The Genus of a Digital Image Boundary Is Determined by Its Foreground, Background, and Reeb Graphs
Discrete Comput Geom
Geometry Discrete & Computational
Lowell Abrams 1 2
Donniell E. Fishkind 0 2
0 Department of Applied Mathematics and Statistics, Johns Hopkins University , Baltimore, MD 212182682 , USA
1 Department of Mathematics, George Washington University , Washington, DC 20052 , USA
2 The Genus of a Digital Image Boundary Is Determined by Its Foreground , Background, and Reeb Graphs
We prove that the genus of the boundary of a digital image is precisely half of the sum of the cycle ranks of three particular graphs: the “foreground graph” and “background graph,” which capture topological information about the digital image and its complement, respectively, and the Reeb graph, relative to the natural height function, associated with the digital image's boundary. We prove several additional results, including a characterization of when the cycle rank of the Reeb graph fails to equal the genus of the digital image's boundary (which can happen by virtue of the failure of the natural height function on the boundary of the digital image to be a Morse function).

A digital image I is a union of unit cubes in R3, each unit cube being of the form
[i − 21 , i + 12 ] × [ j − 21 , j + 12 ] × [k − 21 , k + 21 ] for integers i, j, k. We require, moreover,
that the boundary ∂ I is a surface, i.e. a 2manifold without boundary. In Section 3 we
define a “foreground graph” GI , a “background graph” GIC , and a “boundary graph”
G∂I associated with I, the complement of I, and ∂ I, respectively. These graphs play a
critical role in what is to follow.
We begin by briefly mentioning a current application: The human cerebral cortex,
when viewed as closed at the brain stem, is topologically a sphere and divides the cranial
cavity into two regions, interior and exterior. Magnetic resonance imaging (MRI) of the
brain can distinguish between tissue that is interior to the cerebral cortex and tissue that
is exterior to it. Because of the finiteness of resolution, an MR image can be interpreted
as a digital image, with the cubes representing tissue interior to the cerebral cortex. The
boundary of the digital image may be taken as an approximation of the cerebral cortex
itself.
Although topologically spherical, the cerebral cortex is densely folded. Thus, the
finite resolution, as well as signal noise, may lead to “detection” of topological
“handles” that do not actually exist. It is important to “correct” the topology of the digital
image, since the physiological and neurological function of regions of the cerebral
cortex, as well as the relationship between the regions, is dictated by the spherical
topology rather than just spatial proximity. A number of different strategies are currently
used [
5
], [
6
].
The strategy of Shattuck and Leahy [
6
] is fundamentally based on the foreground and
background graphs; they conjectured that the digital image boundary is topologically
spherical if and only if both foreground and background graphs are trees. In situations
where one or both of the graphs are not trees, these graphs serve as very direct guides to
their topologycorrection algorithm. At the conclusion of the algorithm, both foreground
and background graphs are trees and, by their conjecture, the boundary of the resulting
digital image is topologically spherical.
The Shattuck–Leahy conjecture was proven and generalized by Abrams et al. in [
2
]
and [
3
]. A further generalization by the present authors [
1
], mentioned here as Theorem 1,
establishes that the genus of a digital image boundary is bounded below and above by the
maximum and sum, respectively, of the cycle ranks of the foreground and background
graphs. (It is shown there that these bounds are, in a strong sense, best possible.) In
particular, these bounds also prove the Shattuck–Leahy conjecture, since they imply that
the genus of the digital image boundary is 0 if and only if both foreground and background
graphs have cycle rank 0, which means precisely that foreground and background graphs
are both trees.
If M is a closed surface embedded in R3 with a “height function” f : M → R, then
there is an associated onedimensional space called a “Reeb graph.” It is known [
4
] that
if f is a Morse function, then the genus of M is precisely the cycle rank of the Reeb
graph associated with f . We give formal definitions and discuss these ideas further in
Section 2. In our context, the boundary of a digital image I is a closed surface embedded
in R3, the zaxis serves as a natural height function, and the associated Reeb graph is
topologically equivalent to the boundary graph G∂I . Nevertheless, this height function is
not a Morse function (in fact, it is degenerate everywhere), and thus the cycle rank of G∂I
may fail to equal the surface’s genus. Theorem 4 provide several characterizations for
when this failure occurs; in particular, failure occurs precisely when there is inequality
in the upper bound of Theorem 1.
