1 GEOMETRIE Geometrie in der Technik H. Pottmann TU Wien SS 2007

1www.geometrie.tuwien.ac.at

GEOMETRIE

Geometrie in der Technik

H. PottmannTU Wien

SS 2007


GEOMETRIE

Distance function

Given: geometric object F (curve, surface, solid, …)

Assigns to each point the shortest distance from F

F

p


GEOMETRIE

Distance function

Level sets of the distance function are trimmed offsets

Not smooth at the cut locus


GEOMETRIE

Classical Geometry

Studied the graph surfaces of distance functions: developable surfaces of constant slope

Relation to circle and sphere geometry, cyclographic mapping


GEOMETRIE

PDEs

distance function solves the eikonal equation

Efficient numerical solverson a grid (in R2 and R3)

Fast marching (Sethian, Kimmel,…) Fast sweeping (Danielson, Osher, Tsai,

Zhao,…) Distance functions in manifolds

(Hamilton-Jacobi equations) Signed distance function as

level set function in the level set method


GEOMETRIEPDEs: Comparison of algorithms

Zhao Tsai


GEOMETRIE

Computer-Aided Design

Level sets of distance function: offsets

Offsets and generalized offsets important for NC machining


GEOMETRIE

Computer Visionand Image Processing

Central role in Math. Morphology (dilation, erosion, skeleton,…)

normalization of level set function in level set evolution for segmentation


GEOMETRIE

Robotics

Distance functions on manifolds (configuration space) and derived geodesics or splines for motion planning, also in the presence of obstacles

Collision avoidance


GEOMETRIE

Distance functions in a special manifold: feature sensitive metric

Euclidean

Feature sensitive

ECCV ´04


GEOMETRIE

Further application areas

Pattern Classification: distance fields in high

dimensions; separation of clusters relation to methods from

Computational Geometry Computer Graphics:

unifying implicit representation adaptively sampled distance

fields point cloud processing

Scientific Visualization


GEOMETRIE

Distance functions

d(x) is a distance function if it solves the eikonal equation

For a signed distance function, we admit a

sign change at the set S to which the distance is computed; unlike d it is smooth at S

Geometric meaning of the eikonal equation in R2: all tangent planes of the graph surface have slope 1.


GEOMETRIE

Fast sweeping

Compute distance function on a grid Fast sweeping algorithm (Tsai, Zhao…)

in R2: Grid points (i,j), i=0:Nx-1,j=0:Ny-1 Compute accurate distance values at grid

points close to S Propagate this informtation by sweeping

through the grid.


GEOMETRIE

Fast sweeping

(x+,y+) sweeping: for j=0:Ny-1 for i=0:Nx-1 update d(i,j) Correctly propagates distance

information in directions to the first quadrant

y

x

x


GEOMETRIE

Fast sweeping

(x-,y+) sweeping: for j=0:Ny-1 for i=Nx-1:0 update d(i,j) Correctly propagates distance

information in directions to the second quadrant

y

x

x


GEOMETRIE

Tsai‘s closest point solver

Computes the distance function to a set S on a grid.

Uses 4-neighborhood of (i,j) 1. Initialization: For grid points g close to

S compute and store the exact distance d(g) and a closest point g* on S. These grid points are marked and not updated anymore. The other grid points get distance value ∞


GEOMETRIE

Tsai‘s closest point solver

2. Sweeping: in each of the four sweeps, visit each grid point e that can be updated: A) For each neighbor pl of e compute

B) If set (enforces monotonicity)

C) distance of current grid point e is set to d2(e) =minldl

tmp=:dmtmp and the closest point

e* to e is set to e*=pm*.


GEOMETRIE

Zhao‘s fast sweeping

Computes the distance value at a grid point e only from the distance values of the 4 neighbors (no closest points used)

Key idea: use only two neighbors p1,p2 (depending on the sweep) and estimate a distance value d(e) from their distances d1,d2 by a local approximation of the distance function by the distance function of a straight line L.


GEOMETRIE


For a (x+,y+) sweep:

ep1

p2L


GEOMETRIE


Updating formulae (h=gridsize) (a) if , set

(b) if , set


GEOMETRIE

Comparison of algorithms

Zhao Tsai


GEOMETRIE

Distance fields in the presence of obstacles

The fast sweeping algorithm of Zhao can compute the distance function, considering given obstacles

Set a flag to grid points inside obstacles and do not use them for updating


GEOMETRIE

L. 11: Geodesics

Computation Applications in civil engineering image

processing 3D vision Robotics

Documents

1 GEOMETRIE Geometrie in der Technik H. Pottmann TU Wien SS 2007