match between. P and an image. I(Q) of Q the distance between the cen- troids .... A linear time algorithm for it in the case of convex polygons has been given by ... cell of a vertex from the cell of an edge). Vor(P) has O(p) edges and vertices .... i.e. if the corresponding endpoint of 1A(e) is contained in C !a,c(J) is defined.
Approximate
Matching
of Polygonal
(Extended
Helmut Bernd Johannes Institut
fiir
Alt
Behrends Blomer
Informatik, Fachbereieh Mathematik, Freie Universit5t Arnimallee 2-6, W-1OOO Berlin 33, Germany
Abstract
Given
two objects
resemble For
two
given
determine match
simple
a rigid
between
distance
cases, namely
that
to translations out that that
P
only
timal
in
determining
differ
from
more In
fact
P and an image troids
ejjiciently
multiple
and I(Q).
It is also shown two
O(n log n],
a constant
polygons
turns factor
determining
op-
case the algorithm
hulls
between
between
the cen-
of P and I(Q]
that
n is the total
P
the Hausdorff-distance number
in time
of this
tern
paper
recognition
which
is to present solving
methods
standard
from
problems
can be intuitively
plane.
formulated
similar
to P.
Q are simple the
P Here
polygons
problem
above
in
can be
P and
1(Q)
minimal
distance).
Here
in the plane (see [M])
I such
is minimized
which
that
(and
an isometry preserves
any isometry
the
distance
determine
that
is an affine
map-
distances.
As is well
1 can be represented
I=ropot
or
I=
as
pot,
where
r is the
about
the
we will
reflexion
origion
wlog.
and
mean
(also
at
by isometry
called
rigid
can easily the better
sense,
x —axis, only
motions
p a rot at ion this
isometrics
by first
P and r(Q)
of the two
any isometry
In
or even
be included
P and Q and then
taking
the
t a translation.
article without
isometrics). matching
by rigid
op-
motions
matches.
1 is of the form
I(z)=
(1)
M.z+t
as
research was support ed by the Deutsche Forschungsgemeinschaft under graut Al 253/1–3, Schwerpunktprogramm “Dat enstrukt~en und effiziente Algorithmen” Permission to copy without fee all or part of this material is granted prowded that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of lhe publication and Its date appear, and notice is given that copying is by permmsion of the To copy otherwise, or to Association for Computing Machine~. republish, requires a fee and/or specific permission.
M
$1.50
(
Cos p
sin p
— sin 9
Cos p
and
t E R2 is some fixed a
use
the
fined ~H(X, Y.
distance
Y)
measure
so-called
by 6H(A,
B)
= ~~
that
=
~~
is always 1 shows 6H(P,
for
translation
P
two I(Q))
and 6H
B), ~H(l?,
y), is the
distance defined
p
E [0, 2m[
vector.
between
max(~H(A, d(z,
some
)
Hausdorff-metric
d is the Euclidean
8H (A, B) such
186
=
As
Figure 86
is most
and
recognition)
of a set of samples
Com-
“This
0-89791-426-0/91/0006/01
character
an isometry
between
where
ACM
do they
up to some
as follows:
P, Q, find
known
identical
Q out
So geometrically
formulated Given
that
in pat-
follows:
@ 1991
that which
and
Geometry
(e.g.
and
assume
In this aim
input
P
the
they
?)
we will
reflexions
of vertices.
Introduction
The
P and Q, how much Are
has to be determined
timally
putational
(or:
applications
be the
Reflexions
1
(shapes)
is
between
can be determined
6>0
In many
ping
match
of the Hausdorff-distance
simple where
It
i.e. ones
possible
of the edges of the convex
between
direction.
of Q the distance
a constant
or even
is based on the quite sur-
the optimal
I(Q)
as
special
solutions, than
general
solution
that for
for
Berlin,
each other?
tolerance will
algorithms
to translations
by just
the most
the pseudo-optimal
prising
Faster
one specified
optimum
the Hausdorff-
are obtained
pseudo-optimal
the
solutions.
