Objects as volumes: A stochastic geometry view of opaque solids

Project website

We explore the design space of attenuation coefficients for volumetric representations of opaque solids and demonstrate improvements from our theory in proof-of-concept surface reconstruction experiments. Primarily, we compare the parameters controlling the attenuation coefficient such as the implicit function distribution and the normal distribution. When applicable to the normal distirbution, we allow an anisotropy parameter to either be annealed on a fixed schedule or learned spatially. For the implementation of these distributions see the attenuation coefficient model.

Note

This code was designed to evaluate different attenuation coefficient formulations on equal terms (i.e., equal quadrature and other ancillary settings), rather than for optimal reconstruction performance. See Section 5 of the paper for details. We recommend checking out Chen et al. 2024 (project page / code) for a follow-up that uses our theory on attenuation coefficient and is optimized for performance.

Configuration Options

The main parameters for our experiments are defined below. Across methods we mainly vary the implicit distribution and normal distribution. Refer to an example configuration for full set of training options.

point_sampler {
  n_sdf_pts = 1024         # num evaluation points to find 0-level set intersection
  n_fg_samples = 21        # num samples prior to intersection interval
  n_surf_samples = 22      # num samples inside intersection interval
  n_bg_samples = 21        # num samples behind intersection interval
}

attenuation_coefficient {
  implicit_distribution = gaussian     # e.g. Laplace, loogistic, or Gaussian
  normal_distribution = linearmixture  # e.g. delta, uniform, or linear mixture
}

For normal distributions that take an anisotropy or mixture parameter, we provide additional control over whether the parameter is annealed or learned spatially.

train {
  anneal_end = 0          # anisotropy decays to 0 by anneal_end iters
}

# overrides annealed anisotropy
anisotropy_network {
  d_feature = 256         # dim of feature from SDF net
}

Finally, based on the dataset we use a different background. The DTU dataset uses a black background, the NeRF dataset uses a white background, and blended MVS learns a background color using NeRF++.

# controls constant background color (black or white)
train {
  use_white_bkgd = False
}

# if outside samples > 0 overrides constant background color with learned background
point_sampler {
  n_outside = 32          # num samples used for background net
}

Datasets

The following datasets can be readily ingested by our training pipeline. (See below for the data convention.)

• DTU Dataset

• Blended MVS Dataset

• NeRF Dataset

Data Convention

The data is organized as follows:

exp_runner.py
public_data
|-- <case_name>
    |-- cameras_xxx.npz    # camera parameters
    |-- image
        |-- 000.png        # target image for each view
        |-- 001.png
        ...
    |-- mask
        |-- 000.png        # target mask each view (For unmasked setting, set all pixels as 255)
        |-- 001.png
        ...

Here the cameras_xxx.npz follows the data format in IDR, where world_mat_xx denotes the world to image projection matrix, and scale_mat_xx denotes the normalization matrix.

Usage

Setup

cd volumetric_opaque_solids
pip install -r requirements.txt

Dependencies (click to expand)

trimesh==3.9.8 numpy==1.26.2 pyhocon==0.3.57 opencv_python==4.8.1.78 tqdm==4.50.2 torch==1.13.0 scipy==1.11.3 PyMCubes==0.1.2 tensorboard

Running

• Training

python exp_runner.py \
  --conf ./confs/<config name>.conf \
  --case <case_name> \
  --mode train

• Extract surface from trained model

python exp_runner.py \
  --conf ./confs/<config name>.conf \
  --case <case_name> \
  --mode validate_mesh \
  --is_continue # use latest checkpoint

The corresponding mesh can be found in exp/<case_name>/<exp_name>/meshes/<iter_steps>.ply.

• Render Image

python exp_runner.py \
  --conf ./confs/<config name>.conf \
  --case <case_name> \
  --mode render \
  --image_idx 0 \ # image index
  --is_continue   # use latest checkpoint

Acknowledgement

This codebase is simplified adaption of NeuS; the original codebase makes use of code snippets borrowed from IDR and NeRF-pytorch. Thanks for all of these great projects.