Matching Hierarchical Structures
Using Association Graphs
Marcello Pelillo, Member, IEEE, Kaleem Siddiqi, Member, IEEE, and
Steven W. Zucker, Fellow, IEEE
AbstractÐIt is well-known that the problem of matching two relational structures can be posed as an equivalent problem of finding a
maximal clique in a (derived) ªassociation graph.º However, it is not clear how to apply this approach to computer vision problems
where the graphs are hierarchically organized, i.e., are trees, since maximal cliques are not constrained to preserve the partial order.
Here, we provide a solution to the problem of matching two trees by constructing the association graph using the graph-theoretic
concept of connectivity. We prove that, in the new formulation, there is a one-to-one correspondence between maximal cliques and
maximal subtree isomorphisms. This allows us to cast the tree matching problem as an indefinite quadratic program using the Motzkin-
Straus theorem, and we use ªreplicatorº dynamical systems developed in theoretical biology to solve it. Such continuous solutions to
discrete problems are attractive because they can motivate analog and biological implementations. The framework is also extended to
the matching of attributed trees by using weighted association graphs. We illustrate the power of the approach by matching articulated
and deformed shapes described by shock trees.
Index TermsÐMaximal subtree isomorphisms, association graphs, maximal cliques, replicator dynamical systems, shock trees,
shape recognition.
æ
1INTRODUCTION
T
HE relationships between discrete and continuous
mathematics have always been a subject of intensive
study since the discovery of the irrationals by the
Pythagorean school. Apart from the underlying philoso-
phical implications, the interaction between the two
domains can provide new insights into old problems and
often allows techniques from one side to be profitably
imported into the other. Entire branches of modern
mathematics have been created with the specific motivation
of exploring such connections, examples of which are
singularity theory, combinatorial topology, and spectral
graph theory. In more recent years, with the introduction of
the ellipsoid and the interior point methods for linear
programming, there has also been a tremendous interest in
computer science and operations research in solving
combinatorial optimization problems using continuous
methods [21], [44].
Following the seminal works of Hopfield and Tank [24],
and Durbin and Willshaw [14], the neural network
community also became interested in using continuous
approaches for combinatorial optimization. The basic idea
consists of deriving a continuous ªenergyº function whose
minimizers are in correspondence with the solutions of the
discrete problem, and then minimizing it using continuous-
or discrete-time dynamical systems, typically embedded in
a parallel network of locally interacting processing ele-
ments. Such continuous solutions to discrete problems are
attractive not only because they offer the advantage of
biological plausibility, but also because they can motivate
parallel, analog VLSI implementations. Examples of
problems attacked within this framework include the
traveling salesman problem [14], [24], graph bipartitioning
[16], the maximum clique problem [26], [46], the linear
assignment problem [29], the knapsack problem [43], and
the graph/subgraph isomorphism problems [20], [53].
Thus far, the focus has been on ªflatº problems in the
sense that there is no partial ordering imposed on the data.
In many practical problems, however, data are organized in
a hierarchical manner, i.e., are trees, and the problem of
matching such representations is of interest for pattern
recognition. Applications in domains like computer vision
[32], [36], [55], [57], [62], [66], molecular biology [58], and
natural language processing [41] abound, and many
traditional, discrete algorithms have been developed [31],
[34], [37], [54], [59]. On the other hand, no attempt has yet
been made to approach such problems within a continuous
framework, using analog continuous-time dynamics. The
main difficulty is that it is not clear how to map the
hierarchy embedded in the representations onto a ªflatº
optimization network.
The matching of relational structures is a related (but
different) problem which has also received considerable
attention in computer vision and pattern recognition
because of its applications in such problems as object
recognition, motion, and stereo analysis, etc. [2]. A classical
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 21, NO. 11, NOVEMBER 1999 1105
. M. Pelillo is with the Dipartimento di Informatica, Universita
Á
Ca' Foscari
di Venezia, Via Torino 155, 30172 Venezia Mestre, Italy.
E-mail: pelillo@dsi.unive.it.
. K. Siddiqi is with the School of Computer Science and Center for Intelligent
Machines, McGill University, 3480 University Street, Montreal, PQ, H3A
2A7, Canada. E-mail: siddiqi@cim.mcgill.ca.
. S.W. Zucker is with the Departments of Computer Science and Electrical
Engineering and the Center for Computational Vision and Control, Yale
University, PO Box 208285, New Haven, CT 06520-8285.
