Advances in imaging and sequencing techniques provide increasingly informative but complex structural data for understanding the 3D genome. Segmenting 3D conformations of chromosomes into their topologically associated domains (TADs) – structurally homogeneous parts of a chromosome – is an essential tool in shape analysis. TAD segmentation allows one to divide complex chromosomal structures into smaller, simpler geometries and facilitates statistical analysis of their shapes. There are several TAD segmentation procedures in the current literature, but they often lack consistency and interpretability. We propose a novel algorithm for determining TADs, called TADBay, directly from contact matrices, which avoids the difficult step of estimating 3D conformations. While consistent with some current top-performing existing callers, it has an added ability to provide a hierarchical organization of TADs and subTADs. To validate TADBay, we utilize simulated contact matrices with ground truth TADs which can be used to benchmark TAD callers. We demonstrate the strengths of our caller through both simulated data and the IMR90 real dataset.

Elastic Riemannian metrics have been used successfully for statistical treatments of functional and curve shape data. However, this usage suffers from a significant restriction: the function boundaries are assumed to be fixed and matched. In practice, functional data often comes with unmatched boundaries. It happens, for example, in dynamical systems with variable evolution rates, such as COVID-19 infection rate curves associated with different geographical regions. Here, we develop a Riemannian framework that allows for partial matching, comparing, and clustering of functions with phase variability and uncertain boundaries. We extend past work by (1) Defining a new diffeomorphism group G over the positive reals that is the semidirect product of a time-warping group and a time-scaling group; (2) Introducing a metric that is invariant to the action of G; (3) Imposing a Riemannian Lie group structure on G to allow for an efficient gradient-based optimization for elastic partial matching; and (4) Presenting a modification that, while losing the metric property, allows one to control the amount of boundary disparity in the registration. We illustrate this framework by registering and clustering shapes of COVID-19 rate curves, identifying basic patterns, minimizing mismatch errors, and reducing variability within clusters compared to previous methods.

Despite enormous structural variability exhibited in 3D chromosomal conformations at a global scale, there is a significant commonality of structures visible at smaller, local levels. We hypothesize that chromosomal conformations are representable as concatenations of a handful of prototypical shapelets, termed shape letters. This is akin to expressing complicated sentences in a language using only a small set of letters. Our goal is to organize the vast variability of 3D chromosomal conformation by constructing a set of predominant shape letters, termed a shape alphabet, using statistical shape analysis of curvelets taken from training conformations. This paper utilizes conformations generated from Integrative Genome Modeling to develop a shape alphabet as follows: it first segments 3D conformations into curvelets according to their Topologically Associated Domains. It then clusters these segments, estimates mean shapes, and refines and reorders these shapes into a Chromosome Shape Alphabet. The paper demonstrates effectiveness of this construction by successfully representing independent test conformations taken from IGM and other methods such as SIMBA3D, both symbolically and structurally, using the constructed alphabet.

A common problem in genomics is to test for associations between two or more genomic features, typically represented as intervals interspersed across the genome. Existing methodologies can test for significant pairwise associations between two genomic intervals; however, they cannot test for associations involving multiple sets of intervals. This limits our ability to uncover more complex, yet biologically important associations between multiple sets of genomic features. We introduce GINOM (Genomic INterval Overlap Model), a new method that enables testing of significant associations between multiple genomic features. We demonstrate GINOM's ability to identify higher-order associations with both simulated and real data. In particular, we used GINOM to explore L1 retrotransposable element insertion bias in lung cancer and found a significant pairwise association between L1 insertions and heterochromatic marks. Unlike other methods, GINOM also detected an association between L1 insertions and gene bodies marked by a facultative heterochromatic mark, which could explain the observed bias for L1 insertions towards cancer-associated genes.

Active contour, especially in conjunction with prior-shape models, has become an important tool in image segmentation. However, most contour methods use shape priors based on similarity-shape analysis, i.e. analysis that is invariant to rotation, translation, and scale. In practice, the training shapes used for prior-shape models may be collected from viewing angles different from those for the test images and require invariance to a larger class of transformation. Using an elastic, affine-invariant shape modeling of planar curves, we propose an active contour algorithm in which the training and test shapes can be at arbitrary affine transformations, and the resulting segmentation is robust to perspective skews. We construct a shape space of affine-standardized curves and derive a statistical model for capturing class-specific shape variability. The active contour is then driven by the true gradient of a total energy composed of a data term, a smoothing term, and an affine-invariant shape-prior term. This framework is demonstrated using a number of examples involving the segmentation of occluded or noisy images of targets subject to perspective skew.

Current techniques for shape analysis tend to seek invariance to similarity transformations (rotation, translation, and scale), but certain imaging situations require invariance to larger groups, such as affine or projective groups. Here we present a general Riemannian framework for shape analysis of planar objects where metrics and related quantities are invariant to affine and projective groups. Highlighting two possibilities for representing object boundaries-ordered points (or landmarks) and parameterized curves-we study different combinations of these representations (points and curves) and transformations (affine and projective). Specifically, we provide solutions to three out of four situations and develop algorithms for computing geodesics and intrinsic sample statistics, leading up to Gaussian-type statistical models, and classifying test shapes using such models learned from training data. In the case of parameterized curves, we also achieve the desired goal of invariance to re-parameterizations. The geodesics are constructed by particularizing the path-straightening algorithm to geometries of current manifolds and are used, in turn, to compute shape statistics and Gaussian-type shape models. We demonstrate these ideas using a number of examples from shape and activity recognition.

We present a Riemannian framework for analyzing shapes of planar contours in which metrics and other analyses are invariant to affine transformations and re-parameterizations of contours. Current methods that are affine invariant are restricted to point sets and do not handle full curves, while methods that analyze parameterized curves are restricted to equivalence under similarity transformation (rigid motion and scale). We construct a pre-shape manifold of standardized curves - curves whose centroid is at the origin, are of unit length, and their x and y coordinates are uncorrelated - and develop a path-straightening technique for computing geodesics on this nonlinear manifold under the elastic Riemannian metric. The removal of the rotation and the re-parameterization groups results in a quotient space, termed affine elastic shape space, and the resulting geodesic paths exhibit an improved matching of features across curves. These geodesics are used for shape comparison, retrieval, and statistical modeling of given curves. Experimental results using both simulated and real data, and an application involving pose-invariant activity recognition, demonstrate the success of this framework.