Our main results are stated in Section 4. The central result is Theorem 2, which
expresses the genus of a digital image boundary in terms of the digital image’s foreground,
background, and boundary graphs. Specifically, the genus of a digital image boundary
is precisely half the sum of the respective cycle ranks of the foreground, background,
and boundary graphs. Theorems 2 and 3 are proven in Section 6, after the development
in Section 5.
Morse Functions and Reeb Graphs
In this manuscript the term surface refers to a subset M of R3 that is compact, connected,
and locally homeomorphic to an open disk in R2; in particular, M is orientable and has
no boundary. We let g(M ) denote the genus of M .
Suppose M is a surface. A function f : M → R is a Morse function if all critical
points of f are nondegenerate, i.e. have nonsingular Hessian. Given such a function f on
M , define an equivalence relation on M as follows: x ∼ y if and only if f (x ) = f (y) and
x , y lie in the same connected component of f −1( f (x )). The Reeb graph of f on M is
the quotient space associated with this equivalence relation, under the quotient topology.
It may be viewed as a combinatorial (multi)graph by considering the points of the Reeb
graph not locally homeomorphic to an open interval in R as graphtheoretic vertices, and
considering the connected components of the complement in the Reeb graph of the set
of vertices as graphtheoretic edges.
The cycle rank of any combinatorial (multi)graph G = (V , E ) is the value E  −
V  + c(G), where c(G) is the number of connected components of G; the cycle rank
r (G) is the first Betti number of G when G is viewed as a onedimensional space.
The article [
4
] shows that, given a Morse function f on a surface M , the genus of M
is equal to the cycle rank of the Reeb graph of f on M . (For convenience, an additional
condition imposed on f in [
4
] is that f is injective on the set of critical points.)
Typically, one thinks of Morse functions as “height” functions, and it is true that in our
context of the boundary ∂I of a digital image I (defined in Section 3), the zcoordinate
z: ∂I → R is a very natural height function. Unfortunately, it is not a Morse function. In
fact, it is degenerate everywhere; at each point that the Hessian is defined, it is 0. (Also
worthy of mention is that there are level sets of the zcoordinate with infinitely many
critical points.) As a consequence, the associated Reeb graph, which is topologically
equivalent to G∂I , will not necessarily satisfy r (G∂I ) = g(∂I). In Theorem 3 we
characterize when r (G∂I ) = g(∂I). We also define a foreground graph associated with
the interior of ∂I and an analogous background graph associated with the exterior of ∂I
and, in Theorem 2, we characterized g(∂I) in terms of the Reeb graph of ∂I together
with these foreground and background graphs. In so doing, the information of the genus
of the surface is recovered, despite the height function not being a Morse function.
3.
Definitions
We begin by recalling some definitions from [
1
]. Let N be a fixed positive integer. For
any triplet of integers (i, j, k) ∈ {1, 2, . . . , N }3, define vi, j,k to be the closed Euclidean
unitcube [i − 21 , i + 21 ] × [ j − 21 , j + 21 ] × [k − 21 , k + 21 ] ⊆ R3.
For any A ⊆ {1, 2, . . . , N }3, define the digital preimage IA := (i, j,k)∈A vi, j,k . We
call IA a digital image if its boundary ∂IA is a surface, and also none of i, j, or k equals 1
or N . We define IAC as IAC , where AC := {1, 2, . . . , N }3\ A. When there is no confusion,
we write I in place of IA.
For each k ∈ {1, 2, . . . , N }, the kth level is Lk := i, j∈{1,2,...,N} vi, j,k , and the (k, k +
1)th sheet is Sk,k+1 := Lk ∩ Lk+1. Associated to a digital preimage I is the (multi)graph
GI with vertex set VI and edge set EI defined as follows: For each k ∈ {1, 2, . . . , N },
we declare each connected component of I ∩ Lk to be a vertex in VI . For any two vertices
u and v on adjacent levels, say Lk and Lk+1, we declare each connected component of
u ∩ v ⊆ Sk,k+1 to be an edge in EI whose graphtheoretic endpoints are u and v. When
referring to a vertex u ∈ VI or an edge ε ∈ EI , context will dictate whether we are
viewing u or ε as Euclidean subsets, i.e. subsets of R3, or as discrete graphtheoretic
objects. If I is a digital image, we refer to GI as the “foreground graph” and to GIC as
the “background graph,” and to their respective vertices and edges correspondingly. We
refer to those vertices in GIC which intersect the line (1, 1, z) ∈ R3 as spine vertices,
and to the subgraph of GIC induced by the spine vertices as the spine.
