def anglerotation(R):
"""
compute the angle of the rotation matrix R
:param R: the rotation matrix
:return: the angle of rotation in degree
"""
theta = np.rad2deg(np.arccos((np.trace(R) - 1) / 2))
return theta
def rot_matrix_error(R0, R1, method = 'unit_quaternion_product'):
""" R0, R1 are 3x3 or 4x4 homogeneous Rotation matrixes
returns: the value of the error depending on the method """
if ((R0.shape != (4,4)) and (R0.shape != (3,3))):
print ("Error in the R0 input rotation matrix shape, must be 3x3 or 4x4")
print R0
return -1
if ((R1.shape != (4,4)) and (R1.shape != (3,3))):
print ("Error in the R1 input rotation matrix shape, must be 3x3 or 4x4")
print R1
return -1
if R0.shape == (3,3):
R = np.eye(4)
R[:3,:3] = R0
R0 = R
if R1.shape == (3,3):
R = np.eye(4)
R[:3,:3] = R1
R1 = R
if(method == 'unit_quaternion_product' ):
## From the paper "Metrics for 3D Rotations: Comparison and Analysis" D. Huynh
# The 3D rotation error is computed using the inner product of unit quaterions
#We use the ros library TF to convert rotation matrix into unit quaternions
from tf import transformations
q0 = transformations.quaternion_from_matrix(R0)
q1 = transformations.quaternion_from_matrix(R1)
# We convert into unit quaternions
q0 = q0 / np.linalg.norm(q0)
q1 = q1 / np.linalg.norm(q1)
#Find the error as defined in the paper
rot_error = 1 - np.linalg.norm(np.dot(q0,q1))
if(method == 'angle'):
#option 2 find the angle of this rotation. In particular, the above is invalid
#for very large error angles (error > 90 degrees) and is imprecise for large
#error angles (angle > 45 degrees).
E = R1.dot(R0.T)
from cv2 import Rodrigues
rot_vector, J = Rodrigues(E[:3,:3])
angle = np.linalg.norm(rot_vector)
rot_error = np.rad2deg(angle)
return rot_error
def rotate(arr, theta, axis=0, backend='autograd', device=None):
"""
A rotate function that allows taking gradient with regards to theta.
:param arr: a 3D object in [len_y, len_x, len_z, n_channels].
"""
if backend == 'autograd':
warnings.warn('Rotate (with grad) in Autograd is not yet implemented. Use Pytorch backend instead.')
axes = []
for i in range(3):
if i != axis:
axes.append(i)
return scipy.ndimage.rotate(arr, -anp.rad2deg(theta), reshape=False, axes=axes, mode='nearest', order=1)
elif backend == 'pytorch':
try:
theta = theta.view(1)
except:
theta = tc.tensor(theta, requires_grad=False, device=device)
theta = theta.view(1)
axis_arrangement = [0, 1, 2, 3]
# Move channel to the 2nd dimension.
axis_arrangement[1], axis_arrangement[3] = axis_arrangement[3], axis_arrangement[1]
