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biharmonic-monte-carlo

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Reference Implementation and scene data for the SIGGRAPH 2026 journal paper

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# Monte Carlo Rendering of Biharmonic Diffusion Curves

![BDC](https://raw.githubusercontent.com/Paul-Hi/biharmonic-monte-carlo/main/images/teaser.png)
![language](https://img.shields.io/badge/language-C++-blue)
![license](https://img.shields.io/badge/license-MIT-red)

## [Project Page 🌐](https://paul-hi.github.io/biharmonic_wos.html) | [Preprint 📄](https://vc.tf.fau.de/publications/Himmler26Siggraph/Himmler26Siggraph.pdf) | [doi <img src="images/open_access.png" alt="oa" width="12">](https://doi.org/10.1145/3811397)

This is the official small reference implementation for the paper [**Monte Carlo Rendering of Biharmonic Diffusion Curves**](https://vc.tf.fau.de/publications/Himmler26Siggraph/Himmler26Siggraph.pdf) accepted as journal article at ACM SIGGRAPH 2026.

Stochastic Monte Carlo solvers for partial differential equations (PDEs) recently gained popularity in computer graphics, finding applications in geometry processing, rendering, simulation, and visualization. At present, there exists no Monte Carlo solver for the rendering of biharmonic diffusion curves, an artist-friendly smooth vector graphics primitive. The fourth-order biharmonic equation of biharmonic diffusion curves can be split into two second-order PDEs, namely a Laplace and a Poisson equation. However, since biharmonic diffusion curves set Dirichlet and inhomogeneous Neumann conditions at the same time, these two second-order PDEs are tightly coupled and can hence not be solved directly. We propose to treat the rendering of biharmonic diffusion curves as an inverse problem, in which the Dirichlet data of the Laplace equation is unknown. We formulate a variational energy optimization, such that the user-defined boundary conditions are met. Thereby, the necessary gradients are estimated stochastically by solving two second-order problems with Dirichlet boundary conditions only.

### Small Biharmonic Diffusion Curves

#### Disclaimer

This code is only used to demonstrate the algorithm. It is not as sophisticated as the implementation used for the comparisons and results presented in the paper.

#### Output

The two 2D examples produce the following output:

- the intial result

![initial](https://raw.githubusercontent.com/Paul-Hi/biharmonic-monte-carlo/main/images/initial.bmp)

- the optimized result

![optimized](https://raw.githubusercontent.com/Paul-Hi/biharmonic-monte-carlo/main/images/optimized.bmp)

- and a 2D hermite spline

![hermite](https://raw.githubusercontent.com/Paul-Hi/biharmonic-monte-carlo/main/images/hermite2D.bmp)

#### Build

To execute the small example you need a C++ 17 compatible compiler and CMake 3.17.

#### Options

There are some **#defines** you can vary:

Scene settings:

- #define SAMPLES_PER_PIXEL 1024 : Change for more or less samples per pixel
- #define MAX_STEPS 128 : Change for a lower or higher number of maximum steps
- #define EPSILON 1e-3 : Epsilon Value for the Epsilon Shell
- #define RESOLUTION 128 : Resolution of the output will be RESOLUTION x RESOLUTION

Only **one** of these should be uncommented at the same time:

- #define OPTIMIZATION : If defined the Walk-on-Spheres implementation will optimize the h field to converge towards the biharmonic solution to a given set of boundary conditions.
- #define HERMITE_SPLINE : If defined the Walk-on-Spheres implementation will converge towards a Hermite spline with boundary conditions:
  - Dirichlet data f at x0 = 0 and x1 = 1
  - Neumann data g at x0 = 0 and x1 = 0
  - Ground truth Laplace data h at x0 and x1

For the optimization there are additional settings to play around with:

- #define OPTIMIZATION_STEPS 128 : Change the number of iterations in the optimization
- #define LEARNING_RATE 0.1 : Change the learning rate
- #define BOUNDARY_SAMPLES 512 : Change the nuber of samples on the boundary each iteration
- #define UB_SAMPLES 16 : Change the number of samples for one of the estimators for the Poisson equation
- #define UH_SAMPLES 4 : Change the number of samples for the other one of the estimators for the Poisson equation

### Small 1D Example

In ```./biharmonic-1d``` you can find an additional analytical 1D example, which was also included for review. It does not utilize Monte Carlo walks or path replay derivatives and considers only the simplest possible case for demonstration purposes.

### Scenes

The ```./scenes``` directory contains human-readable representations of the scenes used to create the visualizations in the paper, structured as follows:

```
int point_count;   // number of points in list of points
int segment_count; // number of segments in list of segments

struct Point{
    float2 position;             // 2D vector position
    float3 dirichlet_data_lr[2]; // left and right rgb values
    float3 neumann_data_lr[2];   // left and right (3D) normal derivatives
} points[point_count];           // list of points

struct Segment {
    int point_index_0;       // index of the segments start point in list of points
    int point_index_0;       // index of the segments end point in list of points
    bool is_domain_boundary; // true if the segment is a domain boundary, else false
} segments[segment_count];   // list of segments
```

### BibTex

If you use the code base or the scenes, please cite

```
@article{Himmler26Siggraph,
  author = {Himmler, Paul and G{\"u}nther, Tobias},
  title = {Monte Carlo Rendering of Biharmonic Diffusion Curves},
  journal = {ACM Transactions on Graphics (SIGGRAPH)},
  volume = {45},
  number = {4},
  pages = {107:1-107:12},
  year = {2026},
  publisher = {ACM},
  address = {Los Angeles, United States},
  doi = {10.1145/3811397},
}
```

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