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 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_mattes_densities():
# Test the computation of the joint intensity distribution
# using a dense and a sparse set of values
seed = 1246592
nbins = 32
nr = 30
nc = 35
ns = 20
nvals = 50
for dim in [2, 3]:
if dim == 2:
shape = (nr, nc)
static, moving = create_random_image_pair(shape, nvals, seed)
else:
shape = (ns, nr, nc)
static, moving = create_random_image_pair(shape, nvals, seed)
# Initialize
mbase = MattesBase(nbins)
mbase.setup(static, moving)
# Get distributions computed by dense sampling
mbase.update_pdfs_dense(static, moving)
actual_joint_dense = mbase.joint
actual_mmarginal_dense = mbase.mmarginal
actual_smarginal_dense = mbase.smarginal
# Get distributions computed by sparse sampling
sval = static.reshape(-1)
mval = moving.reshape(-1)
mbase.update_pdfs_sparse(sval, mval)
actual_joint_sparse = mbase.joint
actual_mmarginal_sparse = mbase.mmarginal
actual_smarginal_sparse = mbase.smarginal
# Compute the expected joint distribution with dense sampling
expected_joint_dense = np.zeros(shape=(nbins, nbins))
for index in ndindex(shape):
sv = mbase.bin_normalize_static(static[index])
mv = mbase.bin_normalize_moving(moving[index])
sbin = mbase.bin_index(sv)
# The spline is centered at mv, will evaluate for all row
spline_arg = np.array([i - mv for i in range(nbins)])
contribution = cubic_spline(spline_arg)
expected_joint_dense[sbin, :] += contribution
# Compute the expected joint distribution with sparse sampling
expected_joint_sparse = np.zeros(shape=(nbins, nbins))
for index in range(sval.shape[0]):
sv = mbase.bin_normalize_static(sval[index])
mv = mbase.bin_normalize_moving(mval[index])
sbin = mbase.bin_index(sv)
# The spline is centered at mv, will evaluate for all row
spline_arg = np.array([i - mv for i in range(nbins)])
contribution = cubic_spline(spline_arg)
expected_joint_sparse[sbin, :] += contribution
# Verify joint distributions
expected_joint_dense /= expected_joint_dense.sum()
expected_joint_sparse /= expected_joint_sparse.sum()
assert_array_almost_equal(actual_joint_dense, expected_joint_dense)
assert_array_almost_equal(actual_joint_sparse, expected_joint_sparse)
# Verify moving marginals
expected_mmarginal_dense = expected_joint_dense.sum(0)
expected_mmarginal_dense /= expected_mmarginal_dense.sum()
expected_mmarginal_sparse = expected_joint_sparse.sum(0)
expected_mmarginal_sparse /= expected_mmarginal_sparse.sum()
assert_array_almost_equal(actual_mmarginal_dense,
expected_mmarginal_dense)
assert_array_almost_equal(actual_mmarginal_sparse,
expected_mmarginal_sparse)
# Verify static marginals
expected_smarginal_dense = expected_joint_dense.sum(1)
expected_smarginal_dense /= expected_smarginal_dense.sum()
expected_smarginal_sparse = expected_joint_sparse.sum(1)
expected_smarginal_sparse /= expected_smarginal_sparse.sum()
assert_array_almost_equal(actual_smarginal_dense,
expected_smarginal_dense)
assert_array_almost_equal(actual_smarginal_sparse,
expected_smarginal_sparse)
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 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_mattes_densities():
# Test the computation of the joint intensity distribution
# using a dense and a sparse set of values
seed = 1246592
nbins = 32
nr = 30
nc = 35
ns = 20
nvals = 50
for dim in [2, 3]:
if dim == 2:
shape = (nr, nc)
static, moving = create_random_image_pair(shape, nvals, seed)
else:
shape = (ns, nr, nc)
static, moving = create_random_image_pair(shape, nvals, seed)
# Initialize
mbase = MattesBase(nbins)
mbase.setup(static, moving)
# Get distributions computed by dense sampling
mbase.update_pdfs_dense(static, moving)
actual_joint_dense = mbase.joint
actual_mmarginal_dense = mbase.mmarginal
actual_smarginal_dense = mbase.smarginal
# Get distributions computed by sparse sampling
sval = static.reshape(-1)
mval = moving.reshape(-1)
mbase.update_pdfs_sparse(sval, mval)
actual_joint_sparse = mbase.joint
actual_mmarginal_sparse = mbase.mmarginal
actual_smarginal_sparse = mbase.smarginal
# Compute the expected joint distribution with dense sampling
expected_joint_dense = np.zeros(shape=(nbins, nbins))
for index in ndindex(shape):
sv = mbase.bin_normalize_static(static[index])
mv = mbase.bin_normalize_moving(moving[index])
sbin = mbase.bin_index(sv)
# The spline is centered at mv, will evaluate for all row
spline_arg = np.array([i - mv for i in range(nbins)])
contribution = cubic_spline(spline_arg)
expected_joint_dense[sbin, :] += contribution
# Compute the expected joint distribution with sparse sampling
expected_joint_sparse = np.zeros(shape=(nbins, nbins))
for index in range(sval.shape[0]):
sv = mbase.bin_normalize_static(sval[index])
mv = mbase.bin_normalize_moving(mval[index])
sbin = mbase.bin_index(sv)
# The spline is centered at mv, will evaluate for all row
spline_arg = np.array([i - mv for i in range(nbins)])
contribution = cubic_spline(spline_arg)
expected_joint_sparse[sbin, :] += contribution
# Verify joint distributions
expected_joint_dense /= expected_joint_dense.sum()
expected_joint_sparse /= expected_joint_sparse.sum()
assert_array_almost_equal(actual_joint_dense, expected_joint_dense)
assert_array_almost_equal(actual_joint_sparse, expected_joint_sparse)
# Verify moving marginals
expected_mmarginal_dense = expected_joint_dense.sum(0)
expected_mmarginal_dense /= expected_mmarginal_dense.sum()
expected_mmarginal_sparse = expected_joint_sparse.sum(0)
expected_mmarginal_sparse /= expected_mmarginal_sparse.sum()
assert_array_almost_equal(actual_mmarginal_dense,
expected_mmarginal_dense)
assert_array_almost_equal(actual_mmarginal_sparse,
expected_mmarginal_sparse)
# Verify static marginals
expected_smarginal_dense = expected_joint_dense.sum(1)
expected_smarginal_dense /= expected_smarginal_dense.sum()
expected_smarginal_sparse = expected_joint_sparse.sum(1)
expected_smarginal_sparse /= expected_smarginal_sparse.sum()
assert_array_almost_equal(actual_smarginal_dense,
expected_smarginal_dense)
assert_array_almost_equal(actual_smarginal_sparse,
expected_smarginal_sparse)
