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Cell boundary information is crucial for analyzing cell behaviors from time-lapse microscopy videos. Existing supervised cell segmentation tools, such as ImageJ, require tuning various parameters and rely on restrictive assumptions about the shape of the objects. While recent supervised segmentation tools based on convolutional neural networks enhance accuracy, they depend on high-quality labeled images, making them unsuitable for segmenting new types of objects not in the database. We developed a novel unsupervised cell segmentation algorithm based on fast Gaussian processes for noisy microscopy images without the need for parameter tuning or restrictive assumptions about the shape of the object. We derived robust thresholding criteria adaptive for heterogeneous images containing distinct brightness at different parts to separate objects from the background, and employed watershed segmentation to distinguish touching cell objects. Both simulated studies and real-data analysis of large microscopy images demonstrate the scalability and accuracy of our approach compared with the alternatives. 

We introduce theshapemodule of the Python package Geomstats to analyze shapes of objects represented as landmarks, curves, and surfaces across fields of natural sciences and engineering. Theshapemodule first implements widely used shape spaces, such as the Kendall shape space, as well as elastic spaces of discrete curves and surfaces. Theshapemodule further implements the abstract mathematical structures of group actions, fiber bundles, quotient spaces, and associated Riemannian metrics which allow users to build their own shape spaces. The Riemannian geometry tools enable users to compare, average, interpolate between shapes inside a given shape space. These essential operations can then be leveraged to perform statistics and machine learning on shape data. We present the object-oriented implementation of theshapemodule along with illustrative examples and show how it can be used to perform statistics and machine learning on shape spaces. 

We present a method for computing an approximate Rieman-nian barycenter of a collection of points lying on a Riemannian mani-fold. Our approach relies on the use of theoretically proven under- and over-approximations of the Riemannian distance function. We compare it to Riemannian steepest descent on the exact objective function of the Riemannian barycenter and to an approach that approximates the Rie-mannian logarithm using lifting maps. Experiments are conducted on the Stiefel manifold. 

Elastic metrics can provide a powerful tool to study the heterogeneity arising from cellular morphology. To assess their potential application (e.g. classifying cancer treated cells), we consider a specific instance of the elastic metric, the Square Root Velocity (SRV) metric and evaluate its performance against the linear metric for two datasets of osteosarcoma (bone cancer) cells including pharmacological treatments, and normal and cancerous breast cells. Our comparative statistical analysis shows superior performance of the SRV at capturing cell shape heterogeneity when comparing distance to the mean shapes, with better separation and interpretation between different cell groups. Secondly, when using multidimensional scaling (MDS) to find a low-dimensional embedding for unrescaled contours, we observe that while the linear metric better preserves original pairwise distances, the SRV yields better classification. 

Image datasets in specialized fields of science, such as biomedicine, are typically smaller than traditional machine learning datasets. As such, they present a problem for training many models. To address this challenge, researchers often attempt to incorporate priors, i.e., external knowledge, to help the learning procedure. Geometric priors, for example, offer to restrict the learning process to the manifold to which the data belong. However, learning on manifolds is sometimes computationally intensive to the point of being prohibitive. Here, we ask a provocative question: is machine learning on manifolds really more accurate than its linear counterpart to the extent that it is worth sacrificing significant speedup in computation? We answer this question through an extensive theoretical and experimental study of one of the most common learning methods for manifold-valued data: geodesic regression.