Also associated to a digital image I is the (multi)graph G∂I with vertex set V∂I
and edge set E∂I defined as follows: For each sheet Sk,k+1, we declare the connected
components of ∂I ∩ Sk,k+1 to be vertices of G∂I . Now let Lko denote the open kth level, i.e.
Lko := Lk \(Sk−1,k ∪ Sk,k+1) ⊂ R3. For each level Lk , we declare the respective closures
of the connected components of Lo
k ∩ ∂I to be edges of G∂I . (It becomes important
in Section 5 to note that, as topological subspaces of R3, these edges are cylinders.)
The graphtheoretic ends of such an edge are the two vertices of G∂I which contain
a [onedimensional] boundary of that edge. We refer to G∂I as the “boundary graph,”
and to its vertices and edges correspondingly. As mentioned above, the graph G∂I is
topologically equivalent to the Reeb graph of the zcoordinate height function on ∂I.
Figure 1 displays several examples of digital images I and their respective foreground,
background, and boundary graphs. Figure 2 displays two adjacent levels and the sheet
in between, to illustrate vertices and edges of foreground, background, and boundary
graphs further.
GI
GIc
G∂I
GI
GIc
G∂I
GI GIc G∂I
Fig. 1. Some digital images with their respective foreground, background, and boundary graphs. (Note that
background vertices corresponding to levels which have no foreground vertices have been deleted from the
graphs GIC .)
Two adjacent levels, with the sheet between. In this sheet, the five connected components of white
are foreground edges, the three shaded connected components are background edges, and the seven
connected components of black are boundary vertices. The degrees of the boundary vertices in this sheet are
=
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Let
I
Results
note that each of G , G
proved in [
1
].
I
I
C
, and G
∂
I
Theorem 1.
For any digital image ,
be a digital image. Recall that the genus g(∂ ) of the surface ∂
is the first Betti
number of
and that, for any graph G
(V , E ), the cycle rank r (G) :
is the first Betti number of G; in particular, r (G)
0 if and only if G is acyclic. Also
I
E
=  − +
V
c(G)
I
=
is connected. The following result, Theorem 1, was
max r (G ), r (G
I
I
C )
} ≤
g(∂ )
I ≤
r (G )
I
+
r (G
I
C ).
Moreover, this is best possible in the sense that, for any nonnegative integers a, b, and
c such that max a, b
r (G )
b, there exists a standard digital image
It should be pointed out that Theorem 1 implies the Shattuck–Leahy conjecture that ∂I
is topologically a sphere if and only if both GI and GIC are trees.
Incorporating data from the graph G∂I allows for an exact expression of g(∂I) in
terms of GI , GIC , and G∂I ; Theorem 2 following is the central result of this manuscript.
Theorem 2. For any digital image I,
g(∂I) = 21 (r (GI ) + r (GIC ) + r (G∂I )).
Theorem 2 is proved in Section 6.
As pointed out in Section 2, the genus of a surface is the cycle rank of its Reeb graph
relative to any Morse function. In our situation with ∂ I , where the zcoordinate height
function is not a Morse function, the following result—Theorem 3—characterizes how
much the genus of ∂I differs from the cycle rank of G∂I , this graph being topologically
equivalent to the Reeb graph of the surface ∂I relative to the zcoordinate height function.
Let VI and EI denote the vertex set and edge set, respectively, of GI , let VIk denote
the set of vertices of GI lying in level Lk , let E Ik,k+1 denote the set of edges of GI lying in
sheet Sk,k+1, and let Gk,k+1 denote the induced subgraph of G on vertex set VIk ∪ VIk+1.
