(c) Moo K. Chung

mkchung@wisc.edu

Department of Biostastics and Medical Infomatics

Waisman Laboratory for Brain Imaging and Behavior

University of Wisconsin-Madison

Description

July 14, 2010

We will
show how to construct persistence diagrms [1] and
min-max diagrams
[2] [3] for 1D (simulated functional signal) and 2D data (cortical
thickness). The persistence and min-max diagrams are motivated by
topological data analysis where the underlying topology of data and
images is quantied.

The
codes
have been tested
under a Macbook pro with 4GB
memory and MATLAB 7.5. If
you are using the Matlab codes below for your publication,
please reference [1] or [2]. The detailed description of cortical
thickness data is given in [6].

Persistence & Min-Max diagrams for 1D functions

July 14, 2010

The construction for the persistence diagrm for 1D function is based on the iterative pairing and deleation algorithm [2]. Note that the min-max diagram [2] and the persistence diagram [1] are identical for 1D functions since there is no saddle points in 1D functions. Let us explain 1D example given in the MICCAI 2009 paper [2]. In the interval [0, 1], we construct a signal s and add noise e to obtain simulated signal Y.

x=[0:0.002:1]';

s= x + 7*(x -
0.5).^2 + cos(8*pi*x)/2;

e=normrnd(0,0.2,length(x),1);

Y=s+e;

plot(x,Y,'ko','MarkerEdgeColor',[0.5 0.5 0.5], 'MarkerFaceColor',[0.7 0.7 0.7], 'MarkerSize',4)

This produces a scatter points so we can't obtain critical values directly. We smooth out the scatter plot using heat kernel smoothing with bandwidth 0.0001 and cosine series expansion upto degree 100 [2] [3]. The resulting smoothed signal is stored in the variable wfs.

k=100; sigma=0.0001;

[wfs, beta]=WFS_COS(Y,x,k,sigma);

hold on;

plot(x,wfs,'k','LineWidth',5);

Then using the iterative pairing and deleation algorithm pairing_1D, we determine the pairing of minimums and maximums. pairing_1D also plots critical values as white dots (Figure 1).

pairs=pairing_1D(x,wfs);

set(gcf,'Color','w')

This produces 3 pairs:

>>
pairs

pairs =

1.3126 1.6816

0.5948 1.1636

0.2130 0.8886

```
The
result
will
be
slightly
different each time you run the code since the
added noise is always changing. The pairing rule is explained in Figure
1.
```

Figure 1.
The
births and deaths of components in sublevel sets. We have critical
values a,b,c,d,e,f, where a < b < d <f are minimums and c<
e are maximums. At y=a, we have a single component marked by a single
gray area. When we increase the level to y=b, we have the birth of a
new component in addition to the existing component born at a. At the
maximum y=c, the two components merge together to form a single
component. Following the pairing rule given in Edelsbrunner (2008), we
pair (c,b) and (e,d). Other critical values are paired similarly. See
[1] or [2].

Cortical thickness data

July 17, 2010

We will
use the cortical surface data set used in [2]. There are total 16
autistic and 11 control subjects. Cortical thickness is computed as the
L2 distance between the two surfaces. autism_coord
is the gray matter surface for autistic subjects and autism_coordw
is the white matter surface for control subjects.

load AUTISM.coordinates.mat

thick_au=squeeze(sqrt(sum((autism_coord-autism_coordw).^2,2)));

thick_co=squeeze(sqrt(sum((control_coord-control_coordw).^2,2)));

We will display cortical thickness of the 1st autistic subject on a
unit sphere.

load unitsphere.mat

figure_trimesh(sphere,thick_au(1,:),
'rwb')

Since the cortical thickness data is fairly noisy, we will smooth using
the weighted-sphearicl
harmonic
representation with degree 42 and bandwidth 0.001 [4]
[5]. The detailed explanation of SPHARMsmooth is given here.
The smoothed cortical thickness is given in Figure 2.

directory='/basis/';

SPHARMconstruct(directory,42);

L=42;

sigma=0.001;

coord_co=zeros(11,3,40962);

coord_co(:,1,:)=thick_co;

[coord_sm_co,fourier_coeff_co]=
SPHARMsmooth(coord_co,L,sigma);

coord_au=zeros(16,3,40962);

coord_au(:,1,:)=thick_au;

[coord_sm_au,fourier_coeff_au]=
SPHARMsmooth(coord_au,L,sigma);

Figure 2.
Weighted spherical harmonic representation of cortical thickness.
Critical values are identified as crosses.

Min-max diagram of a function on a sphere

April 28, 2010

The
min-max diagram is constructed by pairing minimums and maximums in a
particular fashion [2]. To simplify the problem, we identify how
critical values are connected using the Delaunay triangulation.

Figure 3.
Delaunay triangulation of few minimums and maximums.

Let us map
cortical thickness given in Figure 2 onto a plane for better
visualization. fourier_coeff_au.x(:,:,1)is the spherical harmonic
coefficients of the 1st autistic subject.

square=SPHARM2square(fourier_coeff_au.x(:,:,1),42,0.001);

[lmax,
lmin]
= figure_extrema(square);

Figure 4. Flatmap represention of cortical thickness given in Figure 2. Critical values are identified as crosses.

The minimum and maximums in the cortical thickness data are paired using the iterative pairing and deleation algorithm [2]. The pairing for all critical values identified in Figure 2 and 4 can be done by

value=squeeze(coord_au(1,1,:));

pairs= pairing_mesh(sphere, value, L, sigma);

The code requires FINDnbr. Since we are expected to obtain topologically invariant pairing, we constructed the pairing in the spherical mesh.

References

April
22,
2010;
October 8, 2010

- Chung, M.K., Bubenik, P., Kim, P.T. 2009. Persistence diagrams of cortical surface data. Information Processing in Medica Imaging (IPMI). Lecture Notes in Computer Science (LNCS). 5636:386-397.
- Chung, M.K., Singh, V., Kim, P.T., Dalton, K.M., Davidson, R.J. 2009. Topological characterization of signal in brain images using the min-max diagram. 12th International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI). Lecture Notes in Computer Science (LNCS). 5762:158-166.
- Pachauri, D., Hinrichs, C., Chung, M.K., Johnson, S.C., Singh, V. and ADNI. 2010. Cortical surface topology based kernels with application to alzheimer's disease. IEEE Transactions on Medical Imaging. under revision.
- Chung, M.K., Dalton, K.M., Davidson, R.J. 2008. Tensor-based cortical surface morphometry via weighed spherical harmonic representation. IEEE Transactions on Medical Imaging. 27:1143-1151.
- Chung, M.K., Dalton, K.M., Shen, L., L., Evans, A.C., Davidson, R.J. 2007. Weighted Fourier series representation and its application to quantifying the amount of gray matter. IEEE Transactions on Medical Imaging 26:566-581.
- Chung, M.K., Robbins,S., Dalton, K.M., Davidson, Alexander, A.L.,
R.J., Evans, A.C. 2005. Cortical
thickness analysis in autism via heat kernel smoothing. NeuroImage 25:1256-1265