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## Results: 12

1.

2.

Illustration of lateral ventricular surface registration using holomorphic one-forms. An example left ventricular surface is shown in the middle with its segmentation results. After this partitioning into several components, a new canonical holomorphic one-form is computed on each piece and each piece is conformally mapped to a rectangle. The new conformal parameterization is visualized using conformal grids. Surface registration is performed by using a constrained harmonic map on the computed conformal coordinates.

3.

Comparison of two lateral ventricular surface parameterization results. (a) is its conformal parameterization with the canonical holomorphic one-form; (b) is its spherical parameterization result (Friedel et al., 2005). Both parameterizations are visualized using texture mapping of a checker board onto the surface. The parameterization in (a) is more uniform than the one in (b), which is very helpful for surface registration.

4.

Illustration of conformal net on various surfaces. (a) is

*w*-plane, where*w*=*z*^{2}; (b) is the*z*-plane. (c) and (d) illustrate the conformal nets on genus 0 surfaces with 2 and 4 open boundaries, respectively. Horizontal trajectories (blue) and vertical trajectories (red) are illustrated in (b)–(d). (c) is topologically equivalent to the ventricular surface after the topology change.5.

Statistical

*p*-map showing multivariate tensor-based morphometry results based on a group of lateral ventricular surfaces from 11 HIV/AIDS patients and 8 matched control subjects. The overall (corrected) statistical significance values are 0.0066 and 0.0028 for the right and left ventricular surfaces, respectively.6.

Illustration of various holomorphic one-forms on a left ventricular surface, generated by using different linear combinations of elements of the holomorphic one-form basis. The holomorphic one-form and its induced conformal parametrization (by Equation 2) are visualized by the texture mapping of a checkerboard onto the surface, such that the checkerboard pattern is uniform in the parameter space. (a) is the canonical conformal parameterization. We use it for surface registration. (b)–(d) show highly irregular areas around the ends of different horns of the ventricles.

7.

Comparison of various tensor-based morphometry results on a group of lateral ventricular surfaces from 11 HIV/AIDS patients and 8 matched control subjects. Multivariate statistics on the full metric tensor detected anatomical differences more powerfully than other scalar statistics. A comparison of the overall (corrected) statistical significance values is given in Table 1.

8.

Comparison of various tensor-based morphometry results on a group of lateral ventricular surfaces from 31 HIV/AIDS patients and 20 matched control subjects (Thompson et al., 2006). The statistical results are consistent with those in the selected smaller dataset Figure 10 and 11. The overall (corrected) statistical significance values are shown in Table 2.

9.

Illustration of automatic lateral ventricular surface segmentation via holomorphic one-forms. (a). Topology change. Three cuts were made on each ventricular surface “extreme points”. (b). One computed exact harmonic one-form, which is visualized by integrating the one-form on the open boundary surface. (c). Conjugate one-form of the one-form in (b). This is computed using our the new algorithm, and is locally perpendicular to the one-form in (b). (d). The canonical holomorphic one-form. The zero point locations are consistent across surfaces. (e). The conformal net induced by the canonical holomorphic one-form. This conformal net separates each ventricular surface into three pieces, each of which is color-coded in (f).

10.

A flow chart shows how canonical holomorphic one-forms are used to model ventricular shape, and how the resulting surfaces are analyzed using multivariate tensor-based morphometry. After ventricular surfaces are extracted from MRI images either manually or automatically (Thompson et al., 2006), surfaces are automatically partitioned into three pieces by computing canonical holomorphic one-forms. For each element of the partitioned surface, we compute a new conformal coordinates and register surfaces with a constrained harmonic map. The statistics of multivariate TBM are computed at each point on the resulting matching surfaces, revealing regions with systematic anatomical differences between groups.

11.

The cumulative distribution of

*p*-values*versus*the corresponding cumulative*p*-value that would be expected from a null distribution for multivariate TBM (magenta) and various tensor-based morphometry results, including the pair of eigenvalues of Jacobian matrix (green), the Jacobian determinant (blue), the largest eigenvalue (blue) and the smallest eigenvalue (black) of the Jacobian matrix, on a group of lateral ventricular surfaces from 11 HIV/AIDS patients and 8 matched control subjects. In FDR methods, any cumulative distribution plot that rises steeply is a sign of a significant signal being detected, with curves that rise faster denoting higher effect sizes. The steep rise of the cumulative plot relative to*p*-values that would be expected by chance can be used to compare the detection sensitivity of different statistics derived from the same data.12.

Detection of a synthetic group difference applied to the left ventricular surfaces of control subjects. The synthetic group of surfaces is generated by applying a geometric deformation (a known mathematical function) to a selected “bump” area (shown in (a)) for each left ventricular surface in a group of control subjects (8 CTLs). The goal is to test the algorithm’s ability to detect the group difference between the original set of left ventricular surfaces and the new set of left ventricular surfaces with deliberately introduced shape changes in a prescribed area. (b)–(d) illustrate the areas that show statistical significance at the voxel level when varying user-selected parameters that control the severity of the introduced shape difference. For the frontal horn, which contains the selected area, the overall significance of the group difference maps is 0.1642, 0.0248 and 0.0004, respectively, when increasing the magnitude of the shape difference. This shows that the effects are detected in an intuitive way when the anatomical scope and magnitude of the shape change is systematically controlled.