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

1Numerical geometry of non-rigid shapes Shortest Path Problems

Shortest Path Problems

As regards obstacles, the shortest distance between two points can

be a curve.

B. Brecht

How to compute the intrinsic metric?

So far, we represented itself.

Our model of rigid shapes as metric spaces involves

the intrinsic metric

Sampling procedure requires as well.

We need a tool to compute geodesic distances on .

Shortest path problem

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Shapes as graphs

Sample the shape at vertices .

Represent shape as an undirected graph

set of edges

representing adjacent vertices.

Define length function

measuring local distances as

Euclidean ones,

Shapes as graphs

Path between is an ordered set of connected edges

where and .

Path length = sum of edge lengths

Geodesic distance

Shortest path between

Length metric in graph

Approximates the geodesic distance on the shape.

Shortest path problem: compute and

between any .

Alternatively: given a source point , compute the

distance map .

Bellman’s principle of optimality

Let be shortest path between

and a point on the path.

Then, and are

shortest sub-paths between , and .

Suppose there exists a shorter path .

Contradiction to being shortest path.

Richard Bellman(1920-1984)

Dynamic programming

How to compute the shortest path between source and on ?

Bellman principle: there exists such that

has to minimize path length

Recursive dynamic programming equation.

Edsger Wybe Dijkstra (1930–2002)

[‘ɛtsxər ‘wibə ‘dɛɪkstra]

Dijkstra’s algorithm

Initialize and for the rest of the graph;

Initialize queue of unprocessed vertices .

Find vertex with smallest value of ,

For each unprocessed adjacent vertex ,

Remove from .

Return distance map .

Dijkstra’s algorithm

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Munich

Prague

Vienna

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Dijkstra’s algorithm – complexity

While there are still unprocessed vertices

Find and remove minimum

For each unprocessed adjacent vertex

Perform update

Every vertex is processed exactly once: outer iterations.

Minimum extraction straightforward complexity:

Can be reduced to using binary or Fibonacci heap.

Updating adjacent vertices is in general .

In our case, graph is sparsely connected, update in .

Total complexity: .

Troubles with the metric

Grid with 4-neighbor connectivity.

True Euclidean distance

Shortest path in graph (not unique)

Increasing sampling density does

not help.

Metrication error

4-neighbor topology

Manhattan distance

Continuous

Euclidean distance

8-neighbor topology

Graph representation induces an inconsistent metric.

Increasing sampling size does not make it consistent.

Neither does increasing connectivity.

Metrication error

How to approximate the metric consistently?

Solution 1

Stick to graph representation.

Change connectivity and sampling.

Under certain conditions consistency is

guaranteed.

Solution 2

Stick to given sampling (and connectivity).

Compute distance map on a surface in some

representation (e.g., mesh).

Requires a new algorithm.

Fast marching algorithms

Imagine a forest fire…

Forest fire

Fire starts at a source at

Propagates with constant velocity

Arrives at time to a point

Fermat’s (least action) principle:

The fire chooses the quickest path

to travel.

Governs refraction laws in optics

(Snell’s law) and acoustics.

Fire arrival time =

distance map from source.

Distance maps on surfaces

Distance map on surface

Mapped locally to the tangent space

A small step in the direction changes the distance by

is directional derivative in the direction .

Intrinsic gradient

For some direction ,

The perpendicular direction is the direction of steepest

change of the distance map.

is referred to as the intrinsic gradient.

Formally, the intrinsic gradient of

function at a point

is a map

satisfying for any

Extrinsic gradient

Consider the distance map as a function .

The extrinsic gradient of at a point is a map

satisfying for any direction

In the standard Euclidean basis

Usually called “the gradient” of .

What is the connection between intrinsic and extrinsic gradients?

Intrinsic gradient = projection of

extrinsic gradient on tangent plane

In coordinates of a parametrization

is the Jacobian matrix

whose columns span .

Intrinsic and extrinsic gradients

Eikonal equation

Let be a minimal geodesic between and .

The derivative

is the fire front propagation direction.

In arclength parametrization .

Fermat’s principle:

Propagation direction = direction of steepest increase of .

Geodesic is perpendicular to the level sets of on .

Eikonal equation

Eikonal equation (from Greek εικων)

Hyperbolic PDE with boundary

condition

Minimal geodesics are

characteristics.

Describes propagation of waves

in medium.

Eikonal equation

Let be a minimal geodesic between and .

The derivative

is the fire front propagation direction.

In arclength parametrization .

Fermat’s principle:

Propagation direction = direction of steepest increase of .

Geodesic is perpendicular to the level sets of on .

Uniqueness of solution

In classic PDE theory, a solution

is a continuous differentiable

function satisfying

PDE theory guarantees existence

and uniqueness of solution.

Distance map is not everywhere

differentiable.

Solution is not unique!

1D example

We need assistance of a super-creature…

Sub- and super-derivatives (1D case)

Superderivative: the set of all slopes above the graph

Subderivative: the set of all slopes below the graph

where is differentiable.

Viscosity solution

is a viscosity solution of the

1D eikonal equation if

Monotonicity: viscosity solution

does not have local maxima.

The largest among all

Existence and uniqueness

guaranteed.

Not a viscosity solution

Viscosity solution

Fast marching methods (FMM)

A family of numerical methods for

solving eikonal equation.

Finds the viscosity solution =

distance map.

Simulates wavefront propagation

from a source set.

A continuous variant of Dijkstra’s

algorithm.

