[Paper Review] The Weighted Euler Curve Transform for Shape and Image Analysis

Qitong Jiang, Sebastian Kurtek+1 more|arXiv (Cornell University)|Apr 23, 2020

Topological and Geometric Data Analysis38 references4 citations

TL;DR

This paper introduces the Weighted Euler Curve Transform (WECT), a complete topological invariant for weighted simplicial complexes that extends the Euler Curve Transform to incorporate intensity or grayscale data. By mapping segmented tumor regions in MRI scans to weighted simplicial complexes and computing their WECT representations, the method enables effective clustering of glioblastoma multiforme tumors based on shape and texture, with clusters strongly correlating with patient survival time—demonstrating its utility in biomedical image analysis and survival prediction.

ABSTRACT

The Euler Curve Transform (ECT) of Turner et al.\ is a complete invariant of an embedded simplicial complex, which is amenable to statistical analysis. We generalize the ECT to provide a similarly convenient representation for weighted simplicial complexes, objects which arise naturally, for example, in certain medical imaging applications. We leverage work of Ghrist et al.\ on Euler integral calculus to prove that this invariant---dubbed the Weighted Euler Curve Transform (WECT)---is also complete. We explain how to transform a segmented region of interest in a grayscale image into a weighted simplicial complex and then into a WECT representation. This WECT representation is applied to study Glioblastoma Multiforme brain tumor shape and texture data. We show that the WECT representation is effective at clustering tumors based on qualitative shape and texture features and that this clustering correlates with patient survival time.

Motivation & Objective

• To develop a topological representation that integrates both geometric shape and intensity information in medical imaging data.
• To extend the Euler Curve Transform (ECT) to weighted simplicial complexes, enabling analysis of grayscale images as topological objects with value-weighted vertices.
• To establish the WECT as a complete invariant for weighted simplicial complexes using results from Euler integral calculus.
• To apply the WECT to real-world glioblastoma multiforme (GBM) tumor imaging data to assess its utility in clustering tumors and predicting patient survival.
• To provide open-source code for WECT computation and analysis, supporting reproducibility and broader adoption in shape analysis.

Proposed method

• Convert segmented grayscale MRI tumor regions into weighted simplicial complexes using Delaunay triangulation, where pixel intensities define vertex weights.
• Compute the weighted Euler curve for each direction on the unit sphere by evaluating the Euler characteristic of sublevel sets of the height function weighted by the intensity function.
• Smooth the resulting Euler curves using a Gaussian kernel with a window size of ten to reduce noise and improve stability.
• Compute pairwise L² distances between smoothed WECT representations, accounting for rotational alignment via registration over SO(2).
• Apply hierarchical clustering with Ward linkage to the distance matrix to identify tumor subtypes based on shape and texture similarity.
• Normalize all complexes to unit diameter and unit maximum weight to ensure comparability across subjects with varying image intensity distributions.

Experimental results

Research questions

• RQ1 Can the WECT effectively represent both shape and intensity features in segmented medical images, such as brain tumor MRIs?
• RQ2 Is the WECT a complete invariant for weighted simplicial complexes, meaning it uniquely determines the underlying complex and weight function?
• RQ3 Can WECT-based clustering of GBM tumors reveal biologically meaningful subtypes correlated with clinical outcomes?
• RQ4 Does the WECT representation enable improved survival prediction compared to traditional shape descriptors?
• RQ5 How robust is the WECT to variations in image intensity distributions and tumor geometry across subjects?

Key findings

• The WECT is a complete invariant for weighted simplicial complexes, as proven using results from Ghrist, Levanger, and Mai on Euler integral calculus.
• WECT-based clustering of 63 GBM tumor images revealed three distinct clusters with mean survival times of 6.7, 12.9, and 20.2 months, respectively, indicating strong correlation with patient prognosis.
• The low-survival cluster (mean 6.7 months) consisted of tumors with irregular shapes and heterogeneous intensity distributions, visually confirmed in Figure 6.
• After removing the low-survival cluster, hierarchical clustering on the remaining 58 tumors identified six sub-clusters, with the cyan and blue clusters showing ring-like intensity patterns near tumor boundaries.
• The blue cluster exhibited higher irregularity in both shape and intensity, with median survival of 14.9 months, while the cyan cluster had a median survival of 22.3 months, suggesting distinct biological or morphological features.
• The WECT representation enabled effective, data-driven clustering purely from imaging data, with survival differences observed even after controlling for image normalization and registration.