Measuring hidden phenotype: quantifying the shape of barley seeds using the Euler characteristic transform

Shape plays a fundamental role in biology. Traditional phenotypic analysis methods measure some features but fail to measure the information embedded in shape comprehensively. To extract, compare and analyse this information embedded in a robust and concise way, we turn to topological data analysis (TDA), specifically the Euler characteristic transform. TDA measures shape comprehensively using mathematical representations based on algebraic topology features. To study its use, we compute both traditional and topological shape descriptors to quantify the morphology of 3121 barley seeds scanned with X-ray computed tomography (CT) technology at 127 μm resolution. The Euler characteristic transform measures shape by analysing topological features of an object at thresholds across a number of directional axes. A Kruskal–Wallis analysis of the information encoded by the topological signature reveals that the Euler characteristic transform picks up successfully the shape of the crease and bottom of the seeds. Moreover, while traditional shape descriptors can cluster the seeds based on their accession, topological shape descriptors can cluster them further based on their panicle. We then successfully train a support vector machine to classify 28 different accessions of barley based exclusively on the shape of their grains. We observe that combining both traditional and topological descriptors classifies barley seeds better than using just traditional descriptors alone. This improvement suggests that TDA is thus a powerful complement to traditional morphometrics to comprehensively describe a multitude of 'hidden' shape nuances which are otherwise not detected.形状在生物学中具有基础性作用。传统表型分析方法虽可测量部分特征，但无法全面捕捉形状所蕴含的信息。为了以稳健且简洁的方式提取、对比与分析这类蕴含于形状中的信息，我们转向拓扑数据分析（topological data analysis, TDA），具体采用欧拉特征变换（Euler characteristic transform）。拓扑数据分析基于代数拓扑特征构建数学表征，可全面量化形状信息。为探究其应用潜力，我们计算了传统形状描述子与拓扑形状描述子，以量化3121颗经127μm分辨率X射线计算机断层扫描（CT）成像的大麦种子的形态特征。欧拉特征变换通过分析物体在多方向轴阈值下的拓扑特征来表征形状。针对拓扑特征签名所编码的信息开展克鲁斯卡尔-沃利斯检验后发现，欧拉特征变换可有效捕捉大麦种子的棱纹与底部形状。此外，传统形状描述子可依据种子所属的品系对其进行聚类，而拓扑形状描述子还能进一步基于种子对应的圆锥花序进行细分聚类。随后我们成功训练了支持向量机（support vector machine），仅基于大麦籽粒形状即可对28个不同大麦品系进行分类。我们发现，将传统描述子与拓扑描述子结合使用时，大麦种子的分类效果优于仅使用传统描述子的情况。这一改进表明，拓扑数据分析可作为传统形态计量学的有力补充，能够全面捕捉诸多此前无法被识别的“隐藏”形状细微特征。