E-mail: xucker-steven@cs.yale.edu.
Manuscript received 15 Dec. 1998; revised 20 July 1999.
Recommended for acceptance by K. Bowyer.
For information on obtaining reprints of this article, please send e-mail to:
tpami@computer.org, and reference IEEECS Log Number 108453.
0162-8828/99/$10.00 ß 1999 IEEE
solution to this problem consists of transforming it into the
equivalent problem of finding a maximum clique in an
auxiliary graph structure, known as the association graph [2],
[3]. The idea goes back to Ambler et al. [1] and has since
been successfully employed in a variety of different tasks,
e.g., [6], [25], [42], [51], [52], [65]. This framework is
attractive because it casts relational structure matching as
a pure graph-theoretic problem for which a solid theory and
powerful algorithms have been developed. Although the
maximum clique problem is known to be NP-complete [17],
powerful heuristics have been developed which efficiently
find good approximate solutions [9] and there exist many
classes of graphs for which the problem is solvable in
polynomial-time [21], [9].
Since, in the standard association graph formulation, the
solutions are not constrained to preserve the required
partial order, it is not clear how to apply the framework for
matching hierarchical structures. The extension of associa-
tion graph techniques to tree matching problems is there-
fore of considerable interest. To illustrate the difficulties
with the standard formulation, consider the problem of
finding the largest subtree in the left tree of Fig. 1, which is
isomorphic to a subtree in the right tree. Up to permuta-
tions, the correct solution is clearly given by 3 ! a, 4 ! b,
5 ! c, 6 ! d, 7 ! f, and 8 ! g. In other words, the
subtree rooted at node 3 is matched against that rooted at
node a in the tree on the right. However, using the standard
association graph formulation (cf. [2, p. 366]), it is easily
verified that the solutions induced by the maximum cliques
correspond (up to permutations) to the following: 2 ! h,
3 ! a, 4 ! b, 5 ! c, 6 ! d, 7 ! f, and 8 ! g, which,
while perfectly in accordance with the usual subgraph
isomorphism constraints, do violate the requirement that the
matched subgraphs be trees (note, in fact, that nodes 2 and
h are isolated from the rest of the matched subtrees).
In this paper, we introduce a solution to this problem by
providing a novel way of deriving an association graph
from two (rooted) trees, based on the graph-theoretic
notions of connectivity and the distance matrix. We prove
that, in the new formulation, there is a one-to-one
correspondence between maximal (maximum) cliques in
the derived association graph and maximal (maximum)
subtree isomorphisms. As an obvious corollary, the
computational complexity of finding a maximum clique in
such graphs is therefore the same as that of the subtree
isomorphism problem, which is known to be polynomial in
the number of nodes [17]. This formulation allows us to
map the hierarchical information contained in the trees onto
a flat structure and this turns out to be the key to the
proposed framework.
Following the development in [48], [49], we use the
Motzkin-Straus theorem [40] to formulate the maximum
clique problem on the association graph as a (continuous)
quadratic program whose solutions are in one-to-one
correspondence with the solutions of the original tree
matching problem. To solve it, we employ replicator
equations, a class of continuous- and discrete-time dynami-
cal systems developed and studied in various branches of
mathematical biology [23], [63], which are also closely
related to parallel relaxation labeling networks [56]. In
addition, we extend the framework to handle the matching
of attributed trees by casting the problem as that of finding
a maximum weight clique in a weighted association graph.
A recent generalization of the Motzkin-Straus theorem
applies [19], allowing the use of the same replicator
dynamics as in the unweighted case. We illustrate the
power of the proposed approach via several examples of
matching articulated and deformed shapes described by
shock trees [62].
2TREE ISOMORPHISM AND MAXIMAL CLIQUES
2.1 Notations and Definitions
Before going into the details of the proposed framework, we
need to introduce some graph-theoretical notations and
definitions. More details can be found in standard textbooks
of graph theory, such as [22]. Let G V;E be a graph,
where V is the set of nodes and E is the set of (undirected)
edges. The order of G is the number of nodes in V , while its
size is the number of edges. Two nodes u; v 2 V are said to
be adjacent (denoted u v) if they are connected by an edge.
