def __Rf(I, x, r, y, T):
Y, rY, rT, MY = cuda.local.array((DIM, ), f4), cuda.local.array(
(DIM, ), f4), cuda.local.array((DIM, ), f4), cuda.local.array((DIM, ),
f4)
for i in range(DIM):
Y[i] = y[i] - x[i]
if r.shape[0] == 1:
for i in range(DIM):
rY[i] = r[0] * Y[i]
rT[i] = r[0] * T[i]
else:
rY[0] = r[0] * Y[0] + r[2] * Y[1]
rY[1] = r[1] * Y[1]
rT[0] = r[0] * T[0] + r[2] * T[1]
rT[1] = r[1] * T[1]
# Normalise rT:
n = 0
for i in range(DIM):
n += rT[i] * rT[i]
n = 1 / c_sqrt(n)
for i in range(DIM):
rT[i] *= n
# Final vector:
IP = 0
for i in range(DIM):
IP += rT[i] * rY[i]
m = 0
for i in range(DIM):
MY[i] = rY[i] - rT[i] * IP
m += MY[i] * MY[i]
return I * __Rg(m) * n
def _bess(X, R, H, J, scale, out): # pragma: no cover
if scale.size == 1:
for i in numba.prange(X.shape[0]):
rad = c_sqrt(X[i, 0] * X[i, 0] + X[i, 1] * X[i, 1]) * R
ind = int(rad * H)
if ind < J.size:
out[i] = J[ind]
else:
out[i] = 0
else:
for i in numba.prange(X.shape[0]):
rad = c_sqrt(X[i, 0] * X[i, 0] + X[i, 1] * X[i, 1]) * R
ind = int(rad * H)
if ind < J.size:
out[i] = scale[i] * J[ind]
else:
out[i] = 0
def __atom_pw_gpu(x0, x1, x2, pc, h):
n = x0 * x0 + x1 * x1 + x2 * x2
if n >= h[0]:
return 0
else:
n = h[1] * c_sqrt(n)
i = int(n)
n -= i
return __quadratic_interp_gpu(n, i, pc)
def __atom_pw_cpu(x0, x1, x2, pc, h): # pragma: no cover
n = x0 * x0 + x1 * x1 + x2 * x2
if n >= h[0]:
return 0
else:
n = h[1] * c_sqrt(n)
i = int(n)
n -= i
# return __linear_interp(n, i, pc)
return __quadratic_interp(n, i, pc)
def __projnorm(x, T, DIM):
tmp0, tmp1 = 0, 0
for i in range(DIM):
tmp0 += x[i] * x[i]
for i in range(T.size):
tmp1 += x[i] * T[i]
tmp0 = tmp0 - tmp1 * tmp1
if tmp0 > 0:
return c_sqrt(tmp0)
else:
return 0
def __index(x, y, p, R, ind):
IP = 0
n = 0
for i in range(x.size):
IP += x[i] * y[i]
n += y[i] * y[i]
IP /= n
R = R / c_sqrt(n)
# Assume p[i] = p[0] + i*(p[1]-p[0])
ind[0] = int((IP - R - p[0]) / (p[1] - p[0]))
ind[1] = int((IP + R - p[0]) / (p[1] - p[0])) + 1
ind[0] = max(0, ind[0])
ind[1] = min(p.size, ind[1] + 1) # [i,j] = range(i,j+1)
def __rf2(I, x, r, y, T, W0, s):
DIM = 2
X = cuda.local.array((2, ), f4)
for i in range(DIM):
X[i] = x[i] - y[i]
R, n = __Radius(r), __projnorm(X, T, DIM)
if n + R < s:
return __total(I, r)
elif n - R > s:
return 0
rT = cuda.local.array((2, ), f4)
rT[0] = r[0] * T[0] + r[2] * T[1]
rT[1] = r[1] * T[1]
# Normalise rT:
n = 0
for i in range(DIM):
