![]() In this paper, following the approach of Fock, we model multiple disconnected 3D objects in terms of the 4D HSH by stereographically projecting each object’s surface coordinates onto a 4D hypersphere, and label such a representation HyperSPHARM. However, as of yet, they have remained elusive for medical imaging. Recently, the HSH have been utilized in a wider array of fields than just quantum chemistry, including computer graphics visualization and crystallography. In his classic paper, Fock stereographically projected 3D momentum-space onto the surface of a 4D unit hypersphere, and after this mapping was made, he was able to show that the eigenfuctions were the 4D HSH. Fock solved the Schrödinger equation for the hydrogen atom directly in momentum-space. It had been solved in position-space by Schrödinger, himself, but not in momentum-space, which is related to position-space via the Fourier transform. The HSH have been mainly confined to quantum chemistry, where their utility first became evident with respect to solving the Schrödinger equation for the hydrogen atom. Then the disconnected 3D objects can be represented as the linear combination of 4D hyperspherical harmonics (HSH), which are the multidimensional analogues of the 3D spherical harmonics. Extending the concept further, any two or more disconnected 3D objects can be projected onto a single connected surface in 4D hypersphere. By performing simple stereographic projection on a 3D volume, it is possible to embed the 3D volume onto the surface of a 4D hypersphere, which bypasses the difficulty of flattening 3D surface to 3D sphere. ![]() Note that a 3D volume is a surface in 4D. Then the surface coordinates can be mapped onto the sphere and each coordinate is represented as a linear combination of spherical harmonics. The angles serve as coordinates for representing the surface using spherical harmonics. The surface flattening is used to parameterize the surface using two spherical angles. Since SPHARM requires a smooth map from surfaces to a 3D sphere, various computationally intensive surface flattening techniques have been proposed: diffusion mapping, conformal mapping, and area preserving mapping. SPHARM, however, can’t represent multiple disconnected structures with a single parameterization. The method has been used to model various brain structures such as ventricles, hippocampi and cortical surfaces. The main global geometric features are encoded in low degree coefficients while the noise will be in high degree spherical harmonics. ![]() Probably the most widely applied shape parameterization technique for cortical structures is the spherical harmonic (SPHARM) representation, which has been mainly used as a data reduction technique for compressing global shape features into a small number of coefficients. The difficulty is mainly caused by the lack of a single, coherent mathematical parameterization for multiple, disconnected objects. This type of topological difference and changes cannot be incorporated directly into the processing and analysis pipeline with existing shape models that assume topological invariance and mainly work on a single connected component. These bones fuse together as the infant grows. For example, an infant may have about 300–350 bones at birth but an adult has 206 bones. There are numerous such examples from longitudinal child development to cancer growth. Many shape modeling frameworks in computational anatomy assume topological invariance between objects and are not applicable for objects with different or changing topology. ![]()
0 Comments
Leave a Reply. |
Details
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |