def test_mi_gradient_dense():
# Test the gradient of mutual information
h = 1e-5
for ttype in factors:
transform = regtransforms[ttype]
dim = ttype[1]
if dim == 2:
nslices = 1
warp_method = vf.warp_2d_affine
else:
nslices = 45
warp_method = vf.warp_3d_affine
# Get data (pair of images related to each other by an known transform)
factor = factors[ttype]
static, moving, static_g2w, moving_g2w, smask, mmask, M = \
setup_random_transform(transform, factor, nslices, 5.0)
smask = None
mmask = None
# Prepare a MattesBase instance
# The computation of the metric is done in 3 steps:
# 1.Compute the joint distribution
# 2.Compute the gradient of the joint distribution
# 3.Compute the metric's value and gradient using results from 1 and 2
metric = MattesBase(32)
metric.setup(static, moving, smask, mmask)
# 1. Update the joint distribution
metric.update_pdfs_dense(static.astype(np.float64),
moving.astype(np.float64))
# 2. Update the joint distribution gradient (the derivative of each
# histogram cell w.r.t. the transform parameters). This requires
# among other things, the spatial gradient of the moving image.
theta = transform.get_identity_parameters().copy()
grid_to_space = np.eye(dim + 1)
spacing = np.ones(dim, dtype=np.float64)
shape = np.array(static.shape, dtype=np.int32)
mgrad, inside = vf.gradient(moving.astype(np.float32), moving_g2w,
spacing, shape, grid_to_space)
metric.update_gradient_dense(theta, transform,
static.astype(np.float64),
moving.astype(np.float64),
grid_to_space, mgrad, smask, mmask)
# 3. Update the metric (in this case, the Mutual Information) and its
# gradient, which is computed from the joint density and its gradient
metric.update_mi_metric(update_gradient=True)
# Now we can extract the value and gradient of the metric
# This is the gradient according to the implementation under test
val0 = metric.metric_val
actual = np.copy(metric.metric_grad)
# Compute the gradient using finite-diferences
n = transform.get_number_of_parameters()
expected = np.empty_like(actual)
for i in range(n):
dtheta = theta.copy()
dtheta[i] += h
M = transform.param_to_matrix(dtheta)
shape = np.array(static.shape, dtype=np.int32)
warped = np.array(warp_method(moving.astype(np.float32), shape, M))
metric.update_pdfs_dense(static.astype(np.float64),
warped.astype(np.float64))
metric.update_mi_metric(update_gradient=False)
val1 = metric.metric_val
expected[i] = (val1 - val0) / h
dp = expected.dot(actual)
enorm = np.linalg.norm(expected)
anorm = np.linalg.norm(actual)
nprod = dp / (enorm * anorm)
assert(nprod >= 0.999)
def test_joint_pdf_gradients_dense():
# Compare the analytical and numerical (finite differences) gradient of
# the joint distribution (i.e. derivatives of each histogram cell) w.r.t.
# the transform parameters. Since the histograms are discrete partitions
# of the image intensities, the finite difference approximation is
# normally not very close to the analytical derivatives. Other sources of
# error are the interpolation used when transforming the images and the
# boundary intensities introduced when interpolating outside of the image
# (i.e. some "zeros" are introduced at the boundary which affect the
# numerical derivatives but is not taken into account by the analytical
# derivatives). Thus, we need to relax the verification. Instead of
# looking for the analytical and numerical gradients to be very close to
# each other, we will verify that they approximately point in the same
# direction by testing if the angle they form is close to zero.
h = 1e-4
# Make sure dictionary entries are processed in the same order regardless
# of the platform. Otherwise any random numbers drawn within the loop
# would make the test non-deterministic even if we fix the seed before
# the loop. Right now, this test does not draw any samples, but we still
# sort the entries to prevent future related failures.
for ttype in sorted(factors):
dim = ttype[1]
if dim == 2:
nslices = 1
transform_method = vf.transform_2d_affine
else:
nslices = 45
transform_method = vf.transform_3d_affine
transform = regtransforms[ttype]
factor = factors[ttype]
theta = transform.get_identity_parameters()
static, moving, static_g2w, moving_g2w, smask, mmask, M = \
setup_random_transform(transform, factor, nslices, 5.0)
parzen_hist = ParzenJointHistogram(32)
parzen_hist.setup(static, moving, smask, mmask)
# Compute the gradient at theta with the implementation under test
M = transform.param_to_matrix(theta)
shape = np.array(static.shape, dtype=np.int32)
moved = transform_method(moving.astype(np.float32), shape, M)
moved = np.array(moved)
parzen_hist.update_pdfs_dense(static.astype(np.float64),
moved.astype(np.float64))
# Get the joint distribution evaluated at theta
J0 = np.copy(parzen_hist.joint)
grid_to_space = np.eye(dim + 1)
spacing = np.ones(dim, dtype=np.float64)
mgrad, inside = vf.gradient(moving.astype(np.float32), moving_g2w,
spacing, shape, grid_to_space)
id = transform.get_identity_parameters()
parzen_hist.update_gradient_dense(id, transform,
static.astype(np.float64),
moved.astype(np.float64),
grid_to_space, mgrad, smask, mmask)
actual = np.copy(parzen_hist.joint_grad)
# Now we have the gradient of the joint distribution w.r.t. the
# transform parameters
# Compute the gradient using finite-diferences
n = transform.get_number_of_parameters()
expected = np.empty_like(actual)
for i in range(n):
dtheta = theta.copy()
dtheta[i] += h
# Update the joint distribution with the transformed moving image
M = transform.param_to_matrix(dtheta)
shape = np.array(static.shape, dtype=np.int32)
moved = transform_method(moving.astype(np.float32), shape, M)
moved = np.array(moved)
parzen_hist.update_pdfs_dense(static.astype(np.float64),
moved.astype(np.float64))
J1 = np.copy(parzen_hist.joint)
expected[..., i] = (J1 - J0) / h
# Dot product and norms of gradients of each joint histogram cell
# i.e. the derivatives of each cell w.r.t. all parameters
P = (expected * actual).sum(2)
enorms = np.sqrt((expected**2).sum(2))
anorms = np.sqrt((actual**2).sum(2))
prodnorms = enorms * anorms
# Cosine of angle between the expected and actual gradients.
# Exclude very small gradients
P[prodnorms > 1e-6] /= (prodnorms[prodnorms > 1e-6])
P[prodnorms <= 1e-6] = 0
# Verify that a large proportion of the gradients point almost in
# the same direction. Disregard very small gradients
mean_cosine = P[P != 0].mean()
std_cosine = P[P != 0].std()
assert (mean_cosine > 0.9)
assert (std_cosine < 0.25)
def test_joint_pdf_gradients_dense():
# Compare the analytical and numerical (finite differences) gradient of the
# joint distribution (i.e. derivatives of each histogram cell) w.r.t. the
# transform parameters. Since the histograms are discrete partitions of the
# image intensities, the finite difference approximation is normally not
# very close to the analytical derivatives. Other sources of error are the
# interpolation used when warping the images and the boundary intensities
# introduced when interpolating outside of the image (i.e. some "zeros" are
# introduced at the boundary which affect the numerical derivatives but is
# not taken into account by the analytical derivatives). Thus, we need to
# relax the verification. Instead of looking for the analytical and
# numerical gradients to be very close to each other, we will verify that
# they approximately point in the same direction by testing if the angle
# they form is close to zero.
h = 1e-4
for ttype in factors:
dim = ttype[1]
if dim == 2:
nslices = 1
warp_method = vf.warp_2d_affine
else:
nslices = 45
warp_method = vf.warp_3d_affine
transform = regtransforms[ttype]
factor = factors[ttype]
theta = transform.get_identity_parameters()
static, moving, static_g2w, moving_g2w, smask, mmask, M = \
setup_random_transform(transform, factor, nslices, 5.0)
metric = MattesBase(32)
metric.setup(static, moving, smask, mmask)
# Compute the gradient at theta with the implementation under test
M = transform.param_to_matrix(theta)
shape = np.array(static.shape, dtype=np.int32)
warped = warp_method(moving.astype(np.float32), shape, M)
warped = np.array(warped)
metric.update_pdfs_dense(static.astype(np.float64),
warped.astype(np.float64))
# Get the joint distribution evaluated at theta
J0 = np.copy(metric.joint)
grid_to_space = np.eye(dim + 1)
spacing = np.ones(dim, dtype=np.float64)
mgrad, inside = vf.gradient(moving.astype(np.float32), moving_g2w,
spacing, shape, grid_to_space)
id = transform.get_identity_parameters()
metric.update_gradient_dense(id, transform, static.astype(np.float64),
warped.astype(np.float64), grid_to_space,
mgrad, smask, mmask)
actual = np.copy(metric.joint_grad)
# Now we have the gradient of the joint distribution w.r.t. the
# transform parameters
# Compute the gradient using finite-diferences
n = transform.get_number_of_parameters()
expected = np.empty_like(actual)
for i in range(n):
dtheta = theta.copy()
dtheta[i] += h
# Update the joint distribution with the warped moving image
M = transform.param_to_matrix(dtheta)
shape = np.array(static.shape, dtype=np.int32)
warped = warp_method(moving.astype(np.float32), shape, M)
warped = np.array(warped)
metric.update_pdfs_dense(static.astype(np.float64),
warped.astype(np.float64))
J1 = np.copy(metric.joint)
expected[..., i] = (J1 - J0) / h
# Dot product and norms of gradients of each joint histogram cell
# i.e. the derivatives of each cell w.r.t. all parameters
P = (expected * actual).sum(2)
enorms = np.sqrt((expected ** 2).sum(2))
anorms = np.sqrt((actual ** 2).sum(2))
