Datasets:

FelixOtt
/

industry_datasets

"""
Copyright (C) 2018 NVIDIA Corporation.  All rights reserved.
Licensed under the CC BY-NC-SA 4.0 license (https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode).
"""
import numpy as np
import scipy.linalg as slin
import open3d as o3d
import math
import transforms3d.quaternions as txq
import transforms3d.euler as txe
from scipy.spatial.transform import Rotation as Rot
# see for formulas:
# https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-801-machine-vision-fall-2004/readings/quaternions.pdf
# and "Quaternion and Rotation" - Yan-Bin Jia, September 18, 2016
# from IPython.core.debugger import set_trace
import argparse

parser = argparse.ArgumentParser()
parser.add_argument('--train', help='train number', required=True)
parser.add_argument('--test', help='test_number', required=True)
args = parser.parse_args()
train_num = int(args.train)
test_num = int(args.test)

# runs pgo in batches of size base_lim until all time steps are handled. consecutive batches have an overlap
base_lim = 100  # number of steps to simultaneously consider for pgo (low for faster runtime)
overlapping_percent = 0.2  # percent of overlapping time steps to the previous pgo run
# path to save the created values
save_base_path = "..."
# adds the test and train num to the save path (for better structure)
save_base_path_ext = str(test_num) + "_" + str(train_num)

# ------------ Rel vals ------------ #
rel_vals_base_folder = "..." % train_num
txt_file_exts = {1: ..., 2: ...}  # One extension for every test case
txt_file_ext = txt_file_exts[test_num]
rel_vals_file = rel_vals_base_folder + txt_file_ext  # file containing the predicted relative poses/orientation
# ---------------------------------- #

# ------------ GT + Abs vals ------------ #
gt_abs_vals_base_folder = "..." % train_num
gt_abs_txt_file_exts = {1: ..., 2: ...}  # One extension for every test case
gt_abs_txt_file_ext = gt_abs_txt_file_exts[test_num]
# file containing the predicted absolute poses/orientation
gt_abs_vals_file = gt_abs_vals_base_folder + gt_abs_txt_file_ext
# ---------------------------------- #


def getRelDiff(posx, posq, actualx, actualq):
    """ Calculates the relative difference in position and rotation. """
    rel_x = posx - actualx
    m1 = Rot.from_quat(posq).as_matrix()
    m2 = Rot.from_quat(actualq).as_matrix()
    m_rel = m1 @ m2.T
    #m_rel = m2.T @ m1
    rel_q = Rot.from_matrix(m_rel).as_quat()
    return rel_x, rel_q


## NUMPY
# see: https://github.com/NVlabs/geomapnet/blob/master/common/pose_utils.py         ----------------------------- #
def qlog(q):
    """
    Applies logarithm map to q
    :param q: (4,)
    :return: (3,)
    """
    if all(q[1:] == 0):
        q = np.zeros(3)
    else:
        q = np.arccos(q[0]) * q[1:] / np.linalg.norm(q[1:])
    return q


def qexp(q):
    """
    Applies the exponential map to q
    :param q: (3,)
    :return: (4,)
    """
    n = np.linalg.norm(q)
    q = np.hstack((np.cos(n), np.sinc(n / np.pi) * q))
    return q


def log_quaternion_angular_error(q1, q2):
    return quaternion_angular_error(qexp(q1), qexp(q2))


def quaternion_angular_error(q1, q2):
    """
    angular error between two quaternions
    :param q1: (4, )
    :param q2: (4, )
    :return:
    """
    d = abs(np.dot(q1, q2))
    d = min(1.0, max(-1.0, d))
    theta = 2 * np.arccos(d) * 180 / np.pi
    return theta


def skew(x):
    """
    returns skew symmetric matrix from vector
    :param x: 3 x 1
    :return:
    """
    s = np.asarray([[0, -x[2], x[1]], [x[2], 0, -x[0]], [-x[1], x[0], 0]])
    return s


def dpq_q(p):
    """
    returns the jacobian of quaternion product pq w.r.t. q
    :param p: 4 x 1
    :return: 4 x 4
    """
    J = np.zeros((4, 4))
    J[0, 0] = p[0]
    J[0, 1:] = -p[1:].squeeze()
    J[1:, 0] = p[1:].squeeze()
    J[1:, 1:] = p[0] * np.eye(3) + skew(p[1:])
    return J


def dpsq_q(p):
    """
    returns the jacobian of quaternion product (p*)q w.r.t. q
    :param p: 4 x 1
    :return: 4 x 4
    """
    J = np.zeros((4, 4))
    J[0, 0] = p[0]
    J[0, 1:] = -p[1:].squeeze()
    J[1:, 0] = -p[1:].squeeze()
    J[1:, 1:] = p[0] * np.eye(3) - skew(p[1:])
    return J


def dpsq_p(q):
    """
    returns the jacobian of quaternion product (p*)q w.r.t. p
    :param q: 4 x 1
    :return: 4 x 4
    """
    J = np.zeros((4, 4))
    J[0, 0] = q[0]
    J[0, 1:] = q[1:].squeeze()
    J[1:, 0] = q[1:].squeeze()
    J[1:, 1:] = -q[0] * np.eye(3) + skew(q[1:])
    return J


