KIM TAEHO
PARK YOUNGMIN

Curve Reconstruction problem
𝑆𝑒𝑡 𝑜𝑓 𝑝𝑜𝑖𝑛𝑡𝑠
𝑃𝑜𝑙𝑦𝑔𝑜𝑛𝑎𝑙 𝑐𝑢𝑟𝑣𝑒

Image processing

Geographic information system
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Pattern recognition

Mathematical modeling
Which one is better?
How can we say one is better than another?
 Mainly refers distribution of distance between vertices

There are several methods to solve curve
reconstruction problem
◦ Mainly start from idea of VD(DT)
 Crust
 𝜷-skeleton
 NN-Crust
 Conservative-Crust
Add another strategy to the VD

Definition
Let S be a finite set of points in the plane
Let V be the vertices of the Voronoi diagram of S.
Let S’ be the union of S ∪ V.
An edge of the Delaunay triangulation of S’ belongs to the crust of
S if both of its endpoints belong to S

Intuition
◦ The intuition behind the definition of the crust is
that the vertices V of Voronoi diagram of S
approximate the medial axis of F and Voronoi
disks of S’ approximate empty circles between F
and its medial axis

Definition
Let s1, s2 be a pair of points in the plane at distance d(s1, s2)
The forbidden region of s1, s2 is the union of the two disk of radius
βd(s1, s2) / 2 touching s1, s2
Let S be a finite set of points in the plane. An edge between s1, s2 ∈
S belongs to β-Skeleton of S if forbidden region of is empty

Implementation and
examples
◦ The minimum required sample
density of β-Skeleton is better
than the density for the crust
◦ The crust tends to error on the
side of adding edges (which can
be useful)
◦ β-Skeleton could be biased toward
adding edges at the cost of
increasing required sampling
density, by tuning the parameter β

… Which is better? When?

Proved that …
◦ Let S be an r-sample from a smooth curve F
◦ For r ≤ 1, the Delaunay triangulation of S
contains the polygonal reconstruction of F
◦ For r ≤ 0.40 the Crust of S contains the polygonal
reconstruction of F
◦ For r ≤ 0.297, the β-Skeleton of S contains the
polygonal reconstruction of F


Their condition is based on local feature size
function which in some sense quantifies the local
‘level of detail’ at a point on smooth curve
Definition
The Medial axis of a curve F is closure of the set of points in the
plane which have two or more closest points in F
The local feature size of LFS(p), of a point p ∈ F is the euclidean
distance from p to the closes point m on the medial axis
F is r-sampled by a set of sample points S if every p ∈ F is within
distance r LFS(p) of sample s ∈ S (here, r ≤ 1)

Medial Axis
LFS(p)

To handle non-smooth curves
◦ For non-smooth curve reconstruction algorithm
needs infinite sample density near the corners
With condition..
 Any point 𝑝 on 𝐾 must have a sample point with ε < 1
times the radius of the larger circle between 𝐶1 and 𝐶2

Based on nearest-neighbor
◦ With three more observations
Angle condition
 An edge 𝑝𝑞 qualifies for the nearest neighbor test only if its dual
Voronoi edge makes an angle less than a user defined
parameter 𝛼
Ratio condition
 The ratio of the length of the dual Voronoi edge to the length of
the edge is more than a preset threshold 𝜌
◦ Proposed acceptable constant value
 𝛼 = 35°~40°
 𝜌 = 1.7~2.0
Topology condition
 For point 𝑝 which has more than 3 adjacent edges, by assuming
input is sampled from set of 1-manifolds we can delete longest
edge from 𝑝

Algorithm
THANK YOU 
[1] Amenta et al, A New Voronoi-Based Surface
Reconstruction Algorithm, 1998
[2] Amenta et al, The Crust and the 𝜷-skeleton:
Combinational Curve Reconstruction, 1998
[3] A Simple Provable Algorithm for Curve
Reconstruction, Dey et al, 1999
[4] Reconstructing Curves with Sharp Corners, Dey et al,
2000