A path is any sequence of distinct nodes u
0
u
1
...u
n
such
that, for all i 1...n, u
iÿ1
u
i
; in this case, the length of the
path is n.Ifu
0
u
n
, the path is called a cycle. A graph is said
to be connected if any pair of nodes is joined by a path. The
distance between two nodes u and v, denoted by du; v,is
the length of the shortest path joining them (by convention,
du; v1 if there is no such path). Given a subset of
nodes C V , the induced subgraph GC is the graph having
C as its node set and two nodes are adjacent in GC if and
only if they are adjacent in G.
A connected graph with no cycles is called a tree.A
rooted tree is one which has a distinguished node, called
the root. The level of a node u in a rooted tree, denoted by
levu, is the length of the path connecting the root to u.
Note that there is an obvious equivalence between rooted
trees and directed trees, where the edges are assumed to
be oriented. We shall therefore use the same terminology
typically used for directed trees to define the relation
between two adjacent nodes. In particular, if u v and
levvÿlevu1, we say that u is the parent of v and,
conversely, v is a child of u. Trees have a number of
interesting properties. One which turns out to be very
useful for our characterization is that in a tree any two
nodes are connected by a unique path.
2.2 Deriving the Association Graph
Let T
1
V
1
;E
1
and T
2
V
2
;E
2
be two rooted trees. Any
bijection : H
1
! H
2
, with H
1
V
1
and H
2
V
2
, is called a
1106 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 21, NO. 11, NOVEMBER 1999
Fig. 1. An example of matching two trees. In the standard formulation of
the association graph, the maximum cliques do not preserve the
hierarchical structure of the two trees (see text for details).
subtree isomorphism if it preserves the adjacency and
hierarchical relationships between the nodes and, in
addition, the subgraphs obtained when we restrict our-
selves to H
1
and H
2
, i.e., T
1
H
1
and T
2
H
2
, are trees. The
former condition amounts to stating that, given u; v 2 H
1
,
we have u v if and only if uv, and u is the parent
of v if and only if u is the parent of v. A subtree
isomorphism is maximal if there is no other subtree
isomorphism
0
: H
0
1
! H
0
2
with H
1
a strict subset of H
0
1
,
and maximum if H
1
has largest cardinality. The maximal
(maximum) subtree isomorphism problem is to find a
maximal (maximum) subtree isomorphism between two
rooted trees.
We now introduce the notion of a path-string, which will
be central to the subsequent development.
Definition 1. Let u and v be two distinct nodes of a rooted tree T,
and let u x
0
x
1
...x
n
v be the (unique) path joining them.
The path-string of u and v, denoted by stru; v, is the string
s
1
s
2
...s
n
on the alphabet fÿ1; 1g where, for all i 1...n,
s
i
levx
i
ÿlevx
iÿ1
. By convention, when u v we
define stru; v", where " is the null string (i.e., the string
having zero length).
The path-string concept has a very intuitive meaning.
Suppose that you stand on a particular node in a rooted tree
and want to move to another adjacent node. Because of the
orientation induced by the root, only two types of moves
can be done, i.e., going down to one of the children (if one
exists) or going up to the parent (if you are not on the root).
Let us assign to the first move the label 1 and, to the
second, the label ÿ1. Now, suppose that you want to move
from node u to v, following the unique path joining them.
Then, the path-string of u and v is simply the string of
elementary moves required to reach v, starting from u.It
may be thought of as the degree of relationship between
two relatives in a ªfamilyº tree. As an illustrative example,
referring to Fig. 1, we have str2; 8ÿ1 1 1 1. Note
that if stru; vs
1
...s
nÿ1
s
n
, then strv; us
n
s
nÿ1
...s
1
,
where
s
i
ÿ1 if s
i
1 and s
i
1 otherwise (i 1...n).
Definition 2. The tree association graph (TAG) of two rooted
trees T
1
V
1
;E
1
and T
2
V
2
;E
2
is the graph G
V;E where
V V
1
V
2
and, for any two nodes u; w and v; z in V , we have
u; wv; z,stru; vstrw; z:
Intuitively, two nodes u; w and v; z are adjacent in the
TAG if and only if the relationship between u and v in T
1
is
the same as that between w and z in T
2
. Note that this
definition of the association graph is stronger than the
standard one used for matching arbitrary relational
structures [2], [3]. A subset of vertices of G is said to be a
clique if all its nodes are mutually adjacent. A maximal clique
is one which is not contained in any larger clique, while a
maximum clique is a clique having largest cardinality. The
maximum clique problem is to find a maximum clique of G.
