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2www.geometrie.tuwien.ac.at
GEOMETRIE
Distance function
Given: geometric object F (curve, surface, solid, …)
Assigns to each point the shortest distance from F
F
p
3www.geometrie.tuwien.ac.at
GEOMETRIE
Distance function
Level sets of the distance function are trimmed offsets
Not smooth at the cut locus
4www.geometrie.tuwien.ac.at
GEOMETRIE
Classical Geometry
Studied the graph surfaces of distance functions: developable surfaces of constant slope
Relation to circle and sphere geometry, cyclographic mapping
5www.geometrie.tuwien.ac.at
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
7www.geometrie.tuwien.ac.at
GEOMETRIE
Computer-Aided Design
Level sets of distance function: offsets
Offsets and generalized offsets important for NC machining
8www.geometrie.tuwien.ac.at
GEOMETRIE
Computer Visionand Image Processing
Central role in Math. Morphology (dilation, erosion, skeleton,…)
normalization of level set function in level set evolution for segmentation
9www.geometrie.tuwien.ac.at
GEOMETRIE
Robotics
Distance functions on manifolds (configuration space) and derived geodesics or splines for motion planning, also in the presence of obstacles
Collision avoidance
10www.geometrie.tuwien.ac.at
GEOMETRIE
Distance functions in a special manifold: feature sensitive metric
Euclidean
Feature sensitive
ECCV ´04
11www.geometrie.tuwien.ac.at
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
12www.geometrie.tuwien.ac.at
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.
13www.geometrie.tuwien.ac.at
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.
14www.geometrie.tuwien.ac.at
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
15www.geometrie.tuwien.ac.at
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
16www.geometrie.tuwien.ac.at
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 ∞
17www.geometrie.tuwien.ac.at
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*.
18www.geometrie.tuwien.ac.at
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.
20www.geometrie.tuwien.ac.at
GEOMETRIE
Zhao‘s fast sweeping
Updating formulae (h=gridsize) (a) if , set
(b) if , set
22www.geometrie.tuwien.ac.at
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