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level_set.py
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import taichi as ti
import taichi_glsl as ts
import utils
from utils import *
from priority_queue import PriorityQueue
from functools import reduce
@ti.data_oriented
class LevelSet:
def __init__(self, dim, res, dx, real):
self.dim = dim
self.res = res
self.dx = dx
self.real = real
self.valid = ti.field(dtype=ti.i32, shape=res) # indices to the closest points / reuse as visit sign
self.phi = ti.field(dtype=real, shape=res)
self.phi_temp = ti.field(dtype=real, shape=res)
self.eps = 5 * self.dx # the "thickness" of the interface, O(∆x) and is smaller than the local feature size
@ti.func
def theta(self, phi): # smoothed step Heaviside function
theta = ti.cast(0, self.real)
if phi <= -self.eps: theta = 0
elif phi >= self.eps: theta = 1
else: theta = 1/2 + phi/(2*self.eps) + 1/(2*ts.pi) * ti.sin(ts.pi*phi/self.eps)
return theta
@ti.func
def delta(self, phi): # smoothed regular Dirac delta function
delta = ti.cast(0, self.real)
if phi <= -self.eps or phi >= self.eps: delta = 0
else: delta = (1 + ti.cos(ts.pi*phi/self.eps)) / (2*self.eps)
return delta
@ti.func
def distance_of_aabb(self, x, x0, x1):
phi = ti.cast(0, self.real)
if all(x > x0) and all(x < x1): # (inside)
phi = (ti.max(x0 - x, x - x1)).max()
else: # (outside)
# Find the closest point (p,q,r) on the surface of the box
p = ti.zero(x)
for k in ti.static(range(self.dim)):
if x[k] < x0[k]: p[k] = x0[k]
elif x[k] > x1[k]: p[k] = x1[k]
else: p[k] = x[k]
phi = (x - p).norm()
return phi
@ti.kernel
def initialize_with_aabb(self, x0 : ti.template(), x1 : ti.template()):
for I in ti.grouped(self.phi):
self.phi[I] = self.distance_of_aabb((I + 0.5) * self.dx, x0, x1)
@ti.kernel
def initialize_with_sphere(self, x0 : ti.template(), r : ti.template()):
for I in ti.grouped(self.phi):
self.phi[I] = ((I + 0.5) * self.dx - x0).norm() - r
@ti.kernel
def target_surface(self):
for I in ti.grouped(self.phi):
sign_change = False
est = ti.cast(1e20, self.real)
for k in ti.static(range(self.dim)):
for s in ti.static((-1, 1)):
offset = ti.Vector.unit(self.dim, k) * s
I1 = I + offset
if I1[k] >= 0 and I1[k] < self.res[k] and \
ts.sign(self.phi[I]) != ts.sign(self.phi[I1]):
theta = self.phi[I] / (self.phi[I] - self.phi[I1])
est0 = ts.sign(self.phi[I]) * theta * self.dx
est = est0 if ti.abs(est0) < ti.abs(est) else est
sign_change = True
if sign_change:
self.phi_temp[I] = est
self.valid[I] = 0
else:
self.phi_temp[I] = ti.cast(1e20, self.real) # an upper bound for all possible distances
@ti.func
def update_from_neighbor(self, I):
# solve the Eikonal equation
nb = ti.Vector.zero(self.real, self.dim)
for k in ti.static(range(self.dim)):
o = ti.Vector.unit(self.dim, k)
if I[k] == 0 or (I[k] < self.res[k] - 1 and ti.abs(self.phi_temp[I + o]) < ti.abs(self.phi_temp[I - o])): nb[k] = ti.abs(self.phi_temp[I + o])
else: nb[k] = ti.abs(self.phi_temp[I - o])
# sort
for i in ti.static(range(self.dim-1)):
for j in ti.static(range(self.dim-1-i)):
if nb[j] > nb[j + 1]: nb[j], nb[j + 1] = nb[j + 1], nb[j]
# (Try just the closest neighbor)
d = nb[0] + self.dx
if d > nb[1]:
# (Try the two closest neighbors)
d = (1/2) * (nb[0] + nb[1] + ti.sqrt(2 * (self.dx ** 2) - (nb[1] - nb[0]) ** 2))
if ti.static(self.dim == 3):
if d > nb[2]:
# (Use all three neighbors)
d = (1/3) * (nb[0] + nb[1] + nb[2] + ti.sqrt(ti.max(0, (nb[0] + nb[1] + nb[2]) ** 2 - 3 * (nb[0] ** 2 + nb[1] ** 2 + nb[2] ** 2 - self.dx ** 2))))
return d
@ti.kernel
def distance_to_markers(self, markers : ti.template(), total_mk : ti.template()):
# (Initialize the arrays near the input geometry)
for p in range(total_mk):
I = (markers[p] / self.dx).cast(int)
d = (markers[p] - (I + 0.5) * self.dx).norm()