n += rT[i] * rT[i]
n = 1 / c_sqrt(n)
for i in range(DIM):
rT[i] *= n
# Square of points is: x-s -> x+s
buf0, buf1, buf2 = cuda.local.array((2, ), f4), cuda.local.array(
(2, ), f4), cuda.local.array((2, ), f4)
limits, Y = cuda.local.array((2, ), f4), cuda.local.array((2, ), f4)
# <x,w_i> \not\in (limits[i,0], limits[i,1]) => Rf(x)=0 or x\not\in [-s,s]
limits[0] = X[0] * W0[0] + X[1] * W0[1]
limits[1] = min(limits[0] + R, s)
limits[0] = max(limits[0] - R, -s)
for i in range(DIM):
Y[i] = limits[0] * W0[i]
limits[0] = .1 * (limits[1] - limits[0])
count, sum = 0, 0
for _ in range(11):
for i in range(DIM):
Y[i] += limits[0] * W0[i]
sum += __Rf(X, r, Y, rT, n, buf0, buf1, buf2, 2)
count += 1
return I * sum / count * 10 * limits[0]
def __f(x, r, y, DIM):
Y, rY = cuda.local.array((3, ), f4), cuda.local.array((3, ), f4)
for i in range(DIM):
Y[i] = y[i] - x[i]
if DIM == 2:
rY[0] = r[0] * Y[0] + r[2] * Y[1]
rY[1] = r[1] * Y[1]
else:
rY[0] = r[0] * Y[0] + r[3] * Y[1] + r[5] * Y[2]
rY[1] = r[1] * Y[1] + r[4] * Y[2]
rY[2] = r[2] * Y[2]
n = 0
for i in range(DIM):
n += rY[i] * rY[i]
return c_exp(-30 * n) * c_sqrt(30 / pi)
def __g(n):
if n > 1:
return 0
else:
return c_exp(-30 * n) * c_sqrt(30 / pi)
def __dRf_aniso(I, x, r, y, tt, R, dR, ddR, order):
# __Rf = I*__Rg(|M(y-x)|^2)/|rT|
Y, rY, rT, MY = cuda.local.array((DIM, ), f4), cuda.local.array(
(DIM, ), f4), cuda.local.array((DIM, ), f4), cuda.local.array((DIM, ),
f4)
index = cuda.local.array((6, 2), dtype=i4)
index[0, 0], index[0, 1] = 0, 0
index[1, 0], index[1, 1] = 1, 1
index[2, 0], index[2, 1] = 2, 2
index[3, 0], index[3, 1] = 0, 1
index[4, 0], index[4, 1] = 1, 2
index[5, 0], index[5, 1] = 0, 2
lenT = tt.shape[0]
for i in range(DIM):
Y[i] = y[i] - x[i]
rY[0] = r[0] * Y[0] + r[3] * Y[1] + r[5] * Y[2]
rY[1] = r[1] * Y[1] + r[4] * Y[2]
rY[2] = r[2] * Y[2]
T = cuda.local.array((DIM, ), f4)
T[0], T[1] = tt[0], tt[1]
if lenT == 2:
T[2] = 0
rT[0] = r[0] * T[0] + r[3] * T[1]
rT[1] = r[1] * T[1]
rT[2] = 0
else:
T[2] = tt[2]
rT[0] = r[0] * T[0] + r[3] * T[1] + r[5] * T[2]
rT[1] = r[1] * T[1] + r[4] * T[2]
rT[2] = r[2] * T[2]
# Normalise rT:
n = 0
for i in range(lenT):
n += rT[i] * rT[i]
n = 1 / c_sqrt(n)
for i in range(lenT):
rT[i] *= n
# Final vector:
IP = 0
for i in range(lenT):
IP += rT[i] * rY[i]
m = 0
for i in range(DIM):
MY[i] = rY[i] - rT[i] * IP
m += MY[i] * MY[i]
# Preliminary derivatives:
dMydT = cuda.local.array((DIM, DIM), f4)
