Asymmetry Analysis of Bilateral Shapes

TL;DR

This paper develops a landmark-based framework for quantifying departures from bilateral symmetry within a size-and-shape context, using rigid-body registration and a midplane to align shapes. It introduces two complementary hypothesis-testing strategies—combine-then-compare with composite asymmetry scores like $\phi_{L1}$ and $\phi_{L2}$, and compare-then-combine via per-feature tests and UIT—with meta-analysis to aggregate evidence. Applied to pre-registered Smile Data 1 (cleft lip) and Smile Data 2 (orthognathic surgery), the methods detect significant asymmetry differences, identify central landmarks (the so-called 'Y'-shape) as especially informative, and reveal that cleft lip subjects are generally more asymmetric than controls while orthognathic surgery can increase or normalize asymmetry across frames. The framework demonstrates robust discriminative power across both univariate and multivariate testing approaches and provides a path toward a clinically useful smile index derived from symmetry measures. The work thus offers a rigorous, registration-based toolkit for symmetry assessment with direct medical relevance and broad applicability to registered shape data.

Abstract

Many biological objects possess bilateral symmetry about a midline or midplane, up to a ``noise'' term. This paper uses landmark-based methods to measure departures from bilateral symmetry, especially for the two-group problem where one group is more asymmetric than the other. In this paper, we formulate our work in the framework of size-and-shape analysis including registration via rigid body motion. Our starting point is a vector of elementary asymmetry features defined at the individual landmark coordinates for each object. We introduce two approaches for testing. In the first, the elementary features are combined into a scalar composite asymmetry measure for each object. Then standard univariate tests can be used to compare the two groups. In the second approach, a univariate test statistic is constructed for each elementary feature. The maximum of these statistics lead to an overall test statistic to compare the two groups and we then provide a technique to extract the important features from the landmark data. Our methodology is illustrated on a pre-registered smile dataset collected to assess the success of cleft lip surgery on human subjects. The asymmetry in a group of cleft lip subjects is compared to a group of normal subjects, and statistically significant differences have been found by univariate tests in the first approach. Further, our feature extraction method leads to an anatomically plausible set of landmarks for medical applications.

Figures (8)

• Figure 1: Coordinates system used for registration of human face used in our illustrative smile examples: with three principal planes. In the figure at top row, the $x$-$y$ plane is in green, the $y$-$z$ plane is in red and the $x$-$z$ plane is in blue. For the three sub-figures in the bottom row, the $x$-axis is in green, $y$-axis is in red and $z$-axis is in blue.
• Figure 2: The left and right figures in the top row show the original configurations of symmetric square $X_1$ and asymmetric square $X_2$ (in black) respectively. The left and right figures in the bottom row show the original objects (in black solid lines) together with their respectively reflections $X_1^{\text{(refl)}}$ and $X_2^{\text{(refl)}}$ (in red dotted lines) respectively, in order to illustrate the coordinatewise elementary feature vector.
• Figure 3: Landmark indices for $x$-$y$ coordinates on the lip periphery of a control subject at first frame from Smile Data 1.
• Figure 4: Photo of a cleft patient with central pair landmarks and solo landmarks. This figure is reproduced from photo taken from the website https://www.nhs.uk/conditions/cleft-lip-and-palate/.
• Figure 5: Plots for three composite asymmetry scores. From top to bottom, we have $\phi^*_{L_1} (\bm a)$ (\ref{['as1']}) (in solid dots), $\phi_{L_1}(\bm a)$ (\ref{['eq:L1score']}) (in triangles) and $\phi_{L_2}(\bm a)$ (\ref{['eq:L2score']}) (in crosses) at the three frames computed on the Smile Data 1. Plot (a) is at first frame, while (b) is at middle frame, whereas (c) is at last frame. In plot (a), $\phi_{L_1}(\bm a)$ and $\phi_{L_2}(\bm a)$ are divided by 2. In plot (b) and (c), $\phi_{L_2}(\bm a)$ is divided by 3. The first, third and fifth rows from top to bottom are cleft while others are control.