Define VIC , EIC , V kC ,IE k,k+1, Gk,Ck+1 and V∂I , E∂I , V∂kI,k+1, IE∂kI analogously.
I IC I
Theorem 3. For any digital image I,
Theorem 4. For any digital image I, the following statements are equivalent:
21.. IFtoirs atrlluek,thGakt,kg+(1∂aIn)d=Grk(,kG+∂1Ia)r.e acyclic.
IC
3. It is true thaIt g(∂I) = r (GI ) + r (GIC ); i.e. equality holds in the upper bound
of Theorem 1.
Proof. The equivalence of statements 1 and 2 follows from Theorem 3, since Gk,k+1 and
I
Gk,Ck+1 are acyclic precisely when r (Gk,k+1) = 0 and r (Gk,k+1) = 0. The equivalence
ofIstatements 1 and 3 follows from the Irephrasing of TheorIeCm 2 as g(∂I) − r (G∂I ) =
r (GI ) + r (GIC ) − g(∂I).
The next result, Theorem 5, may be thought of as a generalization of the theorem
establishing the Shattuck–Leahy conjecture.
Theorem 5. For any digital image I, the following statements are equivalent:
1. The surface ∂I is topologically equivalent to a sphere.
2. Each of the graphs GI , GIC , and G∂I is a tree.
3. At least two of the graphs GI , GIC , and G∂I are trees.
Proof. The equivalence of statements 1 and 2 is an immediate consequence of
Theorem 2. Of course statement 2 implies statement 3, and we showed in [
2
] that if GI and
GIC are both trees then ∂I is topologically spherical. So all that remains to be shown
is that if G∂I is a tree and one of GI and GIC is a tree, then the other is a tree as well.
Indeed, suppose without loss of generality that G∂I and GIC are trees. By Theorem 2
we have 2g(∂I) = r (GI ), yet by Theorem 1 we have r (GI ) ≤ g(∂I), which together
imply r (GI ) = 0, i.e. GI is also a tree.
The following result, Proposition 6, is proved in [
2
]. A multipath is a graph which is
a path when multiplicities of edges are ignored.
Proposition 6. Let I be a digital image. Then every foreground vertex is topologically
a ball if and only if GIC is a multipath of length N − 1. If this indeed occurs, then
g(∂I) = r (GI ).
For a digital image in which every foreground vertex is topologically a ball, we have
the following simplification of Theorem 3.
Corollary 7. If ∂I is a digital image in which every foreground vertex is topologically
a ball, then
and, moreover,
g(∂I) = r (G∂I ) + r (GIC ),
N−1
i=1
r (GIC ) =
r (Gk,Ck+1) =
I
N−1
i=1
r (Gk,k+1).
I
iN=−11 r (Gk,k+1) =
C
far with ThIeorem 3.
Proof. By Proposition 6 we have here that g(∂I) = r (GI ); substituting into Theorem 2
yields the first assertion of this corollary. Also by Proposition 6 we have here that GIC
is a multipath, so its cycle rank is the number of extra edges beyond the number of
edges in an underlying path, which yields r (GIC ) = iN=−11 r (GkI,Ck+1). The equality
iN=−11 r (Gk,k+1) follows from comparing what we have proven so
I
5. Lemmas
In this section we prove various lemmas that will be used in Section 6 to prove Theorems 2
and 3.
By the Jordan Curve Theorem, any simple, closed curve in R2 partitions R2 into an
interior, exterior, and boundary. A simple closed curve γ in R2 is called the hull of a
subset ⊆ R2 if γ is contained in and is also contained in the closure of the interior
of γ . Let P be any plane in R3 perpendicular to the zaxis, and let π : R3 → P denote
the orthogonal projection operator.
For topological space X , let χ (X ) denote the Euler characteristic of X . Recall the
Inclusion–Exclusion formula
for any spaces X, Y , and the Euler–Poincare´ formula
χ (X ∪ Y ) = χ (X ) + χ (Y ) − χ (X ∩ Y )
χ (X ) = 2 − 2g(X )
for any surface X .