# Missing a factor of n
for i in range(DIM):
for j in range(DIM):
dMydT[i, j] = rT[i] * (IP * rT[j] - MY[j])
dMydT[i, i] -= IP
dg, dJdy, dJdT = cuda.local.array((3, ), f4), cuda.local.array(
(DIM, ), f4), cuda.local.array((DIM, ), f4)
__dRg(m, dg)
if (dg[0] == 0) and (dg[1] == 0) and (dg[2] == 0):
R[0] = 0
for i in range(10):
dR[i] = 0
for j in range(10):
ddR[i, j] = 0
else:
tmp = dg[1] * 2 * n
for i in range(DIM):
dJdy[i] = tmp * MY[i]
tmp = -n * n
for i in range(DIM):
dJdT[i] = tmp * (2 * IP * MY[i] * dg[1] + rT[i] * dg[0])
ddJdyy, ddJdyT, ddJdTT = cuda.local.array(
(DIM, DIM), f4), cuda.local.array(
(DIM, DIM), f4), cuda.local.array((DIM, DIM), f4)
if order > 1:
tmp = 2 * n
for i in range(DIM):
for j in range(i):
ddJdyy[i, j] = ddJdyy[j, i]
ddJdyy[i, i] = tmp * (2 * MY[i] * MY[i] * dg[2] +
(1 - rT[i] * rT[i]) * dg[1])
for j in range(i + 1, DIM):
ddJdyy[i, j] = tmp * (2 * MY[i] * MY[j] * dg[2] -
rT[i] * rT[j] * dg[1])
tmp = 2 * n * n
for i in range(DIM):
for j in range(DIM):
ddJdyT[i,
j] = tmp * ((dMydT[i, j] - MY[i] * rT[j]) * dg[1] -
2 * IP * MY[i] * MY[j] * dg[2])
tmp = 2 * n * n * n
for i in range(DIM):
for j in range(i):
ddJdTT[i, j] = ddJdTT[j, i]
ddJdTT[i, i] = tmp * (
(1.5 * rT[i] * rT[i] - .5) * dg[0] +
(2 * IP * MY[i] * rT[i] + IP * rT[i] * MY[i] -
IP * dMydT[i, i] - MY[i] * MY[i]) * dg[1] +
2 * IP * IP * MY[i] * MY[i] * dg[2])
for j in range(i + 1, DIM):
ddJdTT[i, j] = tmp * (
1.5 * rT[i] * rT[j] * dg[0] +
(2 * IP * MY[i] * rT[j] + IP * rT[i] * MY[j] -
IP * dMydT[i, j] - MY[i] * MY[j]) * dg[1] +
2 * IP * IP * MY[i] * MY[j] * dg[2])
# Fill in values:
R[0] = I * dg[0] * n
# dI
dR[0] = dg[0] * n
ddR[0, 0] = 0
# dIdx
# ddR[0, 1 + i] = - r_{j,i}dJdy[j]
ddR[0, 1 + 0] = -r[0] * dJdy[0]
ddR[0, 1 + 1] = -(r[3] * dJdy[0] + r[1] * dJdy[1])
ddR[0, 1 + 2] = -(r[5] * dJdy[0] + r[4] * dJdy[1] + r[2] * dJdy[2])
# dIdr
# ddR[0, 4 + (I, i)] = dJdy[I] * Y[i] + dJdT[I] * T[i]
for i in range(6):
i0, i1 = index[i]
ddR[0, 4 + i] = dJdy[i0] * Y[i1] + dJdT[i0] * T[i1]
# dr, dx
for i in range(9):
dR[1 + i] = I * ddR[0, 1 + i]
if order > 1:
# d^2r
# ddR[1+(I,i), 1+(J,j)] = I*( ddJdyy[I,J]Y[i]Y[j] + ddJdYT[I,J]Y[i]T[j]
# + ddJdYT[J,I]Y[j]T[i] + ddJdTT[I,J]T[i]T[j])
for i in range(6):
i0, i1 = index[i]
for j in range(i, 6):
j0, j1 = index[j]
ddR[4 + i, 4 + j] = I * (ddJdyy[i0, j0] * Y[i1] * Y[j1] +
ddJdyT[i0, j0] * Y[i1] * T[j1] +
