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)