Run in Google Colab# Copyright (c) Meta Platforms, Inc. and affiliates. All rights reserved.
This tutorial shows how to fit a volume given a set of views of a scene using differentiable volumetric rendering.
More specifically, this tutorial will explain how to:
Volumes class).Ensure torch and torchvision are installed. If pytorch3d is not installed, install it using the following cell:
import os
import sys
import torch
need_pytorch3d=False
try:
import pytorch3d
except ModuleNotFoundError:
need_pytorch3d=True
if need_pytorch3d:
if torch.__version__.startswith("2.2.") and sys.platform.startswith("linux"):
# We try to install PyTorch3D via a released wheel.
pyt_version_str=torch.__version__.split("+")[0].replace(".", "")
version_str="".join([
f"py3{sys.version_info.minor}_cu",
torch.version.cuda.replace(".",""),
f"_pyt{pyt_version_str}"
])
!pip install fvcore iopath
!pip install --no-index --no-cache-dir pytorch3d -f https://dl.fbaipublicfiles.com/pytorch3d/packaging/wheels/{version_str}/download.html
else:
# We try to install PyTorch3D from source.
!pip install 'git+https://github.com/facebookresearch/pytorch3d.git@stable'
import os
import sys
import time
import json
import glob
import torch
import math
from tqdm.notebook import tqdm
import matplotlib.pyplot as plt
import numpy as np
from PIL import Image
from IPython import display
# Data structures and functions for rendering
from pytorch3d.structures import Volumes
from pytorch3d.renderer import (
FoVPerspectiveCameras,
VolumeRenderer,
NDCMultinomialRaysampler,
EmissionAbsorptionRaymarcher
)
from pytorch3d.transforms import so3_exp_map
# obtain the utilized device
if torch.cuda.is_available():
device = torch.device("cuda:0")
torch.cuda.set_device(device)
else:
device = torch.device("cpu")
!wget https://raw.githubusercontent.com/facebookresearch/pytorch3d/main/docs/tutorials/utils/plot_image_grid.py
!wget https://raw.githubusercontent.com/facebookresearch/pytorch3d/main/docs/tutorials/utils/generate_cow_renders.py
from plot_image_grid import image_grid
from generate_cow_renders import generate_cow_renders
OR if running locally uncomment and run the following cell:
# from utils.generate_cow_renders import generate_cow_renders
# from utils import image_grid
The following cell generates our training data.
It renders the cow mesh from the fit_textured_mesh.ipynb tutorial from several viewpoints and returns:
Note: For the purpose of this tutorial, which aims at explaining the details of volumetric rendering, we do not explain how the mesh rendering, implemented in the generate_cow_renders function, works. Please refer to fit_textured_mesh.ipynb for a detailed explanation of mesh rendering.
target_cameras, target_images, target_silhouettes = generate_cow_renders(num_views=40)
print(f'Generated {len(target_images)} images/silhouettes/cameras.')
The following initializes a volumetric renderer that emits a ray from each pixel of a target image and samples a set of uniformly-spaced points along the ray. At each ray-point, the corresponding density and color value is obtained by querying the corresponding location in the volumetric model of the scene (the model is described & instantiated in a later cell).
The renderer is composed of a raymarcher and a raysampler.
NDCMultinomialRaysampler which follows the standard PyTorch3D coordinate grid convention (+X from right to left; +Y from bottom to top; +Z away from the user).EmissionAbsorptionRaymarcher which implements the standard Emission-Absorption raymarching algorithm.# render_size describes the size of both sides of the
# rendered images in pixels. We set this to the same size
# as the target images. I.e. we render at the same
# size as the ground truth images.
render_size = target_images.shape[1]