def dqstq_q(q, t):
    """
    jacobian of q* t q w.r.t. q
    :param q: 4 x 1
    :param t: 3 x 1
    :return: 3 x 4
    """
    J = np.zeros((3, 4))
    J[:, :1] = q[0] * t - np.cross(q[1:], t, axis=0)
    J[:, 1:] = -np.dot(t, q[1:].T) + np.dot(t.T, q[1:]) * np.eye(3) + \
               np.dot(q[1:], t.T) + q[0] * skew(t)
    J *= 2
    return J


def dqstq_t(q):
    """
    jacobian of q* t q w.r.t. t
    :param q: 4 x 1
    :return: 3 x 3
    """
    J = (q[0] * q[0] - np.dot(q[1:].T, q[1:])) * np.eye(3) + 2 * np.dot(q[1:], q[1:].T) - \
        2 * q[0] * skew(q[1:])
    return J


def m_rot(x):
    """
    returns Jacobian of exponential map w.r.t. manifold increment
    :param x: part of state vector affected by increment, 4 x 1
    :return: 4 x 3
    """
    # jacobian of full q wrt qm (quaternion update on manifold),
    # evaluated at qv = (0, 0, 0)
    # full q is derived using either the exponential map or q0 = sqrt(1-qm^2)
    jm = np.vstack((np.zeros((1, 3)), np.eye(3)))  # 4 x 3
    m = np.dot(dpq_q(p=x), jm)
    return m


class PoseGraph:
    def __init__(self):
        """
        implements pose graph optimization from
        "Hybrid Hessians for Optimization of Pose Graphs" - Y. LeCun et al
        and "A Tutorial on Graph-Based SLAM" - W. Burgard et al
        """
        self.N = 0
        self.z = np.zeros((0, 0))

    def jacobian(self, L_ax, L_aq, L_rx, L_rq):
        J = np.zeros((0, 6 * self.N))  # 6 because updates for rotation are on manifold

        # unary constraints
        for i in np.arange(self.N):
            # translation constraint
            jt = np.zeros((3, J.shape[1]))
            jt[:, 6 * i: 6 * i + 3] = np.eye(3)
            J = np.vstack((J, np.dot(L_ax, jt)))

            # rotation constraint
            jr = np.zeros((4, J.shape[1]))
            jr[:, 6 * i + 3: 6 * i + 6] = m_rot(x=self.z[7 * i + 3: 7 * i + 7])
            J = np.vstack((J, np.dot(L_aq, jr)))

        # pairwise constraints
        for i in np.arange(self.N - 1):
            # translation constraint
            jt = np.zeros((3, J.shape[1]))
            dt = dqstq_t(q=self.z[7 * i + 3: 7 * i + 7])
            jt[:, 6 * i: 6 * i + 3] = -dt
            jt[:, 6 * (i + 1): 6 * (i + 1) + 3] = dt
            J = np.vstack((J, np.dot(L_rx, jt)))

            # rotation constraint
            jr = np.zeros((4, J.shape[1]))
            m = m_rot(x=self.z[7 * i + 3: 7 * i + 7])
            a = dpsq_p(q=self.z[7 * (i + 1) + 3: 7 * (i + 1) + 7])
            jr[:, 6 * i + 3: 6 * i + 6] = np.dot(a, m)
            m = m_rot(x=self.z[7 * (i + 1) + 3: 7 * (i + 1) + 7])
            b = dpsq_q(p=self.z[7 * i + 3: 7 * i + 7])
            jr[:, 6 * (i + 1) + 3: 6 * (i + 1) + 6] = np.dot(b, m)
            J = np.vstack((J, np.dot(L_rq, jr)))

        return J

    def residuals(self, poses, vos, L_ax, L_aq, L_rx, L_rq):
        """
        computes the residuals
        :param poses: N x 7
        :param vos: (N-1) x 7
        :param L_ax: 3 x 3
        :param L_aq: 4 x 4
        :param L_rx: 3 x 3
        :param L_rq: 4 x 4
        :return:
        """
        r = np.zeros((0, 1))

        # unary residuals
        L = np.zeros((7, 7))
        L[:3, :3] = L_ax
        L[3:, 3:] = L_aq
        for i in np.arange(self.N):
            rr = self.z[7 * i: 7 * (i + 1)] - np.reshape(poses[i], (-1, 1))
            r = np.vstack((r, np.dot(L, rr)))

        # pairwise residuals
        for i in np.arange(self.N - 1):
            # translation residual
            v = self.z[7 * (i + 1):7 * (i + 1) + 3, 0] - self.z[7 * i:7 * i + 3, 0]
            q = txq.qinverse(self.z[7 * i + 3:7 * i + 7, 0])
            rt = txq.rotate_vector(v, q)
            rt = rt[:, np.newaxis] - vos[i, :3].reshape((-1, 1))
            r = np.vstack((r, np.dot(L_rx, rt)))