prodnorms = enorms * anorms
# Cosine of angle between the expected and actual gradients.
# Exclude very small gradients
P[prodnorms > 1e-6] /= (prodnorms[prodnorms > 1e-6])
P[prodnorms <= 1e-6] = 0
# Verify that a large proportion of the gradients point almost in
# the same direction. Disregard very small gradients
mean_cosine = P[P != 0].mean()
std_cosine = P[P != 0].std()
assert(mean_cosine > 0.9)
assert(std_cosine < 0.25)
def test_gradient_3d():
np.random.seed(3921116)
shape = (25, 32, 15)
# Create grid coordinates
x_0 = np.asarray(range(shape[0]))
x_1 = np.asarray(range(shape[1]))
x_2 = np.asarray(range(shape[2]))
X = np.zeros(shape + (4, ), dtype=np.float64)
O = np.ones(shape)
X[..., 0] = x_0[:, None, None] * O
X[..., 1] = x_1[None, :, None] * O
X[..., 2] = x_2[None, None, :] * O
X[..., 3] = 1
transform = regtransforms[("RIGID", 3)]
theta = np.array([0.1, 0.05, 0.12, -12.0, -15.5, -7.2])
T = transform.param_to_matrix(theta)
TX = X.dot(T.T)
# Eval an arbitrary (known) function at TX
# f(x, y, z) = ax^2 + by^2 + cz^2 + dxy + exz + fyz
# df/dx = 2ax + dy + ez
# df/dy = 2by + dx + fz
# df/dz = 2cz + ex + fy
a, b, c = 2e-3, 3e-3, 1e-3
d, e, f = 1e-3, 2e-3, 3e-3
img = (a * TX[..., 0]**2 + b * TX[..., 1]**2 + c * TX[..., 2]**2 +
d * TX[..., 0] * TX[..., 1] + e * TX[..., 0] * TX[..., 2] +
f * TX[..., 1] * TX[..., 2])
img = img.astype(floating)
# Test sparse gradient: choose some sample points (in space)
sample = sample_domain_regular(100, np.array(shape, dtype=np.int32), T)
sample = np.array(sample)
# Compute the analytical gradient at all points
expected = np.empty((sample.shape[0], 3), dtype=floating)
expected[...,
0] = (2 * a * sample[:, 0] + d * sample[:, 1] + e * sample[:, 2])
expected[...,
1] = (2 * b * sample[:, 1] + d * sample[:, 0] + f * sample[:, 2])
expected[...,
2] = (2 * c * sample[:, 2] + e * sample[:, 0] + f * sample[:, 1])
# Get the numerical gradient with the implementation under test
sp_to_grid = np.linalg.inv(T)
img_spacing = np.ones(3)
actual, inside = vfu.sparse_gradient(img, sp_to_grid, img_spacing, sample)
img_d = cupy.asarray(img)
img_spacing_d = cupy.asarray(img_spacing)
sp_to_grid_d = cupy.asarray(sp_to_grid)
sample_d = cupy.asarray(sample)
actual_gpu, inside_gpu = sparse_gradient(img_d, sp_to_grid_d,
img_spacing_d, sample_d.T)
atol = rtol = 1e-5
cupy.testing.assert_allclose(
actual * inside[..., np.newaxis],
actual_gpu * inside_gpu[..., np.newaxis],
atol=atol,
rtol=rtol,
)
cupy.testing.assert_array_equal(inside, inside_gpu)
# TODO: test invalid inputs
# # Verify exception is raised when passing invalid affine or spacings
# invalid_affine = np.eye(3)
# invalid_spacings = np.ones(2)
# assert_raises(ValueError, vfu.sparse_gradient, img, invalid_affine,
# img_spacing, sample)
# assert_raises(ValueError, vfu.sparse_gradient, img, sp_to_grid,
# invalid_spacings, sample)
# Test dense gradient
# Compute the analytical gradient at all points
expected = np.empty(shape + (3, ), dtype=floating)
expected[..., 0] = 2 * a * TX[..., 0] + d * TX[..., 1] + e * TX[..., 2]
expected[..., 1] = 2 * b * TX[..., 1] + d * TX[..., 0] + f * TX[..., 2]
expected[..., 2] = 2 * c * TX[..., 2] + e * TX[..., 0] + f * TX[..., 1]
# Get the numerical gradient with the implementation under test
sp_to_grid = np.linalg.inv(T)
img_spacing = np.ones(3)
actual, inside = vfu.gradient(img, sp_to_grid, img_spacing, shape, T)