I(Q).
problem
is to
the best possible
minimizing
I is restricted
can be done much for
and
the general
P, Q the problem
I of Q giving
P and Q, i.e.
between
the one for
polygons
motion
Shapes*
Abstract)
distance
in the plane.
I(Q)
that A)), from
we
is
de-
where X
Notice
to
that
if A, B C R2 are bounded.
polygons
P, Q and an isometry
is minimized.
1
~H (P, I(Q))
Figure
(Note
that
polygon
this
paper,
P as a set, we always
throughout
mean
the edges of P, not some
special
above
which
the
POby P2:
problem
then
Only
a on
There
problem
interest.
we define
are
P2,
are allowed,
i.e. in(1)
Al
Pl,
is the
allowed,
i.e. A4 = Id
one fixed
and
direction
isometrics
(~=
~d,t
except =
(~)),
i.e. the
the Hausdorff-distance A standard Pl)
example
is again
a very
for it in the Atallah
of polygons and more lated
7 and
9.
efficient
more,
that
Similar
isometry,
to the
ones
measure
imation
algorithms
rithm
[AKMSW]
and
here
with
have been in this
point [HK].
respect considered
context
to
given
2
of
for PO, where
an O((p
algorithm
of the
edges
a constant
is based between
of the
on the
P and distance
convex
the best
opti-
Hausdorff-
by
the
This
the
an be-
hulls
possible
of
match
17t.
two
the polygons
Hausdorff-distance (Problem
PO)
sets instead and [S],
Problems
Let
re-
an alternative in [G].
have been
P, Q be In
tively.
Approxdeveloped
+ q) log(p
p and q are the numbers
+ q))
Hausdorff-distance
the
Voronoi-diagram Vor(P)
algo-
of vertices
187
two
order
the
Voronoi-cell, give
only
distance
of Q giving
5 algorithms
resulting
of Q is ~ then
Determining
by
[G].
2 we will
1(Q)
copy
the
P3 the
very
compute
optimum
minimum
centroids
of the
where
are
is not
as an alternative.
necessarily
the
problem
image
the
one
from
if the
is at most
algorithm
in [AMWW] and
For
P and
but
differs
that
do not
P3 al-
of course,
time
has been
for
have been investigated
distance
In section
questions
they
and
runtimes that
in section
solutions
algo-
comput-
to DavenportP2
Since
we present
means
tween
Q.
PO is,
any
pseudo-optimal
isometric
of P2 (or even
A linear polygons
allowed
related
of degree
giving
for
problems
polynomials
factor.
is to measure
Pand
recognition.
case of convex
are
problem
application
problem.
recently,
in [ABGWl
the
character
[At 1].
identity
between
for
fundamental
the
for
approximate
distance
tO are
t c {A .tol~E R}
for
of functions Next
log(pq))
techniques
whose
fact PO: No
will
an 0( A66(pq)
using
be given
mal along
envelopes
gorithms
matrixld translations
Then
be presented
Schinzel-sequences.
P3 denote
problems
will
ing upper
formulated Let
of P, Q, respectively. rithm
restrictions:
translations
identity PI:
general
are of independent
the following Only
when
the ones in the interior.)
cases of the
general
considering
the set of points
1:
assigns i.e.
polygons to
solve
p, q vertices, PO, P
between of
set
to this
element
any
one (see Figure
respec-
i, e. determine Q we consider
and
P, Vor(P).
to each edge and each vertex the
are closer other
with problem
of points (i.e. 2).
in
the
edge or vertex) The
edges
of P its
plane
which than
c)f Vor(P)
to are
Figure2:
either
line
edges
or
they
segments two
separate
edge).
the
Vor(P) a finite consider
with
Q (see Figure
As easily first
decreases
and
than
that
or
at
the
edge bounding It
that
the
mono-
i.e. (or
of a point with
some
distance
&
monotonically to the elements
edge is a bitonic
(Q, P)
of P.
whose
function
cells
is
must
sumed
1
e of P the
point the
intersection
distance
at some
of Q with smallest points
of Q
vertex some
to P,
~H(Q,
Voronoi-edge
largest
z—coordinate
of Q with
e (see Figure
or
that
is we have the
by the
possibly
In order
to obtain
only
with left
there
and
we
their $H(Q,
distance P).
the the
neighcells
of
elements
points.
across edges
the arrangement of Vor(P)
and
intersection
delete
e from
Q.
points the
data
as the first
intersection
found.
