def periodic_discrete_laplacian(
grid: RectangularGrid,
mask: np.ndarray,
dtype: np.dtype = _dtype_float64) -> csr_matrix:
"""Return a laplacian operator with periodic boundary conditions.
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)
z_extent, y_extent, x_extent = grid.shape
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 offset in [(-1, 0, 0), (1, 0, 0), (0, -1, 0), (0, 1, 0),
(0, 0, -1), (0, 0, 1)]:
# voxel coordinate displacements
dk, dj, di = offset
# find the neighbor for periodic boundary conditions
neighbor: Voxel = Voxel(x=(i + di) % x_extent,
y=(j + dj) % y_extent,
z=(k + dk) % z_extent)
# but maybe it isn't in the mask (i.e. air)
if not mask[neighbor.z, neighbor.y, neighbor.x]:
continue
neighbor_index = grid.get_flattened_index(neighbor)
# continuous space displacements
dx = delta_x[k, j, i] * di
dy = delta_y[k, j, i] * dj
dz = delta_z[k, j, i] * dk
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 discrete_laplacian(grid: RectangularGrid,
mask: np.ndarray,
dtype: np.dtype = np.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)
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)
normalization = 0
for neighbor in grid.get_adjecent_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)
distance2 = 1 / (dx * dx + dy * dy + dz * dz)
normalization -= distance2
laplacian[voxel_index, neighbor_index] = distance2
laplacian[voxel_index, voxel_index] = normalization
return laplacian.tocsr()
def test_get_flattened_index(voxel, index, grid: RectangularGrid):
assert grid.get_flattened_index(voxel) == index
assert grid.voxel_from_flattened_index(index) == voxel