Hamidian et al. 2016 - Quantifying Shape Deformations by Variation of Geometric Spectrum (Paper Summary)
Paper summary for:
Hamidian, H., Hu, J., Zhong, Z., & Hua, J. (2016, October). Quantifying Shape Deformations by Variation of Geometric Spectrum. In International Conference on Medical Image Computing and Computer-Assisted Intervention (pp. 150-157). Springer International Publishing.
I am quite excited about this paper. It provides a way to determine local shape differences using the Laplace Beltrami spectrum. The Laplace Beltrami spectrum is typically a global shape analysis method. Here is the abstract:
This paper presents a registration-free method based on geometry spectrum for mapping two shapes. Our method can quantify and visualize the surface deformation by the variation of Laplace-Beltrami spectrum of the object. In order to examine our method, we employ synthetic data that has non-isometric deformation. We have also applied our method to quantifying the shape variation between the left and right hippocampus in epileptic human brains. The results on both synthetic and real patient data demonstrate the effectiveness and accuracy of our method.
Two types of methods for mapping shapes: spatial registration methods and spectral methods
spatial registration methods:
require landmarks to map two shapes, often labor intensive, landmarks can be difficult to define on some objects
do not require landmarks, or alignment, invariant to isometric deformation
inspired by Fourier transform in signal processing
can only describe the global difference between shapes
In this paper, "they focus on spectrum alignment of general shapes using the eigenvalue variation and present a spectral geometry method for localizing and quantifying non-isometric deformations between surface shapes."
"A recent study shows that the shape spectrum can be controlled using a function on the Riemann metric"
The method described in this paper uses the laplace-beltrami spectrum to detect and localize small deformations in addition to global difference of shape. it is registration free and does not need landmarks. It is invariant to spacial translation, rotation, scaling, and isometric deformation.
see paper for detailed description of the methods
Experiments and Results
They tested their method with hippocampi surface meshes. The surfaces have about 5000 vertices. They used the values k = 100 and K = 10 (see paper for details). (100 eigenvalues).
They first test their method with synthetic data by generating non-isometric deformations on a single hippocampus. A hippocampus is segmented from a brain MRI image. The surface was then deformed using Blender. They apply their spectrum alignment to the first 100 non-zero eigenvalues. Figure 1 describes the results of these tests:
They compare their methods to the non-rigid Iterative Closest Point method and found their method to have higher accuracy (92% vs 81%) (time <20s vs >60s )
Epilepsy imaging test
They also apply their method to an epilepsy study, again looking at the hippocampus. Epilepsy may cause shrinkage of the affected hippocampus (one side vs. the other). They applied the method to 20 subjects with epilepsy to quantify the shape variation between left and right hippocampus.
The right hippocampi were then mirrored to have the same direction as the left ones. Figure 2 shows the results of the shape comparisons. Columns 1 and 2 show the left and right hippocampi. A and B show examples of 2 different subjects. The third column shows the computed scale function distributions on the left hippocampus surface when mapping from the left to the right. Red means dilating and blue means contraction. White means no distortion.
A is for a patient with a left abnormal hippocampus, so mapping from left to right should show expansion. B is for a patient with a right abnormal hippocampus. so it should show contraction.
C is showing how the eigenfunction vary after the eigenvalues are mapped to the right one.
This paper presents a registration free method to quantify the deformation between shapes. It can handle complicated meshes with more than 5000 or 10000 vertices.
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