Differential geometric approximation of the gradient and Hessian on a triangulated manifold

Abstract

In a number of medical imaging modalities, including measurements or estimates of electrical activity on cortical or cardiac surfaces, it is often useful to estimate spatial derivatives of data on curved anatomical surfaces represented by triangulated meshes. Assuming the triangle vertices are points on a smooth manifold, we derive a method for estimating gradients and Hessians on locally 2D surfaces embedded in 3D directly in the global coordinate system. Accuracy of the method is validated through simulations on both smooth and corrugated surfaces.

Spatiotemporal estimation of activation times of fractionated ECGs on complex heart surfaces

“…Maximum spatial gradient methods were shown to work well, but required samples on a regularly sampled grid to estimate the gradient [1]. In a recent paper, we introduced a way to estimate the gradient of scalar functions defined on the nodes of triangulated surface geometries [18]. In this work, we used that approach to formulate a spatiotemporal method that selects simultaneous occurrence of large gradient magnitude and minimum negative temporal derivative.…”

“…Furthermore, let Df i,t denote the approximated spatial gradient and ∂f i,t /∂t denote the approximated temporal derivative (as described in [18] and [1], respectively). As a spatiotemporal method of detecting activation, we estimate the activation time τ i at the point x i as the index t which minimizes the function $$R(i,t)={\Vert D{f}_{i,t}\Vert }_2\cdot \frac{\partial f_{i,t}}{\partial t}$$ over the available temporal samples.…”

Using Transmural Regularization and Dynamic Modeling for Noninvasive Cardiac Potential Imaging of Endocardial Pacing With Imprecise Thoracic Geometry

“…Our group has previously described a method to estimate derivatives of functions defined on discretized manifold surfaces embedded in ℝ 3 [25]. We have extended this method here to estimate derivatives transmurally across cardiac ventricular surfaces.…”

“…Specifically, given our estimated heart surface potentials, we first obtained a naive estimate of the activation time at each node as the time sample with the most negative derivative (estimated numerically) [30], [31], collected into a vector τ. We then extracted the dominant propagation pattern from the resulting activation time estimates by smoothing their spatial distribution as follows: given a vector of activation times, τ, and a surface Laplacian approximation matrix, L , [25] (note this is different from the transmural gradient estimator described in Sec. III-A), we solved the following problem: $$\underset{{\tau }_D}{\text{min}}{\Vert \tau -{\tau }_D\Vert }_2^2+\gamma {\Vert L{\tau }_D\Vert }_2^2.$$ The parameter γ controls the smoothness of the resulting propagation pattern.…”

Spatiotemporal Activation Time Estimation Improves Noninvasive Localization of Cardiac Electrical Activity

“…The second term, Lτ D 2 2 , minimizes the surface Laplacian of the estimates. This is obtained through multiplication of τ D by a numerical approximation of the Laplace operator L obtained as in [4].…”