# Move invariant axis to front.
if axis != 0:
q = axis_arrangement.index(axis)
axis_arrangement[0], axis_arrangement[q] = axis_arrangement[q], axis_arrangement[0]
if axis_arrangement[2] < axis_arrangement[3]:
theta = -theta
arr = permute_axes(arr, axis_arrangement, override_backend='pytorch')
naught = cast(tc.tensor([0.], device=device), pytorch_dtype_query_mapping_dict[theta.dtype], override_backend='pytorch')
m0 = tc.cat([tc.cos(theta), -tc.sin(theta), naught])
m1 = tc.cat([tc.sin(theta), tc.cos(theta), naught])
m = tc.stack([m0, m1]).view(1, 2, 3)
m = cast(tile(m, [arr.shape[0], 1, 1], override_backend='pytorch'), pytorch_dtype_query_mapping_dict[arr.dtype], override_backend='pytorch')
g = tc.nn.functional.affine_grid(m, arr.shape, align_corners=False)
arr = tc.nn.functional.grid_sample(arr, g, padding_mode='border', align_corners=False)
arr = permute_axes(arr, [axis_arrangement.index(0), axis_arrangement.index(1),
axis_arrangement.index(2), axis_arrangement.index(3)], override_backend='pytorch')
return arr
def fov(self):
""" Calculate field of view angles (grads) from camera matrix """
fovx = np.rad2deg(2 * atan(self.img_width / (2. * self.fx)))
fovy = np.rad2deg(2 * atan(self.img_height / (2. * self.fy)))
return fovx, fovy
def test_rad2deg():
fun = lambda x: 3.0 * np.rad2deg(x)
check_grads(fun)(10.0 * npr.rand())
def create_cam_distribution_in_YZ(cam=None,
plane_size=(0.3, 0.3),
theta_params=(0, 180, 10),
r_params=(0.25, 1.0, 4),
plot=False):
"""
cam distritubution in YZ plane
:param cam:
:param plane_size:
:param theta_params:
:param phi_params:
:param r_params:
:param plot:
:return:
"""
if cam == None:
# Create an initial camera on the center of the world
cam = Camera()
f = 800
cam.set_K(fx=f, fy=f, cx=320, cy=240) # Camera Matrix
cam.img_width = 320 * 2
cam.img_height = 240 * 2
# we create a default plane with 4 points with a side lenght of w (meters)
plane = Plane(origin=np.array([0, 0, 0]),
normal=np.array([0, 0, 1]),
size=plane_size,
n=(2, 2))
# We extend the size of this plane to account for the deviation from a uniform pattern
# plane.size = (plane.size[0] + deviation, plane.size[1] + deviation)
d_space = np.linspace(r_params[0], r_params[1], r_params[2])
t_list = []
for d in d_space:
xx, yy, zz = uniform_halfCircle_in_YZ(theta_params, d,
False) # YZ plane
sphere_points = np.array(
[xx.ravel(), yy.ravel(), zz.ravel()], dtype=np.float32)
t_list.append(sphere_points)
t_space = np.hstack(t_list)
acc_row = r_params[2]
acc_col = theta_params[2]
accuracy_mat = np.zeros([
acc_row, acc_col
]) # accuracy_mat is used to describe accuracy degree for marker area
cams = []
for t in t_space.T:
cam = cam.clone()
cam.set_t(-t[0], -t[1], -t[2])
cam.set_R_mat(Rt_matrix_from_euler_t.R_matrix_from_euler_t(0.0, 0, 0))
cam.look_at([0, 0, 0])
radius = sqrt(t[0] * t[0] + t[1] * t[1] + t[2] * t[2])
angle = np.rad2deg(np.arccos(t[1] / radius))
cam.set_radius(radius)
cam.set_angle(angle)
plane.set_origin(np.array([0, 0, 0]))
plane.uniform()
objectPoints = plane.get_points()
# print "objectPoints",objectPoints
imagePoints = cam.project(objectPoints)
# print "imagePoints\n",imagePoints
if ((imagePoints[0, :] < cam.img_width) &
(imagePoints[0, :] > 0)).all():
if ((imagePoints[1, :] < cam.img_height) &
(imagePoints[1, :] > 0)).all():
cams.append(cam)
if plot:
planes = []
plane.uniform()
planes.append(plane)
# plot3D(cams, planes) #TODO comment because of from mayavi import mlab
return cams, accuracy_mat
# ==============================Test=================================================
#cams = create_cam_distribution(cam = None, plane_size = (0.3,0.3), theta_params = (0,360,10), phi_params = (0,70,5), r_params = (0.25,1.0,4), plot=True)
