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\begin{document}


\title{\small 10-th Annual Meeting of the Organization for Human Brain Mapping 2004\\
\LARGE \bf Subpixel Curvature Estimation of the Corpus Callosum\\
via Splines and its Application to Autism}
\author{Thomas J. Hoffmann$^1$, Moo K. Chung$^{1,2,3}$, Kim D.
Dalton$^3$, Andrew L. Alexander$^{3,4,5}$, Grace Wahba$^{1,2}$,
Richard J. Davidson$^{3,4,6}$\\
\small $^1$Department of Statistics, $^2$Department of
Biostatistics and Medical Informatics, $^3$Keck Laboratory for
Functional Brain Imaging and Behavior, Waisman Center\\
\small $^4$Department of Psychiatry, $^5$Department of Medical
Physics, $^6$Department of Psychology\\
University of Wisconsin, Madison, WI 53706}
\date{June 16, 2004}
 \maketitle \thispagestyle{empty}




\begin{multicols}{3}
\section*{Introduction} Autism is a neurodevelopmental disorder
with abnormal corpus callosum (CC) size [1]. Most previous studies
used the area of predefined Witelson partition [5] as a
morphometric measure but other shape metrics have not been
considered. We present a new computational technique for curvature
estimation via piecewise quintic splines and use it in both CC
nonlinear dynamic time warping algorithm [4] and detecting the
regions of curvature difference.


\begin{figure}
\centering
\includegraphics[width=180mm]{twoBrains}
\caption{Left: level set segmentation showing partial volume
effect. Right: spline smoothing. A similar approach has been taken
in \cite{Carew}.\label{fig:curvature}}
\end{figure}

\section*{Methods}

A group of 2D mid sagittal cross section images of the corpus
callosum was taken from males of similar age, 15 autistic, and 12
normal controls. The level set method was used to extract the
boundary $\Psi$ of the corpus callosum automatically by solving
$$\frac{\partial \Psi}{\partial t} + F|\nabla \Psi| = 0$$
where $F$ is the given boundary propagation velocity
\cite{Malladi}. Then the pixelated CC contour was reconstructed
into a rough closed curve in Euclidean space (Figure 1. red).
Smoothing of this zigzag contour was necessary to account for the
partial volume effect (Figure 1. blue). Two different methods were
used to smooth and estimate the curvature function. The first
method uses Taubin's smoothing \cite{Taubin}, a Gaussian filtering
without shrinkage, followed by the least-squares estimation. The
second method uses quintic splines to estimate the first and
second derivatives to compute the curvature:
$$\min_g \frac{1}{n}\sum_{j=1}^n [Y_j -g(t_j)]^2 + \lambda \int
[g^{'''}(t)]^2 \;dt.$$

Afterwards a curve from the control group was chosen as a template
and all other curves were registered to the template. First an
affine registration was used to normalize the global CC size
differences. Second the fast nonlinear dynamic-time warping
algorithm was used \cite{Wang}. The algorithm penalized against
large deformation and curvature difference, thereby matching the
extrema of curvature while maintaining a smooth and stable
deformation. So we estimate the deformation $d$ by minimizing
$$ \frac{1}{n}\sum_{j=1}^n \|X(t_j)-X(d(t_j))\|+\lambda\int \big|\kappa_1(t) -\kappa_2(d(t))\big|  \;dt  $$
where $X$ is a parameterization of CC and $\kappa_j$ are the
curvature functions. After registering the curves, a local
estimation of curvature could be compared across subjects, using
Welch's $t$-test at each point to correct for the somewhat unequal
variance in a few areas (Figure 4).

\begin{figure}
\centering
\includegraphics[width=190mm]{avg_curvature}
\caption{Mean curvatures of autistic (blue) and control (red)
groups showing different curvature patterns. R1 is the posterior
midbody. \label{fig:curvature}}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=170mm]{dtw_maps}
\caption{Outer curve: template, inner curve: individual CC
boundary. The template has been enlarged to show the pattern of
curve registration.\label{fig:curvature}}
\end{figure}

\section*{Results}
Both methods provided effective estimates of curvature for the
entire CC contours. The smoothing splines performed better in
terms of ease of fit and more stable results. The results of the
comparison of curvatures between the autistic and control subjects
are shown in Figure 2 where the sample mean curvature functions
are plotted (blue: autistic, red: control). Figure 5 is the
uncorrected P-value map where the blue areas indicate more
significant curvature difference. Most significant curvature
difference is detected at the posterior midbody (R1).
\begin{figure}
\centering
\includegraphics[width=180mm]{curvature_tmap}
\caption{$t$-statistic map of curvature difference between
autistic and control groups. \label{fig:curvature}}
\end{figure}
\begin{figure}
\centering

\includegraphics[width=180mm]{curvature_pvalue}
\caption{$P$-value map of curvature difference between two groups.
\label{fig:curvature}}
\end{figure}

\section*{Acknowledgements} The original level set and dynamic
time warping code were due to Fan Ding and Yuefeng Lu and have
been modified extensively. This work was funded by WARF, NIMH U54
MH066398-1A1, NICHD U19 HD035476, NSF Grant DMS0072292 and by a
NARSAD Distinguished Investigator Award to RJD.
\begin{thebibliography}{10}
\small
\bibitem{Harden}
Harden, A.Y. et al. Corpus Callosum Size in Autism. Neurology,
{\bf 55}:1033-1036, 2000.

\bibitem{Malladi}
Malladi, R. and Sethian, J.A. An o(n log(n)) Algorithm for Shape
Modeling. Applied Mathematics {\bf 18}:9389--9392, 1996.

\bibitem{Taubin}
Taubin, G. A Signal Processing Approach to Fair Surface Design.
Computer Graphics, {\bf 29}:351--358, 1995.

\bibitem{Wang} Wang, K. and Wang, Y. Alignment of Curves by Dynamic Time
Warping. Annals of Statistics, {\bf 25}:1251--1276, 1997.

\bibitem{Witelson} Witelson, S.F. Hand and sex differences in the isthmus and genu
of the human corpus callosum. Brain {\bf 112}:799--835. 1989.

\bibitem{Carew} Carew, J.D., Dalal, R.K., Wahba, G. and Fain, S.B.
Estimating arterior wall shear stress.
Tech. Report 1088, University of Wisconsin.


\end{thebibliography}

\end{multicols}
\end{document}
