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This paper proposes to use the Laplace-Beltrami spectrum (LBS) as a global shape descriptor for medical shape analysis, allowing for shape comparisons using minimal shape preprocessing: no registration, mapping, or remeshing is necessary. The discriminatory power of the method is tested on a population of female caudate shapes of normal control subjects and of subjects with schizotypal personality disorder. Motivation and BackgroundMorphometric studies of brain structures have classically been based on volume measurements. More recently, shape studies of gray matter brain structures have become popular. Methodologies for shape comparison may be divided into global and local shape analysis approaches. While local shape comparisons [1,2] yield powerful, spatially localized results that are relatively straightforward to interpret, they usually rely on a number of preprocessing steps: one-to-one correspondences between surfaces need to be established, shapes need to be registered and resampled, possibly influencing shape comparisons. While global shape comparison methods cannot spatially localize shape changes they may be used to indicate the existence of shape differences between populations. Global approaches may be formulated with a significantly reduced number of assumptions and preprocessing steps in comparison to local approaches, staying as true as possible to the original data. This paper describes a methodology for global shape comparison based on the LaplaceBeltrami spectrum (LBS) [3,4] of a Riemannian manifold (of closed surfaces in space). Previous approaches for global shape analysis in medical imaging include the use of invariant moments [5], the shape index [6], and global shape descriptors based on spherical harmonics [7]. The proposed methodology based on the LBS differs in the following ways from these previous approaches: 2 Shape-DNA: The Laplace-Beltrami SpectrumIn this section we introduce the necessary background for the computation of the LBS beginning sequence (also called "Shape-DNA"). The "Shape-DNA" is a signature computed only from the intrinsic geometry of an object. 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. This methodology was first introduced in [8] with a first description of basic ideas given in [9]. 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.) Let us review the basic theory in the general case (see [3,4] for details).Let f be a real-valued function, with f ∈ C 2 , defined on a Riemannian manifold M. The LaplaceBeltrami Operator Δ is: Δf := div(grad f) with grad f the gradient of f and div the divergence on the manifold. The ...

In the area of image retrieval from data bases and for copyright protection of large image collections there is a growing demand for unique but easily computable fingerprints for images. These fingerprints can be used to quickly identify every image within a larger set of possibly similar images. This paper introduces a novel method to automatically obtain such fingerprints from an image. It is based on a reinterpretation of an image as a Riemannian manifold. This representation is feasible for gray value images and color images. We discuss the use of the spectrum of eigenvalues of different variants of the Laplace operator as a fingerprint and show the usability of this approach in several use cases. Contrary to existing works in this area we do not only use the discrete Laplacian, but also with a particular emphasis the underlying continuous operator. This allows better results in comparing the resulting spectra and deeper insights in the problems arising. We show how the well known discrete Laplacian is related to the continuous Laplace-Beltrami operator. Furthermore, we introduce the new concept of solid height functions to overcome some potential limitations of the method.

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