#create_cam_distribution_in_YZ(cam = None, plane_size = (0.3,0.3), theta_params = (0,180,3), r_params = (0.3,0.9,3), plot=False)
# print "cams size: ",len(cams)
# -----------------------------Test for cam look at method------------------------------
# cam = Camera()
# f = 800
# cam.set_K(fx = f, fy = f, cx = 320, cy = 240) #Camera Matrix
# cam.img_width = 320*2
# cam.img_height = 240*2
# cam.set_t(1,1,1,"world")
# cam.set_R_mat(Rt_matrix_from_euler_t.R_matrix_from_euler_t(0,np.deg2rad(0),0))
# cam.look_at([0,0,0])
# plane_size = (0.3,0.3)
# plane = Plane(origin=np.array([0, 0, 0] ), normal = np.array([0, 0, 1]), size=plane_size, n = (2,2))
# plane.set_origin(np.array([0, 0, 0]))
# plane.uniform()
# planes = []
# planes.append(plane)
# cams = []
# cams.append(cam)
# plot3D(cams,planes)
#
# print "cam.R",cam.R
# print "cam.Rt",cam.Rt
# print "cam.P",cam.P
# ------------------Code End-----------Test for cam look at method------------------------------
# create_cam_distribution_rotation_around_Z(cam=None, plane_size=(0.3, 0.3), theta_params=(0, 360, 10), phi_params=(45, 45, 1),
# r_params=(3.0, 3.0, 1), plot=False)
# create_cam_distribution_square_in_XY(cam=None, plane_size=(0.3, 0.3), theta_params=(0, 360, 5), phi_params=(45, 45, 1),
# r_params=(3.0, 3.0, 1), plot=False)
def rotate(im, angle):
return _rotate(im, np.rad2deg(angle), reshape=False)
def test_rad2deg():
fun = lambda x : 3.0 * np.rad2deg(x)
d_fun = grad(fun)
check_grads(fun, 10.0*npr.rand())
check_grads(d_fun, 10.0*npr.rand())
def test_rad2deg():
unary_ufunc_check(lambda x: np.rad2deg(x) / 50.0)
theta = keypoints[0][j].angle
dx_, dy_ = np.cos(np.deg2rad(theta)), -np.sin(np.deg2rad(theta))
x_0, y_0, _ = tuple(
h_apply(H[0, i], (x0 + s * dx_, y0 + s * dy_, 1)))
x_1, y_1, _ = tuple(
h_apply(H[0, i], (x0 - s * dy_, y0 + s * dx_, 1)))
x_2, y_2, _ = tuple(
h_apply(H[0, i], (x0 + s * dy_, y0 - s * dx_, 1)))
x_3, y_3, _ = tuple(
h_apply(H[0, i], (x0 - s * dx_, y0 - s * dy_, 1)))
s_new = np.mean([
np.linalg.norm((x_0 - x_3, y_0 - y_3)) / 2,
np.linalg.norm(((x_1 - x_2, y_1 - y_2))) / 2
])
angle_new = np.arctan2(-y_0 + y_3, x_0 - x_3)
angle_new = np.rad2deg(angle_new + 2 * np.pi * (angle_new < 0))
myplot.append((s_new, s, octave, layer, scale, theta, angle_new))
# Angles
a_orig = np.array([x[5] for x in myplot])
a_new = np.array([x[6] for x in myplot])
plt.scatter(a_orig, a_new)
plt.title('Angle differences')
plt.xlabel('Original')
plt.ylabel('New')
plt.show()
sys.exit()
# Plot sizes and octaves
z = np.array([0 for x in myplot])
sorig = np.array([x[1] for x in myplot])
snew = np.array([x[0] for x in myplot])
c_octaves = np.array([x[2] for x in myplot])
def invar(mdl, args, inputs_train, targets_train, prev_hess, prev_eigval,
prev_eigvec, coeff, ang_np, ang_sb, p_angles):
# calculating hessian
hess = mdl.hessian(mdl.params_flat) # Calculating Hessian
# Converting the Hessian to Tensor
hess = torch.tensor(hess).float()
# Extracting the eigenvalues and Eigen Vectors from the Calculated Hessian
eigenvalues, eigenvec = torch.symeig(hess, eigenvectors=True)
top = args.top # This decides how many top eigenvectors are considered
# |The reason for negative top :: torch.symeig outputs eigen vectors in the increasing order and as a result |
dom = eigenvec[:, -top:]
# | mdl the top (maximum) eigenvectors will be atlast. |
dom = dom.float()
# A random vector which is of the dim of variable "top" is being initialized
alpha = torch.rand(top)
# Finding the top vector
# Representing alpha onto dominant eigen vector
vec = (alpha * dom.float()).sum(1)
vec = vec / torch.sqrt((vec * vec).sum()) # Normalization of top vector
# Dummy Model for calculating gradient
mdl_test = model.create_model(args, inputs_train, targets_train)