            # rotation residual
            q0 = self.z[7 * i + 3: 7 * i + 7].squeeze()
            q1 = self.z[7 * (i + 1) + 3: 7 * (i + 1) + 7].squeeze()
            qvo = txq.qmult(txq.qinverse(q0), q1).reshape((-1, 1))
            rq = qvo - vos[i, 3:].reshape((-1, 1))
            r = np.vstack((r, np.dot(L_rq, rq)))

        return r

    def update_on_manifold(self, x):
        """
        Updates the state vector on manifold
        :param x: manifold increment, column vector
        :return:
        """
        for i in np.arange(self.N):
            # update translation
            t = x[6 * i: 6 * i + 3]
            self.z[7 * i: 7 * i + 3] += t

            # update rotation
            qm = x[6 * i + 3: 6 * i + 6]  # quaternion on the manifold
            dq = np.zeros(4)
            # method in Burgard paper
            # dq[1:] = qm.squeeze()
            # dq[0] = math.sqrt(1 - sum(np.square(qm)))  # incremental quaternion
            # method of exponential map
            n = np.linalg.norm(qm)
            dq[0] = math.cos(n)
            dq[1:] = np.sinc(n / np.pi) * qm.squeeze()
            q = self.z[7 * i + 3: 7 * i + 7].squeeze()
            q = txq.qmult(q, dq).reshape((-1, 1))
            self.z[7 * i + 3: 7 * i + 7] = q

    def optimize(self, poses, vos, sax=1, saq=1, srx=1, srq=1, n_iters=10):
        """
        run PGO, with init = poses
        :param poses:
        :param vos:
        :param sax: sigma for absolute translation
        :param saq: sigma for absolute rotation
        :param srx: sigma for relative translation
        :param srq: sigma for relative rotation
        :param n_iters:
        :return:
        """
        self.N = len(poses)
        # init state vector with the predicted poses
        self.z = np.reshape(poses.copy(), (-1, 1))

        # construct the information matrices
        L_ax = np.linalg.cholesky(np.eye(3) / sax)
        L_aq = np.linalg.cholesky(np.eye(4) / saq)
        L_rx = np.linalg.cholesky(np.eye(3) / srx)
        L_rq = np.linalg.cholesky(np.eye(4) / srq)

        for n_iter in np.arange(n_iters):
            J = self.jacobian(L_ax.T, L_aq.T, L_rx.T, L_rq.T)
            r = self.residuals(poses.copy(), vos.copy(), L_ax.T, L_aq.T, L_rx.T,
                               L_rq.T)
            H = np.dot(J.T, J)  # hessian
            b = np.dot(J.T, r)  # residuals

            # solve Hx = -b for x
            R = slin.cholesky(H)  # H = R' R
            y = slin.solve_triangular(R.T, -b)
            x = slin.solve_triangular(R, y)

            self.update_on_manifold(x)

        return self.z.reshape((-1, 7))


def print_poses(poses):
    for i in np.arange(poses.shape[0]):
        a = txe.mat2euler(txq.quat2mat(poses[i, 3:]))
        print([np.rad2deg(aa) for aa in a])
# --------------------------------------------------------------------------------------------------------------- #


def getError(posx, posq, actualx, actualq):
    """ Calculates the position error (euclidean distance) and the rotation error. """
    q1 = actualq / np.linalg.norm(actualq)
    q2 = posq / np.linalg.norm(posq)
    d = abs(np.sum(np.multiply(q1, q2)))
    theta = 2 * np.arccos(d) * 180 / np.pi
    errx = np.linalg.norm(actualx - posx)
    return errx, theta


def run_pgo():
    # read in the prediction and ground truth values for the absolute position and rotation
    with open(gt_abs_vals_file, "r") as f:
        gt_abs_vals = f.readlines()
    pred_poses = []
    position_gt = []
    rotation_gt = []
    gt_abs_vals_names = []
    for gav in gt_abs_vals:
        pred_poses.append([float(x.split(",")[0]) for x in gav.split("\n")[0].split(": ")[-7:]])
        position_gt.append([float(x.split(",")[0]) for x in gav.split("\n")[0].split(" Prediction: ")[0].split(": ")[-7:-4]])
        rotation_gt.append([float(x.split(",")[0]) for x in gav.split("\n")[0].split(" Prediction: ")[0].split(": ")[-4:]])
        gt_abs_vals_names.append(int(gav.split(" ")[1].split("/")[1].split("_")[2][:-5]))
    pred_poses = np.array(pred_poses)
    position_gt = np.array(position_gt)
    rotation_gt = np.array(rotation_gt)
    gt_abs_vals_names = np.array(gt_abs_vals_names)
    # for any positions outside of the area replace them with the position of the previous step
    for i in range(len(pred_poses)):
        for j in range(2):
            if pred_poses[i, j] < -1 or pred_poses[i, j] > 46:
                pred_poses[i, j] = pred_poses[i - 1, j]
        if pred_poses[i, 2] < -1 or pred_poses[i, 2] > 8:
            pred_poses[i, 2] = pred_poses[i - 1, 2]
    # read in the predicted relative translation/rotation
    with open(rel_vals_file, "r") as f:
        r_vals = f.readlines()
    r_vals_rel = []
    r_vals_names = []
    for rv in r_vals:
        r_vals_rel.append([float(x.split(",")[0]) for x in rv.split("\n")[0].split(": ")[-7:]])
        r_vals_names.append(int(rv.split(" ")[1].split("/")[1].split("_")[1][:-5]))
    r_vals_names = np.array(r_vals_names)
    r_vals_rel = np.array(r_vals_rel)
    removed_images = []
    removed_images_pt = []
    cur_ind = 0
    # makes sure that each absolute value has the correct corresponding relative value
    for xi, x in enumerate(gt_abs_vals_names):
        if cur_ind >= len(r_vals_names):
            removed_images.append(xi)
            continue
        while x > r_vals_names[cur_ind]:
            removed_images_pt.append(cur_ind)
            cur_ind += 1
            if cur_ind >= len(r_vals_names):
                break
        if cur_ind >= len(r_vals_names) or (x != r_vals_names[cur_ind]):
            removed_images.append(xi)
        else:
            cur_ind += 1
    valid_images = np.array(list(set(np.arange(len(gt_abs_vals_names))) - set(removed_images))).astype(int)
    valid_images_pt = np.array(list(set(np.arange(len(r_vals_names))) - set(removed_images_pt))).astype(int)
    print(len(position_gt), len(rotation_gt), len(pred_poses), len(valid_images), len(valid_images_pt))
    position_gt = position_gt[valid_images]
    rotation_gt = rotation_gt[valid_images]
    pred_poses = pred_poses[valid_images]
    vos = r_vals_rel[valid_images_pt]

    # --- correct rel xyz ---
    # rotate the relative orientation back to the world coordinate system (before it is based on the previous view)
    for i in range(len(pred_poses)):
        vos[i][:3] = Rot.from_quat(pred_poses[i][-4:]).as_matrix().T @ vos[i][:3]
    # -------
    # calculates the position/orientation error before applying pgo
    position_errors_base = []
    rotation_errors_base = []
    for i in range(len(position_gt)):
        pos_err, rot_err = getError(pred_poses[i, :3], pred_poses[i, 3:], position_gt[i], rotation_gt[i])
        position_errors_base.append(pos_err)
        rotation_errors_base.append(rot_err)
    position_errors_base = np.array(position_errors_base)
    rotation_errors_base = np.array(rotation_errors_base)
    # -------- overlapping PGO -------- #
    # applies pgo on overlapping batches of a above defined size
    start_poses = np.arange(0, len(pred_poses), int(base_lim*(1-overlapping_percent)))
    all_optimized_poses = np.zeros((len(valid_images), 7))
    for si, sp in enumerate(start_poses):
        print(si, sp)
        pgo = PoseGraph()
        optimized_poses = pgo.optimize(pred_poses[sp:sp+base_lim], vos[sp:sp+base_lim])
        if si == 0:
            all_optimized_poses[sp:sp+base_lim] = optimized_poses
        elif si == len(start_poses) - 1:
            if len(optimized_poses) > int(base_lim * overlapping_percent):
                all_optimized_poses[sp + int(base_lim * overlapping_percent):sp + len(optimized_poses)] = optimized_poses[
                                                                                                          int(
                                                                                                              base_lim * overlapping_percent):]
        else:
            all_optimized_poses[sp + int(base_lim * overlapping_percent):sp + base_lim] = optimized_poses[int(
                base_lim * overlapping_percent):]
    # --------------------------------- #
    # calculates the position/orientation error after applying pgo
    position_errors = []
    rotation_errors = []
    for i in range(len(all_optimized_poses)):
        pos_err, rot_err = getError(all_optimized_poses[i, :3], all_optimized_poses[i, 3:], position_gt[i], rotation_gt[i])
        position_errors.append(pos_err)
        rotation_errors.append(rot_err)
    position_errors = np.array(position_errors)
    rotation_errors = np.array(rotation_errors)
    all_errors_arr = np.vstack([position_errors_base, rotation_errors_base, position_errors, rotation_errors])
    np.save(save_base_path + save_base_path_ext + "_all_errors.npy", all_errors_arr)
    with open(save_base_path + save_base_path_ext + "_medians.txt", "w") as f:
        f.write("Before PGO: %.5f\t%.5f\n" % (np.nanmedian(position_errors_base), np.nanmedian(rotation_errors_base)))
        f.write("After PGO: %.5f\t%.5f\n" % (np.nanmedian(position_errors), np.nanmedian(rotation_errors)))
    print("Before PGO: %.5f\t%.5f" % (np.nanmedian(position_errors_base), np.nanmedian(rotation_errors_base)))
    print("After PGO: %.5f\t%.5f" % (np.nanmedian(position_errors), np.nanmedian(rotation_errors)))


if __name__ == '__main__':
    run_pgo()

	"""
	Copyright (C) 2018 NVIDIA Corporation. All rights reserved.
	Licensed under the CC BY-NC-SA 4.0 license (https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode).
	"""
	import numpy as np
	import scipy.linalg as slin
	import open3d as o3d
	import math
	import transforms3d.quaternions as txq
	import transforms3d.euler as txe
	from scipy.spatial.transform import Rotation as Rot
	# see for formulas:
	# https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-801-machine-vision-fall-2004/readings/quaternions.pdf
	# and "Quaternion and Rotation" - Yan-Bin Jia, September 18, 2016
	# from IPython.core.debugger import set_trace
	import argparse

	parser = argparse.ArgumentParser()
	parser.add_argument('--train', help='train number', required=True)
	parser.add_argument('--test', help='test_number', required=True)
	args = parser.parse_args()
	train_num = int(args.train)
	test_num = int(args.test)

	# runs pgo in batches of size base_lim until all time steps are handled. consecutive batches have an overlap
	base_lim = 100 # number of steps to simultaneously consider for pgo (low for faster runtime)
	overlapping_percent = 0.2 # percent of overlapping time steps to the previous pgo run
	# path to save the created values
	save_base_path = "..."
	# adds the test and train num to the save path (for better structure)
	save_base_path_ext = str(test_num) + "_" + str(train_num)

	# ------------ Rel vals ------------ #
	rel_vals_base_folder = "..." % train_num
	txt_file_exts = {1: ..., 2: ...} # One extension for every test case
	txt_file_ext = txt_file_exts[test_num]
	rel_vals_file = rel_vals_base_folder + txt_file_ext # file containing the predicted relative poses/orientation
	# ---------------------------------- #

	# ------------ GT + Abs vals ------------ #
	gt_abs_vals_base_folder = "..." % train_num
	gt_abs_txt_file_exts = {1: ..., 2: ...} # One extension for every test case
	gt_abs_txt_file_ext = gt_abs_txt_file_exts[test_num]
	# file containing the predicted absolute poses/orientation
	gt_abs_vals_file = gt_abs_vals_base_folder + gt_abs_txt_file_ext
	# ---------------------------------- #


	def getRelDiff(posx, posq, actualx, actualq):
	""" Calculates the relative difference in position and rotation. """
	rel_x = posx - actualx
	m1 = Rot.from_quat(posq).as_matrix()
	m2 = Rot.from_quat(actualq).as_matrix()
	m_rel = m1 @ m2.T
	#m_rel = m2.T @ m1
	rel_q = Rot.from_matrix(m_rel).as_quat()
	return rel_x, rel_q


	## NUMPY
	# see: https://github.com/NVlabs/geomapnet/blob/master/common/pose_utils.py ----------------------------- #
	def qlog(q):
	"""
	Applies logarithm map to q
	:param q: (4,)
	:return: (3,)
	"""
	if all(q[1:] == 0):
	q = np.zeros(3)
	else:
	q = np.arccos(q[0]) * q[1:] / np.linalg.norm(q[1:])
	return q


	def qexp(q):
	"""
	Applies the exponential map to q
	:param q: (3,)
	:return: (4,)
	"""
	n = np.linalg.norm(q)
	q = np.hstack((np.cos(n), np.sinc(n / np.pi) * q))
	return q


	def log_quaternion_angular_error(q1, q2):
	return quaternion_angular_error(qexp(q1), qexp(q2))


	def quaternion_angular_error(q1, q2):
	"""
	angular error between two quaternions
	:param q1: (4, )
	:param q2: (4, )
	:return:
	"""
	d = abs(np.dot(q1, q2))
	d = min(1.0, max(-1.0, d))
	theta = 2 * np.arccos(d) * 180 / np.pi
	return theta


	def skew(x):
	"""
	returns skew symmetric matrix from vector
	:param x: 3 x 1
	:return:
	"""
	s = np.asarray([[0, -x[2], x[1]], [x[2], 0, -x[0]], [-x[1], x[0], 0]])
	return s


	def dpq_q(p):
	"""
	returns the jacobian of quaternion product pq w.r.t. q
	:param p: 4 x 1
	:return: 4 x 4
	"""
	J = np.zeros((4, 4))
	J[0, 0] = p[0]
	J[0, 1:] = -p[1:].squeeze()
	J[1:, 0] = p[1:].squeeze()
	J[1:, 1:] = p[0] * np.eye(3) + skew(p[1:])
	return J


	def dpsq_q(p):
	"""
	returns the jacobian of quaternion product (p*)q w.r.t. q
	:param p: 4 x 1
	:return: 4 x 4
	"""
	J = np.zeros((4, 4))
	J[0, 0] = p[0]
	J[0, 1:] = -p[1:].squeeze()
	J[1:, 0] = -p[1:].squeeze()
	J[1:, 1:] = p[0] * np.eye(3) - skew(p[1:])
	return J


