From e0c07ee5f947d940f4ad2b6f50419cee9641ecc4 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marcus=20Valtonen=20=C3=96rnhag?= Date: Mon, 12 Sep 2022 09:40:48 +0200 Subject: [PATCH 1/2] Tutorial: Non-Rigid Structure-from-Motion --- docs/source/conf.py | 4 +- tutorials/nrsfm.py | 323 ++++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 326 insertions(+), 1 deletion(-) create mode 100644 tutorials/nrsfm.py diff --git a/docs/source/conf.py b/docs/source/conf.py index 6a25d27..83312dd 100644 --- a/docs/source/conf.py +++ b/docs/source/conf.py @@ -66,7 +66,9 @@ # Modules for which function level galleries are created. 'doc_module': 'pyproximal', # Insert links to documentation of objects in the examples - 'reference_url': {'pyproximal': None} + 'reference_url': {'pyproximal': None}, + # Allow animations + 'matplotlib_animations': True, } # Always show the source code that generates a plot diff --git a/tutorials/nrsfm.py b/tutorials/nrsfm.py new file mode 100644 index 0000000..d36fb26 --- /dev/null +++ b/tutorials/nrsfm.py @@ -0,0 +1,323 @@ +r""" +Non-rigid structure-from-motion (NRSfM) +======================================= +In computer vision, structure-from-motion (SfM) is an imaging technique for +estimating three-dimensional structures from two-dimensional images. Theoretically, +the problem is generally well-posed when considering rigid objects, meaning +that the objects do not move or deform in the scene. However, non-static scenes +are still relevant and have gained increased popularity among researchers in +recent years. This is known as *non-rigid structure-from-motion*. + +In this tutorial, we will consider motion capture (MOCAP). This is a special +case, where we use images from multiple camera views to compute the 3D positions +of specifically designed markers that track the motion of a person (or object) +performing various tasks. + +Non-rigid shapes +**************** +To make the problem well-posed, one has to control the complexity of +the deformations using some minor assumptions on the possible space of object +shapes. This is not a weird thing to do: consider e.g. the human body, +we have different joints that bend and turn in a finite amount of ways; the +skeleton itself is rigid and not capable of such deformations. For this reason, +Bregler et al. [1]_ suggested that all movements (or shapes) can be represented +by a low-dimensional basis. In the context of motion capture, this means that +every movement a person does can be considered a combination of core movements +(or basis shapes). + +Mathematically speaking, this translates to any motion being a linear combination +of the basis shapes, i.e. assuming there are :math:`K` basis shapes, any non-rigid +shape :math:`X_i` can be written as + + .. math:: + X_i = \sum_{i=1}^K c_{ik}B_k + +where :math:`c_{ik}` are the basis coefficients and :math:`B_k` are the basis shapes. +Here, :math:`X_i` is a :math:`3\times N` matrix where each column is a point in +3D space. + +The CMU MOCAP dataset +********************* +Let us first try to understand the data we are given. We will use the *Pickup* +instance from the CMU MOCAP dataset, which depicts a person picking something +up from the floor. +""" + +import matplotlib.pyplot as plt +import matplotlib.animation as animation +import numpy as np +import scipy as sp + + +plt.close('all') +np.random.seed(0) +data = np.load('../testdata/mocap.npz', allow_pickle=True) +X_gt = data['X_gt'] +markers = data['markers'].item() + +############################################################################### +# First we view the first 3D poses. In order to easily visualize the person, we +# draw a skeleton between the markers corresponding to certain body parts. Note +# that these are not used in any other way. + + +def plot_first_3d_pose(ax, X, color='b', marker='o', linecolor='k'): + ax.scatter(X[0, :], X[1, :], X[2, :], color, marker=marker) + for j, ind in enumerate(markers.values()): + ax.plot(X[0, ind], X[1, ind], X[2, ind], '-', color=linecolor) + ax.set_box_aspect(np.ptp(X[:3, :], axis=1)) + ax.view_init(20, 25) + + +fig = plt.figure() +ax = fig.add_subplot(projection='3d') +plot_first_3d_pose(ax, X_gt) +plt.tight_layout() + +############################################################################### +# Now, we turn the attention to the data the algorithm is given, which is a +# sequence of 2D images from varying views. The goal is to recreate the 3D +# points, such as in the example above, from all timestamps. + +M = data['M'] +F = int(X_gt.shape[0] / 3) + + +def _update(f: int): + X = M[2 * f:2 * f + 2, :] + lines[0].set_data(X[0, :], X[1, :]) + for j, ind in enumerate(markers.values()): + lines[j + 1].set_data(X[0, ind], X[1, ind]) + return lines + + +fig, ax = plt.subplots() +lines = ax.plot([], [], 'r.') +for _ in range(len(markers)): + lines.append(ax.plot([], [], 'k-')[0]) +ax.set(xlim=(-2.5, 2.5), ylim=(-3.5, 3.5)) +ax.set_aspect('equal') + +ani = animation.FuncAnimation(fig, _update, F, interval=25, blit=True) + + +############################################################################### +# Note that these are the 2D image correspondences and that the image view is +# constantly changing as it is spinning around. Such motion can be modeled +# with orthographic cameras, which we will discuss next. +# +# Orthographic projections +# ************************ +# Assuming that we know the pose of the camera from which the image was taken +# the corresponding 2D image point can be obtained. In the case of rotations, +# we assume the cameras are orthographic, meaning that the image points :math:`x_i` +# are obtained from the relation +# +# .. math:: +# x_i = R_iX_i +# +# where :math:`R_i` is a :math:`2\times 3` matrix fulfilling :math:`R_iR_i^T=I`. +# Essentially, the :math:`R_i` matrices consists of the top two rows of the +# corresponding rotation matrix. +# +# Now, the task at hand is essentially the inverse problem: to reconstruct the +# 3D points for each point in time from these 2D images. +# +# Treating NRSfM as a low-rank factorization problem +# ************************************************** +# One of the main novelties of the paper by Dai et al. [2]_ is to reshape and +# stack the non-rigid shapes :math:`X_i` in a way that allows us to treat the +# problem using methods from low-rank factorization. This is done in the following +# way: first concatenate all rows of :math:`X_i` creating a vector +# :math:`X^\sharp_i` of size :math:`1\times 3N`. Secondly, assuming there are in +# total :math:`F` non-rigid shapes we create the matrix :math:`X^\sharp` of size +# :math:`F\times 3N` by stacking all :math:`X^\sharp_i`. This enables us to decompose +# the newly created matrix :math:`X^\sharp=CB^\sharp` in the low-rank factors +# consisting of the shape coefficients :math:`C` and the basis shapes :math:`B^\sharp` +# (constructed in the same way as :math:`X^\sharp`). +# +# Now, let us implement and visualize this approach. + + +def stack(X: np.ndarray): + return np.hstack((X[::3, :], X[1::3, :], X[2::3, :])) + + +Xi = np.arange(15).reshape((3, 5)) +fig, ax = plt.subplots(1, 2) +ax[0].matshow(Xi) +ax[0].set_title(r'$X_i$') +ax[0].axis('off') +ax[1].matshow(stack(Xi)) +ax[1].set_title(r'$X_i^\sharp$') +ax[1].axis('off') +plt.tight_layout() + +############################################################################### +# Furthermore, we introduce the inverse +# operation, such that :math:`X=\mathcal{U}(X^\sharp)`, where +# :math:`\mathcal{U}` is the "unstacking" operator. + + +def unstack(Xs: np.ndarray): + """Inverse operation of stack.""" + m, n = Xs.shape + m *= 3 + n //= 3 + X = np.zeros((m, n), dtype=Xs.dtype) + X[::3] = Xs[:, :n] + X[1::3] = Xs[:, n:2*n] + X[2::3] = Xs[:, 2*n:3*n] + return X + +############################################################################### +# In many cases, the necessary amount of basis shapes is not known +# *a priori*. Therefore, a suitable objective to try to minimize is +# +# .. math:: +# \argmin_X \mu \rank(X^\sharp) + \frac{1}{2}\sum_{i=1}^F\|R_iX_i - x_i\|_2^2 +# +# or, equivalently, +# +# .. math:: +# \argmin_X \mu \rank(X^\sharp) + \frac{1}{2}\|RX - M\|_F^2 +# +# where :math:`R` is a block-diagonal matrix with :math:`R_i` on the main diagonal, +# whereas :math:`X` and :math:`M` are the concatenations of :math:`X_i` and +# :math:`x_i`, respectively. +# +# Since the rank function is non-convex and discontinuous, it is often replaced +# by a relaxation. In [2]_ the *nuclear norm* :math:`\|\cdot\|_{*}` was used, i.e. +# we seek to minimize +# +# .. math:: +# \argmin_X \mu \|X^\sharp\|_{*} + \frac{1}{2}\|RX - M\|_F^2 \; . +# +# There are some theoretical justifications for this specific choice of relaxation, +# e.g. the nuclear norm is the convex envelope of the rank function under +# curtain assumptions [3]_. +# +# Solving the relaxed problem +# *************************** +# We will now show how to solve this problem using splitting schemes. Specifically, +# we will use :class:`pyproximal.ADMM` and re-write the objective as +# +# .. math:: +# \argmin_{X, Z} \mu \|Z^\sharp\|_{*} + \frac{1}{2}\|RX - M\|_F^2, +# +# and add the constraint :math:`X=Z`. This way, the proximal operators +# are simply those of the (stacked) nuclear norm and the Frobenius norm. +# We implement these next: + +from pyproximal.ProxOperator import _check_tau +from pyproximal import Nuclear, ProxOperator + + +class BlockDiagFrobenius(ProxOperator): + r"""Proximal operator for 1/2 * ||RX - M||_F^2 where R is block-diagonal. + Note: You could also wrap pyproximal.L2, but this class is used here in + this tutorial to increase legibility. + """ + def __init__(self, dim, R, M): + super().