# Our rendered scene is centered around (0,0,0)
# and is enclosed inside a bounding box
# whose side is roughly equal to 3.0 (world units).
volume_extent_world = 3.0
# 1) Instantiate the raysampler.
# Here, NDCMultinomialRaysampler generates a rectangular image
# grid of rays whose coordinates follow the PyTorch3D
# coordinate conventions.
# Since we use a volume of size 128^3, we sample n_pts_per_ray=150,
# which roughly corresponds to a one ray-point per voxel.
# We further set the min_depth=0.1 since there is no surface within
# 0.1 units of any camera plane.
raysampler = NDCMultinomialRaysampler(
image_width=render_size,
image_height=render_size,
n_pts_per_ray=150,
min_depth=0.1,
max_depth=volume_extent_world,
)
# 2) Instantiate the raymarcher.
# Here, we use the standard EmissionAbsorptionRaymarcher
# which marches along each ray in order to render
# each ray into a single 3D color vector
# and an opacity scalar.
raymarcher = EmissionAbsorptionRaymarcher()
# Finally, instantiate the volumetric render
# with the raysampler and raymarcher objects.
renderer = VolumeRenderer(
raysampler=raysampler, raymarcher=raymarcher,
)
Next we instantiate a volumetric model of the scene. This quantizes the 3D space to cubical voxels, where each voxel is described with a 3D vector representing the voxel's RGB color and a density scalar which describes the opacity of the voxel (ranging between [0-1], the higher the more opaque).
In order to ensure the range of densities and colors is between [0-1], we represent both volume colors and densities in the logarithmic space. During the forward function of the model, the log-space values are passed through the sigmoid function to bring the log-space values to the correct range.
Additionally, VolumeModel contains the renderer object. This object stays unaltered throughout the optimization.
In this cell we also define the huber loss function which computes the discrepancy between the rendered colors and masks.
class VolumeModel(torch.nn.Module):
def __init__(self, renderer, volume_size=[64] * 3, voxel_size=0.1):
super().__init__()
# After evaluating torch.sigmoid(self.log_colors), we get
# densities close to zero.
self.log_densities = torch.nn.Parameter(-4.0 * torch.ones(1, *volume_size))
# After evaluating torch.sigmoid(self.log_colors), we get
# a neutral gray color everywhere.
self.log_colors = torch.nn.Parameter(torch.zeros(3, *volume_size))
self._voxel_size = voxel_size
# Store the renderer module as well.
self._renderer = renderer
def forward(self, cameras):
batch_size = cameras.R.shape[0]
# Convert the log-space values to the densities/colors
densities = torch.sigmoid(self.log_densities)
colors = torch.sigmoid(self.log_colors)
# Instantiate the Volumes object, making sure
# the densities and colors are correctly
# expanded batch_size-times.
volumes = Volumes(
densities = densities[None].expand(
batch_size, *self.log_densities.shape),
features = colors[None].expand(
batch_size, *self.log_colors.shape),
voxel_size=self._voxel_size,
)
# Given cameras and volumes, run the renderer
# and return only the first output value
# (the 2nd output is a representation of the sampled
# rays which can be omitted for our purpose).
return self._renderer(cameras=cameras, volumes=volumes)[0]
# A helper function for evaluating the smooth L1 (huber) loss
# between the rendered silhouettes and colors.
def huber(x, y, scaling=0.1):
diff_sq = (x - y) ** 2
loss = ((1 + diff_sq / (scaling**2)).clamp(1e-4).sqrt() - 1) * float(scaling)
return loss
Here we carry out the volume fitting with differentiable rendering.
In order to fit the volume, we render it from the viewpoints of the target_cameras
and compare the resulting renders with the observed target_images and target_silhouettes.
The comparison is done by evaluating the mean huber (smooth-l1) error between corresponding
pairs of target_images/rendered_images and target_silhouettes/rendered_silhouettes.
# First move all relevant variables to the correct device.
target_cameras = target_cameras.to(device)
target_images = target_images.to(device)
target_silhouettes = target_silhouettes.to(device)
# Instantiate the volumetric model.
# We use a cubical volume with the size of
# one side = 128. The size of each voxel of the volume
# is set to volume_extent_world / volume_size s.t. the
# volume represents the space enclosed in a 3D bounding box
# centered at (0, 0, 0) with the size of each side equal to 3.
volume_size = 128
volume_model = VolumeModel(
renderer,
volume_size=[volume_size] * 3,
voxel_size = volume_extent_world / volume_size,
).to(device)
# Instantiate the Adam optimizer. We set its master learning rate to 0.1.
lr = 0.1
optimizer = torch.optim.Adam(volume_model.parameters(), lr=lr)
# We do 300 Adam iterations and sample 10 random images in each minibatch.
batch_size = 10
n_iter = 300
for iteration in range(n_iter):
# In case we reached the last 75% of iterations,
# decrease the learning rate of the optimizer 10-fold.
if iteration == round(n_iter * 0.75):
print('Decreasing LR 10-fold ...')