	def dpsq_p(q):
	"""
	returns the jacobian of quaternion product (p*)q w.r.t. p
	:param q: 4 x 1
	:return: 4 x 4
	"""
	J = np.zeros((4, 4))
	J[0, 0] = q[0]
	J[0, 1:] = q[1:].squeeze()
	J[1:, 0] = q[1:].squeeze()
	J[1:, 1:] = -q[0] * np.eye(3) + skew(q[1:])
	return J


	def dqstq_q(q, t):
	"""
	jacobian of q* t q w.r.t. q
	:param q: 4 x 1
	:param t: 3 x 1
	:return: 3 x 4
	"""
	J = np.zeros((3, 4))
	J[:, :1] = q[0] * t - np.cross(q[1:], t, axis=0)
	J[:, 1:] = -np.dot(t, q[1:].T) + np.dot(t.T, q[1:]) * np.eye(3) + \
	np.dot(q[1:], t.T) + q[0] * skew(t)
	J *= 2
	return J


	def dqstq_t(q):
	"""
	jacobian of q* t q w.r.t. t
	:param q: 4 x 1
	:return: 3 x 3
	"""
	J = (q[0] * q[0] - np.dot(q[1:].T, q[1:])) * np.eye(3) + 2 * np.dot(q[1:], q[1:].T) - \
	2 * q[0] * skew(q[1:])
	return J


	def m_rot(x):
	"""
	returns Jacobian of exponential map w.r.t. manifold increment
	:param x: part of state vector affected by increment, 4 x 1
	:return: 4 x 3
	"""
	# jacobian of full q wrt qm (quaternion update on manifold),
	# evaluated at qv = (0, 0, 0)
	# full q is derived using either the exponential map or q0 = sqrt(1-qm^2)
	jm = np.vstack((np.zeros((1, 3)), np.eye(3))) # 4 x 3
	m = np.dot(dpq_q(p=x), jm)
	return m


	class PoseGraph:
	def __init__(self):
	"""
	implements pose graph optimization from
	"Hybrid Hessians for Optimization of Pose Graphs" - Y. LeCun et al
	and "A Tutorial on Graph-Based SLAM" - W. Burgard et al
	"""
	self.N = 0
	self.z = np.zeros((0, 0))

	def jacobian(self, L_ax, L_aq, L_rx, L_rq):
	J = np.zeros((0, 6 * self.N)) # 6 because updates for rotation are on manifold

	# unary constraints
	for i in np.arange(self.N):
	# translation constraint
	jt = np.zeros((3, J.shape[1]))
	jt[:, 6 * i: 6 * i + 3] = np.eye(3)
	J = np.vstack((J, np.dot(L_ax, jt)))

	# rotation constraint
	jr = np.zeros((4, J.shape[1]))
	jr[:, 6 * i + 3: 6 * i + 6] = m_rot(x=self.z[7 * i + 3: 7 * i + 7])
	J = np.vstack((J, np.dot(L_aq, jr)))

	# pairwise constraints
	for i in np.arange(self.N - 1):
	# translation constraint
	jt = np.zeros((3, J.shape[1]))
	dt = dqstq_t(q=self.z[7 * i + 3: 7 * i + 7])
	jt[:, 6 * i: 6 * i + 3] = -dt
	jt[:, 6 * (i + 1): 6 * (i + 1) + 3] = dt
	J = np.vstack((J, np.dot(L_rx, jt)))

	# rotation constraint
	jr = np.zeros((4, J.shape[1]))
	m = m_rot(x=self.z[7 * i + 3: 7 * i + 7])
	a = dpsq_p(q=self.z[7 * (i + 1) + 3: 7 * (i + 1) + 7])
	jr[:, 6 * i + 3: 6 * i + 6] = np.dot(a, m)
	m = m_rot(x=self.z[7 * (i + 1) + 3: 7 * (i + 1) + 7])
	b = dpsq_q(p=self.z[7 * i + 3: 7 * i + 7])
	jr[:, 6 * (i + 1) + 3: 6 * (i + 1) + 6] = np.dot(b, m)
	J = np.vstack((J, np.dot(L_rq, jr)))

	return J

	def residuals(self, poses, vos, L_ax, L_aq, L_rx, L_rq):
	"""
	computes the residuals
	:param poses: N x 7
	:param vos: (N-1) x 7
	:param L_ax: 3 x 3
	:param L_aq: 4 x 4
	:param L_rx: 3 x 3
	:param L_rq: 4 x 4
	:return:
	"""
	r = np.zeros((0, 1))

	# unary residuals
	L = np.zeros((7, 7))
	L[:3, :3] = L_ax
	L[3:, 3:] = L_aq
	for i in np.arange(self.N):
	rr = self.z[7 * i: 7 * (i + 1)] - np.reshape(poses[i], (-1, 1))
	r = np.vstack((r, np.dot(L, rr)))

	# pairwise residuals
	for i in np.arange(self.N - 1):
	# translation residual
	v = self.z[7 * (i + 1):7 * (i + 1) + 3, 0] - self.z[7 * i:7 * i + 3, 0]
	q = txq.qinverse(self.z[7 * i + 3:7 * i + 7, 0])
	rt = txq.rotate_vector(v, q)
	rt = rt[:, np.newaxis] - vos[i, :3].reshape((-1, 1))
	r = np.vstack((r, np.dot(L_rx, rt)))