Two
such
sweeps,
one from
right
to left
are neces-
points
we obtain
are O(p
+ q) event
an O((P + g) log(p + q))-algorithm candidates in the sense of Lemma
if we
It remains
as soon
Q has been
Since
split
the that
nearest
of Q and
intersection
at
Notice
to determine
sweep
e of Vor(P),
to right
and their
hav-
pieces
occurs.)
the extreme
2-3-tree)
segments
two
1 is O(p+q).
vertices
obtained
of an dis-
in Lemma
by a plane
(e.g.
into
tangent
to the critical
edge
parabolic
cut
these points
containing
of each
that
are
vertical
of points
We do this
to P and Analogously,
taking $H(P,
their Q)
one
for determining all 1. By determining maximum and
thus
we get C$H(P, Q)
can be determined.
by this Summa-
P)
of Q or at some
the
on P,
Vor(P)
An
3
The
either
where
sarY.
as-
we assume tangent
the number
point
be
F’
to show how to find
from
we have
Lemma
point
of the
point
before.
a vertical
structure
Voronoi-
are separated
as described
the lemma
of Q on
Furthermore
on a Voronoi-edge
(In
of P closest
decreasing).
at the endpoints
point
Polygon
ing
C.
monotonically
distance
a Voronoi-edge
tance
ezther
C
function,
monotone
ofa
bors
Voronoi-cell
of Q or an intersection
move
tion
to
to the corresponding
be assumed
at a vertex
edge of Q with
rizing
can be
In order
cell C’.
follows
sumed
of an
we move
this
or just
intersection
(if
Voronoi-cell
C is a bit onic
the maximal
edge to P must
cell and
,[F]).
that
increases
increasing
of two
the following:
of Q within cell
the
of a fixed
Suppose
cells
Diagram
segments
vertices
can be seen the distance
monotone
edge
and
(see[Y]
the intersection 3).
the
from
we observe
of P defining
It follows this
edges
O(p logp)
on an edge
element
separate or parabolic
of a vertex
problem
First
just
cell
in time
tonically
they
of P)
has O(p)
constructed obtain
(if
vertices
Voronoi
is as-
intersec-
among
For problem of the
e of P having the
3).
allowed
for
PI
P 1 we can assume translations
wlog.
is parallel
translation vectors are of the form ~ c R and an egde e of Q we denote of
188
algorithm
e.
that
the direction
to the x-axis,
i.e.
(A, O), A E R. For by 1A(e) the image
# : : ..* %. %. %.*
●✎✌ ✞✚✎ ‘. .
. . . . . . . . . . . . . . . ...*.-
.’ .“
.’ ~“
: : : : : : :“
Figure Vor(P)
3:
● - extreme
and Q
intersection
ure
. .. .. . . . . . .. .. .. . . . . . .. . . . . . .. .. / . . .. ... .. .. . . .. ..
4).
points.
For
~ E ~j denote
intersection
point.
signs
to A E lj
edge
A
- each
of P
pair
6H of the image
the maximum that
algebraic
functions
of these
consists
4:
O (~66 (pq) the
Iog(pq)), distance
A. By merging it is easily
three incide) an
intervals
over,
11,12,13
such that
intersection while
seen that
bounding, for point
~ moves
the intersection
point
for e an edge of Q and e’ an say, cell C there
(that
any
may
overlap
exist
and even co-
A c 13, j“ = 1, 2, 3, there
between continuously
moves
1A(e)
and
through
continuously
at most
e’ and,
the
exists more-
Ii, j = 1,2, 3, on e’ (see Fig-
189
cf.
