StippleDiffusion: Capacity-Constrained Stippling using Controlled Diffusion

arXiv:2605.15816 · cs.GR, cs.CV, cs.LG · Submitted 2026-05-15 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: We're looking at the paper: "StippleDiffusion: Capacity-Constrained Stippling using Controlled Diffusion".

Jane: Here's the gist of the paper. The gist: This paper presents StippleDiffusion, a diffusion-based sampler that simultaneously satisfies a learned local point-distribution prior and a continuous, image-defined capacity constraint at inference, enabling high-quality, spatially controlled stippling without requiring per-density optimization.

Tom: Classical approaches for generating 2D point sets with prescribed spatial properties are often computationally expensive because they rely on iterative optimization or carefully designed sampling strategies that require a fresh run for each new target density The generation of high-quality 2D point sets with prescribed spatial properties is a fundamental problem in computer graphics, with applications in rendering, object placement, and image stippling Recently, denoising diffusion probabilistic models have emerged as a powerful alternative for learning complex geometric distributions directly from data However, applying diffusion models to scattered point sets remains challenging due to their convolutional structure, which is inherently designed for grid-based data rather than unordered sets Extending diffusion-based samplers to spatially controlled settings—such as grayscale-conditioned stippling or capacity-constrained point distributions—introduces additional challenges.

Jane: The method is a ControlNet branch built on top of an optimal-transport-grid point-set diffusion baseline, conditioned on the target density map and a high-resolution image The denoising process involves a sequence of denoising operations which operate at given timesteps.

Tom: The baseline model corresponds to a U-Net [Ronneberger et al.

Paper summary: Tom: So we're looking at this paper called "StippleDiffusion: Capacity-Constrained Stippling using Controlled Diffusion". Basically, it tackles how to generate those point sets—those stipple patterns—that follow a target image's density map.

Jane: It’s about moving past the old way where you had to run these iterative optimizers every single time you wanted a new density profile.

Tom: Exactly. The paper claims they’ve created a diffusion-based sampler that handles both learning a local point distribution and having that capacity constraint defined by an image at inference time.

Lu: It’s interesting because they address something where learned alternatives were only good for unconditional generation, but image-conditioned stippling was still out of reach.

Meng: What does this actually mean for deployment? If it’s truly conditioned on the image and doesn't require re-running an optimization for every new target density, that cuts down on a ton of computational overhead at scale.

Lalam: I think that’s a big win because traditional methods are slow and non-differentiable, which means you can't easily train them with deep learning techniques.

Jane: It seems they frame this as solving the problem of balancing two things: getting the points to follow the local density map, and making sure there aren't too many or too few points overall based on that image constraint.

Tom: That sounds like a tough trade-off for any sampler to manage correctly.

Lu: The core idea they’re pushing is using a ControlNet branch built on top of an optimal transport grid point-set diffusion baseline to achieve this. They are taking an existing baseline and adding learned spatial conditioning without retraining the whole thing from scratch.

Meng: I wonder how robust that learned prior initialization is, especially when you're dealing with these hard constraints.

Jane: The paper says they start with a structured initialization obtained via density-weighted rejection sampling, which gives the point locations a spatial prior right at the beginning of the diffusion process.

Tom: That’s clever because it bypasses starting from pure noise, which usually causes problems when you have strict geometric rules to follow.

Meng: And they only focus on late-stage denoising during inference, which is smart for speed. They apply a truncated diffusion process focused primarily on late-stage denoising, inspired by editing-based diffusion strategies Men.

Jane: That way, it’s fast at generating the final pattern without having to run the full thousand steps of the original diffusion schedule every time.

Lu: They also introduce a specific mechanism where they replace standard zero convolutions with a sigmoid-gated one to inject those density signals into the base model.

Tom: I like that—it’s an architectural choice that makes sense because it preserves the blue-noise structure of the base model even when there are hard density signals present.

Paper summary: Meng: From a production standpoint, they mention that a single trained checkpoint can accept arbitrary target densities at inference, which means you don't need a new training run for every image you want to stipple.

Jane: That’s huge for applications where you might have thousands of different images coming in and needing stippling quickly.

Tom: The results show that this single trained checkpoint reaches parity with per-density-optimized baselines on every reported metric, even when comparing against Weighted Voronoi Stippling, Blue Noise Through Optimal Transport, and Gaussian Blue Noise.

Jane: So they’re claiming this is the first learned sampler to match those optimized baselines on capacity-constrained stippling without needing a re-run for each new target density.

Lu: They also mentioned that they reinterpret structured point generation through an optimal transport formulation to enable learning of blue-noiselike distributions under constraints, and they say inference cost is nearly independent of the output point count.

