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HIPPOCAMPAL SURFACE ANALYSIS USING SPHERICAL HARMONIC FUNCTION
APPLIED TO SURFACE CONFORMAL MAPPING
Boris Gutman
, Yalin Wang
, Lok Ming Lui
, Tony F. Chan
, Paul M. Thompson
Department of Mathematics, University of California, Los Angeles, CA USA 2 Department of Neurology, UCLA Medical School, Los Angeles, CA USA
[email protected]
ABSTRACT
Using spherical harmonics of an inverse conformal map, we com-
pared hippocampal surfaces of sixteen Alzheimer (AD) and fourteen
control subjects. Hippocampal surfaces were conformally mapped
to a sphere. Maps were regularly sampled and exact, high-degree
spherical harmonic transforms of the inverse maps were computed.
Using the transforms shape descriptors corresponding to the degree
of the harmonics and invariant to translation, rotation, and scale were
obtained and normalized against sample mean. Two-dimensional vi-
sualizations of the shape descriptors were indicative of global as well
as local shape features of hippocampal surfaces. These descriptors
are potentially useful for visual detection of global patterns and cre-
ation of population-based, probabilistic, disease-specific digital at-
lases, especially for comparison of global shape features.
1. INTRODUCTION
Recent studies have confirmed a long-observed correlation be-
tween changes in hippocampal shape and volume, and Alzheimer
disease. Csernansky et al. [1] have found shape analysis methods
that could potentially predict the onset of symptoms using high di-
mensional diffeomorphic transformations of a neuroanatomical tem-
plate. Goldman et al. [2] have found that some patients with con-
firmed Alzheimer sometimes show few or no symptoms through-
out much of their lives. Thus, changes in hippocampal shape char-
acteristic of Alzheimer may take place years before patients show
symptoms. Others [3] have even suggested the possibility of drug
treatments capable of preventing or significantly slowing the pro-
gression of the disease. All these studies suggest a future need for
accurate methods of analyzing local as well as global features of
hippocampal shape. In our study, we compared the shapes of 14 left
and 14 right hippocampi of control subjects with 16 left and 16 right
hippocampi of Alzheimer subjects using spherical harmonic trans-
form applied to surface conformal mapping. The procedure went as
follows: (1) Triangulation meshes were reconstructed from 3-D T
weighted SPGR (spoiled gradient) MRI images, by using an active
surface algorithm that deforms a mesh onto the hippocampal surface
[4]. (2) The meshes were then conformally mapped to a 2-sphere ac-
cording to [5] and regularly sampled using linear interpolation, thus
creating a spherical parameterization of the mesh. (3) A fast spheri-
cal harmonic transform algorithm (FST) was then performed on the
regularly sampled meshes according to [6]. (4)Lastly, rotationally
invariant shape descriptors were calculated, normalized for easy vi-
sual analysis and plotted in R
. We hope these two-dimensional
visualizations of global shape descriptors will serve as a guide for
future statistical analysis similar to that in [1] and the creation of
disease-specific brain atlases as in [7].
2. PREVIOUS WORK
Various methods have been employed in the field of brain sub-
manifold shape analysis. Gerig et al. [8] used spherical harmon-
ics to compute mean squared distance between lateral ventricles of
twins as a measure of pairwise shape difference by normalizing co-
efficients with respect to volume and applying Parseval’s equation.
Although mathematically elegant, this method involves much com-
putational error due to irregularity of sample points. This is be-
cause spherical harmonic coefficients are approximated using a least
squares solution. Another method is a high dimensional diffeomor-
phic map directly from a subject manifold onto an exemplar target
manifold following Miller [9]. With this method, manifolds are di-
vided into subregions and an overall mean transformation between
all subjects and the target manifold is found for each of (usually)
many thousands of points. Then, using the mean transformation, the
mean manifold is constructed. Thus, the perpendicular displacement
between each subject’s surface and the overall mean for each subre-
gion is calculated as a measure of shape variation. Csernanksi et al.
[1] have employed this method in their study of Alzheimer’s.
Fig. 1. Low-Pass Filtering: (a) through (d) are hippocampal sur-
faces reconstructed from harmonics up to degree 10, 20, 63 and 127,
respectively. (e) is the original hippocampus
Use of spherical harmonic shape descriptors as initial representa-
tion of shape is advantageous to methods involving neuroanatomical
templates in that the transform is independent of any population-
based averages and that it describes global shape features in addition
to locally detailed features. Figure 1 illustrates this property: lower-
order spherical harmonics correspond to the major shape features of
a hippocampus, while those of higher order correspond to noise and
local features. Further, the analogue of a template in this method is a
fixed target space, the sphere, which eliminates error due to variabil-
ity of subject-based templates. As shown in [5], the conformal map
onto the sphere is invariant to the specifics of triangulation and rota-
tion. That spherical harmonics-based shape descriptors are also rota-
tionally invariant in effect guarantees rotational invariance through-
out the entire procedure of our method. In addition, the regularity
of sampled grid points allows for a fast calculation of spherical har-
monic coefficients which are exact up to numerical error associated
with floating-point implementation.
