Glossary of Common Terms
- Boundary condition
In partial differential equations (PDEs), boundary conditions (BCs) constrain the solution function \(u(\mathbf{x})\) on the boundary \(\partial D\) of a domain \(D\). Common choices include Dirichlet BCs, which prescribe the value of \(u\), and Neumann BCs, which prescribe the normal derivative \(\partial u / \partial n\). In SpharaPy, homogeneous Neumann boundary conditions are used when solving the Laplace–Beltrami eigenproblem on triangulated surfaces.
- Manifold
A topological space that locally resembles Euclidean space. EEG electrode positions on the scalp form a discrete sampling of a two-dimensional manifold in \(\mathbb{R}^3\). SPHARA methods operate on discretised manifolds (triangular meshes) and compute harmonic functions intrinsic to the surface.
- EEG
Electroencephalography (EEG) is an electrophysiological method for measuring the brain’s electrical activity using sensors on the scalp. The recorded potentials typically have amplitudes in the microvolt (µV) range and are spatially irregularly sampled, which motivates SPHARA-based processing.
- Finite Element Method
The Finite Element Method (FEM) approximates the solution of PDEs by representing functions as linear combinations of nodal basis functions defined on a discretized (triangulated) domain. In SpharaPy, FEM is used to construct discrete stiffness and mass matrices, \(\mathbf{S}\) and \(\mathbf{M}\), leading to the generalized eigenproblem
\[\mathbf{S}\,\mathbf{\phi}_i = \tau_i\,\mathbf{M}\,\mathbf{\phi}_i.\]The resulting eigenpairs \((\tau_i, \mathbf{\phi}_i)\) define the FEM-based SPHARA basis.
- Laplace–Beltrami operator
A generalization of the classical Laplacian to curved surfaces and manifolds. For a smooth manifold \(\mathcal{M}\), the Laplace–Beltrami operator \(\Delta_{\mathcal{M}}\) acts on scalar functions \(u\) and encodes intrinsic surface geometry. On triangulated surfaces, several discretisation schemes exist, including the cotangent Laplacian and FEM Laplacian used by SPHARA.
- SPHARA (Spatial Harmonic Analysis)
A technique for spatial Fourier analysis on irregular sampling grids or manifolds. Instead of classical sinusoidal basis functions, SPHARA uses eigenfunctions \(\mathbf{\phi}_i\) of a discrete Laplace–Beltrami operator. These eigenfunctions form an orthonormal basis under the appropriate inner product, enabling decomposition of multichannel sensor data into spatial frequency modes.
- SPHARA basis
The collection of eigenvectors \(\{\mathbf{\phi}_i\}_{i=1}^{n}\) obtained by solving a discrete Laplace–Beltrami eigenproblem on the sensor mesh. Each basis vector defines a spatial mode, ordered by its eigenvalue (spatial frequency). The basis matrix is written as \(\mathbf{\Phi} = [\,\mathbf{\phi}_1,\dots,\mathbf{\phi}_n\,]\).
- SPHARA transform
The analogue of the Fourier transform on irregularly sampled domains. For multichannel data \(\mathbf{x}(t)\) sampled at mesh vertices, the forward SPHARA transform computes modal coefficients
\[\mathbf{c}(t) = \mathbf{\Phi}^{\mathsf{T}} \mathbf{M}\,\mathbf{x}(t),\]where \(\mathbf{\Phi}\) contains SPHARA eigenvectors and \(\mathbf{M}\) is the FEM mass matrix. The inverse transform reconstructs the signal as
\[\mathbf{x}(t) = \mathbf{\Phi}\,\mathbf{c}(t).\]Coefficients \(c_i(t)\) describe the temporal evolution of the \(i\)-th spatial mode.
- Spatial frequency
In SPHARA, the spatial frequency associated with eigenvalue \(\tau_i\) is given by
\[f_i = \frac{\sqrt{\tau_i}}{2\pi},\]with units of \(\mathrm{mm}^{-1}\) when coordinates are in millimetres. Higher \(f_i\) correspond to faster spatial oscillations on the mesh.
- Spatial wavelength
The spatial wavelength corresponding to eigenvalue \(\tau_i\) is
\[\lambda_i = \frac{1}{f_i} = \frac{2\pi}{\sqrt{\tau_i}}.\]It expresses the characteristic spatial scale of the \(i\)-th SPHARA mode.
- Modal energy
The contribution of a SPHARA mode \(i\) to the total signal energy. For coefficients \(c_i(t)\) it is defined as
\[E_i = \sum_t |c_i(t)|^2.\]Modal energy spectra reveal which spatial frequency bands dominate a signal and how filtering affects spatial content.
- Triangular mesh
A triangular mesh is a piecewise linear approximation of a smooth surface in \(\mathbb{R}^3\). It is defined by
a vertex set \(\{\mathbf{v}_i\}_{i=1}^{n}\) with \(\mathbf{v}_i \in \mathbb{R}^3\), and
a triangle list \(\{t_j\}_{j=1}^{m}\), where each triangle \(t_j\) stores three vertex indices.
The mesh provides the geometric domain on which SPHARA basis functions are defined.
- Mass matrix
In FEM, the mass matrix \(\mathbf{M}\) arises from the inner product of nodal basis functions. It ensures that the eigenvectors are orthonormal under the \(\mathbf{M}\)-weighted inner product. It plays the role of a discrete metric on the triangulated surface.
- Stiffness matrix
The FEM stiffness matrix \(\mathbf{S}\) corresponds to the discretised Laplace–Beltrami operator. It encodes local geometric interactions between neighbouring vertices and governs the spatial frequency ordering of SPHARA modes.