We begin by pointing out some of the critical implications of our assumption that
∂I is a surface. In [
3
] it was shown that the boundary of a digital preimage is locally
homeomorphic to an open disk in R2 if and only if it does not contain anywhere one of
the three “forbidden” configurations in Fig. 3. Consequently, as pointed out in [
2
], each
foreground or background vertex v ∈ VI ∪ VIC is such that ∂v is itself a surface.
Suppose vertex v ∈ VI ∪ VIC is from level Lk . Note that the respective closures of the
connected components of ∂v\(Sk−1,k ∪ Sk,k+1) are topologically cylinders, and are edges
in the boundary graph. We can decompose ∂v as a union of ∂v ∩ Sk−1,k , ∂v ∩ Sk,k+1, and
the cylinders mentioned above. Each of ∂v ∩ Sk−1,k , ∂v ∩ Sk,k+1 is topologically a disk
with punctures, and has Euler characteristic equal to 2 minus the number of cylinders.
Since each cylinder has Euler characteristic 0 and has a boundary of Euler characteristic
0, formulas (1) and (2) show that the genus of ∂v is one less than the number of cylinders,
i.e. one less than the number of connected components of ∂v\(Sk−1,k ∪ Sk,k+1).
Proposition 8. For any digital image I and each index k, E∂kI  = VIk  + VIkC  − 1.
Proof. To each vertex v ∈ V k
I ∪ V kC other than the spine vertex in V kC , we assign the
boundary edge ε ∈ E∂kI for which Iπ(ε) is the hull of π(v). This corIrespondence is a
bijection, and the result follows.
Another consequence of ∂I containing no “forbidden” configuration, also pointed
out in [
2
], is that for each edge ε ∈ E k,k+1 ∪ E k,Ck+1, each component of the boundary in
I I
(1)
(2)
Sk,k+1 of ε is a simple, closed curve; one of these components is the hull of ε in Sk,k+1.
Let h(ε) denote the number of “holes” in ε; i.e. h(ε) is one less than the number of
connected components of the boundary in Sk,k+1 of ε. Since ε is topologically a disk
with h(ε) punctures, we have χ (ε) = 1 − h(ε).
Proposition 9. For any digital image I and each index k,
V k,k+1
 ∂I
 =
ε∈EIk,k+1
Proof. Let γ be a component of the boundary in Sk,k+1 of a foreground or background
edge ε, but which is not the hull of ε. The curve γ must be the hull in Sk,k+1 of a unique
boundary vertex v ∈ V k,k+1. The function γ → v gives a bijection between the set of
“holes” of edges in Ek,k∂+I1 ∪ EIk,Ck+1 and the vertices V∂kI,k+1, so the result follows.
I
For any graph G, let c(G) denote the number of connected components in G.
Proposition 10. For any digital image I and each index k,
c(Gk,k+1) =
I
Proof. Let H denote a connected component of Gk,k+1, let denote the union, as
I
Euclidean subsets, of all of the foreground vertices and foreground edges in H , and let
γ denote the hull of π( ). The curve γ is equal to π(γ ) for a unique curve γ in Sk,k+1.
The curve γ is necessarily a connected component of the boundary in Sk,k+1 of some
background edge ε ∈ Ek,Ck+1, but it cannot be the hull of ε since it is in the boundary
of a foreground graph coImponent. The correspondence H → γ is a bijection between
components of Gk,k+1 and “holes” of edges in Ek,Ck+1, and the result is established.
I I
Proposition 11. For any digital image I and each index k,
c(Gk,Ck+1) = 1 +
I
ε∈EIk,k+1
h(ε).
The proof of Proposition 11 is analogous to the proof of Proposition 10, the only
difference in the argument being consideration of the connected component of Gk,Ck+1 which
contains vertices of the spine. I
Proposition 12. For any digital image I and each index k, V∂kI,k+1 = c(Gk,k+1) +
c(Gk,k+1) − 1. I
IC
Proposition 12 is an immediate corollary of Propositions 9–11.
Lastly, we mention that, by definition of cycle rank, for any digital image I and each
index k,
r (GkI,k+1) = E Ik,k+1 − VIk  − VIk+1 + c(GkI,k+1)
r (Gk,Ck+1) = E Ik,Ck+1 − VIkC  − VIkC+1 + c(Gk,Ck+1).