__init__(None, False) + self.dim = dim + self.R = R + self.M = M + + def __call__(self, x): + X = x.reshape(self.dim) + return 0.5 * np.linalg.norm(self.R @ X - self.M, 'fro') ** 2 + + @_check_tau + def prox(self, x, tau): + X = x.reshape(self.dim) + Y = np.zeros_like(X) + for f, Rf in enumerate(self.R): + Y[3 * f: 3 * f + 3, :] = np.linalg.solve( + tau * Rf.T @ Rf + np.eye(3), + tau * Rf.T @ self.M[2 * f:2 * f + 2, :] + X[3 * f: 3 * f + 3, :] + ) + return Y.flatten() + + +class StackedNuclear(Nuclear): + r"""Proximal operator for the stacked nuclear norm.""" + def __init__(self, dim, sigma=1.): + super().__init__(dim, sigma) + self.unstacked_dim = (dim[0] * 3, dim[1] // 3) + + def __call__(self, x): + X = stack(x.reshape(self.unstacked_dim)) + return super().__call__(X.ravel()) + + def prox(self, x, tau): + X = stack(x.reshape(self.unstacked_dim)) + x = super().prox(X.ravel(), tau) + X = unstack(x.reshape(self.dim)) + return X.ravel() + + +############################################################################### +# Now we are ready to solve the problem using ADMM. + +from pyproximal.optimization.primal import ADMM + + +mu = 1 +R = data['R'] +Rblk = sp.linalg.block_diag(*R) +M = Rblk @ X_gt +N = M.shape[1] +normal_size = (3*F, N) +stacked_size = (F, 3*N) +X0 = np.zeros(normal_size) +proxf = BlockDiagFrobenius(normal_size, R, M) +proxg = StackedNuclear(stacked_size, mu) +X_rec = ADMM(proxf, proxg, x0=X0.flatten(), tau=0.9, niter=200)[0] +X_rec = X_rec.reshape(normal_size) + +############################################################################### +# Let us compare the results with the ground truth data. +fig = plt.figure() +ax = fig.add_subplot(projection='3d') +plot_first_3d_pose(ax, X_gt) +plot_first_3d_pose(ax, X_rec, color='r', marker='v', linecolor='r') +plt.tight_layout() + +############################################################################### +# Furthermore, we compute some statistics on the reconstruction performance. You +# can vary the regulation strength :math:`\mu` to see if you can achieve better +# performance yourself! +print(f'Datafit: {np.linalg.norm(Rblk @ X_rec - M, "fro")}') +print(f'Distance to GT: {np.linalg.norm(X_rec - X_gt, "fro")}') + +############################################################################### +# One issue with the nuclear norm is that you have the hyperparameter +# :math:`\mu` that regulates the impact of the datafit vs. model assumption. +# When :math:`\mu=0` the datafit is perfect, but there would be no enforcement +# of the basis shapes, leading to severe over-fitting. On the other end, as +# :math:`\mu\to\infty` the reconstruction turns towards the zero matrix, thus +# completely ignoring the given data. +# +# Later publications have tried to mitigate this issue, and have shown that +# the weighted nuclear norm performs even better when the weights are selected +# with care [4]_. Other non-convex relaxations show similar results [5]_. +# +# **References** +# +# .. [1] C. Bregler, A. Hertzmann, and H. Biermann. Recovering non-rigid 3d shape +# from image streams. In The IEEE Conference on Computer Vision and Pattern +# Recognition (CVPR), 2000. +# .. [2] Y. Dai, H. Li, and M. He. A simple prior-free method for non-rigid +# structure-from-motion factorization. International Journal of Computer Vision, +# 107(2):101–122, 2014. +# .. [3] M. Fazel, H. Hindi, and S. P. Boyd. A rank minimization heuristic with +# application to minimum order system approximation. In the Proceedings of the +# American Control Conference (ACC), 2001. +# .. [4] S. Kumar. Non-rigid structure from motion: Prior-free factorization method +# revisited. In The IEEE Winter Conference on Applications of Computer Vision +# (WACV), 2020. +# .. [5] M. Valtonen Örnhag and C. Olsson. A unified optimization framework for +# low-rank inducing penalties. 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