The Genus of a Digital Image Boundary is Determined by Its Foreground, Background and Reeb Graphs

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The Genus of a Digital Image Boundary is
Determined by Its Foreground, Background and Reeb
Graphs

• Presented by Donniell E. Fishkind (JHU)
• Project is joint with Lowell Abrams (GWU) and
  Carey Priebe (JHU)

V 3/07/07
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Background
A (multi)graph G(V,E) consists of vertices V,
edges E we allow parallel edges.
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Background
A (multi)graph G(V,E) consists of vertices V,
edges E we allow parallel edges.
A tree is an acyclic, connected graph.
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Background
A (multi)graph G(V,E) consists of vertices V,
edges E we allow parallel edges. The cycle rank
of (connected) G is rGE-V1.
rG9-614
A tree is an acyclic, connected graph.
rG5-610
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Background
A (multi)graph G(V,E) consists of vertices V,
edges E we allow parallel edges. The cycle rank
of (connected) G is rGE-V1.
rG9-614
A tree is an acyclic, connected graph.
rG5-610
Proposition Suppose G(V,E) is a connected
graph. Then rG ? 0, with rG 0 iff G is a tree.
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is a surface if is
connected, compact, and locally homeomorphic to
an open disc in
sphere
torus
double torus
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is determined up to homeomorphism by its
genus g (the number of handles that are
added to a sphere)
g 2
g 1
g 0
g 0
g 2
g 1
(human cerebral cortex)
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In our brain MRI application The images volume
is subdivided into voxels.
MRI classifies each voxel as foreground (interior
to cerebral cortex) or background (exterior to
cerebral cortex).
foreground
background
individual voxel
(split open)
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Define union of foreground voxels
union of background voxels
Image
(Assume F strictly contained in B)
? cerebral cortex should be
topologically spherical but may not be due to
noise/resolution. Assume is a surface.
ith level
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We next construct the foreground graph
Gf(Vf,Ef) and the background graph Gb(Vb,Eb)
The foreground vertices Vf are the connected
components of foreground in the levels. The
background vertices Vb are the connected
components of background in the levels.
foreground vtcs
ith level
(outermost vtx)
background vtcs
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If two foreground vertices in adjacent levels
have a nonempty intersection, (each connected
component of) this intersection is an edge in
Ef. If two background vertices in adjacent
levels an edge in Eb.
?1
?1
?1
? ?1 ?2
?
ith level
i1th level
?2
?2
?2
foreground edge
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rf 0
rb 0
(top. spherical)
(tree)
(tree)
(top. spherical)
(tree)
(tree)
Gf
Gb
Gf
Gb
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rf 1
rb 0
(not a tree)
(top. toroidal)
(tree)
Gf
Gb
Gf
Gb
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rf 0
rb 1
(tree)
(top. toroidal)
(not a tree)
Gf
Gb
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rf 1
rb 1
(top. toroidal)
(not a tree)
(not a tree)
Gf
Gb
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Conjecture (Spherical Homeomorphism
Conjecture, Shattuck and Leahy , IEEE TMI, 2001)
is topologically equivalent to a sphere
iff both Gf and Gb are trees.
This conjecture plays a central role in their
algorithm to restore to being
topologically equivalent to the spherical
cerebral cortex that approximates.
Theorem (Spherical Homeomorphism
Theorem, Abrams, F, Priebe IEEE TMI, 2002) Under
the assumption that is a surface, is
topologically equivalent to a sphere iff both Gf
and Gb are trees.
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Theorem (Genus Bound Theorem, Abrams, F. SIDMA
2004) Under the assumption that is a
surface, max rf, rb g( ) rf rb.
The Genus Bound Theorem implies the Spherical
Homeomorphism Theorem
The bounds in the Genus Bound Theorem are best
possible in a strong sense.
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Very Special Case Is when, for all
, it holds that is topologically spherical
example
Proposition In Very Special Case it holds that
.
(In example above, Gf is , with
cycle rank 1.)
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Tools
A 2-cell embedding of a graph G on surface S is
an embedding where no edges cross, and each face
may be contracted to a point.
n 32, e 60 f 30, g 0
Theorem (Euler-Poincare) For 2-cell embedding of
graph G on surface S of genus g, having n
vertices, e edges, and f faces
n - e f 2 - 2 g.
, the Euler characteristic of surface S
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VSC cont
?1
?1
?1
? ?1 ?2
?
?2
?2
?2
foreground edge
Boundary of each is a topological sphere w/ Euler
characteristic 2. Note In particular, 2 2 -
2 2 .
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VSC cont
?1
?1
?1
? ?1 ?2
?
?2
?2
?2
foreground edge
Boundary of each is a topological sphere w/ Euler
characteristic 2. Note In particular, 2 2 -
2 2 . Globally,