Our main goal in this section is to establish a one-to-one
correspondence between maximal cliques in the TAG and
maximal subtree isomorphisms. To this end, we need the
following result.
Lemma 1. Let u
1
;v
1
;w
1
;z
1
2 V
1
and u
2
;v
2
;w
2
;z
2
2 V
2
be
distinct nodes of rooted trees T
1
V
1
;E
1
and
T
2
V
2
;E
2
, and suppose that the following conditions hold
(see Fig. 2):
1. w
1
is on the u
1
v
1
-path and w
2
is on the u
2
v
2
-path
2. stru
1
;w
1
stru
2
;w
2
3. strw
1
;v
1
strw
2
;v
2
4. stru
1
;z
1
stru
2
;z
2
5. strv
1
;z
1
strv
2
;z
2
Then, strw
1
;z
1
strw
2
;z
2
.
Proof. See the Appendix. tu
The following theorem, which is the basis of the work
reported here, establishes a one-to-one correspondence
between the maximum subtree isomorphism problem and
the maximum clique problem.
Theorem 1. Any maximal (maximum) subtree isomorphism
between two rooted trees induces a maximal (maximum) clique
in the corresponding TAG, and vice versa.
Proof. Let : H
1
! H
2
be a maximal subtree isomorphism
between rooted trees T
1
and T
2
, and let G V;E
PELILLO ET AL.: MATCHING HIERARCHICAL STRUCTURES USING ASSOCIATION GRAPHS 1107
Fig. 2. An illustration of the hypotheses of Lemma 1. Each curved line represents a path between two nodes; when two paths are labeled by the
same symbol, the corresponding path-strings are assumed to be the same. The lemma states that strw
1
;z
1
strw
2
;z
2
.
denote the corresponding tree association graph. Let
C
V be defined as:
C
fu; u : u 2 H
1
g:
From the definition of a subtree isomorphism, it follows
that maps the path between any two nodes u; v 2 H
1
onto the path joining u and v. This clearly implies
that stru; vstru;v for all u 2 H
1
and, there-
fore, C
is a clique. Trivially, C
is a maximal clique
because is maximal. This proves the first part of the
theorem.
Suppose now that C fu
1
;w
1
; ; u
n
;w
n
g is a
maximal clique of G, and let H
1
fu
1
; ;u
n
gV
1
and H
2
fw
1
; ;w
n
gV
2
.Define : H
1
! H
2
as
u
i
w
i
, for all i 1...n. From the definition of a
tree association graph and the hypothesis that C is a
clique, it is simple to see that is a one-to-one and onto
correspondence between H
1
and H
2
, which trivially
preserves both the adjacency and the hierarchical
relationships between nodes. The fact that is a maximal
isomorphism is a straightforward consequence of the
maximality of C.
To conclude the proof we have to show that the
subgraphs that we obtain when we restrict ourselves to
H
1
and H
2
, i.e., T
1
H
1
and T
2
H
2
, are trees and this is
equivalent to showing that they are connected. Suppose,
by contradiction, that this is not the case and let u
i
;u
j
2
H
1
be two nodes which are not joined by a path in T
1
H
1
.
Since both u
i
and u
j
are nodes of T
1
, however, there must
exist a path u
i
x
0
x
1
...x
m
u
j
joining them in T
1
. Let
x
x
k
, for some k 1...m, be a node on this path
which is not in H
1
. Moreover, let y
y
k
be the kth node
on the path w
i
y
0
y
1
...y
m
w
j
which joins w
i
and w
j
in
T
2
(remember that stru
i
;u
j
strw
i
;w
j
and, hence,
dw
i
;w
j
m). We now show that the set fx
;y
g [
C V is a clique. To this end, let u; w2C. Since u
i
;w
i
and u
j
;w
j
are also nodes in C,wehave
stru
i
;ustrw
i
;w, and stru
j
;ustrw
j
;w. Further-
more, we have that x
and y
are on the u
i
u
j
- and w
i
w
j
-
paths, respectively, and, clearly, stru
i
;x
strw
i
;y
and strx
;u
j
stry
;w
j
. Therefore, all the hypotheses
of Lemma 1 are satisfied and this implies that
strx
;ustry
;w, which amounts to stating that
node x
;y
is adjacent to u; w, for all u; w2C. This
means that fx
;y
g [ C is a clique, thereby contra-
dicting the hypothesis that C is a maximal clique and
proving the second part of the theorem.
