def init_path_geod(beta1, beta2, T=100, k=5):
"""
Initializes a path in \cal{C}. beta1, beta2 are already
standardized curves. Creates a path from beta1 to beta2 in
shape space, then projects to the closed shape manifold.
:param beta1: numpy ndarray of shape (2,M) of M samples (first curve)
:param beta2: numpy ndarray of shape (2,M) of M samples (end curve)
:param T: Number of samples of curve (Default = 100)
:param k: number of samples along path (Default = 5)
:rtype: numpy ndarray
:return alpha: a path between two q-functions
:return beta: a path between two curves
:return O: rotation matrix
"""
alpha = zeros((2, T, k))
beta = zeros((2, T, k))
dist, pathq, O = geod_sphere(beta1, beta2, k)
for tau in range(0, k):
alpha[:, :, tau] = cf.project_curve(pathq[:, :, tau])
x = cf.q_to_curve(alpha[:, :, tau])
a = -1 * cf.calculatecentroid(x)
beta[:, :, tau] = x + tile(a, [T, 1]).T
return (alpha, beta, O)
def sample_shapes(mu, K, mode='O', no=3, numSamp=10):
"""
Computes sample shapes from mean and covariance
:param betamean: numpy ndarray of shape (n, M) describing the mean curve
:param mu: numpy ndarray of shape (n, M) describing the mean srvf
:param K: numpy ndarray of shape (M, M) describing the covariance
:param mode: Open ('O') or closed curve ('C') (default 'O')
:param no: number of direction (default 3)
:param numSamp: number of samples (default 10)
:rtype: tuple of numpy array
:return samples: sample shapes
"""
n, T = mu.shape
modes = ['O', 'C']
mode = [i for i, x in enumerate(modes) if x == mode]
if len(mode) == 0:
mode = 0
else:
mode = mode[0]
U, s, V = svd(K)
if mode == 0:
N = 2
else:
N = 10
epsilon = 1./(N-1)
samples = empty(numSamp, dtype=object)
for i in range(0, numSamp):
v = zeros((2, T))
for m in range(0, no):
v = v + randn()*sqrt(s[m])*vstack((U[0:T, m], U[T:2*T, m]))
q1 = mu
for j in range(0, N-1):
normv = sqrt(cf.innerprod_q2(v, v))
if normv < 1e-4:
q2 = mu
else:
q2 = cos(epsilon*normv)*q1+sin(epsilon*normv)*v/normv
if mode == 1:
q2 = cf.project_curve(q2)
# Parallel translate tangent vector
basis2 = cf.find_basis_normal(q2)
v = cf.parallel_translate(v, q1, q2, basis2, mode)
q1 = q2
samples[i] = cf.q_to_curve(q2)
return(samples)
def curve_karcher_cov(betamean, beta, mode='O'):
"""
This claculates the mean of a set of curves
:param betamean: numpy ndarray of shape (n, M) describing the mean curve
:param beta: numpy ndarray of shape (n, M, N) describing N curves
in R^M
:param mode: Open ('O') or closed curve ('C') (default 'O')
:rtype: tuple of numpy array
:return K: Covariance Matrix
"""
n, T, N = beta.shape
modes = ['O', 'C']
mode = [i for i, x in enumerate(modes) if x == mode]
if len(mode) == 0:
mode = 0
else:
mode = mode[0]
# Compute Karcher covariance of uniformly sampled mean
betamean = cf.resamplecurve(betamean, T)
mu = cf.curve_to_q(betamean)
