Niethammer et al. 2007 - Global Medical Shape Analysis Using the Laplace-Beltrami Spectrum (Paper Summary)
Summary notes for:
Niethammer, M., Reuter, M., Wolter, F. E., Bouix, S., Peinecke, N., Koo, M. S., & Shenton, M. E. (2007, October). Global medical shape analysis using the Laplace-Beltrami spectrum. In International Conference on Medical Image Computing and Computer-Assisted Intervention(pp. 850-857). Springer, Berlin, Heidelberg.
This is a paper from the MICCAI 2007 conference proposing using the Laplace Beltrami Spectrum as a global shape descriptor for medical shape analysis. This method is particularly useful because it allows for shape comparisons with minimal preprocessing. No registration, mapping, or remeshing is required. They demonstrate the usefulness on a population of female caudate shapes, comparing normal control to subjects with schizotypal personality disorder.
Motivation and Background
Before this time (2007), morphometric studies of the brain focused on volumetric measurements. Now the method in this paper and other non volumetric methods are popular.
This is a GLOBAL method. There is a difference between local shape differences and global shape differences.
Local shape comparisons: compare spatially localized areas, straightforward to interpret, but rely on a lot of preprocessing steps. there must be one-to-one correspondence between the two surfaces/areas being compared. The shapes have to be registered and aligned to each other, possibly resampled.
Global shape comparisons: cannot necessarily show spatially localized shape changes, but can be used to indicate shape differences between groups. can reduce the preprocessing steps needed.
This paper describes a methodology for global shape comparison using the Laplace Beltrami Specturm (LBS) of a Riemannian manifold.
Previous methods for global shape analysis include: use of invariant moments, shape index, spherical harmonics (check paper for citations).
This method differs from those in these ways:
- it works for any reimannian manifold. spherical harmonics requires surfaces with spherical topology
- the only preprocessing is the extraction of a surface approximation from a manually segmented binary volume. NO registration, remeshing, or additional mapping required. Invariant to isometries and remeshing.
Isometries include rotation, reflection, translation, meaning that it doesnt matter if the object is rotated or reflected etc. the result is the same.
Shape DNA - the Laplace Beltrami Spectrum
They refer to the LBS as the shape dna of an object.
The LBS is a signature computer from the intrinsic geometry of an object. The description in the paper:
It is the beginning sequence of the spectrum of the LB operator defined for real valued functions on Riemannian manifolds and can be used to identify and compare objects like surfaces and solids independent of their representation, position and, if desired, independent of their size. The LBS can be regarded as the set of squared frequencies (the so called natural or resonant frequencies) that are associated to the eigenmodes of an oscillating membrane defined on the manifold. (E.g., the eigenmodes of a sphere are the spherical harmonics.)
This method was first described in:
Reuter, M., Wolter, F.E., Peinecke, N.: Laplace-spectra as fingerprints for shape matching. In: SPM ’05: ACM Symposium on Solid and Physical Modeling, ACM Press (2005) 101–106
and a first description of the basic ideas were given in:
Wolter, F.E., Friese, K.: Local and global geometric methods for analysis, interrogation, reconstruction, modification and design of shape. In: CGI’00. (2000) 137–151
See the paper for a review of the 'basic theory' behind the LBS.
The spectrum itself is defined to be the family of eigenvalues of the Laplacian Eigenvalue Problem consisting of a diverging sequence 0 < ev1 < ev2 < etc. with each ev repeated according to its multiplicity.
The spectrum is ISOMETRIC INVARIANT. it only depends on the gradient and divergence which are depending only on the Riemannian structure of the manifold (the intrinsic geometry).
Scaling an n-dimensional manifold by factor a results in the eigenvalues being scaled by a factor 1/a^2. The eigenvalues can be normalized, allowing shape to be compared regardless of scale and position.
Normalization and comparison:
When unnormalized spectra are compared, higher eigenvalues will lead to better discrimination between the two groups, but most of that difference will be due to the difference in surface area.
By normalizing by surface area you can see if there are additional shape differences. Identical noise levels are require for accurate results. For shapes acquired using the same method (manual segmentation etc.) identical noise levls is a safe assumption.
Statistical Analysis of groups of LB spectra
Here they used a maximum t-statistic permutation test. see paper for more details.
MR images of 32 females with Schizotypal Personality Disorder (SPD) and 29 female normal controls.
Caudate nucleus delineated manually, isosurfaces extracted using marching cubes. analysis was performed on the surfaces directed and on 'smoothed' surfaces that were smoothed using spherical harmonics.
Volume and Area Analysis:
SPD subjects have less volume and surface area. smoothing the surfaces did not affect volume much, but did reduce the surface area. however the relationship between the groups was preserved.
they normalized by surface area. A maximum t-statistic permutation test on a 100 dimensional spectral shape descriptor shows significant shape differences for surace area normalization for the unsmoothed surfaces, but not the smoothed surfaces.
The use of too many evs can be detrimental as it introduces noise. Higher order modes correspond to higher frequencies and are thus more likely noise.
I will be using this blog space as a repository of notes on various articles and lectures related to my research. Click on the categories below to organize the posts by topic.