ddJdyT[j0, i0] * Y[j1] * T[i1] +
ddJdTT[i0, j0] * T[i1] * T[j1])
# dxdr
# ddR[1+j, 4+(I,i)] = -I(ddJdyy[I,k]Y[i]r[k,j] + ddJdYT[k,I]T[i]r[k,j]
# + dJdY[I](i==j))
# 0 -> 0,0
# 1 -> 1,1
# 2 -> 2,2
# 3 -> 0,1
# 4 -> 1,2
# 5 -> 0,2
for i in range(6):
i0, i1 = index[i]
ddR[1 + 0, 4 + i] = -I * (
(ddJdyy[i0, 0] * Y[i1] + ddJdyT[0, i0] * T[i1]) * r[0])
ddR[1 + 1, 4 + i] = -I * (
(ddJdyy[i0, 1] * Y[i1] + ddJdyT[1, i0] * T[i1]) * r[1] +
(ddJdyy[i0, 0] * Y[i1] + ddJdyT[0, i0] * T[i1]) * r[3])
ddR[1 + 2, 4 + i] = -I * (
(ddJdyy[i0, 2] * Y[i1] + ddJdyT[2, i0] * T[i1]) * r[2] +
(ddJdyy[i0, 1] * Y[i1] + ddJdyT[1, i0] * T[i1]) * r[4] +
(ddJdyy[i0, 0] * Y[i1] + ddJdyT[0, i0] * T[i1]) * r[5])
# j == i1
ddR[1 + i1, 4 + i] -= I * dJdy[i0]
# d^2x
# ddR[1+i,1+j] = I(ddJdYY[I,J]r[I,i]r[J,j])
ddR[1 + 0, 1 + 0] = I * (ddJdyy[0, 0] * r[0] * r[0])
ddR[1 + 0, 1 + 1] = I * (ddJdyy[0, 1] * r[0] * r[1] +
ddJdyy[0, 0] * r[0] * r[3])
ddR[1 + 0,
1 + 2] = I * (ddJdyy[0, 2] * r[0] * r[2] + ddJdyy[0, 1] *
r[0] * r[4] + ddJdyy[0, 0] * r[0] * r[5])
ddR[1 + 1, 1 + 1] = I * (
ddJdyy[1, 1] * r[1] * r[1] + ddJdyy[1, 0] * r[1] * r[3] +
ddJdyy[0, 1] * r[3] * r[1] + ddJdyy[0, 0] * r[3] * r[3])
ddR[1 + 1, 1 + 2] = I * (
ddJdyy[1, 2] * r[1] * r[2] + ddJdyy[1, 1] * r[1] * r[4] +
ddJdyy[1, 0] * r[1] * r[5] + ddJdyy[0, 2] * r[3] * r[2] +
ddJdyy[0, 1] * r[3] * r[4] + ddJdyy[0, 0] * r[3] * r[5])
ddR[1 + 2, 1 + 2] = I * (
ddJdyy[2, 2] * r[2] * r[2] + ddJdyy[2, 1] * r[2] * r[4] +
ddJdyy[2, 0] * r[2] * r[5] + ddJdyy[1, 2] * r[4] * r[2] +
ddJdyy[1, 1] * r[4] * r[4] + ddJdyy[1, 0] * r[4] * r[5] +
ddJdyy[0, 2] * r[5] * r[2] + ddJdyy[0, 1] * r[5] * r[4] +
ddJdyy[0, 0] * r[5] * r[5])
# Symmetrise the Hessian
for i in range(10):
for j in range(i):
ddR[i, j] = ddR[j, i]
def __drf3(I, x, r, Y, T, W0, W1, s, R, dR, ddR, order):
DIM = 3
X = cuda.local.array((3, ), f4)
for i in range(DIM):
X[i] = x[i] - Y[i]
R[0] = 0
for j0 in range(10):
dR[j0] = 0
for j1 in range(10):
ddR[j0, j1] = 0
rr, n = __Radius(r), __projnorm(X, T, DIM)
if n + rr < s:
R[0] = __total(I, r)
return
elif n - rr > s:
return
rT = cuda.local.array((3, ), f4)
rT[0] = r[0] * T[0] + r[3] * T[1] + r[5] * T[2]
rT[1] = r[1] * T[1] + r[4] * T[2]
rT[2] = r[2] * T[2]
# Normalise rT:
n = 0
for i in range(DIM):
n += rT[i] * rT[i]
n = 1 / c_sqrt(n)
for i in range(DIM):
rT[i] *= n
# Square of points is: x-s -> x+s
buf0, buf1, buf2 = cuda.local.array((3, ), f4), cuda.local.array(