Consistently approximate the

intrinsic metric on the surface.

Fast marching algorithm

Initialize and mark it as black.

Initialize for other vertices and mark them as green.

Initialize queue of red vertices .

Repeat

Mark green neighbors of black vertices as red (add to )

For each red vertex

For each triangle sharing the vertex

Update from the triangle.

Mark with minimum value of as black (remove from )

Until there are no more green vertices.

Return distance map .

Update step

Dijkstra’s update

Vertex updated from

adjacent vertex

Distance computed

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

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

Assume w.l.o.g. and .

is given by the point-to-plane distance

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

In vector notation

where , , and .

In a non-degenerate triangle matrix is full-rank

Apparently, we have two equations with three variables.

However, is a unit vector, hence .

where .

Substitute and obtain a quadratic equation

Causality condition

Quadratic equation is satisfied by both

Two solutions for

Causality: front can propagate only

forward in time.

Causality condition

In other words

has to form obtuse angles with both

triangle edges .

Causality is required to obtain

consistent

approximation of the distance map.

Smallest solution for is inconsistent

and is discarded.

If largest solution is consistent, live the

largest solution!

Monotonicity condition

Viscosity solution has to be a monotonically increasing function.

Monotonicity condition: increase when or increase.

In other words:

Differentiate

w.r.t obtaining

Substitute

Monotonicity satisfied when both coordinates of

have the same sign.

is positive definite

Causality condition:

Monotonicity condition:

At least one coordinate

of is negative

Since we have

Rows of are orthogonal to triangle edges

Monotonicity condition:

Geometric interpretation:

must form obtuse angles with normals

to triangle edges.

Said differently:

must come from within the triangle.

Monotonicity condition: update direction

must come from within the triangle.

If it does not, project inside the triangle.

will coincide with one of the edges.

Update will reduce to Dijkstra’s update

One-sided update

Solve for the quadratic equation

Compute propagation direction

If monotonicity condition is violated,

Fast marching update

Acute triangle

All directions in the triangle

satisfy consistency and

monotonicity conditions.

Consistency and monotonicity encore

Monotonicity

Consistency

Obtuse triangle

Some directions in the triangle

violate consistency condition!

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

MATLAB® intermezzo

Parametrization of over .

Compute distance map ,

from source .

Chain rule

Extrinsic gradient in parametrization coordinates

Intrinsic gradient in parametrization coordinates

Eikonal equation on parametric surfaces

Eikonal equation in parametrization coordinates

Solve eikonal equation in parametrization domain

March on discretized parametrization domain.

We need to express update step in parametrization coordinates.

Fast marching on parametric surfaces

Cartesian sampling of with unit step.

Some connectivity (e.g. 4- or 8-neighbor).

Vertex updated from triangle

Assuming w.l.o.g.

or in matrix form

Inner product matrix

Describes triangle geometry.

lengths of the edges.

cosine of the angle.

Substitute into the update quadratic equation

Only first fundamental form coefficients and grid connectivity are

required for update.

Can measure distances when only surface gradients are known.

Heap-based grid update

Fast marching and Dijkstra’s algorithm use heap-based grid update.

Next vertex to be updated is decided by extracting the smallest .

Update order is unknown and data-dependent.

Inefficient use of memory system and cache.

Inherently sequential algorithm – next update depends on previous

Can we do better?

Regular access to memory (known in advance).

Vectorizable (parallelizable) algorithm.

Marching even faster

Danielsson’s algorithm: update the grid in a raster scan order

In Euclidean case, parametrization is trivial.

Geodesics are straight lines in parametrization domain.

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

Euclidean distance map computed by four alternating raster scans.

Raster scan fast marching

Generally, geodesics are curved in

parametrization domain.

Raster scans have to be repeated to

produce a convergent solution.

Iterative algorithm.

Number of iterations depends on

geometry and parametrization.

Practically, few iterations are

required.

1 iteration

2 iterations

3 iterations

4 iterations

5 iterations

6 iterations

Raster scan fast marching

What we lost:

No more a one-pass algorithm.

Computational complexity is data-dependent.

What we found:

Coherent memory access, efficient use of cache.

No heap, each iteration is .

Raster scans can be parallelized.

BBK, "Parallel algorithms for approximation of distance maps on parametric surfaces”, 2007

Parallellization

Rotate scan directions by 450.

All updates performed along a row or column can be parallelized.

Constant CPU load – suitable for SIMD architecture and GPUs.

Parallel marching

Rotate scan directions by 450.

All updates performed along a row or column can be parallelized.

Constant CPU load.

Suitable for SIMD architecture and GPUs.

GPU implementation computes geodesic on grid with 10,000,000

vertices in less than 50 msec.

About 200 million distances per second!

Minimal geodesics

We have a numerical tool to compute geodesic distance.

Sometimes, the shortest path itself is needed.

Minimal geodesics are characteristics of the eikonal equation.

In other words:

Along geodesic, eikonal equation becomes an ODE

with initial condition .

Solve the ODE for .

Minimal geodesics

To find a minimal geodesic between two points

Compute distance map from to all other points.

Starting at , follow the direction of until is reached.

Steepest descent on the distance map.

In the parametrization coordinates

Let be the preimage of in

Minimal geodesics

Substitute into characteristic equation

Steepest descent on surface = scaled steepest descent in

parametrization domain.

Uses of fast marching

Geodesic distances

Minimal geodesics

Voronoi tessellation &

sampling

Offset curves

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