	# rotation residual
	q0 = self.z[7 * i + 3: 7 * i + 7].squeeze()
	q1 = self.z[7 * (i + 1) + 3: 7 * (i + 1) + 7].squeeze()
	qvo = txq.qmult(txq.qinverse(q0), q1).reshape((-1, 1))
	rq = qvo - vos[i, 3:].reshape((-1, 1))
	r = np.vstack((r, np.dot(L_rq, rq)))

	return r

	def update_on_manifold(self, x):
	"""
	Updates the state vector on manifold
	:param x: manifold increment, column vector
	:return:
	"""
	for i in np.arange(self.N):
	# update translation
	t = x[6 * i: 6 * i + 3]
	self.z[7 * i: 7 * i + 3] += t

	# update rotation
	qm = x[6 * i + 3: 6 * i + 6] # quaternion on the manifold
	dq = np.zeros(4)
	# method in Burgard paper
	# dq[1:] = qm.squeeze()
	# dq[0] = math.sqrt(1 - sum(np.square(qm))) # incremental quaternion
	# method of exponential map
	n = np.linalg.norm(qm)
	dq[0] = math.cos(n)
	dq[1:] = np.sinc(n / np.pi) * qm.squeeze()
	q = self.z[7 * i + 3: 7 * i + 7].squeeze()
	q = txq.qmult(q, dq).reshape((-1, 1))
	self.z[7 * i + 3: 7 * i + 7] = q

	def optimize(self, poses, vos, sax=1, saq=1, srx=1, srq=1, n_iters=10):
	"""
	run PGO, with init = poses
	:param poses:
	:param vos:
	:param sax: sigma for absolute translation
	:param saq: sigma for absolute rotation
	:param srx: sigma for relative translation
	:param srq: sigma for relative rotation
	:param n_iters:
	:return:
	"""
	self.N = len(poses)
	# init state vector with the predicted poses
	self.z = np.reshape(poses.copy(), (-1, 1))

	# construct the information matrices
	L_ax = np.linalg.cholesky(np.eye(3) / sax)
	L_aq = np.linalg.cholesky(np.eye(4) / saq)
	L_rx = np.linalg.cholesky(np.eye(3) / srx)
	L_rq = np.linalg.cholesky(np.eye(4) / srq)

	for n_iter in np.arange(n_iters):
	J = self.jacobian(L_ax.T, L_aq.T, L_rx.T, L_rq.T)
	r = self.residuals(poses.copy(), vos.copy(), L_ax.T, L_aq.T, L_rx.T,
	L_rq.T)
	H = np.dot(J.T, J) # hessian
	b = np.dot(J.T, r) # residuals

	# solve Hx = -b for x
	R = slin.cholesky(H) # H = R' R
	y = slin.solve_triangular(R.T, -b)
	x = slin.solve_triangular(R, y)

	self.update_on_manifold(x)

	return self.z.reshape((-1, 7))


	def print_poses(poses):
	for i in np.arange(poses.shape[0]):
	a = txe.mat2euler(txq.quat2mat(poses[i, 3:]))
	print([np.rad2deg(aa) for aa in a])
	# --------------------------------------------------------------------------------------------------------------- #


	def getError(posx, posq, actualx, actualq):
	""" Calculates the position error (euclidean distance) and the rotation error. """
	q1 = actualq / np.linalg.norm(actualq)
	q2 = posq / np.linalg.norm(posq)
	d = abs(np.sum(np.multiply(q1, q2)))
	theta = 2 * np.arccos(d) * 180 / np.pi
	errx = np.linalg.norm(actualx - posx)
	return errx, theta