(3, ), f4), cuda.local.array((3, ), f4)
buf3, buf4, buf5 = cuda.local.array((1, ), f4), cuda.local.array(
(10, ), f4), cuda.local.array((10, 10), f4)
limits = cuda.local.array((2, 2), f4)
# <x,w_i> \not\in (limits[i,0], limits[i,1]) => Rf(x)=0 or x\not\in [-s,s]
limits[0, 0] = X[0] * W0[0] + X[1] * W0[1] + X[2] * W0[2]
limits[0, 1] = min(limits[0, 0] + rr, s)
limits[0, 0] = max(limits[0, 0] - rr, -s)
limits[1, 0] = X[0] * W1[0] + X[1] * W1[1] + X[2] * W1[2]
limits[1, 1] = min(limits[1, 0] + rr, s)
limits[1, 0] = max(limits[1, 0] - rr, -s)
for i in range(DIM):
Y[i] = limits[0, 0] * W0[i] + limits[1, 0] * W1[i]
for i in range(2):
limits[i, 0] = .1 * (limits[i, 1] - limits[i, 0])
count = 0
for _ in range(11):
for i in range(DIM):
Y[i] += limits[0, 0] * W0[i]
for __ in range(11):
for i in range(DIM):
Y[i] += limits[1, 0] * W1[i]
__dRf3(I, X, r, Y, T, rT, n, buf0, buf1, buf2, buf3, buf4, buf5,
order)
R[0] += buf3[0]
for j0 in range(10):
dR[j0] += buf4[j0]
for j1 in range(j0, 10):
ddR[j0, j1] += buf5[j0, j1]
count += 1
for i in range(DIM):
Y[i] -= 11 * limits[1, 0] * W1[i]
scale = 100 * limits[0, 0] * limits[1, 0] / count
R[0] *= scale
for j0 in range(10):
dR[j0] *= scale
for j1 in range(j0, 10):
ddR[j0, j1] *= scale
ddR[j1, j0] = ddR[j0, j1]
def __drf2(I, x, r, y, T, W0, s, R, dR, ddR, order):
DIM = 2
X = cuda.local.array((2, ), f4)
for i in range(DIM):
X[i] = x[i] - y[i]
rr, n = __Radius(r), __projnorm(X, T, DIM)
if n + rr < s:
R[0] = __total(I, r)
for j0 in range(6):
dR[j0] = 0
for j1 in range(j0, 6):
ddR[j0, j1] = 0
ddR[j1, j0] = 0
return
elif n - rr > s:
R[0] = 0
for j0 in range(6):
dR[j0] = 0
for j1 in range(j0, 6):
ddR[j0, j1] = 0
ddR[j1, j0] = 0
return
rT = cuda.local.array((2, ), f4)
rT[0] = r[0] * T[0] + r[2] * T[1]
rT[1] = r[1] * T[1]
# Normalise rT:
n = 0
for i in range(DIM):
n += rT[i] * rT[i]
n = 1 / c_sqrt(n)
for i in range(DIM):
rT[i] *= n
# Square of points is: x-s -> x+s
buf0, buf1, buf2 = cuda.local.array((2, ), f4), cuda.local.array(
(2, ), f4), cuda.local.array((2, ), f4)
buf3, buf4, buf5 = cuda.local.array((1, ), f4), cuda.local.array(
(6, ), f4), cuda.local.array((6, 6), f4)
limits, Y = cuda.local.array((2, ), f4), cuda.local.array((2, ), f4)
limits[0] = X[0] * W0[0] + X[1] * W0[1]
limits[1] = min(limits[0] + rr, s)
limits[0] = max(limits[0] - rr, -s)
for i in range(DIM):
Y[i] = limits[0] * W0[i]
