Analysis of Joint Shape Variation from Multi-Object Complexes

Abstract

Shape correlation of multi-object complexes in the human body can have significant implications in understanding the development of disease. While there exist geometric and statistical methods that aim for multi-object shape analysis, very little research can effectively extract shape correlation. It is especially difficult to extract the correlation when the involved objects have different variability in separate non-Euclidean spaces. To address these difficulties, this paper proposes geometric and statistical methods to extract the shape correlation from multi-object complexes. In particular, we focus on the shape correlation of the hippocampus and the caudate subject to the development of autism. The proposed methods are designed (1) to capture objects’ shape features (2) to capture shape correlation regardless of different variability between the two objects and (3) to provide interpretable shape correlation in multi-object complexes. In our experiments on synthetic data and autism data, the quantitative results and the qualitative visualization suggest that our methods are effective and robust.

A Non-Euclidean Joint and Individual Variation Explained

Fig. 11

Joint and individual structures from NEUJIVE on two correlated non-Euclidean blocks. The first row shows the simulation where the joint structures are designed as circular variables along Small Circles (SC) in the two blocks (see Eq. (11)). Moreover, the two blocks share the Equal Noise Level (ENL) in which $\epsilon _k$ share the same standard deviation. The second row shows a different simulation study. In this simulation, the joint component is designed to be a Great Circle (GC) in block 1. But this component follows a small circle in block 2. Moreover, the two blocks have Different Noise Level (DNL). On each sphere, the black dots are simulated data. The magenta dots are joint structures from each algorithm. The blue dots are individual structures from NEUJIVE and residual structures from AJIVE, respectively

Fig. 12

Root Mean Square Error (RMSE) between the actual joint structures and NEUJIVE estimated joint structures as increasing the noise level. Left panel: Simulated data as increasing the standard deviation $\sigma (\epsilon _k)$ of noise from 0.01 to 0.05 and to 0.1. The two circular blocks are of different radii, i.e., $a_1=0.25$ while $a_2=0.35$. Right Panel: RMSE of the NEUJIVE joint structures of the two blocks as increasing standard deviation of noise. The red and the blue curve are, respectively, the RMSE of the block 1 and the block 2

In this section, we show more comprehensive experimental analysis of NEUJIVE.

First, it is of interest to show all NEUJIVE components of the toy example discussed in Sect. 7.2, which has homogeneous circular data in each block. In Fig. 11 the first row shows both the joint and the individual structures from NEUJIVE. The individual structures are designed as random variables from multivariate Gaussian (see Eq. (11)). Moreover, the top left cell in Fig. 11 shows that the individual structures (shown as blue dots) from NEUJIVE are distributed around the PNS mean as expected. As a comparison, AJIVE fails to find significant individual components in either block, as shown in the top right cell. Instead, the blue dots are the residual components resulting from AJIVE.

Second, we show the NEUJIVE components when the two blocks are of notably different variability in the bottom row of Fig. 11. Different from Sect. 7.2, the first block is generated via

$$\begin{aligned} X_1 = g_1(Exp(p(\theta ) + \epsilon _1)) \end{aligned}$$

(13)

where $p(\theta )$ is a straight line in the tangent space at the north pole. A point on this line has coordinates $(\theta , 0.3\theta )$. The second block is still generated via Eq. (11). Moreover, to have Different Noise Levels (DNL) across the two blocks, we set different standard deviations of $\epsilon _k$ across the two blocks. The results show that NEUJIVE can still give effective estimation of the joint components regardless of the different variability between the two blocks.

Third, we investigate the robustness of the joint structures estimated by NEUJIVE as we increase noise. We simulate two blocks of data via Eq. (11), increasing the standard deviation of $\epsilon _k$ from 0.01 to 0.1. The simulated data under 3 different noise levels can be seen in Fig. 12 left, i.e., $\sigma (\epsilon _k)=\{0.01,\, 0.05,\, 0.1\}$. For each noise level, we compute the joint structure from NEUJIVE. We measure the Root of Mean Square Error (RMSE) between the NEUJIVE joint structures and the actual joint structures. The curves of RMSE versus noise levels of the two blocks are shown in Fig. 12 right. This figure shows that the RMSE is almost linearly increasing along with the increasing noise level.