14 research outputs found
Hypothesis testing for medical imaging analysis via the smooth Euler characteristic transform
Shape-valued data are of interest in applied sciences, particularly in
medical imaging. In this paper, inspired by a specific medical imaging example,
we introduce a hypothesis testing method via the smooth Euler characteristic
transform to detect significant differences among collections of shapes. Our
proposed method has a solid mathematical foundation and is computationally
efficient. Through simulation studies, we illustrate the performance of our
proposed method. We apply our method to images of lung cancer tumors from the
National Lung Screening Trial database, comparing its performance to a
state-of-the-art machine learning model
A Sheaf-Theoretic Construction of Shape Space
We present a sheaf-theoretic construction of shape space -- the space of all
shapes. We do this by describing a homotopy sheaf on the poset category of
constructible sets, where each set is mapped to its Persistent Homology
Transform (PHT). Recent results that build on fundamental work of Schapira have
shown that this transform is injective, thus making the PHT a good summary
object for each shape. Our homotopy sheaf result allows us to "glue" PHTs of
different shapes together to build up the PHT of a larger shape. In the case
where our shape is a polyhedron we prove a generalized nerve lemma for the PHT.
Finally, by re-examining the sampling result of Smale-Niyogi-Weinberger, we
show that we can reliably approximate the PHT of a manifold by a polyhedron up
to arbitrary precision.Comment: 17 pages, 6 figure
Detecting Temporal shape changes with the Euler Characteristic Transform
Organoids are multi-cellular structures that are cultured in vitro from stem cells to resemble specific organs (e.g., brain, liver) in their three-dimensional composition. Dynamic changes in the shape and composition of these model systems can be used to understand the effect of mutations and treatments in health and disease. In this paper, we propose a new technique in the field of topological data analysis for DEtecting Temporal shape changes with the Euler Characteristic Transform (DETECT). DETECT is a rotationally invariant signature of dynamically changing shapes. We demonstrate our method on a data set of segmented videos of mouse small intestine organoid experiments and show that it outperforms classical shape descriptors. We verify our method on a synthetic organoid data set and illustrate how it generalizes to 3D. We conclude that DETECT offers rigorous quantification of organoids and opens up computationally scalable methods for distinguishing different growth regimes and assessing treatment effects
Euler Characteristics and Homotopy Types of Definable Sublevel Sets
Given a continuous definable function on a definable
set , we study sublevel sets of the form
for all . Using o-minimal structures, we prove that the Euler
characteristic of is right continuous with respect to . Furthermore,
when is compact, we show that deformation retracts to
for all sufficiently small . Applying these results, we
also characterize the relationship between the concepts of Euler characteristic
transform and smooth Euler characteristic transform in topological data
analysis.Comment: 14 pag
Efficient Graph Reconstruction and Representation Using Augmented Persistence Diagrams
Persistent homology is a tool that can be employed to summarize the shape of
data by quantifying homological features. When the data is an object in
, the (augmented) persistent homology transform ((A)PHT) is a
family of persistence diagrams, parameterized by directions in the ambient
space. A recent advance in understanding the PHT used the framework of
reconstruction in order to find finite a set of directions to faithfully
represent the shape, a result that is of both theoretical and practical
interest. In this paper, we improve upon this result and present an improved
algorithm for graph -- and, more generally one-skeleton -- reconstruction. The
improvement comes in reconstructing the edges, where we use a radial binary
(multi-)search. The binary search employed takes advantage of the fact that the
edges can be ordered radially with respect to a reference plane, a feature
unique to graphs.Comment: This work originally appeared in the 2022 proceedings of the Canadian
Conference on Computational Geometry (CCCG). We have updated the proof of
Theorem 2 in Appendix A for clarity and correctness. We have also corrected
and clarified Section 3.2, as previously, it used slightly stricter general
position assumptions than those given in Assumption
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.</p