4D hyperspherical harmonic (HyperSPHARM) representation of surface anatomy: a holistic treatment of multiple disconnected anatomical structures

Abstract

Image-based parcellation of the brain often leads to multiple disconnected anatomical structures, which pose significant challenges for analyses of morphological shapes. Existing shape models, such as the widely used spherical harmonic (SPHARM) representation, assume topological invariance, so are unable to simultaneously parameterize multiple disjoint structures. In such a situation, SPHARM has to be applied separately to each individual structure. We present a novel surface parameterization technique using 4D hyperspherical harmonics in representing multiple disjoint objects as a single analytic function, terming it HyperSPHARM. The underlying idea behind HyperSPHARM is to stereographically project an entire collection of disjoint 3D objects onto the 4D hypersphere and subsequently simultaneously parameterize them with the 4D hyperspherical harmonics. Hence, HyperSPHARM allows for a holistic treatment of multiple disjoint objects, unlike SPHARM. In an imaging dataset of healthy adult human brains, we apply HyperSPHARM to the hippocampi and amygdalae. The HyperSPHARM representations are employed as a data smoothing technique, while the HyperSPHARM coefficients are utilized in a support vector machine setting for object classification. HyperSPHARM yields nearly identical results as SPHARM, as will be shown in the paper. Its key advantage over SPHARM lies computationally; HyperSPHARM possess greater computational efficiency than SPHARM because it can parameterize multiple disjoint structures using much fewer basis functions and stereographic projection obviates SPHARM's burdensome surface flattening. In addition, HyperSPHARM can handle any type of topology, unlike SPHARM, whose analysis is confined to topologically invariant structures.

Keywords: Classification; Hippocampus & Amygdala; Hyperspherical harmonics; SPHARM; Shape analysis.

Figure 1. Holistic Treatment of Multiple Disjoint Structures

The underlying idea of HyperSPHARM is stereographically projecting n-dimensional data onto the (n+1)-dimensional sphere in order to subsequently parameterize the data with the (n+1)-dimensional spherical harmonics. Here we illustrate the n = 2 case. Three disjoint 2D objects are mapped on the 3D sphere. Since each object is unique in 2D, their projections onto the sphere will also be unique. Consequently, all three disjoint objects exist on the same sphere, so according to Fourier analysis they can be simultaneously parameterized by the 3D spherical harmonics. Please note that the shapes’ angles are preserved since stereographic projection is conformal. However, the projected shapes lying on the sphere will experience metric distortion, e.g. the area of the rectangle existing on the sphere is different from that of the rectangle lying on the 2D plane.

Figure 2

The 3D subcortical structures (left) in the coordinates (v¹, v², v³) went through the 4D stereographic projection that resulted in conformally deformed structures (right) in the 4D spherical coordinates (β, θ, ϕ). The 3D subcortical structure is then embedded on the surface of the 4D hypersphere with radius p_o = 23.

Figure 3. HyperSPHARM Interpolation

In this 3D illustration, a 2D object is stereographically projected onto the 3D sphere S². Since the object is finite, its projection will not occupy the entire surface of the sphere; rather, it will lie along a portion of the spherical surface, which in our illustration is denoted as S′. Points residing within S′ can be used for interpolation, whereas outside points will lead to extrapolation.

Figure 4

Plot of ${\mathrm{MSE}}_{\mathrm{HSH}}^{\mathrm{interp}}$ as a function of the hypersphere radius p_o. The HyperSPHARM coefficients are estimated using the population template, and then used to interpolate along the hyperspherical mesh hysph mesh interp. The HSH of truncation order N = 6 are employed. The MSE is minimized at p_o = 23, which we adopt as our radius.

Figure 5

HyperSPHARM (N = 6) representations of amygdala and hippocampus surfaces for subjects 10 and 68. The vertex-wise reconstruction errors are also plotted.

Figure 6

We rotate the MIDAS by some angle to see how the subsequent HyperSPHARM reconstruction is affected. Plot of ${\mathrm{MSE}}_{\mathrm{HSH}}^{\mathrm{interp}}$ as a function of the rotation angle is shown above. The HyperSPHARM coefficients are estimated using the population template, and then used to interpolate along the hyperspherical mesh hysph mesh interp. HyperSPHARM parameters are N = 6 and p_o = 23. The plot confirms the rotational variance of HyperSPHARM, which is due to stereographic projection being dependent on rotation.

Figure 7. Simulation Experiment I

We select the right hippocampus and right amygdala of two subjects that exhibit manifest shape differences, and create two distinct groups by adding Gaussian noise N(0, 0.01) to each subject's surface.

Figure 8. Simulation Experiment I Results

We carry out a Hotelling T² test to see if HyperSPHARM/SPHARM can distinguish between two groups that have manifest shape differences. The p-values after FDR correction (i.e. q-value) are projected back onto the template, which is the average of the 60 simulated surfaces. Group differences are detected using each method, with all voxels statistically significant.

Figure 9. Simulation Experiment II

We select the right hippocampus and right amygdala of a subject. We create two distinct groups that barely have any shape differences. The first group was formed by adding Gaussian noise N(0, 0.01) to the surface, while the second was created using N(0, 0.16).

Figure 10. Simulation Experiment II Results

We carry out a Hotelling T² test to see if HyperSPHARM/SPHARM can distinguish between two groups that are nearly identical in shape. The p-values after FDR correction (i.e. q-value) are projected back onto the template, which is the average of the 60 simulated surfaces. No group differences are detected using each method, with all voxels statistically insignificant.

Figure 11

Plot of the percentage of statistically significant voxels as a function of truncation order for each simulation experiments using Hyper-SPHARM and SPHARM. Experiment I involves Hotelling T² analysis of two distinct groups characterized by major shape differences between them, while Experiment II looks at two distinct groups characterized by very little shape differences between them.

Figure 12

The SPHARM results for Simulation Experiment I using L = 2 and L = 10. Although consistent with L = 20 SPHARM results, the lower-order SPHARM representations over-smooth the MIDAS, especially L = 2.

Figure 13

Statistical testing for gender effects in the hippocampi and amygdalae thresholded at p < 0.05 (corrected). A T-statistic exceeding 4.8 indicates statistical significance. No statistically significant gender effect was detected using either method.

Figure 14

Statistical testing for age effects in the hippocampi and amygdalae thresholded at p < 0.05 (corrected). A T-statistic exceeding 4.8 indicates statistical significance. Statistically significant age effects were detected, mainly in the tail regions of the hippocampi, using both methods.