The ªmaximumº part of the statement is proven
similarly. tu
The next proposition provides us with a straightforward
criterion to construct the TAG.
Proposition 1. Let T
1
V
1
;E
1
and T
2
V
2
;E
2
be two
rooted trees, u; v 2 V
1
,andw; z 2 V
2
.Then,stru; v
strw; z if and only if the following two conditions hold:
1. du; vdw; z
2. levuÿlevvlevwÿlevz:
Proof. The proposition is a straightforward consequence of
the observation that, given any two nodes u and v in a tree,
with stru; vs
1
s
2
...s
n
, we have levuÿlevv
P
i
s
i
,
and the fact that s
i
1 implies s
j
1 for all j i. tu
This property allows us to efficiently derive the TAG by
using a classical representation for graphs, i.e., the so-called
distance matrix (see, e.g., [22]) which, for an arbitrary graph
G V;E of order n, is the n n matrix D d
ij
where
d
ij
du
i
;u
j
, the distance between nodes u
i
and u
j
.
3ACONTINUOUS FORMULATION OF THE MAXIMUM
CLIQUE PROBLEM
We now exploit the interplay between discrete and
continuous mathematics, alluded to in Section 1. Let G
V;E be an arbitrary graph of order n and let S
n
denote the
standard simplex of IR
n
(see Fig. 3):
S
n
x 2 IR
n
: e
0
x 1 and x
i
0;i 1...nfg;
where e is the vector whose components equal 1 and a
prime denotes transposition. Given a subset of vertices C of
G, we will denote by x
c
its characteristic vector whichisthe
point in S
n
defined as
x
c
i
1=jCj; if u
i
2 C
0; otherwise;
where jCj denotes the cardinality of C.
Now, consider the following quadratic function
fxx
0
Ax; 1
where A a
ij
is the adjacency matrix of G, i.e., the n n
symmetric matrix defined as
a
ij
1; if u
i
u
j
0; otherwise:
A point x
2 S
n
is said to be a global maximizer of f in S
n
if
fx
fx, for all x 2 S
n
. It is said to be a local maximizer
if there exists an >0 such that fx
fx for all x 2 S
n
whose distance from x
is less than and if fx
fx
1108 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 21, NO. 11, NOVEMBER 1999
Fig. 3. The simplex S
3
.
implies x
x, then x
is said to be a strict local maximizer.
Note that fx1 for all x 2 S
n
.
The Motzkin-Straus theorem [40] establishes a remark-
able connection between global (local) maximizers of the
function f in S
n
and maximum (maximal) cliques of G.
Specifically, it states that a subset of vertices C of a graph G
is a maximum clique if and only if its characteristic vector
x
c
is a global maximizer of f on S
n
. A similar relationship
holds between (strict) local maximizers and maximal
cliques [19], [50]. This result has an intriguing computa-
tional significance in that it allows us to shift from the
discrete to the continuous domain. Such a reformulation is
attractive for several reasons: It suggests how to exploit the
full arsenal of continuous optimization techniques, thereby
leading to the development of new algorithms, and may
also reveal unexpected theoretical properties. Additionally,
continuous optimization methods are often described in
terms of sets of differential equations and are, therefore,
potentially implementable in analog circuitry. The Motzkin-
Straus theorem has served as the basis of several clique-
finding procedures [10], [18], [45], [46] and has also been
used to determine theoretical bounds on the cardinality of
the maximum clique [45], [64].
One drawback associated with the original Motzkin-
Straus formulation relates to the existence of spurious
solutions, i.e., maximizers of f which are not in the form of
characteristic vectors. This was observed empirically by
Pardalos and Phillips [45] and more recently formalized by
Pelillo and Jagota [50]. In principle, spurious solutions
represent a problem since, while providing information
about the cardinality of the maximum clique, they do not
allow us to easily extract its vertices. Fortunately, there is a
solution to this problem which has recently been introduced
and studied by Bomze [7]. Consider the following regular-
ized version of f:
^
fxx
0
Ax
1
2
x
0
x; 2
which is obtained from (1) by substituting the adjacency
matrix A of G with
^
A A
1
2
I
n
;
where I
n
is the n n identity matrix. The following is the
spurious-free counterpart of the original Motzkin-Straus
theorem (see [7] for a proof).
Theorem 2. Let C be a subset of vertices of a graph G, and let x
c
be its characteristic vector. Then the following statements hold:
1. C is a maximum clique of G if and only if x
c
is a global
maximizer of the function
^
f in S
n
. In this case,
jCj1=21 ÿ fx
c
.