if all(I >= 0 and I < self.res) and d < self.phi[I]:
self.phi[I] = d
self.valid[I] = p
@ti.kernel
def target_minus(self):
for I in ti.grouped(self.phi):
self.phi[I] -= (0.99 * self.dx) # the particle radius r (typically just a little less than the grid cell size dx)
for I in ti.grouped(self.phi):
sign_change = False
for k in ti.static(range(self.dim)):
for s in ti.static((-1, 1)):
offset = ti.Vector.unit(self.dim, k) * s
I1 = I + offset
if I1[k] >= 0 and I1[k] < self.res[k] and \
ts.sign(self.phi[I]) != ts.sign(self.phi[I1]):
sign_change = True
if sign_change and self.phi[I] <= 0:
self.valid[I] = 0
self.phi_temp[I] = self.phi[I]
elif self.phi[I] <= 0:
self.phi_temp[I] = ti.cast(-1, self.real)
else:
self.phi_temp[I] = self.phi[I]
self.valid[I] = 0
@ti.kernel
def smoothing(self, phi : ti.template(), phi_temp : ti.template()):
for I in ti.grouped(phi_temp):
phi_avg = ti.cast(0, self.real)
tot = ti.cast(0, int)
for k in ti.static(range(self.dim)):
for s in ti.static((-1, 1)):
offset = ti.Vector.unit(self.dim, k) * s
I1 = I + offset
if I1[k] >= 0 and I1[k] < self.res[k]:
phi_avg += phi_temp[I1]
tot += 1
phi_avg /= tot
phi[I] = phi_avg if phi_avg < phi_temp[I] else phi_temp[I]
# J. Sethian. A fast marching level set method for monotonically ad- vancing fronts. Proc. Natl. Acad. Sci., 93:1591–1595, 1996.
@ti.data_oriented
class FastMarchingLevelSet(LevelSet):
def __init__(self, dim, res, dx, real):
super().__init__(dim, res, dx, real)
self.priority_queue = PriorityQueue(dim, res, real)
self.surface_grid = ti.Vector.field(dim, dtype=ti.i32, shape=reduce(lambda x, y : x * y, res))
self.total_sg = ti.field(dtype=ti.i32, shape=())
@ti.func
def sg_to_pq(self):
self.priority_queue.clear()
cnt = 0
while cnt < self.total_sg[None]:
I = self.surface_grid[cnt]
self.priority_queue.push(self.phi_temp[I], I)
cnt += 1
@ti.kernel
def init_queue(self):
self.total_sg[None] = 0
for I in ti.grouped(self.valid):
if self.valid[I] != -1:
offset = self.total_sg[None].atomic_add(1)
self.surface_grid[offset] = I
self.sg_to_pq()
@ti.kernel
def propagate(self):
while not self.priority_queue.empty():
I0 = self.priority_queue.top()
self.priority_queue.pop()
for k in ti.static(range(self.dim)):
for s in ti.static((-1, 1)):
offset = ti.Vector.unit(self.dim, k) * s
I = I0 + offset
if I[k] >= 0 and I[k] < self.res[k] and \
self.valid[I] == -1:
d = self. update_from_neighbor(I)
if d < ti.abs(self.phi_temp[I]):
self.phi_temp[I] = d * ts.sign(self.phi[I0])
self.valid[I] = 0
self.priority_queue.push(ti.abs(self.phi_temp[I]), I)
def redistance(self):
self.valid.fill(-1)
self.target_surface()
self.init_queue()
self.propagate()
self.phi.copy_from(self.phi_temp)
@ti.kernel
def markers_propagate(self, markers : ti.template(), total_mk : ti.template()):
while not self.priority_queue.empty():
I0 = self.priority_queue.top()
self.priority_queue.pop()
for k in ti.static(range(self.dim)):
for s in ti.static((-1, 1)):
offset = ti.Vector.unit(self.dim, k) * s
I = I0 + offset
if I[k] >= 0 and I[k] < self.res[k] and \
self.valid[I] == -1:
for k1 in ti.static(range(self.dim)):
for s1 in ti.static((-1, 1)):
o = ti.Vector.unit(self.dim, k1) * s1
if I[k1] + s1 >= 0 and I[k1] + s1 < self.res[k1] and self.valid[I + o] != -1:
d = (markers[self.valid[I + o]] - (I + 0.5) * self.dx).norm()
if d < self.phi[I]:
self.phi[I] = d
self.valid[I] = self.valid[I + o]
self.priority_queue.push(self.phi[I], I)
def build_from_markers(self, markers, total_mk):
self.phi.fill(1e20)
self.valid.fill(-1)
self.distance_to_markers(markers, total_mk)
self.init_queue()
self.markers_propagate(markers, total_mk)
self.valid.fill(-1)
self.target_minus()
self.init_queue()
self.propagate()
self.smoothing(self.phi, self.phi_temp)
self.smoothing(self.phi_temp, self.phi)
self.smoothing(self.phi, self.phi_temp)