3. CONFORMAL MAPPING ONTO THE 2-SPHERE
In this section we give the idea behind the conformal mapping
algorithm following X. Gu, Y. Wang, et al. [5]. The idea is to first
find a homeomorphism
f : M! S
(monomorphism between two
topological spaces that is continuous in both directions) and then op-
timize it by minimizing harmonic energy. Here M is the manifold
represented by a triangulation mesh of the object surface embedded
in R
, defined by , (K ; g ) where K is a simplicial complex and
g : jK j! R
is a function mapping the vertices of K to R
. For
simplicity, consider a scalar piecewise-linear continuous function
f : M! R. Let u; v 2 K be vertices, fu; v g 2 K the edge formed
by u; v. (Here, we approximate all functions on M by continuous
piecewise linear (PL) functions. Thus, the range-space of the confor-
mal map is also a triangulation mesh.) Define the inner product on
the space of PL functions by < f ; g >=
P
fu;v g2K ku;v (f (u)
f (v ))(g (u) g (v )), where ku;v is string energy. By choosing
the correct string energy constants, harmonic energy is defined by
E (f ) =< f ; f >=
P
fu;v g2K ku;v jjf (u) f (v )jj
. Vector func-
tions on M to are defined by
f = (f 1 ; f 2 ; f 3 ). Vector harmonic
energy is E (
f ) =
P
i= E (fi ). Minimizing the harmonic en-
ergy ensures that the map is harmonic i.e. that the laplacian is
equal to zero. That the map is harmonic guarantees its conformality.
Here, the initial homeomorphism used is the Gauss map defined by ! f (v ) =
n (v ); v 2 M. For details on the algorithm minimizing
harmonic energy and additional constraints placed on the function
to ensure convergence as well as an explanation in a more general
setting, see [5].
4. SPHERICAL HARMONICS
A function f : S
! C is a spherical harmonic if it is the eigen-
function of the Laplacian operator f = f where is a scalar
multiplier. A countable set of spherical harmonics provides an or-
thonormal basis for the space of square-integrable functions on the
sphere L
(S
). If we parameterize the sphere with a latitudinal co-
ordinate c and a colatitudinal coordinate p, spherical harmonics are
expressed explicitly:
Y
m l (^ ;^ )^ =^
s
(2l + 1)(l m)!
4(l + m)!
P
m l (^ os^ )e
im
for the degree l and order m, where l and m are integers with
jmj < l. Here, P
m l (^ os^ )^ is the associated Legendre polynomial
P
m l (^ os^ )^ =^
( 1)m 2 l^ l! (1 x
m 2 d
m+
dxm+^ (x
l , which is a solution
to the associated Legendre differential equation. Let f be in L
(S
. For a given order l and degree m, a spherical harmonic coefficient
is defined by fb (l ; m) =< f ; Y
m l >^ , where^ <^ f^ ;^ g^ >^ is the usual
L
inner product in spherical coordinates. The spherical harmonic
expansion is the series f ( ; ) =
P 1
l=
Pl m= l fb (l ; m)Y m l (^ ;^ ).
The set of all coefficients b f (l ; m) is called the spherical harmonic
transform of f. In practice, the transform is computed with a fast
algorithm described in [6], which relies on regular mesh sampling.
The transform is only computed up to a certain degree l < B , where
the limit B is called the bandwidth.
A consequence of Parseval’s equation is that any function in
L
(S
) is uniquely determined by its spherical harmonic coeffi-
cients, implying that linear transformations (scaling, rotation, trans-
lation) in the object space alter an object’s spectrum. Thus, further
registration is needed to make each individual coefficient completely
invariant to linear transformations. While translational invariance is
achieved easily, rotational invariance will be the subject of the con-
cluding section. For now, we have achieved a limited rotational in-
variance by simplifying the spectrum as described below.
Fig. 2. Rotational invariance: A hippocampal surface was rotated
45 degrees around each axis, mapped conformally and decomposed
into new spherical harmonics. The plot shows the relative difference
between the new and the original descriptors: [s(l ) s(l
)℄=s(l )
versus degree l. Error is within 1%.
5. SPHERICAL HARMONIC ANALYSIS AND THE
SURFACE CONFORMAL MAP
Let
f : M R
! S
be a conformal homeomorphism defined
discretely by
f = (f 1 ; f 2 ; f 3 ), as described in section (3), where
M is a mesh representing the object. Let
f
: S
! R
be the
inverse map from the sphere onto the hippocampal surface, defined
by the isomorphic property of the homeomorphism. We regularly
sample
f
= (f
1 ;^ f^
2 ;^ f^
3 )^ using a matching area algorithm and linear interpolation, and apply the FST to each scalar component
of the inverse map. This amounts to projecting the inverse of the dis-
crete conformal map onto a finite-dimensional subspace of L
(S
The result is a set of vector spherical harmonic coefficients in C
f
![
f
(l ; m) = f b f
1 (l^ ;^ m);^
b f
2 (l^ ;^ m);^
b f
3 (l^ ;^ m)jjmj^ ^ l^ ^ B^ gg
, where B is the bandwidth. The spectrum simplification neces-
sary for rotational invariance comes from two key observations.
One is that the norm of a function in L
(S
) does not change
with rotation. The other is that for a spherical function pl 2
S panfY
l l
; Y
l+ l
; :::; Y
l l g, given an element of the rotation group g 2 S O (3) and its associated operator (g ), the transformed
function remains in the same subspace of L
(S
) : (g )(pl ) 2
S panfY
l l ;^ Y^
l+ l ;^ :::;^ Y^
l l g^. Thus,
X^ l
m= l
j pbl (l ; m)j
= jjpl jj
2 =^ jj(g^ )(pl^ )jj
2 =^
Xl
m= l
j(g )(pl )(l ; m)j
Further, the linearity of (g ) implies f ( ; ) =
P
B 1
l= pl )
(g )[f ( ; )℄ =
PB 1
l= (g )(pl ). This motivates the following
definition. Returning to our original notation, we define spherical
harmonic shape descriptors s(l ) =
P
i=
P
l m= l jj fbi
(l ; m)jj
to be the squared Euclidean norms of squared L
-norms of the vector
than 122 new hippocampal models. To make the first step in this di-
rection, we are currently transforming the models into mesh format.
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