Meng: I gotta wonder about the limitations here because they acknowledge that the model is trained at a single grid resolution, like thirty-two by thirty-two and generalizing across sample sizes isn't directly demonstrated.

Jane: And they also flag that inference is iterative and noticeably slower than deterministic low-discrepancy samplers like Sobol’ or LDBN, which means for applications where you need super fast output, those traditional methods are still better.

Tom: So the paper shows a powerful way to get high quality at inference cost that isn't necessarily the fastest possible way to generate points overall.

Lu: They also pointed out that while their pipeline works in principle for any dimension, the optimal transport grid storage cost scales as k to the power of d with dimension, which makes it prohibitive beyond two dimensions.

Jane: So it seems like this is a very specific solution optimized for two-dimensional imagery right now.

Tom: Overall, "StippleDiffusion: Capacity-Constrained Stippling using Controlled Diffusion" presents a viable way to use diffusion models for spatially controlled point generation by combining learned priors with image constraints during inference.

Jane: It’s a paper that shows how you can get high-quality geometric outputs by using diffusion and ControlNet architectures in a way that handles capacity constraints directly.

Tom: The title itself points to the main technical challenge they solved: making sure the point sets respect both local density and a global capacity limit simultaneously during the generation process.

Lu: I think what this means for future work is exploring how these learned spatial priors could be integrated with other generative models, maybe something that handles three dee point clouds or more complex volumetric data <ref:2605.15816#pg3>.

Jane: It shows that diffusion models aren't just for pixels anymore; they are being adapted to handle complex geometric distributions like point sets, provided you design the conditioning and the sampling strategy correctly.

Tom: It’s about making those learned spatial statistics work in a practical, usable way for image stippling.

Conclusion: Tom: So we're wrapping up this deep dive into StippleDiffusion, which is about using diffusion models for point sets that have to follow specific image densities and capacity limits at inference time.

Jane: Right, it seems like the main thing here is that they’ve managed to get a single trained model to handle both the spatial prior learning and the hard capacity constraint without needing a new optimization run every time you want a different target density.

Lu: I think what's really interesting is how they structured that combination. They built it on top of an optimal transport grid baseline and added this ControlNet branch for image conditioning, which is pretty clever engineering.

Meng: From where I’m sitting, the fact that one trained checkpoint can handle arbitrary target densities at inference without a re-run means we could potentially deploy this kind of generation much faster in a real application.

Lalam: The conclusion is that they successfully combined a learned spatial statistic prior with an image-defined capacity constraint into one diffusion framework.

Jane: So what does that actually mean for us? It means for applications like rendering or object placement, you don't have to spend hours optimizing the point distribution every time the scene or image changes.

Tom: Exactly. The authors show this single checkpoint matches per-density optimized baselines on every metric reported, which is a big deal when you’re talking about practical performance versus theoretical optimization.

Lu: It suggests that for certain tasks involving spatial control, diffusion models can learn those complex distributions effectively when you give them the right architecture and conditioning branch.

Jane: But there are still caveats they mentioned. They noted that the method is trained at a specific grid resolution, and while it works in principle for higher dimensions, the storage cost scales with the dimension to a power of k, which makes it difficult beyond two dimensions.

Meng: Yeah, I see that too. And they also said inference is iterative and noticeably slower than deterministic methods like Sobol’ or LDBN for time-critical needs. So it's high quality, but maybe not always the fastest way to get those points out of the system.

Tom: The authors are pretty clear about that trade-off between quality and speed. But what they achieved in terms of matching those established baselines on capacity constraints is something we need to keep watching.

Jane: It definitely pushes the boundary for how we condition generative models spatially, showing that diffusion can handle more than just standard image generation when you add these complex geometric constraints.

Tom: Next up, we're going to look at the specific architecture they used for that conditioning branch and how it handles the density signals during the denoising process.

OFIR GILAD, ALEKSANDER PLOCHARSKI, PRZEMYSLAW MUSIALSKI, ANDREI SHARF

Ben Gurion University of the Negev · Warsaw University of Technology · New Jersey Institute of Technology

cs.GR, cs.CV, cs.LG

Submitted: 2026-05-15

Updated: 2026-10-19

Comments: 16 pages, 17 figures

License: http://creativecommons.org/publicdomain/zero/1.0/

Importance score: 89/100

The gist: The gist: This paper presents StippleDiffusion, a diffusion-based sampler that simultaneously satisfies a learned local point-distribution prior and a continuous, image-defined capacity constraint at

Key concepts

Optimal Transport (OT) formulation: This mathematical framework is used to reinterpret structured point generation. It allows the model to learn blue-noiselike distributions by treating the process as moving mass from an initial distribution to a target distribution, enabling the learning of constraints on point placement.
ControlNet Branch: This architecture extends a baseline diffusion model (U-Net) by adding a trainable conditioning branch inspired by ControlNet. This allows the model to be guided by external inputs, specifically the target density map and image features, without needing to retrain the main diffusion network.
Capacity Constraint: This is a continuous, image-defined limit on how many points can be generated in a specific area. Unlike traditional methods that require re-running optimization for every new point count, this constraint is enforced during inference by the learned sampler.
Late-stage Denoising Regime: The training and inference process is restricted to the later stages of the diffusion model's denoising schedule. This strategy helps ensure that the model focuses on refining high-frequency details and spatial structure, leading to better quality stipples.

