Parametrization domain is sampled on a regular grid (geometry

image)

Fast marching can be performed entirely in the parametrization domain

Update step accounts for the surface first fundamental form to

measure geodesic distances on the surface

Surface itself is not required, knowledge of the first fundamental form

and grid connectivity is sufficient

Obtuse triangles are split by adding virtual connections to

non-adjacent grid points


Heap-based grid update

Fast marching on triangular meshes and geometric images both use

Dijstra-type heap-based grid update

Update order is unknown and data-dependent

Inefficient use of memory system and cache

Inherently sequential algorithm – next update depends on previous one


Raster scan fast marching

Update the grid in a raster scan order

Dates back to Danielsson’s algorithm

In Euclidean case, geodesics are straight lines

Each raster scan covers ¼ of the possible directions of the geodesics

Four raster scans covering four quadrants are sufficient to compute a

Euclidean distance map



Generally, geodesics are curved in parametrization domain

Raster scans have to be repeated to produce a convergent solution

Number of iterations depends on geometry and parametrization

Practically, few iterations are required

1 iteration 2 iterations 3 iterations



What we lost:

No more a one-pass algorithm

Computational complexity is data-dependent

What we gained:

Coherent memory access, efficient use of cache

No heap, each iteration is

Raster scans can be parallelized


Marching even faster

All updates performed on a diagonal

are independent

Can be computed simultaneously

Uneven load


Marching even faster

Rotate scan directions by 45 degrees

All updates performed along a row or

column can be parallelized

Balanced load

Suitable for SIMD architecture and GPUs

x100 speedup with comparable accuracy


We have all numerical tools required to approximate the intrinsic

geometry of objects

Graph representation may introduce metrication error

Intrinsic metric approximated using Dijkstra’s algorithm may be

inconsistent

Fast marching is a consistent numerical tool for measuring

geodesic distances

Fast marching can be parallelized

Conclusions so far…

1 numerical geometry of non-rigid shapes shortest path problems numerical geometry of non-rigid...

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