if mode == 1:
mu = cf.project_curve(mu)
basis = cf.find_basis_normal(mu)
v = zeros((n, T, N))
for i in range(0, N):
beta1 = beta[:, :, i]
w, dist = cf.inverse_exp_coord(betamean, beta1)
# Project to the tangent sapce of manifold to obtain v_i
if mode == 0:
v[:, :, i] = w
else:
v[:, :, i] = cf.project_tangent(w, mu, basis)
K = zeros((2*T, 2*T))
for i in range(0, N):
w = v[:, :, i]
wtmp = w.reshape((T*n, 1), order='C')
K = K + wtmp.dot(wtmp.T)
K = K/(N-1)
return(K)
def update_path(alpha, beta, gradE, delta, T=100, k=5):
"""
Update the path along the direction -gradE
:param alpha: numpy ndarray of shape (2,M) of M samples
:param beta: numpy ndarray of shape (2,M) of M samples
:param gradE: numpy ndarray of shape (2,M) of M samples
:param delta: gradient paramenter
:param T: Number of samples of curve (Default = 100)
:param k: number of samples along path (Default = 5)
:rtype: numpy scalar
:return alpha: updated path of srvfs
:return beta: updated path of curves
"""
for tau in range(1, k - 1):
alpha_new = alpha[:, :, tau] - delta * gradE[:, :, tau]
alpha[:, :, tau] = cf.project_curve(alpha_new)
x = cf.q_to_curve(alpha[:, :, tau])
a = -1 * cf.calculatecentroid(x)
beta[:, :, tau] = x + tile(a, [T, 1]).T
return (alpha, beta)
def curve_principal_directions(betamean, mu, K, mode='O', no=3, N=5):
"""
Computes principal direction of variation specified by no. N is
Number of shapes away from mean. Creates 2*N+1 shape sequence
:param betamean: numpy ndarray of shape (n, M) describing the mean curve
:param mu: numpy ndarray of shape (n, M) describing the mean srvf
:param K: numpy ndarray of shape (M, M) describing the covariance
:param mode: Open ('O') or closed curve ('C') (default 'O')
:param no: number of direction (default 3)
:param N: number of shapes (2*N+1) (default 5)
:rtype: tuple of numpy array
:return pd: principal directions
"""
n, T = betamean.shape
modes = ['O', 'C']
mode = [i for i, x in enumerate(modes) if x == mode]
if len(mode) == 0:
mode = 0
else:
mode = mode[0]
U, s, V = svd(K)
qarray = empty((no, 2*N+1), dtype=object)
qarray1 = empty(N, dtype=object)
qarray2 = empty(N, dtype=object)
pd = empty((no, 2*N+1), dtype=object)
pd1 = empty(N, dtype=object)
pd2 = empty(N, dtype=object)
for m in range(0, no):
princDir = vstack((U[0:T, m], U[T:2*T, m]))
v = sqrt(s[m]) * princDir
q1 = mu
epsilon = 2./N
# Forward direction from mean
for i in range(0, N):
normv = sqrt(cf.innerprod_q2(v, v))
if normv < 1e-4:
q2 = mu
else:
q2 = cos(epsilon*normv)*q1 + sin(epsilon*normv)*v/normv
if mode == 1:
q2 = cf.project_curve(q2)
qarray1[i] = q2
p = cf.q_to_curve(q2)
centroid1 = -1*cf.calculatecentroid(p)
beta_scaled, scale = cf.scale_curve(p + tile(centroid1, [T, 1]).T)
pd1[i] = beta_scaled
# Parallel translate tangent vector
basis2 = cf.find_basis_normal(q2)