2. C is a maximal clique of G if and only if x
c
is a local
maximizer of
^
f in S
n
.
3. All local (and, hence, global) maximizers of
^
f in S
n
are
strict.
Unlike the original Motzkin-Straus formulation, the
previous result guarantees that all maximizers of
^
f on S
n
are strict, and are characteristic vectors of maximal/
maximum cliques in the graph. In a formal sense, therefore,
a one-to-one correspondence exists between maximal
cliques and local maximizers of
^
f in S
n
on the one hand
and maximum cliques and global maximizers on the other
hand.
4REPLICATOR EQUATIONS AND TREE MATCHING
We now turn our attention to a class of dynamical systems
that we use for solving our quadratic optimization problem.
Let W be a nonnegative real-valued n n matrix and
consider the following dynamical system:
_
x
i
tx
i
tWxt
i
ÿ xt
0
Wxt
;i 1...n; 3
where a dot signifies derivative w.r.t. time t, and its
discrete-time counterpart
x
i
t 1x
i
t
Wxt
i
xt
0
Wxt
;i 1...n: 4
It is readily seen that the simplex S
n
is invariant under these
dynamics, which means that every trajectory starting in S
n
will remain in S
n
for all future times. Moreover, it turns out
that their stationary points, i.e., the points satisfying
_
x
i
t0
for (3) or x
i
t 1x
i
t for (4), coincide and are the
solutions of the equations:
x
i
Wx
i
ÿ x
0
Wx0 ;i 1...n:
A stationary point x is said to be asymptotically stable if every
solution to (3) or (4) which starts close enough to x
converges to x as t !1.
Both (3) and (4) are called replicator equations in
theoretical biology since they are used to model evolution
over time of relative frequencies of interacting, self-
replicating entities [23]. The discrete-time dynamical equa-
tions turn out to be a special case of a general class of
dynamical systems introduced by Baum and Eagon [5] in
the context of the theory of Markov chains. They also
represent an instance of the original Rosenfeld-Hummel-
Zucker relaxation labeling algorithm [56], whose dynamical
properties have recently been clarified [47] (specifically, it
corresponds to the 1-object, n-label case).
We are now interested in the dynamical properties of
replicator equations; it is these properties that will allow us
to solve our original tree matching problem.
Theorem 3. If W W
0
, then the function xt
0
Wxt is strictly
increasing with increasing t along any nonstationary trajec-
tory xt under both continuous-time (3) and discrete-time (4)
replicator dynamics. Furthermore, any such trajectory con-
verges to a stationary point. Finally, a vector x 2 S
n
is
asymptotically stable under (3) and (4) if and only if x is a
strict local maximizer of x
0
Wx on S
n
.
The previous result is known in mathematical biology as
the fundamental theorem of natural selection [13], [23], [63]
and, in its original form, traces back to Fisher [15]. As far as
the discrete-time model is concerned, it can be regarded as a
straightforward implication of the more general Baum-
Eagon theorem [5]. The fact that all trajectories of the
replicator dynamics converge to a stationary point has been
proven more recently [33], [35].
PELILLO ET AL.: MATCHING HIERARCHICAL STRUCTURES USING ASSOCIATION GRAPHS 1109
![Fig. 4. A coloring of shocks into four types [28]. A 1-shock derives from a protrusion and traces out a curve segment of adjacent 1-shocks, along which the radius function varies monotonically. A 2-shock arises at a neck, where the radius function attains a strict local minimum, and is immediately followed by two 1-shocks flowing away from it in opposite directions. 3-shocks correspond to an annihilation into a curve segment due to a bend, along which the radius function is constant, and a 4-shock is an annihilation into a point or a seed, where the radius function attains a strict local maximum. The loci of these shocks gives Blum's medial axis.](/figures/fig-4-a-coloring-of-shocks-into-four-types-28-a-1-shock-10i7vp5q.webp)
![Fig. 5. Two illustrative examples of the shocks obtained from curve evolution (from [62]). Left: The notation associated with the locus of shock points is of the form shock_type-identifier. Right: The corresponding trees have the shock_type on each node, with the identifier adjacent. The last shocks to form during the curve evolution process appear under a root node labeled #.](/figures/fig-5-two-illustrative-examples-of-the-shocks-obtained-from-35mdtglo.webp)