# Finding gradient at top vec using Dummy network.
mdl_test.params_flat = np.array(vec)
# Find coeff and append.
top_vec = mdl_test.params_flat
c = torch.mv(hess.transpose(0, 1),
torch.tensor(mdl_test.params_flat).float())
if np.size(coeff) == 0:
coeff = c.detach().cpu().numpy()
coeff = np.expand_dims(coeff, axis=0)
else:
coeff = np.concatenate(
(coeff, np.expand_dims(c.detach().cpu().numpy(), axis=0)), 0)
# Statistics of subspaces, (1) Angle between top subpaces
eigenvalues_prev, eigenvec_prev = torch.symeig(prev_hess,
eigenvectors=True)
# Is it not the same as the variable "dom" that was calculated earlier ?
dom_prev = eigenvec_prev[:, -top:]
# calculation 1 norm, which is nothing but angle between subspaces
ang = np.linalg.norm(torch.mm(dom_prev, dom.transpose(0, 1)).numpy(), 1)
ang_sb.append(ang)
ang = np.rad2deg(subspace_angles(dom_prev, dom))
ang_np.append(ang)
# Calculating principal angles
u, s, v = torch.svd(torch.mm(dom.transpose(0, 1), dom_prev))
# Output in radians
s = torch.acos(torch.clamp(s, min=-1, max=1))
s = s * 180 / math.pi
# Attach 's' to p_angles
if np.size(p_angles) == 0:
p_angles = s.detach().cpu().numpy()
p_angles = np.expand_dims(p_angles, axis=0)
else:
p_angles = np.concatenate(
(p_angles, np.expand_dims(s.detach().cpu().numpy(), axis=0)), 0)
prev_hess = hess
prev_eigval = eigenvalues
prev_eigvec = eigenvec
return hess, eigenvalues, eigenvec, coeff, ang_np, ang_sb, p_angles, top_vec
def get_angle_subspaces(U, V, verbose=True):
"""
Gets the angle between two subspaces.
"""
use_simple_meth = True
U = orth(U)
V = orth(V)
len_1_u, len_2_u = U.shape
len_1_v, len_2_v = V.shape
if len_2_u > len_2_v:
a = U.copy()
U = V.copy()
V = a.copy()
len_1_u, len_2_u = U.shape
len_1_v, len_2_v = V.shape
if use_simple_meth:
print(180 * subspace_angles(U, V) /
np.pi) #180 * orthogonality_matrix / np.pi
angle = np.rad2deg(max(subspace_angles(U, V)))
else:
# normalizing subspaces:
for i in range(0, len_2_u):
U[:, i] *= (1 / np.linalg.norm(U[:, i]))
for i in range(0, len_2_v):
V[:, i] *= (1 / np.linalg.norm(V[:, i]))
# constuct M
M = np.zeros((len_2_u, len_2_v))
for i in range(0, len_2_u):
for j in range(0, len_2_v):
M[i, j] = np.dot(V[:, j], U[:, i])
MMt = M.dot(M.transpose())
det_MMt = np.linalg.det(MMt)
# construct u
uu = np.zeros((len_2_u, len_2_u))
for i in range(0, len_2_u):
for j in range(0, len_2_u):
uu[i, j] = np.dot(U[:, j], U[:, i])
#print(i,j,np.dot(V[:,j], U[:,i]))
det_uu = np.linalg.det(uu)
cos_square_theta = det_MMt / (det_uu + 1e-14)
# print("cos_square_theta",cos_square_theta)
theta = np.sqrt(cos_square_theta)
# print("theta",theta)
# print("angle",np.arccos(theta))
angle = np.rad2deg(np.arccos(theta))
if verbose:
print("angle", angle)
return angle
def rotate(im, angle):
return _rotate(im, np.rad2deg(angle), reshape=False)
def test_rad2deg(): unary_ufunc_check(lambda x : np.rad2deg(x)/50.0) def test_sign(): unary_ufunc_check(np.sign)