v = cf.parallel_translate(v, q1, q2, basis2, mode)
q1 = q2
# Backward direction from mean
v = -sqrt(s[m])*princDir
q1 = mu
for i in range(0, N):
normv = sqrt(cf.innerprod_q2(v, v))
if normv < 1e-4:
q2 = mu
else:
q2 = cos(epsilon*normv)*q1+sin(epsilon*normv)*v/normv
if mode == 1:
q2 = cf.project_curve(q2)
qarray2[i] = q2
p = cf.q_to_curve(q2)
centroid1 = -1*cf.calculatecentroid(p)
beta_scaled, scale = cf.scale_curve(p + tile(centroid1, [T, 1]).T)
pd2[i] = beta_scaled
# Parallel translate tangent vector
basis2 = cf.find_basis_normal(q2)
v = cf.parallel_translate(v, q1, q2, basis2, mode)
q1 = q2
for i in range(0, N):
qarray[m, i] = qarray2[(N-1)-i]
pd[m, i] = pd2[(N-1)-i]
qarray[m, N] = mu
centroid1 = -1*cf.calculatecentroid(betamean)
beta_scaled, scale = cf.scale_curve(betamean +
tile(centroid1, [T, 1]).T)
pd[m, N] = beta_scaled
for i in range(N+1, 2*N+1):
qarray[m, i] = qarray1[i-(N+1)]
pd[m, i] = pd1[i-(N+1)]
return(pd)
def curve_karcher_mean(beta, mode='O'):
"""
This claculates the mean of a set of curves
:param beta: numpy ndarray of shape (n, M, N) describing N curves
in R^M
:param mode: Open ('O') or closed curve ('C') (default 'O')
:rtype: tuple of numpy array
:return mu: mean srvf
:return betamean: mean curve
:return v: shooting vectors
:return q: srvfs
"""
n, T, N = beta.shape
q = zeros((n, T, N))
for ii in range(0, N):
q[:, :, ii] = cf.curve_to_q(beta[:, :, ii])
modes = ['O', 'C']
mode = [i for i, x in enumerate(modes) if x == mode]
if len(mode) == 0:
mode = 0
else:
mode = mode[0]
# Initialize mu as one of the shapes
mu = q[:, :, 0]
betamean = beta[:, :, 0]
delta = 0.5
tolv = 1e-4
told = 5*1e-3
maxit = 20
itr = 0
sumd = zeros(maxit+1)
v = zeros((n, T, N))
normvbar = zeros(maxit+1)
while itr < maxit:
print("Iteration: %d" % itr)
mu = mu / sqrt(cf.innerprod_q2(mu, mu))
sumv = zeros((2, T))
sumd[itr+1] = 0
out = Parallel(n_jobs=-1)(delayed(karcher_calc)(beta[:, :, n],
q[:, :, n], betamean, mu, mode) for n in range(N))
v = zeros((n, T, N))
for i in range(0, N):
v[:, :, i] = out[i][0]
sumd[itr+1] = sumd[itr+1] + out[i][1]**2
sumv = v.sum(axis=2)
# Compute average direction of tangent vectors v_i
vbar = sumv/float(N)
normvbar[itr] = sqrt(cf.innerprod_q2(vbar, vbar))
normv = normvbar[itr]
if normv > tolv and fabs(sumd[itr+1]-sumd[itr]) > told:
# Update mu in direction of vbar
mu = cos(delta*normvbar[itr])*mu + sin(delta*normvbar[itr]) * vbar/normvbar[itr]
if mode == 1:
mu = cf.project_curve(mu)
x = cf.q_to_curve(mu)
a = -1*cf.calculatecentroid(x)
betamean = x + tile(a, [T, 1]).T
else:
break
itr += 1
return(mu, betamean, v, q)
def init_path_rand(beta1, beta_mid, beta2, T=100, k=5):
"""
Initializes a path in \cal{C}. beta1, beta_mid beta2 are already
standardized curves. Creates a path from beta1 to beta_mid to beta2 in
shape space, then projects to the closed shape manifold.
