Victor Ostromoukhov

Bastien Doignies, Nicolas Bonneel, David Coeurjolly et al.

Much effort has been put into developing samplers with specific properties, such as producing blue noise, low-discrepancy, lattice or Poisson disk samples. These samplers can be slow if they rely on optimization processes, may rely on a wide range of numerical methods, are not always differentiable. The success of recent diffusion models for image generation suggests that these models could be appropriate for learning how to generate point sets from examples. However, their convolutional nature makes these methods impractical for dealing with scattered data such as point sets. We propose a generic way to produce 2-d point sets imitating existing samplers from observed point sets using a diffusion model. We address the problem of convolutional layers by leveraging neighborhood information from an optimal transport matching to a uniform grid, that allows us to benefit from fast convolutions on grids, and to support the example-based learning of non-uniform sampling patterns. We demonstrate how the differentiability of our approach can be used to optimize point sets to enforce properties.

Nicolas Bonneel, David Coeurjolly, Victor Ostromoukhov

To numerically integrate a function, one may resort to Quasi-Monte Carlo estimators, that average integrand values at pseudo-random well-distributed uniform sampling locations. Better uniformity improves the worst-case integration-error bound. A standard measure of uniformity is given by an integer $t$ value, where $t=0$ yields the best uniformity. Producing sequences of samples with bounded $t$ values can be achieved with Sobol' recursive construction, that uses coefficients of irreducible polynomials. While $b$-dimensional sequences with $t=0$ can be obtained by taking $b$ polynomials of degree $1$ over the Galois Field $\mathrm{GF}(b)$, we show conditions that guarantee $t=0$ for specific higher degree polynomials. In particular, we relate the Sobol' construction to tensorized powers of Pascal matrices when the chosen polynomials only differ by a constant and exhibit simple conditions to guarantee $t=0$ in this case. We then focus on Artin-Schreier irreducible polynomials, in the form $p_i(x) = x^b - x + c_i$, where $i \in \{1, \dots, b-1\}$ and $b$ is prime, and we make explicit conditions that always guarantees $t=0$ in $b-1$ dimensions. Combining $b$-dimensional Sobol' of degree $1$ and our $(b-1)$-dimensional Artin-Schreier sequence of degree $b$, we provide a fast greedy procedure that optimizes the $(2b-1)$-dimensional combined $t$ value, while guaranteeing $t=0$ projection in subspaces.