# H. Zhao. A fast sweeping method for Eikonal equations. Math. Comp., 74:603–627, 2005.
@ti.data_oriented
class FastSweepingLevelSet(LevelSet):
def __init__(self, dim, res, dx, real):
super().__init__(dim, res, dx, real)
self.repeat_times = 2
@ti.func
def propagate_update(self, I, s):
if self.valid[I] == -1:
d = self.update_from_neighbor(I)
if ti.abs(d) < ti.abs(self.phi_temp[I]): self.phi_temp[I] = d * ts.sign(self.phi[I])
return s
@ti.func
def markers_propagate_update(self, markers, lI, o, s):
I, offset = ti.Vector(lI), ti.Vector(o)
if all(I + offset >= 0) and all(I + offset < self.res):
d = (markers[self.valid[I + offset]] - (I + 0.5) * self.dx).norm()
if d < self.phi[I]:
self.phi[I] = d
self.valid[I] = self.valid[I + o]
return s
@ti.kernel
def propagate(self):
if ti.static(self.dim == 2):
for t in ti.static(range(self.repeat_times)):
for i in range(self.res[0]):
j = 0
while j < self.res[1]: j += self.propagate_update([i, j], 1)
for i in range(self.res[0]):
j = self.res[1] - 1
while j >= 0: j += self.propagate_update([i, j], -1)
for j in range(self.res[1]):
i = 0
while i < self.res[1]: i += self.propagate_update([i, j], 1)
for j in range(self.res[1]):
i = self.res[1] - 1
while i >= 0: i += self.propagate_update([i, j], -1)
if ti.static(self.dim == 3):
for t in ti.static(range(self.repeat_times)):
for i, j in ti.ndrange(self.res[0], self.res[1]):
k = 0
while k < self.res[2]: k += self.propagate_update([i, j, k], 1)
for i, j in ti.ndrange(self.res[0], self.res[1]):
k = self.res[2] - 1
while k >= 0: k += self.propagate_update([i, j, k], -1)
for i, k in ti.ndrange(self.res[0], self.res[2]):
j = 0
while j < self.res[1]: j += self.propagate_update([i, j, k], 1)
for i, k in ti.ndrange(self.res[0], self.res[2]):
j = self.res[1] - 1
while j >= 0: j += self.propagate_update([i, j, k], -1)
for j, k in ti.ndrange(self.res[1], self.res[2]):
i = 0
while i < self.res[1]: i += self.propagate_update([i, j, k], 1)
for j, k in ti.ndrange(self.res[1], self.res[2]):
i = self.res[0] - 1
while i >= 0: i += self.propagate_update([i, j, k], -1)
def redistance(self):
self.valid.fill(-1)
self.target_surface()
self.propagate()
self.phi.copy_from(self.phi_temp)
@ti.kernel
def markers_propagate(self, markers : ti.template(), total_mk : ti.template()):
if ti.static(self.dim == 2):
for t in ti.static(range(self.repeat_times)):
for i in range(self.res[0]):
j = 0
while j < self.res[1]: j += self.markers_propagate_update(markers, [i, j], [0, 1], 1)
for i in range(self.res[0]):
j = self.res[1] - 1
while j >= 0: j += self.markers_propagate_update(markers, [i, j], [0, -1], -1)
for j in range(self.res[1]):
i = 0
while i < self.res[1]: i += self.markers_propagate_update(markers, [i, j], [1, 0], 1)
for j in range(self.res[1]):
i = self.res[1] - 1
while i >= 0: i += self.markers_propagate_update(markers, [i, j], [-1, 0], -1)
if ti.static(self.dim == 3):
for t in ti.static(range(self.repeat_times)):
for i, j in ti.ndrange(self.res[0], self.res[1]):
k = 0
while k < self.res[2]: k += self.markers_propagate_update(markers, [i, j, k], [0, 0, 1], 1)
for i, j in ti.ndrange(self.res[0], self.res[1]):
k = self.res[2] - 1
while k >= 0: k += self.markers_propagate_update(markers, [i, j, k], [0, 0, -1], -1)
for i, k in ti.ndrange(self.res[0], self.res[2]):
j = 0
while j < self.res[1]: j += self.markers_propagate_update(markers, [i, j, k], [0, 1, 0], 1)
for i, k in ti.ndrange(self.res[0], self.res[2]):
j = self.res[1] - 1
while j >= 0: j += self.markers_propagate_update(markers, [i, j, k], [0, -1, 0], -1)
for j, k in ti.ndrange(self.res[1], self.res[2]):
i = 0
while i < self.res[1]: i += self.markers_propagate_update(markers, [i, j, k], [1, 0, 0], 1)
for j, k in ti.ndrange(self.res[1], self.res[2]):
i = self.res[0] - 1
while i >= 0: i += self.markers_propagate_update(markers, [i, j, k], [-1, 0, 0], -1)
def build_from_markers(self, markers, total_mk):
self.phi.fill(1e20)
self.valid.fill(-1)
self.distance_to_markers(markers, total_mk)
self.markers_propagate(markers, total_mk)
self.valid.fill(-1)
self.target_minus()
self.propagate()
self.smoothing(self.phi, self.phi_temp)
self.smoothing(self.phi_temp, self.phi)
self.smoothing(self.phi, self.phi_temp)