Terminology

Benchmark — 1× in this paper · explained in 2 episodes

Summary

The gist: This paper presents StippleDiffusion, a diffusion-based sampler that simultaneously satisfies a learned local point-distribution prior and a continuous, image-defined capacity constraint at inference, enabling high-quality, spatially controlled stippling without requiring per-density optimization.

Introduction and Motivation

Classical approaches for generating 2D point sets with prescribed spatial properties are often computationally expensive because they rely on iterative optimization or carefully designed sampling strategies that require a fresh run for each new target density The generation of high-quality 2D point sets with prescribed spatial properties is a fundamental problem in computer graphics, with applications in rendering, object placement, and image stippling Recently, denoising diffusion probabilistic models have emerged as a powerful alternative for learning complex geometric distributions directly from data However, applying diffusion models to scattered point sets remains challenging due to their convolutional structure, which is inherently designed for grid-based data rather than unordered sets Extending diffusion-based samplers to spatially controlled settings—such as grayscale-conditioned stippling or capacity-constrained point distributions—introduces additional challenges.

Methodology and Architecture

The method is a ControlNet branch built on top of an optimal-transport-grid point-set diffusion baseline, conditioned on the target density map and a high-resolution image The denoising process involves a sequence of denoising operations which operate at given timesteps. The baseline model corresponds to a U-Net [Ronneberger et al. 2015], where each level is composed of two convolutional residual blocks (ResNet) and the feature maps are downsampled by a factor of (a) Target (b) WVS (c) BNOT (d) GBN (e) Ours. To enable spatial conditioning without retraining this base model, our network architecture introduces a trainable conditioning branch parallel to the frozen U-Net, inspired by ControlNet [Zhang et al. 2023]. Instead of connecting these conditions to the baseline model with standard Zero Convolutions, we replace them with an Gated Zero Convolutions mechanism. Our condition input is an aggregation of static grid features and dynamically sampled image features. To capture high-resolution context, we employ a geometric feature extractor based on GECCO [Tyszkiewicz et al. 2023]. These extracted features are then concatenated with our static conditioning grid.

Inference and Training Strategy

During inference, we generate a stipple pattern for a novel input image by first extracting the necessary spatial conditions. We compute a target density map and, instead of starting from pure Gaussian noise, generate a structural prior—our initial point distribution—via density-weighted rejection sampling. These initial points are then converted into an OT offset tensor. To provide a continuous spatial prior to the network, we rasterize these initial points into a low-resolution N × N grid. To capture sharp, boundary-aware details, we adapt the continuous feature extraction formulation introduced by GECCO [Tyszkiewicz et al. 2023]. We then utilize bilinear interpolation (with gridsample operation) to dynamically extract 16-channel continuous features directly at the exact floatingpoint coordinates of our current noisy point set. This 19-channel vector is then processed by a Conditioning Encoder consisting of sequential convolutional layers with SiLU activations. Finally, we execute a truncated, SDEdit-style denoising schedule. We apply explicit forward noise to our initial OT offsets, effectively skipping the first 70% of the global diffusion schedule of T = 1000 steps.

Evaluation and Results

The results show that a single trained checkpoint reaches parity with the per-density-optimized baselines across every metric in Table 1. We further demonstrate the quality of the outputs of the proposed model by visually comparing against established stippling methods: Weighted Voronoi Stippling (WVS) [Secord 2002], Blue Noise Through Optimal Transport (BNOT) [de Goes et al. 2012] and Gaussian Blue Noise (GBN) [Ahmed et al. 2022]. The results show that a single trained checkpoint reaches parity with the per-density-optimized baselines across every metric in Table 1. To our knowledge, this is the first learned sampler to match per-density-optimized stipplers on capacity-constrained stippling without re-running an optimization for each new target density.

Key Contributions

The contributions are as follows:

• We propose the first diffusion-based framework for capacityconstrained stippling under explicit spatial control; a single trained model accepts arbitrary target densities at inference, in contrast to optimization-based stipplers that re-run per density.

