def discrete_laplacian(grid: RectangularGrid,
mask: np.ndarray,
dtype: np.dtype = _dtype_float64) -> csr_matrix:
"""Return a discrete laplacian operator for the given restricted grid.
This computes a standard laplacian operator as a scipy linear operator, except it is
restricted to a grid mask. The use case for this is to compute surface diffusion
on a gridded variable. The mask is generated from a category on the lung_tissue
variable.
"""
graph_shape = len(grid), len(grid)
laplacian = dok_matrix(graph_shape, dtype=dtype)
delta_z = grid.delta(0)
delta_y = grid.delta(1)
delta_x = grid.delta(2)
for k, j, i in zip(*(mask).nonzero()):
voxel = Voxel(x=i, y=j, z=k)
voxel_index = grid.get_flattened_index(voxel)
for neighbor in grid.get_adjacent_voxels(voxel, corners=False):
ni = neighbor.x
nj = neighbor.y
nk = neighbor.z
if not mask[nk, nj, ni]:
continue
neighbor_index = grid.get_flattened_index(neighbor)
dx = delta_x[k, j, i] * (i - ni)
dy = delta_y[k, j, i] * (j - nj)
dz = delta_z[k, j, i] * (k - nk)
inverse_distance2 = 1 / (dx * dx + dy * dy + dz * dz
) # units: 1/(µm^2)
laplacian[voxel_index, voxel_index] -= inverse_distance2
laplacian[voxel_index, neighbor_index] += inverse_distance2
return laplacian.tocsr()
def test_get_adjacent_voxels(grid: RectangularGrid, voxel, neighbors):
assert set(grid.get_adjacent_voxels(voxel)) == neighbors
def test_get_corner_adjacent_voxels(grid: RectangularGrid, voxel, neighbors):
assert len(list(grid.get_adjacent_voxels(voxel, corners=True))) == neighbors