optimizer = torch.optim.Adam(
volume_model.parameters(), lr=lr * 0.1
)
# Zero the optimizer gradient.
optimizer.zero_grad()
# Sample random batch indices.
batch_idx = torch.randperm(len(target_cameras))[:batch_size]
# Sample the minibatch of cameras.
batch_cameras = FoVPerspectiveCameras(
R = target_cameras.R[batch_idx],
T = target_cameras.T[batch_idx],
znear = target_cameras.znear[batch_idx],
zfar = target_cameras.zfar[batch_idx],
aspect_ratio = target_cameras.aspect_ratio[batch_idx],
fov = target_cameras.fov[batch_idx],
device = device,
)
# Evaluate the volumetric model.
rendered_images, rendered_silhouettes = volume_model(
batch_cameras
).split([3, 1], dim=-1)
# Compute the silhouette error as the mean huber
# loss between the predicted masks and the
# target silhouettes.
sil_err = huber(
rendered_silhouettes[..., 0], target_silhouettes[batch_idx],
).abs().mean()
# Compute the color error as the mean huber
# loss between the rendered colors and the
# target ground truth images.
color_err = huber(
rendered_images, target_images[batch_idx],
).abs().mean()
# The optimization loss is a simple
# sum of the color and silhouette errors.
loss = color_err + sil_err
# Print the current values of the losses.
if iteration % 10 == 0:
print(
f'Iteration {iteration:05d}:'
+ f' color_err = {float(color_err):1.2e}'
+ f' mask_err = {float(sil_err):1.2e}'
)
# Take the optimization step.
loss.backward()
optimizer.step()
# Visualize the renders every 40 iterations.
if iteration % 40 == 0:
# Visualize only a single randomly selected element of the batch.
im_show_idx = int(torch.randint(low=0, high=batch_size, size=(1,)))
fig, ax = plt.subplots(2, 2, figsize=(10, 10))
ax = ax.ravel()
clamp_and_detach = lambda x: x.clamp(0.0, 1.0).cpu().detach().numpy()
ax[0].imshow(clamp_and_detach(rendered_images[im_show_idx]))
ax[1].imshow(clamp_and_detach(target_images[batch_idx[im_show_idx], ..., :3]))
ax[2].imshow(clamp_and_detach(rendered_silhouettes[im_show_idx, ..., 0]))
ax[3].imshow(clamp_and_detach(target_silhouettes[batch_idx[im_show_idx]]))
for ax_, title_ in zip(
ax,
("rendered image", "target image", "rendered silhouette", "target silhouette")
):
ax_.grid("off")
ax_.axis("off")
ax_.set_title(title_)
fig.canvas.draw(); fig.show()
display.clear_output(wait=True)
display.display(fig)
Finally, we visualize the optimized volume by rendering from multiple viewpoints that rotate around the volume's y-axis.
def generate_rotating_volume(volume_model, n_frames = 50):
logRs = torch.zeros(n_frames, 3, device=device)
logRs[:, 1] = torch.linspace(0.0, 2.0 * 3.14, n_frames, device=device)
Rs = so3_exp_map(logRs)
Ts = torch.zeros(n_frames, 3, device=device)
Ts[:, 2] = 2.7
frames = []
print('Generating rotating volume ...')
for R, T in zip(tqdm(Rs), Ts):
camera = FoVPerspectiveCameras(
R=R[None],
T=T[None],
znear = target_cameras.znear[0],
zfar = target_cameras.zfar[0],
aspect_ratio = target_cameras.aspect_ratio[0],
fov = target_cameras.fov[0],
device=device,
)
frames.append(volume_model(camera)[..., :3].clamp(0.0, 1.0))
return torch.cat(frames)
with torch.no_grad():
rotating_volume_frames = generate_rotating_volume(volume_model, n_frames=7*4)
image_grid(rotating_volume_frames.clamp(0., 1.).cpu().numpy(), rows=4, cols=7, rgb=True, fill=True)
plt.show()
In this tutorial, we have shown how to optimize a 3D volumetric representation of a scene such that the renders of the volume from known viewpoints match the observed images for each viewpoint. The rendering was carried out using the PyTorch3D's volumetric renderer composed of an NDCMultinomialRaysampler and an EmissionAbsorptionRaymarcher.