[At2].
from
P
pieces
the two functions
~ in time
pression
for
bounded
by O (pq log(pq)
this
upper of the
graphs
of
in time
we can deter-
we can determine
is known log* (pq))
theory
envelop e
as a function
O (A66 (pq) log(pq)). function
the
be computed lA(Q)
of any
64 points.
from
Likewise, to
are pieces
at most
t heir
can
A to P it can be
8. Therefore
terminology
it
Hausdorff-
position
above
in
to the site
Now
at most
~~~(pq) and
optimal
in C
point 1 the
defined
sequences,
functions
is contained
of this
intersect
using of at most
single
mine Now
functions
i.e. if
functions.
of degree
edge
fa,c,
to Lemma
these
or for
of some
of Q at some
the functions
of Davenport-Schinzel j
edge of Vor(P)
of all
observed
Consequently,
Figure
According
as-
vertex
we define
the function of 1A(e)
that
to the
Likewise
as the distance
~ell C.
the
C’.
endpoint
is defined
corresponding
a is an endpoint
defining
two a
where
distance is
................. .................... .......L ......
cell
of ej (~)
C a cell of Vor(P),
corresponding
!a,c(J)
the
~~~, (A) is the function
distance
defining
(a, C),
e of Q and the
Now the
by ej (~)
No explicit but
it
certainly
(see e.g. [ASS]).
of the ex-
is
4
Algorithms
We
will
only
P2
and
P3,
for
briefly
two P2
and
6 :=
that
We
between
a and
between
such the
pair
Here, the
that of P
dist ante
and
the
6
vector
d is the shortest
Lemma
2 Either
there
the corresponding
there
are at least
The
proof
ering
three
of Lemma
the different
pairs.
This
stract.
We just
of Q is not
critical
2 is done
will
there
or
segment. specify
be only
constantly
above
5, why
are only
d(as,
7-3)
ab-
O ((pq)3)
tion
for
for
some
time cient
have
possible
for
PI
from
8H-value give
has
sec-
among optimal
a runtime
of
that
such
P3
solution,
but
checks 6H
be solved
in
more effito our so-
Davenport-Schinzel about
methods
which
not
in [EPSS]).
still
Instead,
in the next
would
not
we present
an
section.
solutions
we will
and
So P3 can
algorithms.
(except
determining
cases are treated
Pseudo-optimal section
each time 2.
pairs
algorithm
e functions,
yet (special
efficient
P2
4 critical So the
considering
multivariat
ah ernat ive approach
mal
algorithm
+ q) log(p + g)). Probably can be found analogously
we believe
really
lems
the equa-
checked
at least cases).
problem
However,
In this
pawill
we deter-
algorithm
of section
is known
5
are
two
1, there
of them
minimal
isometrics,
algorithm
for
give
the
if ri
such that
each
with
which
degenerate
structures much
For
and
ri,
+ q) log(p + q)). Problem P3 can be solved Analogously to Lemma 2 any optimal solu-
O ((pq)4(p algorithms
lution
P2 only
isometrics
So the
in
(translation)
by applying
P3 must
by the
pairs.
many
many
point
in problem
satisfied.
P2.
if r-i is a point
ai to any
Since
isometrics
to
O ((pq)4)
the placement critical
the
distance
from
the isometry
6H (P, 1(Q)) 2. Those
O ((Pq)3(P similarly.
2 critical
extended
two
d(a~, TZ)
=
are
tion
solutions
consid-
are only
in this
in Figure
if there
such that
by contradiction
be omitted
illustrate
optimal,
pairs
u, v, u = –v,
pairs.
cases, where
proof
critical
vectors
=
r~)
Euclidean
rameters tions
We claim:
are two
critical
m)
d(a~,
distance
is a line
mine
for
d(al,
that
a, b be points
which
vector.
present
for problem
(translation) Let
equations
problems
Suppose
as an instance
(a, b) a critical
b a critical
for we will
approach.
is minimal.
call
two
section
efficient
respectively,
occurs.
