Surface Reconstruction via Green's 3rd Identity (G3R)

This repository contains the official implementation of the paper:

Surface Reconstruction via Green's 3rd Identity (SIGGRAPH 2026 Conference Paper)

Zhonghao Wu, Dong Xiao, Renjie Chen

Quick Start

toy_g3r.py is a self-contained, single-file toy implementation for quickly trying out the algorithm. It explicitly constructs the full dense matrix and solves it directly, so it is only intended for small point clouds (<10,000 points).

python toy_g3r.py bunny.xyz

Full Implementation

main_g3r.py loads a point cloud and solves the linear system derived from Green's 3rd Identity using iterative solvers. It is built on PyTorch and uses a C++/CUDA extension (wn_treecode) with octree acceleration.

Setup

Make sure PyTorch and the CUDA toolkit are installed, then install the remaining dependencies and compile the extension:

pip install tqdm trimesh psutil
pip install -e ext --no-build-isolation

Usage

Basic usage:

python main_g3r.py INPUT [options]

Example:

python main_g3r.py bunny.xyz \
    --epsilon 1e-3 \
    --test_funcs "poly(:2)" \
    --iter 10 sd 20 cg

Arguments

Argument	Description	Default
INPUT	Path to the input point cloud file.	(required)
--epsilon	Regularization parameter.	1e-3
--test_funcs	Test functions for evaluating reconstruction quality (see below).	"poly(:2)"
--iter	Number of iterations for each solver.	10 sd 20 cg
--out_dir	Output directory for results.	./results
--cpu	Run on CPU instead of GPU.	—

--test_funcs syntax. Test functions are specified as regular solid harmonics with degree $l$ and order $m$:

• "poly(:2)" — $l$ from 0 to 2, all orders.
• "poly(2:4)" — $l$ from 2 to 4, all orders.
• "poly(2, -1:1)" — $l = 2$, $m$ from −1 to 1.
• "poly(0); poly(2)" — $l = 0$ and $l = 2$, all orders.

--iter syntax. Solvers and their iteration counts are specified as alternating number–name pairs:

• sd — steepest descent.
• cg — conjugate gradient.

For example, --iter 10 sd 20 cg runs 10 iterations of steepest descent followed by 20 iterations of conjugate gradient.