Parametric Gauss Reconstruction (PGR)

This repository contains the implementation of the paper:

Surface Reconstruction from Point Clouds without Normals by Parametrizing the Gauss Formula (ACMTOG 2022)

Siyou Lin, Dong Xiao, Zuoqiang Shi, Bin Wang

Project page | Slides (SIGGRAPH 2023)

Parametric Gauss Reconstruction (PGR) takes an unoriented point cloud as input and solves an equation where the normals are the unknowns. The equation is written with the Gauss formula from potential theory, which can be established using only the coordinates of surface point samples. Upon obtaining the solution, PGR further computes an implicit indicator field for extracting the mesh surface.

Instructions on Running the Program

Environment and Dependencies

This program has been tested on Ubuntu 20.04 with Python 3.8 and CUDA 11.1. This program requires these Python packages: numpy, cupy, scipy and tqdm.

Compiling Source Files

This program uses third-party libraries CLI11 and cnpy. We provide copies of them in this repository for convenicence. If you have cmake, you can build with:

mkdir build
cd build
cmake ..
make -j8

Or you can also build with g++ directly:

cd src
g++ PGRExportQuery.cpp Cube.cpp Geometry.cpp MarchingCubes.cpp Mesh.cpp Octnode.cpp Octree.cpp ply.cpp plyfile.cpp cnpy/cnpy.cpp -ICLI11 -o ../apps/PGRExportQuery -lz -O2
g++ PGRLoadQuery.cpp Cube.cpp Geometry.cpp MarchingCubes.cpp Mesh.cpp Octnode.cpp Octree.cpp ply.cpp plyfile.cpp cnpy/cnpy.cpp -ICLI11 -o ../apps/PGRLoadQuery -lz -O2

If successful, this will generate two executables PGRExportQuery and PGRLoadQuery in ParametricGaussRecon/apps. The former builds an octree and exports grid corner points for query; the latter loads query values solved by PGR and performs iso-surfacing.

Run the All-in-one Script

We provide a script run_pgr.py to run the complete pipeline. Usage:

python run_pgr.py point_cloud.xyz [-wk WIDTH_K] [-wmax WIDTH_MAX] [-wmin WIDTH_MIN]
                  [-a ALPHA] [-m MAX_ITERS] [-d MAX_DEPTH] [-md MIN_DEPTH]
                  [--cpu] [--save_r]

Note that point_cloud.xyz should NOT contain normals. The meaning of the options can be found by typing

python run_pgr.py -h

We provide in data directory some example point clouds (from dense to sparse, depending on your accessible computation resources). We recommend running with ~40k points on GPU, which takes about one minute on RTX3090 and uses ~8GB GPU memory:

python run_pgr.py data/Armadillo_40000.xyz		# 8GB memory, 1 min

This will create a folder results/Armadillo_40000 together with three subfolders: recon, samples and solve :

• recon contains the reconstructed meshes in PLY format.
• samples contains the input point cloud in XYZ format, the normalized input point cloud and the octree grid corners as query set in NPY format.
• solve contains the solved Linearized Surface Elements in XYZ and NPY formats, the queried values, query set widths in NPY format, and the iso-value in TXT format.

With fewer samples, you can use larger alpha and k for better smoothness (meanwhile, computation cost becomes much smaller):

python run_pgr.py data/bunny_20000.xyz --alpha 1.2 -wk 16	# 2.8 GB, 12 sec
python run_pgr.py data/bunny_10000.xyz --alpha 1.2 -wk 16	# 1.2 GB, 5 sec

With sparse samples (<5k), we recommend using a large constant width by setting --width_min:

python run_pgr.py data/Utah_teapot_5000.xyz --alpha 2 -wmin 0.04	# 700 MB, 1.7 sec

Moreover, sparse inputs allow running on CPU (using --cpu) within acceptable time:

python run_pgr.py data/Utah_teapot_5000.xyz --alpha 2 -wmin 0.04 --cpu	# 67 sec
python run_pgr.py data/Utah_teapot_3000.xyz --alpha 2 -wmin 0.04 --cpu	# 19 sec
python run_pgr.py data/Utah_teapot_1000.xyz --alpha 2 -wmin 0.04 --cpu	# 1.6 sec

Pipeline Explained

The pipeline is divided into three stages: PGRExportQuery, PGRSolve and PGRLoadQuery. The script run_pgr.py runs all three steps. For clarity, we explain each in detail.

Exporting Query

Usage:

./apps/PGRExportQuery -i point_cloud.xyz -o out_prefix [-m minDepth] [-d maxDepth]

This takes point_cloud.xyz as input and builds an adaptive octree. It outputs two files:

• out_prefix_for_query.npy: the grid corners on the octree for query.
• out_prefix_for_normalized.npy: normalized points from the input point cloud.

Solving for the Implicit Field (Main)

Usage:

python -m apps.PGRSolve [-h] -b BASE -s SAMPLE -q QUERY -o OUTPUT
						 -wk WIDTH_K -wmin WIDTH_MIN -wmax WIDTH_MAX
						 -a ALPHA
						[--max_iters MAX_ITERS] [--save_r] [--cpu]

Explanation of options:

• BASE: the point set $\mathcal{Y}=\mathcal{P}$ for determining basis functions, stored in NPY format.
• SAMPLE: the point set $\mathcal{X}=\mathcal{P}$ for listing equations, stored in NPY format.
• QUERY: the point set $\mathcal{Q}$ of octree grid corners for field queries, stored in NPY format.
• OUTPUT: the output prefix

Other options are straight forward. One can use

python -m apps.PGRSolve -h

to check the help message.

This program outputs (OUTPUT is the argument corresponding to -o):

• OUTPUT_eval_grid.npy: queried values of octree grid corners.
• OUTPUT_grid_width.npy: width function values of octree grid corners.
• OUTPUT_isoval.txt: isovalue.
• OUTPUT_lse.npy: solved linearized surface elements in NPY format.
• OUTPUT_isoval.txt: solved linearized surface elements in XYZ format, can be opened as an oriented point cloud.

Loading Queries and Iso-surfacing

Usage:

./apps/PGRLoadQuery -i point_cloud.xyz -o out_mesh.ply
					--grid_val grid_query.npy --grid_width grid_width.npy
					[-m minDepth] [-d maxDepth] [--isov iso_val]

where

• point_cloud.xyz is the unnormalized point cloud.
• grid_query.npy and grid_width.npy are the output from the last stage.

The meanings of other options should be straight-forward.

Related Research

Surface Reconstruction Based on the Modified Gauss Formula (ACMTOG 2019) Wenjia Lu, Zuoqiang Shi, Jian Sun, Bin Wang