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WESD--Weighted Spectral Distance for measuring shape dissimilarity
Ender Konukoglu1, Ben Glocker, Antonio Criminisi
1Athinoula A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital/Harvard Medical School, Cambridge, MA 02144, USA. ender.konukoglu@gmail.com
Summary
We introduce the Weighted Spectral Distance (WESD), a novel method for measuring shape dissimilarity using eigenvalues of the Laplace operator. WESD offers an intuitive, geometrically grounded approach for shape analysis in computer vision and medical imaging.
Area of Science:
- Computer Vision
- Geometric Analysis
- Medical Image Analysis
Background:
- Eigenvalues of the Laplace operator are emerging as effective shape descriptors.
- Quantifying shape dissimilarity requires robust and geometrically meaningful distance metrics.
Purpose of the Study:
- To introduce a new distance metric, Weighted Spectral Distance (WESD), for shape dissimilarity.
- To provide WESD with an intuitive meaning and mathematical grounding in object geometry.
- To analyze the theoretical properties and practical applications of WESD.
Main Methods:
- Derivation of WESD through analysis of the heat trace.
- Investigation of theoretical properties including convergence and pseudometric nature.
- Approximation of WESD using a finite number of eigenvalues.
Main Results:
- WESD is defined over the entire eigenvalue sequence and converges.
- WESD is a pseudometric, accurately approximated by finite eigenvalues.
- WESD can be mapped to the [0,1) interval, facilitating comparisons.
Conclusions:
- WESD provides a theoretically sound and practically useful measure of shape dissimilarity.
- The method demonstrates benefits in computer vision and medical image analysis applications.
- WESD offers an intuitive link between spectral shape descriptors and intrinsic object geometry.
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