P3
algorithms
next
1 is an isometry 1(Q))
1(Q),
the
P, Q are given
6H(P,
and
in
more
polygons
and
describe
since
an alternative,
P2
give efficient
which
not
algorithms
necessarily
a pseudo-optimal
for prob-
find
the
opti-
one in the following
sense: Definition
t
3 An
pseudo-optimal is a constant Figure
5: Nonoptimal
vector
t will
placement
decrease
the
of Q, a translation
finds
by
c >0
a translation
where
Hausdorff-distance.
Our
algorithm
possibilities
for
P, Q containing scribe
three three
cases can be solved (al,
runtime. rl),
vertices
pairs.
critical
vertices with
So the
(az, rz),
on Lemma
(i.e.
critical
the case here where
(a~, b~) involve same
P2 based
of elements
edges)
will
only
We
pairs
(al,
bl),
al, az, as of P.
similar where
ai
is
all
a vertex
For a polygon
other
Let
and
ri
is
a vertex
i = 1, 2, 3. Clearly, a given between
triple, ai
and
there
or
edge
i =
the
are O ((pq)3)
if we demand ri,
of
that
the
1, 2, 3 are the
other such
or
polygon, triples.
three
distances we obtain
produce iff
a
there
P, Q the algorithm 6H(P,
I(Q))
Hausdorff-distance
solution
length
P let rp
< c6,
determined
for
P2
can
be found
where
zp(yp
very
I(Q))
is minimal.
rp,
its
if
difference
at most
~
6H(P,
190
:= (ZP, yp) (y-coordinate)
) is the
of any point
in P
6).
‘1’herefore,
o b.
onto
For
same
I with
P, Q be two polygons
6 := 6H(P, W
Q
on input
z-coordinate
(see Figure
the P
such that (isometry)
to
P2 (P3),
solution.
pseudo-optimal
smallest
triples of
said
problem
easily:
de-
within
considers
A of
(az, bz),
The
techniques
algorithm
(a~, ra)
2 considers and
is
for
6 is the minimal
by the optimal
all
algorithm
solution
.6.
r
and
I a solution
Obviously
d(rp,
to P2, i.e. rI(Q))
is the translation mapping to the optimal one is a vector
Hence,
T(Q))
< (1+
fi)ti
< rQ
of
For
:
proving
parametrized
: : :.
4 First
tance
measure
called
“Fr6chet-distance”,
Definition
8C(CI,
for
a ranges
over
be determined
in time
if we
also
apply
the algorithm
want
+ 9) M(P course,
problem
Sohlt ion.
the
value for
+ d) the
O(p + q), the
r-p is not
ary of the convex
compute
hull
An
always
mappings
a : [a, b] ~
R2,
val, such that
the image
In particular,
we will
consider a : [0, LF] total
length
~
show
minimizing
wlog. now
that
that
1(Q)
parametrization
his dog, 8C(C1,
the man C2)
the origin,
walk-
is the min-
is possible.
pair
of
of Cl
d (ncl(t),
convex
Cz),
ciosed
in fact
curves
to any natural
there
exists
a(t))
< 6H(C1,
can be shown
a)
Let
Lcl,
Cl
Lc,
and
their
Cz). Then
a parametrizaC2) for
by elementary
C2
be
ali
geo-
convex
circumferences
ILc1
– Lc,
where
inter= a(b).
from
LF
@
be polygons,
then
< 6H(P,
closed
and
6
=
I < 27rti
we can sketch
the proof
Let
R
:=
:=
~,
T
such
a(0)
that d (cr(t),
~(t))
of
1~)
of R,
according
for
the
convex
(=
and
Lemma I(Q)),
~
4: rx
a
natural
a parametrization
to Lemma
of T
6