i.e.
in Very Special Case
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The Reeb Graph of a surface
z-axis is height function
Reeb Graph
Planes perpendicular to z-axis level sets are
drawn in them, connected components of level sets
are identified as points in Reeb Graph
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embedded differently
z-axis is height function
Reeb Graph
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Critical points are points on surface where
plane-perp.-to-z-axis is tangential
Nondegenerate if invertible Hessian (of function
describing surface, z-axis function of x-axis and
y-axis)
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Theorem In the absence of degeneracy, the genus
of a surface is equal to the cycle rank of its
Reeb graph.
g 1
g 1
g 2
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Theorem In the absence of degeneracy, the genus
of a surface is equal to the cycle rank of its
Reeb graph.
g 1
g 1
g 2
Reeb graph of boundary
g 1
Cycle rank is 0
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All critical points are degenerate (worse, not
smooth at many critical points)!!
Reeb graph of boundary
g 1
Cycle rank is 0
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the boundary of a digital image
(conditions?)
Theorem In the absence of degeneracy, the genus
of a surface is equal to the cycle rank of its
Reeb graph.
Is there a way to recover the topological
information of a digital images boundary lost
to degeneracy when the associated Reeb graph is
formulated?
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Theorem (Abrams, F. DCG 2007) For any digital
image ,
the cycle rank of the Reeb graph of
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Theorem (Abrams, F. DCG 2007) For any digital
image ,
the cycle rank of the Reeb graph of
Recall Genus Bound Theorem
Corollary For any digital image , it holds
that if and only if equality holds in the upper
bound
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cycle rank is 1
cycle rank is 0
cycle rank is 0
Gf
cycle rank is 2
sum is 3
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Theorem (Abrams, F. DCG 2007) For any digital
image ,
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Theorem (Abrams, F. DCG 2007) For any digital
image ,

Corollary For any digital image , it holds
that if and only if every subgraph of Gf and Gb
induced by successive levels is acyclic.
?
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Very Special Case Is when, for all
, it holds that is topologically spherical
Gb
Proposition VSC happens iff Gb is a multipath of
length levels-1.
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Very Special Case Is when, for all
, it holds that is topologically spherical
Gb
Proposition VSC happens iff Gb is a multipath of
length levels-1.
Corollary In Very Special Case,
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Very Special Case
Thus far, we have
(VSC)
(VSC)
(general)
(general)
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Very Special Case
Thus far, we have
(VSC)
(VSC)
(general)
(general)
Corollary In Very Special Case we have (by the
above) that
degeneracy term
and
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Very Special Case
Thus far, we have
(VSC)
(VSC)
(general)
(general)
Corollary In Very Special Case we have (by the
above) that
degeneracy term
and
Conjecture For general digital image,
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Theorem (Generalized Spherical Homeomorphism
Theorem, Abrams F., 2007) For any digital image
F, the following are equivalent
The surface is topologically equivalent to
a sphere
Each of foreground graph, background graph, and
Reeb graph is a tree.
At least two of the foreground graph, background
graph, and Reeb graph are trees.
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Proposition is not locally homeomorphic
to a disc if and only if the image contains one
of the following three forbidden configurations.
Two foreground voxels sharing only a voxel edge,
two background voxels sharing same edge.
Two foreground voxels sharing only a voxel
vertex, 6 background voxels sharing same vertex
Reverse roles of fg/bg in previous.