Multivariate Planar Curves: A Statistical Framework for Shape Analysis in Images

Abstract

Recent developments in computer vision have made segmented images widely available across many domains, such as medicine, where segmented radiographs play an important role in diagnosis. As prediction problems are common in image analysis, this work explores the use of the object contours highlighted by such images as predictors in a supervised classification context. To this end, we develop a new statistical learning framework that accounts for the joint shape of the multiple objects contained in an image. We introduce a formalism that extends the study of a single random planar curve to the joint analysis of several planar curves, referred to as a multivariate planar curve. Modeling the contours jointly, rather than separately, preserves the inter-component information, such as their relative position, scale, and orientation, which is often essential to the analysis. Based on this model, we propose a joint alignment procedure and we extend core inferential tools to multivariate shapes: shape dissimilarity, Fréchet mean estimation, and tangent-space representation. These tangent coordinates are then used as predictors in standard functional classification models. A simulation study shows accurate recovery of deformation parameters over increasing noise levels. Then, through a cardiomegaly detection problem on segmented chest X-rays, we show that jointly modeling the contours is robust to misalignment and improves classification accuracy over both a contour-wise univariate analysis and a naive approach based on the raw curves.