limits[0] = .1 * (limits[1] - limits[0])
R[0] = 0
for j0 in range(6):
dR[j0] = 0
for j1 in range(j0, 6):
ddR[j0, j1] = 0
count = 0
for _ in range(11):
for i in range(DIM):
Y[i] += limits[0] * W0[i]
__dRf2(I, X, r, Y, T, rT, n, buf0, buf1, buf2, buf3, buf4, buf5, order)
R[0] += buf3[0]
for j0 in range(6):
dR[j0] += buf4[j0]
for j1 in range(j0, 6):
ddR[j0, j1] += buf5[j0, j1]
count += 1
scale = 10 * limits[0] / count
R[0] *= scale
for j0 in range(6):
dR[j0] *= scale
for j1 in range(j0, 6):
ddR[j0, j1] *= scale
ddR[j1, j0] = ddR[j0, j1]
def __norm(x, DIM):
tmp = 0
for i in range(DIM):
tmp += x[i] * x[i]
return c_sqrt(tmp)
def __total(I, r):
if r.size == 3:
return I / (r[0] * r[1]) * c_sqrt(pi / 30)
else:
return I / (r[0] * r[1] * r[2]) * (pi / 30)
def __rf3(I, x, r, y, T, W0, W1, s):
DIM, lenT = 3, T.shape[0]
X = cuda.local.array((3, ), f4)
for i in range(DIM):
X[i] = x[i] - y[i]
R, n = __Radius(r), __projnorm(X, T, DIM)
if n + R < s:
return __total(I, r)
elif n - R > s:
return 0
rT = cuda.local.array((3, ), f4)
if lenT == 2:
rT[0] = r[0] * T[0] + r[3] * T[1]
rT[1] = r[1] * T[1]
rT[2] = 0
else:
rT[0] = r[0] * T[0] + r[3] * T[1] + r[5] * T[2]
rT[1] = r[1] * T[1] + r[4] * T[2]
rT[2] = r[2] * T[2]
# Normalise rT:
n = 0
for i in range(lenT):
n += rT[i] * rT[i]
n = 1 / c_sqrt(n)
for i in range(lenT):
rT[i] *= n
# Square of points is: x-s -> x+s
buf0, buf1, buf2 = cuda.local.array((3, ), f4), cuda.local.array(
(3, ), f4), cuda.local.array((3, ), f4)
limits, Y = cuda.local.array((2, 2), f4), cuda.local.array((3, ), f4)
# <x,w_i> \not\in (limits[i,0], limits[i,1]) => Rf(x)=0 or x\not\in [-s,s]
limits[0, 0] = X[0] * W0[0] + X[1] * W0[1] + X[2] * W0[2]
limits[0, 1] = min(limits[0, 0] + R, s)
limits[0, 0] = max(limits[0, 0] - R, -s)
limits[1, 0] = X[0] * W1[0] + X[1] * W1[1] + X[2] * W1[2]
limits[1, 1] = min(limits[1, 0] + R, s)
limits[1, 0] = max(limits[1, 0] - R, -s)
for i in range(DIM):
Y[i] = limits[0, 0] * W0[i] + limits[1, 0] * W1[i]
for i in range(2):
limits[i, 0] = .1 * (limits[i, 1] - limits[i, 0])
count, sum = 0, 0
for _ in range(11):
for i in range(DIM):
Y[i] += limits[0, 0] * W0[i]
for __ in range(11):
for i in range(DIM):
Y[i] += limits[1, 0] * W1[i]
sum += __Rf(X, r, Y, rT, n, buf0, buf1, buf2, 3)
count += 1
for i in range(DIM):
Y[i] -= 11 * limits[1, 0] * W1[i]
# return sum
if count == 0:
return 0
else:
return I * sum / count * 100 * limits[0, 0] * limits[1, 0]
