ravetools (version 0.3.2)

mris_curvature: Estimate per-node curvature of a surface mesh

Description

Estimates, at every vertex of a closed triangular mesh, the local mean curvature, Gaussian curvature, and the two principal curvatures, by fitting an osculating quadratic surface to each vertex's neighborhood.

Usage

mris_curvature(mesh, verbose = FALSE)

Value

A named list of four numeric vectors, each of length ncol(mesh$vb) (one entry per vertex, in vertex order):

mean

Mean curvature \(H = (k_1 + k_2)/2\).

gaussian

Gaussian curvature \(K = k_1 k_2\).

k1

First principal curvature (\(k_1 \geq k_2\)).

k2

Second principal curvature.

Arguments

mesh

triangular mesh of class 'mesh3d' (or coercible via ensure_mesh3d). The mesh must be closed, manifold, and genus-0 (watertight, no boundary or non-manifold edges, single connected component); mris_curvature raises an error if such defects are detected

verbose

logical; print progress messages. Default FALSE

Details

For each vertex, the function:

1. builds an orthonormal tangent frame \((e_1, e_2, n)\), where \(n\) is the vertex's (already-computed) unit normal;

2. expresses each neighbor's offset from the vertex in this frame as tangential coordinates \((u, w)\) and a height \(h\) above the tangent plane along \(n\);

3. fits, by least squares over the vertex's 2-ring neighborhood, the osculating paraboloid \(h = a u^2 + b u w + c w^2\);

4. derives the mean curvature \(H = a + c\), the Gaussian curvature \(K = 4ac - b^2\), and the principal curvatures \(k_{1,2} = H \pm \sqrt{\max(H^2 - K,\ 0)}\).

Because the tangent frame is orthonormal and the paraboloid is fitted with zero gradient at the vertex (by construction, since \(n\) is the fitted normal direction), these are exactly the eigenvalues of the second fundamental form, i.e. the principal curvatures.

The sign of the result follows the orientation of the per-vertex outward normal: a locally convex ('gyrus-like') patch, where neighbors lie toward the surface's interior relative to the outward normal, has negative mean curvature, while a locally concave ('sulcus-like') patch has positive mean curvature. For example, a sphere of radius \(r\) with outward-pointing normals has uniform curvature \(H = -1/r\) and \(K = 1/r^2\) everywhere.

Vertices whose 2-ring neighborhood is degenerate (fewer than three neighbors, or neighbor offsets that do not span the tangent plane, e.g. nearly collinear) are reported with all four curvature values set to zero.

References

Cortical surface-based analysis II: Inflation, flattening, and a surface-based coordinate system. NeuroImage, 9(2), 195-207 (1999).

Examples

if (is_not_cran()) {

data("left_hippocampus_mask")
mesh <- vcg_isosurface(left_hippocampus_mask)

plot(mesh)

# Fix defects
mesh <- vcg_fix_defects(mesh, verbose = TRUE)

# Smooth
smoothed <- mris_smooth(mesh, verbose = TRUE)

res <- mris_curvature(smoothed)

range(res$k1)

col <- color_ramp_continuous(
  res$k1, clim = c(-1, 1), alpha = TRUE,
  cmap = c("black", "gray", "red"))

plot(smoothed, col = list(col),
     eye = c(-100, 100, 0), up = c(0, 0, 1))

}