sp_to_grid_d = cupy.asarray(sp_to_grid)
img_spacing_d = cupy.asarray(img_spacing)
T_d = cupy.asarray(T)
actual_gpu, inside_gpu = gradient(img_d, sp_to_grid_d, img_spacing_d,
shape, T_d)
atol = rtol = 1e-5
cupy.testing.assert_allclose(
actual * inside[..., np.newaxis],
actual_gpu * inside_gpu[..., np.newaxis],
atol=atol,
rtol=rtol,
)
cupy.testing.assert_array_equal(inside, inside_gpu)
def test_gradient_2d():
np.random.seed(3921116)
sh = (25, 32)
# Create grid coordinates
x_0 = np.arange(sh[0])
x_1 = np.arange(sh[1])
X = np.empty(sh + (3, ), dtype=np.float64)
O = np.ones(sh)
X[..., 0] = x_0[:, None] * O
X[..., 1] = x_1[None, :] * O
X[..., 2] = 1
transform = regtransforms[("RIGID", 2)]
theta = np.array([0.1, 5.0, 2.5])
T = transform.param_to_matrix(theta)
TX = X.dot(T.T)
# Eval an arbitrary (known) function at TX
# f(x, y) = ax^2 + bxy + cy^{2}
# df/dx = 2ax + by
# df/dy = 2cy + bx
a = 2e-3
b = 5e-3
c = 7e-3
img = (a * TX[..., 0]**2 + b * TX[..., 0] * TX[..., 1] + c * TX[..., 1]**2)
img = img.astype(floating)
# img is an image sampled at X with grid-to-space transform T
# Test sparse gradient: choose some sample points (in space)
sample = sample_domain_regular(20, np.array(sh, dtype=np.int32), T)
sample = np.array(sample)
# Compute the analytical gradient at all points
expected = np.empty((sample.shape[0], 2), dtype=floating)
expected[..., 0] = 2 * a * sample[:, 0] + b * sample[:, 1]
expected[..., 1] = 2 * c * sample[:, 1] + b * sample[:, 0]
# Get the numerical gradient with the implementation under test
sp_to_grid = np.linalg.inv(T)
img_spacing = np.ones(2)
img_d = cupy.asarray(img)
img_spacing_d = cupy.asarray(img_spacing)
sp_to_grid_d = cupy.asarray(sp_to_grid)
sample_d = cupy.asarray(sample)
actual, inside = vfu.sparse_gradient(img, sp_to_grid, img_spacing, sample)
actual_gpu, inside_gpu = sparse_gradient(img_d, sp_to_grid_d,
img_spacing_d, sample_d.T)
atol = rtol = 1e-5
cupy.testing.assert_allclose(
actual * inside[..., np.newaxis],
actual_gpu * inside_gpu[..., np.newaxis],
atol=atol,
rtol=rtol,
)
cupy.testing.assert_array_equal(inside, inside_gpu)
# TODO: verify exceptions
# # Verify exception is raised when passing invalid affine or spacings
# invalid_affine = np.eye(2)
# invalid_spacings = np.ones(1)
# assert_raises(ValueError, vfu.sparse_gradient, img, invalid_affine,
# img_spacing, sample)
# assert_raises(ValueError, vfu.sparse_gradient, img, sp_to_grid,
# invalid_spacings, sample)
# Test dense gradient
# Compute the analytical gradient at all points
expected = np.empty(sh + (2, ), dtype=floating)
expected[..., 0] = 2 * a * TX[..., 0] + b * TX[..., 1]
expected[..., 1] = 2 * c * TX[..., 1] + b * TX[..., 0]
# Get the numerical gradient with the implementation under test
sp_to_grid = np.linalg.inv(T)
img_spacing = np.ones(2)
actual, inside = vfu.gradient(img, sp_to_grid, img_spacing, sh, T)
sp_to_grid_d = cupy.asarray(sp_to_grid)
img_spacing_d = cupy.asarray(img_spacing)
T_d = cupy.asarray(T)
actual_gpu, inside_gpu = gradient(img_d, sp_to_grid_d, img_spacing_d, sh,
T_d)
atol = rtol = 1e-5
cupy.testing.assert_allclose(
actual * inside[..., np.newaxis],
actual_gpu * inside_gpu[..., np.newaxis],
atol=atol,
rtol=rtol,
)
cupy.testing.assert_array_equal(inside, inside_gpu)
static = np.array(static).astype(np.float64)
moving = np.array(moving).astype(np.float64)
import numpy.linalg as npl
moving_world2grid = npl.inv(moving_grid2world)
