4D hyperspherical harmonic (HyperSPHARM) representation of surface anatomy: A holistic treatment of multiple disconnected anatomical structures

Abstract

Image-based parcellation of the brain often leads to multiple disconnected anatomical structures, which pose significant challenges for analyses of morphological shapes. Existing shape models, such as the widely used spherical harmonic (SPHARM) representation, assume topological invariance, so are unable to simultaneously parameterize multiple disjoint structures. In such a situation, SPHARM has to be applied separately to each individual structure. We present a novel surface parameterization technique using 4D hyperspherical harmonics in representing multiple disjoint objects as a single analytic function, terming it HyperSPHARM. The underlying idea behind Hyper-SPHARM is to stereographically project an entire collection of disjoint 3D objects onto the 4D hypersphere and subsequently simultaneously parameterize them with the 4D hyperspherical harmonics. Hence, HyperSPHARM allows for a holistic treatment of multiple disjoint objects, unlike SPHARM. In an imaging dataset of healthy adult human brains, we apply HyperSPHARM to the hippocampi and amygdalae. The HyperSPHARM representations are employed as a data smoothing technique, while the HyperSPHARM coefficients are utilized in a support vector machine setting for object classification. HyperSPHARM yields nearly identical results as SPHARM, as will be shown in the paper. Its key advantage over SPHARM lies computationally; Hyper-SPHARM possess greater computational efficiency than SPHARM because it can parameterize multiple disjoint structures using much fewer basis functions and stereographic projection obviates SPHARM's burdensome surface flattening. In addition, HyperSPHARM can handle any type of topology, unlike SPHARM, whose analysis is confined to topologically invariant structures.

Automatic construction of statistical shape models using deformable simplex meshes with vector field convolution energy

“…We compared our proposed method with the classical spherical harmonics (SPHARM) shape correspondence method [ 8 ]. Although the SPHARM method is not being proposed recently, the reason we chose this method for comparison is that it is still quite efficient and popular, especially for the analysis of brain morphology structures in the neuroimaging community [ 30 ]. Moreover, the SPHARM method is still frequently cited by researchers for comparative experiments [ 30 – 32 ].…”

“…Although the SPHARM method is not being proposed recently, the reason we chose this method for comparison is that it is still quite efficient and popular, especially for the analysis of brain morphology structures in the neuroimaging community [ 30 ]. Moreover, the SPHARM method is still frequently cited by researchers for comparative experiments [ 30 – 32 ]. Since the SPHARM method requires the input shape to be of spherical topology (i.e., objects without holes), all holes in the training images must be filled firstly.…”

NeuVV: Neural Volumetric Videos with Immersive Rendering and Editing

“…Hyperspherical Harmonics are widely used in quantum mechanical and chemistry field to solve few-body systems. It also has been used in computer graphics visualization [Bonvallet et al 2007] and the representation of complicated brain subcortical structures [Pasha Hosseinbor et al 2015] 4D complex Hyperspherical Harmonics can be derived from as complex 3D Shpherical Harmonics [Lombardi et al 2016]…”

Embedding of Functional Human Brain Networks on a Sphere

“…Once we embedded correlation networks onto a sphere, it is possible to algebraically represent such networks as basis expansion involving the hyperspherical harmonics [13,21,23,24,25]. Let ϕ = (θ, ϕ 2 , • • • , ϕ n−1 ) be the spherical coordinates of S n−1 such that θ ∈ [0, π), ϕ i ∈ [0, 2π) where θ is the axial angle.…”