	def run_pgo():
	# read in the prediction and ground truth values for the absolute position and rotation
	with open(gt_abs_vals_file, "r") as f:
	gt_abs_vals = f.readlines()
	pred_poses = []
	position_gt = []
	rotation_gt = []
	gt_abs_vals_names = []
	for gav in gt_abs_vals:
	pred_poses.append([float(x.split(",")[0]) for x in gav.split("\n")[0].split(": ")[-7:]])
	position_gt.append([float(x.split(",")[0]) for x in gav.split("\n")[0].split(" Prediction: ")[0].split(": ")[-7:-4]])
	rotation_gt.append([float(x.split(",")[0]) for x in gav.split("\n")[0].split(" Prediction: ")[0].split(": ")[-4:]])
	gt_abs_vals_names.append(int(gav.split(" ")[1].split("/")[1].split("_")[2][:-5]))
	pred_poses = np.array(pred_poses)
	position_gt = np.array(position_gt)
	rotation_gt = np.array(rotation_gt)
	gt_abs_vals_names = np.array(gt_abs_vals_names)
	# for any positions outside of the area replace them with the position of the previous step
	for i in range(len(pred_poses)):
	for j in range(2):
	if pred_poses[i, j] < -1 or pred_poses[i, j] > 46:
	pred_poses[i, j] = pred_poses[i - 1, j]
	if pred_poses[i, 2] < -1 or pred_poses[i, 2] > 8:
	pred_poses[i, 2] = pred_poses[i - 1, 2]
	# read in the predicted relative translation/rotation
	with open(rel_vals_file, "r") as f:
	r_vals = f.readlines()
	r_vals_rel = []
	r_vals_names = []
	for rv in r_vals:
	r_vals_rel.append([float(x.split(",")[0]) for x in rv.split("\n")[0].split(": ")[-7:]])
	r_vals_names.append(int(rv.split(" ")[1].split("/")[1].split("_")[1][:-5]))
	r_vals_names = np.array(r_vals_names)
	r_vals_rel = np.array(r_vals_rel)
	removed_images = []
	removed_images_pt = []
	cur_ind = 0
	# makes sure that each absolute value has the correct corresponding relative value
	for xi, x in enumerate(gt_abs_vals_names):
	if cur_ind >= len(r_vals_names):
	removed_images.append(xi)
	continue
	while x > r_vals_names[cur_ind]:
	removed_images_pt.append(cur_ind)
	cur_ind += 1
	if cur_ind >= len(r_vals_names):
	break
	if cur_ind >= len(r_vals_names) or (x != r_vals_names[cur_ind]):
	removed_images.append(xi)
	else:
	cur_ind += 1
	valid_images = np.array(list(set(np.arange(len(gt_abs_vals_names))) - set(removed_images))).astype(int)
	valid_images_pt = np.array(list(set(np.arange(len(r_vals_names))) - set(removed_images_pt))).astype(int)
	print(len(position_gt), len(rotation_gt), len(pred_poses), len(valid_images), len(valid_images_pt))
	position_gt = position_gt[valid_images]
	rotation_gt = rotation_gt[valid_images]
	pred_poses = pred_poses[valid_images]
	vos = r_vals_rel[valid_images_pt]

	# --- correct rel xyz ---
	# rotate the relative orientation back to the world coordinate system (before it is based on the previous view)
	for i in range(len(pred_poses)):
	vos[i][:3] = Rot.from_quat(pred_poses[i][-4:]).as_matrix().T @ vos[i][:3]
	# -------
	# calculates the position/orientation error before applying pgo
	position_errors_base = []
	rotation_errors_base = []
	for i in range(len(position_gt)):
	pos_err, rot_err = getError(pred_poses[i, :3], pred_poses[i, 3:], position_gt[i], rotation_gt[i])
	position_errors_base.append(pos_err)
	rotation_errors_base.append(rot_err)
	position_errors_base = np.array(position_errors_base)
	rotation_errors_base = np.array(rotation_errors_base)
	# -------- overlapping PGO -------- #
	# applies pgo on overlapping batches of a above defined size
	start_poses = np.arange(0, len(pred_poses), int(base_lim*(1-overlapping_percent)))
	all_optimized_poses = np.zeros((len(valid_images), 7))
	for si, sp in enumerate(start_poses):
	print(si, sp)
	pgo = PoseGraph()
	optimized_poses = pgo.optimize(pred_poses[sp:sp+base_lim], vos[sp:sp+base_lim])
	if si == 0:
	all_optimized_poses[sp:sp+base_lim] = optimized_poses
	elif si == len(start_poses) - 1:
	if len(optimized_poses) > int(base_lim * overlapping_percent):
	all_optimized_poses[sp + int(base_lim * overlapping_percent):sp + len(optimized_poses)] = optimized_poses[
	int(
	base_lim * overlapping_percent):]
	else:
	all_optimized_poses[sp + int(base_lim * overlapping_percent):sp + base_lim] = optimized_poses[int(
	base_lim * overlapping_percent):]
	# --------------------------------- #
	# calculates the position/orientation error after applying pgo
	position_errors = []
	rotation_errors = []
	for i in range(len(all_optimized_poses)):
	pos_err, rot_err = getError(all_optimized_poses[i, :3], all_optimized_poses[i, 3:], position_gt[i], rotation_gt[i])
	position_errors.append(pos_err)
	rotation_errors.append(rot_err)
	position_errors = np.array(position_errors)
	rotation_errors = np.array(rotation_errors)
	all_errors_arr = np.vstack([position_errors_base, rotation_errors_base, position_errors, rotation_errors])
	np.save(save_base_path + save_base_path_ext + "_all_errors.npy", all_errors_arr)
	with open(save_base_path + save_base_path_ext + "_medians.txt", "w") as f:
	f.write("Before PGO: %.5f\t%.5f\n" % (np.nanmedian(position_errors_base), np.nanmedian(rotation_errors_base)))
	f.write("After PGO: %.5f\t%.5f\n" % (np.nanmedian(position_errors), np.nanmedian(rotation_errors)))
	print("Before PGO: %.5f\t%.5f" % (np.nanmedian(position_errors_base), np.nanmedian(rotation_errors_base)))
	print("After PGO: %.5f\t%.5f" % (np.nanmedian(position_errors), np.nanmedian(rotation_errors)))


	if __name__ == '__main__':
	run_pgo()