• We reinterpret structured point generation through an optimal transport formulation to enable learning of blue-noiselike distributions under constraints, with inference cost nearly independent of the output point count and gradients available end-to-end through the sampler.

• We introduce a quantitative and qualitative evaluation protocol adapted from blue-noise and capacity-constrained sampling literature for comparing capacity-aware and spatially conditioned point generation methods against prior unconditional samplers.

Limitations

Three limitations remain. The model is trained at a single grid resolution (32 × 32, 1024 points); generalization across sample sizes is inherited from the OT-grid representation but is not directly demonstrated here. Inference is iterative and noticeably slower than deterministic low-discrepancy samplers such as Sobol’ or LDBN, so for time-critical applications the latter remain preferable. Finally, while our pipeline transfers to arbitrary dimension in principle, the OT-grid storage cost scales as k d with dimension and is prohibitive beyond the two-dimensional setting we focus on; this scaling is inherited from [Doignies et al. 2023].

Conclusion

We have shown that a controlled diffusion architecture, built on top of the optimal-transport-grid point-set sampler of [Doignies et al. 2023], can produce stipple distributions that simultaneously preserve a learned spatial-statistic prior and adhere to a continuous, image-defined capacity constraint. The combination is workable because of two specific architectural decisions. We have shown that a single trained checkpoint reaches parity with the per-density-optimized baselines across every metric in Table 1. To our knowledge, this is the first learned sampler to match per-density-optimized stipplers on capacity-constrained stippling without re-running an optimization for each new target density. The results show that a single trained checkpoint reaches parity with the per-density-optimized baselines across every metric in Table 1. To our knowledge, this is the first learned sampler to match per-density-optimized stipplers on capacity-constrained stippling without re-running an optimization for each new target density. The combination is workable because of two specific architectural decisions. We have shown that a single trained checkpoint reaches parity with the per-density-optimized baselines across every metric in Table 1. To our knowledge, this is the first learned sampler to match per-density-optimized stipplers on capacity-constrained stippling without re-running an optimization for each new target density. The combination is workable because of two specific architectural decisions <ref:2605.

Improvements for AI systems

Diffusion-based sampling for capacity constrained stippling allows a single trained model to accept arbitrary target densities at inference, unlike optimization-based stipplers that re-run per density. This enables rapid generation of high-quality, spatially controlled point sets without needing a fresh optimization run for every new target density.
The framework combines a learned local point-distribution prior and a continuous, image-defined capacity constraint at inference, which results in stipples that exhibit characteristic blue noise properties—maintaining regular relative spacing to avoid clustering—while closely tracking the underlying global target density. This means the improved AI system can generate distributions that are both spatially structured (blue noise) and precisely match complex grayscale intensity maps.
The use of a Gated Zero Convolutions mechanism, replacing standard Zero Convolutions, prevents the early-stage loss of geometric regularity in the underlying point distribution, ensuring that the model preserves the structure during refinement. This architectural choice stabilizes diffusion-based point generation under structured constraints, preventing residual point clustering and loss of fine geometric structure.
The inference process utilizes a truncated SDEdit schedule and a forward-backward resampling flow, which allows for iterative refinement to enforce strict capacity constraints. This mechanism gives the model multiple opportunities to un-clump the distribution and enforce strict capacity constraints, leading to final point sets that adhere closely to local density requirements.
The system can evaluate performance across varying output resolutions because a single trained checkpoint generalizes across point budgets that were not seen during training. This means a deployed model can generate stipples with 256, 1024, or even 2304 points while preserving the main target structure, allowing for flexible output scaling without retraining.

Abstract

Stipple patterns, point sets whose local density tracks a target image, are traditionally produced by per-density iterative optimizers, which are slow, non-differentiable, and must be re-run from scratch for each new target. Learned alternatives have so far addressed only unconditional point generation; capacity-constrained, image-conditioned stippling has remained out of reach. We present the first diffusion-based sampler that simultaneously satisfies a learned local point-distribution prior and a continuous, image-defined capacity constraint at inference. The method is a ControlNet branch built on top of an optimal-transport-grid point-set diffusion baseline, conditioned on the target density map and a high-resolution image. Two design choices make the combination tractable: training and inference are restricted to the late-stage denoising regime, initialized from a density-weighted rejection sample, and the standard zero-convolution injection is replaced with a sigmoid-gated 1x1 projection that preserves the base model's blue-noise structure under hard density signals. A single trained checkpoint accepts arbitrary target densities at inference, generalizes to point budgets that were not seen during training, and produces stipples in time nearly independent of the output point count. On the Icons-50 benchmark, our learned sampler reaches parity with per-density-optimized baselines on every reported metric while remaining differentiable end-to-end.