:param beta1: numpy ndarray of shape (2,M) of M samples (first curve)
:param betamid: numpy ndarray of shape (2,M) of M samples (mid curve)
:param beta2: numpy ndarray of shape (2,M) of M samples (end curve)
:param T: Number of samples of curve (Default = 100)
:param k: number of samples along path (Default = 5)
:rtype: numpy ndarray
:return alpha: a path between two q-functions
:return beta: a path between two curves
:return O: rotation matrix
"""
alpha = zeros((2, T, k))
beta = zeros((2, T, k))
q1 = cf.curve_to_q(beta1)
q_mid = cf.curve_to_q(beta_mid)
# find optimal rotation of q2
beta2, O1, tau1 = cf.find_rotation_and_seed_coord(beta1, beta2)
q2 = cf.curve_to_q(beta2)
# find the optimal coorespondence
gam = cf.optimum_reparam_curve(q2, q1)
gamI = uf.invertGamma(gam)
# apply optimal reparametrization
beta2n = cf.group_action_by_gamma_coord(beta2, gamI)
# find optimal rotation of q2
beta2n, O2, tau1 = cf.find_rotation_and_seed_coord(beta1, beta2n)
centroid2 = cf.calculatecentroid(beta2n)
beta2n = beta2n - tile(centroid2, [T, 1]).T
q2n = cf.curve_to_q(beta2n)
O = O1.dot(O2)
# Initialize a path as a geodesic through q1 --- q_mid --- q2
theta1 = arccos(cf.innerprod_q2(q1, q_mid))
theta2 = arccos(cf.innerprod_q2(q_mid, q2n))
tmp = arange(2, int((k - 1) / 2) + 1)
t = zeros(tmp.size)
alpha[:, :, 0] = q1
beta[:, :, 0] = beta1
i = 0
for tau in range(2, int((k - 1) / 2) + 1):
t[i] = (tau - 1.0) / ((k - 1) / 2.0)
qnew = (1 / sin(theta1)) * (sin((1 - t[i]) * theta1) * q1 + sin(t[i] * theta1) * q_mid)
alpha[:, :, tau - 1] = cf.project_curve(qnew)
x = cf.q_to_curve(alpha[:, :, tau - 1])
a = -1 * cf.calculatecentroid(x)
beta[:, :, tau - 1] = x + tile(a, [T, 1]).T
i += 1
alpha[:, :, int((k - 1) / 2)] = q_mid
beta[:, :, int((k - 1) / 2)] = beta_mid
i = 0
for tau in range(int((k - 1) / 2) + 1, k - 1):
qnew = (1 / sin(theta2)) * (sin((1 - t[i]) * theta2) * q_mid + sin(t[i] * theta2) * q2n)
alpha[:, :, tau] = cf.project_curve(qnew)
x = cf.q_to_curve(alpha[:, :, tau])
a = -1 * cf.calculatecentroid(x)
beta[:, :, tau] = x + tile(a, [T, 1]).T
i += 1
alpha[:, :, k - 1] = q2n
beta[:, :, k - 1] = beta2n
return (alpha, beta, O)
def karcher_mean(self, parallel=False, cores=-1):
"""
This calculates the mean of a set of curves
:param parallel: run in parallel (default = F)
:param cores: number of cores for parallel (default = -1 (all))
"""
n, T, N = self.beta.shape
modes = ['O', 'C']
mode = [i for i, x in enumerate(modes) if x == self.mode]
if len(mode) == 0:
mode = 0
else:
mode = mode[0]
# Initialize mu as one of the shapes
mu = self.q[:, :, 0]
betamean = self.beta[:, :, 0]
itr = 0
gamma = zeros((T, N))
maxit = 20
sumd = zeros(maxit + 1)
v = zeros((n, T, N))
normvbar = zeros(maxit + 1)
delta = 0.5
tolv = 1e-4
told = 5 * 1e-3
print("Computing Karcher Mean of %d curves in SRVF space.." % N)
while itr < maxit:
print("updating step: %d" % (itr + 1))
if iter == maxit:
print("maximal number of iterations reached")
mu = mu / sqrt(cf.innerprod_q2(mu, mu))
if mode == 1:
self.basis = cf.find_basis_normal(mu)
else:
self.basis = []
sumv = zeros((n, T))
sumd[0] = inf
sumd[itr + 1] = 0
out = Parallel(n_jobs=cores)(delayed(karcher_calc)(
self.beta[:, :, n], self.q[:, :,
n], betamean, mu, self.basis, mode)
for n in range(N))
v = zeros((n, T, N))
for i in range(0, N):
v[:, :, i] = out[i][0]
sumd[itr + 1] = sumd[itr + 1] + out[i][1]**2
sumv = v.sum(axis=2)
# Compute average direction of tangent vectors v_i
vbar = sumv / float(N)
normvbar[itr] = sqrt(cf.innerprod_q2(vbar, vbar))
normv = normvbar[itr]
if normv > tolv and fabs(sumd[itr + 1] - sumd[itr]) > told:
# Update mu in direction of vbar
mu = cos(delta * normvbar[itr]) * mu + sin(
delta * normvbar[itr]) * vbar / normvbar[itr]
if mode == 1:
mu = cf.project_curve(mu)
x = cf.q_to_curve(mu)
a = -1 * cf.calculatecentroid(x)
betamean = x + tile(a, [T, 1]).T
else:
break
itr += 1
self.q_mean = mu
self.beta_mean = betamean
self.v = v
self.qun = sumd[0:(itr + 1)]
self.E = normvbar[0:(itr + 1)]
return
def init_path_rand(beta1, beta_mid, beta2, T=100, k=5):
r"""
Initializes a path in :math:`\cal{C}`. beta1, beta_mid beta2 are already
standardized curves. Creates a path from beta1 to beta_mid to beta2 in
shape space, then projects to the closed shape manifold.