from dipy.align.imwarp import get_direction_and_spacings
dim = len(static.shape)
_, moving_spacing = get_direction_and_spacings(moving_grid2world, dim)
from dipy.align.vector_fields import gradient
mgrad, _ = gradient(moving, moving_world2grid, moving_spacing, static.shape,
static_grid2world)
np.save('sl_aff_par_grad.npy', mgrad)
# out = np.asarray(out, dtype=np.float64)
# out = 255 * (out - out.min()) / (out.max() - out.min())
# slice_indices = np.array(out.shape) // 2
# axial = np.asarray(out[:, :, slice_indices[2]]).astype(np.uint8)
# coronal = np.asarray(out[:, slice_indices[1], :]).astype(np.uint8)
# sagittal = np.asarray(out[slice_indices[0], :, :]).astype(np.uint8)
# import matplotlib.pyplot as plt
# from xvfbwrapper import Xvfb
# with Xvfb() as xvfb:
# fig, ax = plt.subplots(3, 3)
def test_mi_gradient_dense():
# Test the gradient of mutual information
h = 1e-5
for ttype in factors:
transform = regtransforms[ttype]
dim = ttype[1]
if dim == 2:
nslices = 1
warp_method = vf.warp_2d_affine
else:
nslices = 45
warp_method = vf.warp_3d_affine
# Get data (pair of images related to each other by an known transform)
factor = factors[ttype]
static, moving, static_g2w, moving_g2w, smask, mmask, M = \
setup_random_transform(transform, factor, nslices, 5.0)
smask = None
mmask = None
# Prepare a MattesBase instance
# The computation of the metric is done in 3 steps:
# 1.Compute the joint distribution
# 2.Compute the gradient of the joint distribution
# 3.Compute the metric's value and gradient using results from 1 and 2
metric = MattesBase(32)
metric.setup(static, moving, smask, mmask)
# 1. Update the joint distribution
metric.update_pdfs_dense(static.astype(np.float64),
moving.astype(np.float64))
# 2. Update the joint distribution gradient (the derivative of each
# histogram cell w.r.t. the transform parameters). This requires
# among other things, the spatial gradient of the moving image.
theta = transform.get_identity_parameters().copy()
grid_to_space = np.eye(dim + 1)
spacing = np.ones(dim, dtype=np.float64)
shape = np.array(static.shape, dtype=np.int32)
mgrad, inside = vf.gradient(moving.astype(np.float32), moving_g2w,
spacing, shape, grid_to_space)
metric.update_gradient_dense(theta, transform,
static.astype(np.float64),
moving.astype(np.float64), grid_to_space,
mgrad, smask, mmask)
# 3. Update the metric (in this case, the Mutual Information) and its
# gradient, which is computed from the joint density and its gradient
metric.update_mi_metric(update_gradient=True)
# Now we can extract the value and gradient of the metric
# This is the gradient according to the implementation under test
val0 = metric.metric_val
actual = np.copy(metric.metric_grad)
# Compute the gradient using finite-diferences
n = transform.get_number_of_parameters()
expected = np.empty_like(actual)
for i in range(n):
dtheta = theta.copy()
dtheta[i] += h
M = transform.param_to_matrix(dtheta)
shape = np.array(static.shape, dtype=np.int32)
warped = np.array(warp_method(moving.astype(np.float32), shape, M))
metric.update_pdfs_dense(static.astype(np.float64),
warped.astype(np.float64))
metric.update_mi_metric(update_gradient=False)
val1 = metric.metric_val
expected[i] = (val1 - val0) / h
dp = expected.dot(actual)
enorm = np.linalg.norm(expected)
anorm = np.linalg.norm(actual)
nprod = dp / (enorm * anorm)
assert (nprod >= 0.999)
