def factor(W):
"""
This implements rigid factorization as described in
Tomasi, C. & Kanade, T. "Shape and motion from image streams under
orthography: a factorization method International Journal of Computer
Vision, 1992
"""
F = W.shape[0]/2
N = W.shape[1]
# Center W
T = W.mean(axis=1)
W = W - T[:, np.newaxis]
# Factor W
M_hat, B_hat = factor_matrix(W, J=3)
# Where we will build the linear system.
A = np.zeros((3*F, 6))
b = np.zeros((3*F,))
for f in range(F):
# Extract the two rows.
x_f, y_f = M_hat[f*2:f*2+2]
# Both rows must have unit length.
A[f*3] = util.vc(x_f, x_f)
b[f*3] = 1.0
A[f*3+1] = util.vc(y_f,y_f)
b[f*3+1] = 1.0
# And be orthogonal.
A[f*3+2] = util.vc(x_f - y_f, x_f + y_f)
# Recovec vech(Q) and Q
vech_Q = np.linalg.lstsq(A, b)[0]
Q = util.from_vech(vech_Q, 3, 3, sym=True)
# Factor out G recovery matrix
G, Gt = factor_matrix(Q)
# Upgrade M and B.
M = np.dot(M_hat, G)
B = np.linalg.solve(G, B_hat)
# Find actual rotations matrices.
Rs = np.zeros((F, 3, 3))
Rs[:,:2] = M.reshape(F,2,3)
Rs[:,2] = util.normed(np.cross(Rs[:,0], Rs[:,1], axis=-1))
# And 3D translations.
Ts = np.zeros((F,3))
Ts[:,:2] = T.reshape(F,2)
model = BasisShapeModel(Rs, Bs = B[np.newaxis,:,:], Ts = Ts)
return model
def find_G_k(M_hat):
K = M_hat.shape[1] / 3
F = M_hat.shape[0] / 2
# Get the linear basis that vech(Q_k) should reside in.
U = find_linear_basis_for_vech_Q_k(M_hat)
# 2D subspace of vech style.
B = np.asarray([
from_vech(U[:, i], 3 * K, 3 * K, sym=True).flatten()
for i in range(U.shape[1])
]).T
# Coefficients correspond to taking trace.
from cvxopt import matrix, solvers
# Calculate the cost vector.
non_diag = np.where(np.eye(3 * K).flatten() != 1)[0]
c = B.copy()
c[non_diag, :] = 0
c = c.sum(axis=0)
# This matrix should be positive semi-definite.
G = [matrix(-B)]
h = [matrix(np.zeros((3 * K, 3 * K)))]
# We fix the first row of the recovered M_hat to have unit length
# which avoids the trivial solution.
A = np.dot(vc(M_hat[0], M_hat[0]), U)[np.newaxis, :].copy()
A = matrix(A)
b = matrix([[1.0]])
# Disable progress reporting in solver.
solvers.options['show_progress'] = False
# Solve the sdp using cvxopt
sol = solvers.sdp(matrix(c), Gs=G, hs=h, A=A, b=b)
# Extract the solution.
x = np.asarray(sol['x'])
# Upgrade to vec(Q_k)
y = np.dot(B, x)
# And reshape to Q_k
Q_k = y.reshape((3 * K, 3 * K))
# Factor into G_k
Q_k_U, Q_k_s, Q_k_V = np.linalg.svd(Q_k)
G_k = Q_k_U[:, :3] * np.sqrt(Q_k_s)[np.newaxis, :3]
# Squared residual objective function as defined in Dai2012.
def f(g_k):
G_k = g_k.reshape(3 * K, 3)
M_k = np.dot(M_hat, G_k)
M_kx = M_k[0::2]
M_ky = M_k[1::2]
term1 = np.square(1 - (M_kx * M_kx).sum(axis=-1) /
(M_ky * M_ky).sum(axis=-1))
term2 = np.square(2 * (M_ky * M_kx).sum(axis=-1) /
(M_ky * M_ky).sum(axis=-1))
err = np.sum(term1 + term2)
return err
# Minimize this function with LBFGS
from scipy.optimize import fmin_l_bfgs_b
rc = fmin_l_bfgs_b(f, G_k.flatten(), approx_grad=True)
G_k = rc[0].reshape(3 * K, 3)
return G_k
def find_G_k(M_hat):
K = M_hat.shape[1]/3
F = M_hat.shape[0]/2
# Get the linear basis that vech(Q_k) should reside in.
U = find_linear_basis_for_vech_Q_k(M_hat)
# 2D subspace of vech style.
B = np.asarray([from_vech(U[:,i], 3*K, 3*K, sym=True).flatten() for i in range(U.shape[1])]).T
# Coefficients correspond to taking trace.
from cvxopt import matrix, solvers
# Calculate the cost vector.
non_diag = np.where(np.eye(3*K).flatten()!=1)[0]
c = B.copy()
c[non_diag,:] = 0
c = c.sum(axis=0)
# This matrix should be positive semi-definite.
G = [matrix(-B)]
h = [matrix(np.zeros((3*K,3*K)))]
# We fix the first row of the recovered M_hat to have unit length
# which avoids the trivial solution.
A = np.dot(vc(M_hat[0], M_hat[0]), U)[np.newaxis,:].copy()
A = matrix(A)
b = matrix([[1.0]])
# Disable progress reporting in solver.
solvers.options['show_progress'] = False
# Solve the sdp using cvxopt
sol = solvers.sdp(matrix(c), Gs = G, hs = h, A = A, b = b)
# Extract the solution.
x = np.asarray(sol['x'])
# Upgrade to vec(Q_k)
y = np.dot(B, x)
# And reshape to Q_k
Q_k = y.reshape((3*K, 3*K))
# Factor into G_k
Q_k_U, Q_k_s, Q_k_V = np.linalg.svd(Q_k)
G_k = Q_k_U[:,:3] * np.sqrt(Q_k_s)[np.newaxis,:3]
# Squared residual objective function as defined in Dai2012.
def f(g_k):
G_k = g_k.reshape(3*K, 3)
M_k = np.dot(M_hat, G_k)
M_kx = M_k[0::2]
M_ky = M_k[1::2]
term1 = np.square(1 - (M_kx * M_kx).sum(axis=-1)/(M_ky*M_ky).sum(axis=-1 ))
term2 = np.square(2*(M_ky*M_kx).sum(axis=-1)/(M_ky*M_ky).sum(axis=-1))
err = np.sum(term1 + term2)
return err
# Minimize this function with LBFGS
from scipy.optimize import fmin_l_bfgs_b
rc = fmin_l_bfgs_b(f, G_k.flatten(), approx_grad=True)
G_k = rc[0].reshape(3*K,3)
return G_k