:param beta1: numpy ndarray of shape (2,M) of M samples (first curve)
:param betamid: numpy ndarray of shape (2,M) of M samples (mid curve)
:param beta2: numpy ndarray of shape (2,M) of M samples (end curve)
:param T: Number of samples of curve (Default = 100)
:param k: number of samples along path (Default = 5)
:rtype: numpy ndarray
:return alpha: a path between two q-functions
:return beta: a path between two curves
:return O: rotation matrix
"""
alpha = zeros((2, T, k))
beta = zeros((2, T, k))
q1 = cf.curve_to_q(beta1)[0]
q_mid = cf.curve_to_q(beta_mid)[0]
# find optimal rotation of q2
beta2, O1, tau1 = cf.find_rotation_and_seed_coord(beta1, beta2)
q2 = cf.curve_to_q(beta2)[0]
# find the optimal coorespondence
gam = cf.optimum_reparam_curve(q2, q1)
gamI = uf.invertGamma(gam)
# apply optimal reparametrization
beta2n = cf.group_action_by_gamma_coord(beta2, gamI)
# find optimal rotation of q2
beta2n, O2, tau1 = cf.find_rotation_and_seed_coord(beta1, beta2n)
centroid2 = cf.calculatecentroid(beta2n)
beta2n = beta2n - tile(centroid2, [T, 1]).T
q2n = cf.curve_to_q(beta2n)[0]
O = O1 @ O2
# Initialize a path as a geodesic through q1 --- q_mid --- q2
theta1 = arccos(cf.innerprod_q2(q1, q_mid))
theta2 = arccos(cf.innerprod_q2(q_mid, q2n))
tmp = arange(2, int((k - 1) / 2) + 1)
t = zeros(tmp.size)
alpha[:, :, 0] = q1
beta[:, :, 0] = beta1
i = 0
for tau in range(2, int((k - 1) / 2) + 1):
t[i] = (tau - 1.) / ((k - 1) / 2.)
qnew = (1 / sin(theta1)) * (sin(
(1 - t[i]) * theta1) * q1 + sin(t[i] * theta1) * q_mid)
alpha[:, :, tau - 1] = cf.project_curve(qnew)
x = cf.q_to_curve(alpha[:, :, tau - 1])
a = -1 * cf.calculatecentroid(x)
beta[:, :, tau - 1] = x + tile(a, [T, 1]).T
i += 1
alpha[:, :, int((k - 1) / 2)] = q_mid
beta[:, :, int((k - 1) / 2)] = beta_mid
i = 0
for tau in range(int((k - 1) / 2) + 1, k - 1):
qnew = (1 / sin(theta2)) * (sin(
(1 - t[i]) * theta2) * q_mid + sin(t[i] * theta2) * q2n)
alpha[:, :, tau] = cf.project_curve(qnew)
x = cf.q_to_curve(alpha[:, :, tau])
a = -1 * cf.calculatecentroid(x)
beta[:, :, tau] = x + tile(a, [T, 1]).T
i += 1
alpha[:, :, k - 1] = q2n
beta[:, :, k - 1] = beta2n
return (alpha, beta, O)