I I
6. Proof of Theorems 2 and 3
In this section we prove Theorems 2 and 3.
Proof of Theorem 3. Let I be a digital image. The assertion of Theorem 3 is that
We prove this assertion by first calculating the Euler characteristic of ∂I, and then
deducing g(∂I) using the Euler–Poincare´ Theorem.
Since the Euler characteristic of a simple, closed curve is 0, by the Inclusion–Exclusion
formula (1) we have that
χ (∂ I ) =
χ (v) +
χ (ε).
v∈V∂I ε∈E∂I
Since each boundary edge ε, as a Euclidean subset, is topologically a cylinder and
thus has χ (ε) = 0, we have
χ (∂ I ) =
χ (v).
v∈V∂I
We calculate the righthand side of (5) sheetbysheet.
Temporarily fix k; Sk,k+1 is a closed disk, so χ (Sk,k+1) = 1. Note that the complement
in Sk,k+1 of the union of the interiors of the foreground and background edges is precisely
the union of the boundary vertices in Sk,k+1 (and the hull of Sk,k+1). Applying (1) and
keeping in mind that the Euler characteristic of a simple, closed curve is 0, we therefore
have
χ (∂I ∩ Sk,k+1) = 1 −
χ (ε) −
χ (ε),
ε∈EIk,k+1 ε∈EIk,Ck+1
since the union of all sets mentioned in the above displayed equation forms Sk,k+1. As
discussed at the beginning of Section 5, each background edge and foreground edge ε
is topologically a punctured sphere, so has χ (ε) = 1 − h(ε), where h(ε) is the number
of holes in ε. Thus, we have
χ (∂I ∩ Sk,k+1) = 1 −
[1 − h(ε)] −
[1 − h(ε)]
h(ε).
(3)
(4)
(5)
v∈V∂kI,k+1
Using Propositions 10 and 11, and then Proposition 12, we can rewrite this as
χ (∂I ∩ Sk,k+1) = −EIk,k+1 − EIk,Ck+1 + c(Gk,k+1) + c(Gk,Ck+1)
I I
= 2V∂kI,k+1 − EIk,k+1 − EIk,Ck+1 − c(Gk,k+1) − c(Gk,Ck+1) + 2. (6)
I I
By Proposition 8, we have
degree(v) = E∂I + E∂kI+1 = VIk + VIk+1 + VIkC  + VIkC+1 − 2. (7)
k
Subtracting and then adding
plying (7), then simplifying the resulting equation with (3), we obtain
v∈V∂kI,k+1 degree(v) to the righthand side of (6), then
apχ (∂I ∩ Sk,k+1) = 2V∂kI,k+1 −
degree(v) − r (Gk,k+1) − r (Gk,Ck+1).
I I
Summing χ (∂I ∩ Sk,k+1) over all k now yields
χ (∂I) = 2V∂I − 2E∂I −
r (Gk,k+1) −
I
r (Gk,Ck+1).
I
Using the Euler–Poincare´ formula and the previous equation, and then simplifying, yields
1
g(∂I) = 21 (2 − χ (∂I)) = E∂I − V∂I + 1 + 2
N−1
k=1
Proof of Theorem 2. Let I be a digital image; we need to show that
v∈V∂kI,k+1
N−1
k=1
N−1
k=1
N−1
k=1
N−1
k=1
E∂I − V∂I + 1 − VI +
c(Gk,k+1) − VIC  +
I
c(Gk,Ck+1) = 0.
I
(8)
By Proposition 12, and summing over all sheets, we have
g(∂I) = 21 (r (GI) + r (GIC ) + r (G∂I)).
V∂I =
N−1
k=1
N−1
k=1
c(Gk,k+1) +
I
c(Gk,Ck+1) − (N − 1).
I
By Proposition 8, and summing over all levels, we have
Combining these two equations together and rearranging yields
E∂I = VI + VIC  − N .
We can rearrange the equation in Theorem 3 and apply (3) as follows:
1
g(∂ I) = r (G∂I ) + 2
Substituting (8) into this yields
1
g(∂ I) = 2 (E∂I  − V∂I  + 1)
1
+ 2 (EI  − VI  + 1 + EIC  − VIC  + 1),
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