def test_rad2deg():
unary_ufunc_check(lambda x: np.rad2deg(x) / 50.0, test_complex=False)
def test_rad2deg(): unary_ufunc_check(lambda x : np.rad2deg(x)/50.0, test_complex=False) def test_radians(): unary_ufunc_check(np.radians, test_complex=False)
def train_model(args, mdl, mdl_test, results):
coeff = []
ang_sb=[]
ang_np=[]
p_angles = []
all_w=[]
results['args'] = args
init_loss = mdl.loss(mdl.params_flat)
init_grad_norm = np.linalg.norm(mdl.gradient(mdl.params_flat))
print("\n===============================================================================================\n")
print('\nInitial loss: {}, norm grad: {}\n'.format(init_loss, init_grad_norm))
results['init_full_loss'] = init_loss
results['init_full_grad_norm'] = init_grad_norm
results['history1'] = []
results['history1_columns'] = ['iter_no', 'batch_loss', 'batch_grad_norm', 'batch_param_norm']
results['history2'] = []
results['history2_columns'] = ['full_hessian', 'full_hessian_evals']
for iter_no in tqdm(range(args.max_iterations), desc="Training Progress", dynamic_ncols=True):
inputs, targets = get_batch_samples(iter_no, args, mdl)
batch_loss = mdl.loss(mdl.params_flat, inputs, targets)
batch_grad = mdl.gradient(mdl.params_flat, inputs, targets)
batch_grad_norm = np.linalg.norm(batch_grad)
batch_param_norm = np.linalg.norm(mdl.params_flat)
if iter_no % args.freq == 0:
# calculating hessian
layer_interest = 2
hess_len, hess_start = layer_weights(layer_size(args), layer_interest)
hess = mdl.hessian(mdl.params_flat)
new_hess = hess[hess_start:(hess_start+hess_len),hess_start:(hess_start+hess_len)]
# Calculating Hessian
new_hess = torch.tensor(new_hess).float() # Converting the Hessian to Tensor
eigenvalues, eigenvec = torch.symeig(new_hess,eigenvectors=True) # Extracting the eigenvalues and Eigen Vectors from the Calculated Hessian
if iter_no == 0:
prev_hess = new_hess
prev_eigval = eigenvalues
prev_eigvec = eigenvec
top = args.top # This decides how many top eigenvectors are considered
dom = eigenvec[:,-top:] # |The reason for negative top :: torch.symeig outputs eigen vectors in the increasing order and as a result |
# | the top (maximum) eigenvectors will be atlast. |
dom = dom.float()
alpha=torch.rand(top) # A random vector which is of the dim of variable "top" is being initialized
# Finding the top vector
vec=(alpha*dom.float()).sum(1) # Representing alpha onto dominant eigen vector
vec=vec/torch.sqrt((vec*vec).sum()) # Normalization of top vector
# Finding gradient at top vec using Dummy network.
mdl_test.params_flat = mdl.params_flat
mdl_test.params_flat[hess_start:(hess_start+hess_len)] = np.array(vec)
batch_grad_mdl_test = mdl_test.gradient(mdl_test.params_flat, inputs, targets)
# Gradients are updated only for the first layer.
mdl_test.params_flat[hess_start:(hess_start+hess_len)] -= batch_grad_mdl_test[hess_start:(hess_start+hess_len)] * args.learning_rate
# Find coeff and append. But why do we need to find the coeffs ?
c = torch.mv(new_hess.transpose(0,1), torch.tensor(mdl_test.params_flat[hess_start:(hess_start+hess_len)]).float())
if np.size(coeff) == 0:
coeff = c.detach().cpu().numpy()
coeff = np.expand_dims(coeff, axis=0)
else:
coeff = np.concatenate((coeff,np.expand_dims(c.detach().cpu().numpy(),axis=0)),0)
# Statistics of subspaces, (1) Angle between top subpaces
eigenvalues_prev, eigenvec_prev = torch.symeig(prev_hess, eigenvectors = True)
dom_prev = eigenvec_prev[:,-top:] # Is it not the same as the variable "dom" that was calculated earlier ?
# calculation 1 norm, which is nothing but angle between subspaces
ang=np.linalg.norm(torch.mm(dom_prev, dom.transpose(0,1)).numpy(),1)
ang_sb.append(ang)
ang = np.rad2deg(subspace_angles(dom_prev, dom))
ang_np.append(ang)
# Calculating principal angles
u,s,v =torch.svd(torch.mm(dom.transpose(0, 1), dom_prev))
# Output in radians
s = torch.acos(torch.clamp(s,min=-1,max=1))
s = s*180/math.pi
# Attach 's' to p_angles
if np.size(p_angles) == 0:
p_angles = s.detach().cpu().numpy()
p_angles = np.expand_dims(p_angles, axis=0)
else:
p_angles = np.concatenate((p_angles,np.expand_dims(s.detach().cpu().numpy(),axis=0)),0)
prev_hess = new_hess
prev_eigval = eigenvalues
prev_eigvec = eigenvec
# saving weights in all iterations
if batch_grad_norm <= args.stopping_grad_norm:
break
mdl.params_flat -= batch_grad * args.learning_rate
all_w.append(np.power(math.e,mdl.params_flat))
#print('{:06d} {} loss: {:.8f}, norm grad: {:.8f}'.format(iter_no, datetime.now(), batch_loss, batch_grad_norm))
final_loss = mdl.loss(mdl.params_flat)
final_grad_norm = np.linalg.norm(mdl.gradient(mdl.params_flat))
print('\nFinal loss: {}, norm grad: {}'.format(final_loss, final_grad_norm))
args.suffix=args.results_folder+'/coeff.npy'
np.save(args.suffix,coeff)
args.suffix=args.results_folder+'/ang_sb.npy'
np.save(args.suffix,ang_sb)
args.suffix=args.results_folder+'/ang_np.npy'
np.save(args.suffix,ang_np)
args.suffix=args.results_folder+'/p_angles.npy'
np.save(args.suffix,p_angles)
args.suffix=args.results_folder+'/all_weights.npy'
np.save(args.suffix,np.array(all_w))
# Saving png plots
coeff = torch.tensor(coeff)
for i in range(coeff.shape[0]):
a=torch.zeros(coeff[i].shape[0]).long()
b=torch.arange(0, coeff[i].shape[0])
c=torch.where(((coeff[i] > -0.1) & (coeff[i] < 0.1)),b,a)
z = torch.zeros(coeff[i].shape[0]).fill_(0)
z[torch.nonzero(c)] = coeff[i][torch.nonzero(c)]
z = np.array(z)
plt.plot(z)
plt.xlabel('Dimension',fontsize=14)
plt.ylabel('Coefficient',fontsize=14)
pnpy = args.results_folder+'/plot1'
plt.savefig(pnpy, format='png', pad_inches=5)
return args.results_folder
def test_rad2deg():
fun = lambda x : 3.0 * np.rad2deg(x)
d_fun = grad(fun)
check_grads(fun, 10.0*npr.rand())
check_grads(d_fun, 10.0*npr.rand())
def test_rad2deg():
fun = lambda x : 3.0 * np.rad2deg(x)
check_grads(fun)(10.0*npr.rand())