Dithering by Differences of Convex Functions

Motivated by a recent halftoning method which is based on electrostatic principles, we analyse a halftoning framework where one minimizes a functional consisting of the difference of two convex functions (DC). One of them describes attracting forces caused by the image gray values, the other one enforces repulsion between points. In one dimension, the minimizers of our functional can be computed analytically and have the following desired properties: the points are pairwise distinct, lie within the image frame and can be placed at grid points. In the two-dimensional setting, we prove some useful properties of our functional like its coercivity and suggest to compute a minimizer by a forward-backward splitting algorithm. We show that the sequence produced by such an algorithm converges to a critical point of our functional. Furthermore, we suggest to compute the special sums occurring in each iteration step by a fast summation technique based on the fast Fourier transform at non-equispaced knots which requires only Ο(m log(m)) arithmetic operations for m points. Finally, we present numerical results showing the excellent performance of our DC dithering method

A projection method on measures sets

We consider the problem of projecting a probability measure π on a set MN of Radon measures. The projection is defined as a solution of the following variational problem: inf µ∈M N h (µ − π) 2 2 , where h ∈ L 2 (Ω) is a kernel, Ω ⊂ R d and denotes the convolution operator. To motivate and illustrate our study, we show that this problem arises naturally in various practical image rendering problems such as stippling (representing an image with N dots) or continuous line drawing (representing an image with a continuous line). We provide a necessary and sufficient condition on the sequence (MN) N ∈N that ensures weak convergence of the projections (µ * N) N ∈N to π. We then provide a numerical algorithm to solve a discretized version of the problem and show several illustrations related to computer-assisted synthesis of artistic paintings/drawings

Backward Diffusion Methods for Digital Halftoning

We examine using discrete backward diffusion to produce digital halftones. The noise introduced by the discrete approximation to backwards diffusion forces the intensity away from uniform values, so that rounding each pixel to black or white can produce a pleasing halftone. We formulate our method by considering the Human Visual System norm and approximating the inverse of the blurring operator. We also investigate several possible mobility functions for use in a nonlinear backward diffusion equation for higher quality results

A projection algorithm on measures sets

We consider the problem of projecting a probability measure $\pi$ on a set $\mathcal{M}_N$ of Radon measures. The projection is defined as a solution of the following variational problem:\begin{equation*}\inf_{\mu\in \mathcal{M}_N} \|h\star (\mu - \pi)\|_2^2,\end{equation*}where $h\in L^2(\Omega)$ is a kernel, $\Omega\subset \R^d$ and $\star$ denotes the convolution operator.To motivate and illustrate our study, we show that this problem arises naturally in various practical image rendering problems such as stippling (representing an image with $N$ dots) or continuous line drawing (representing an image with a continuous line).We provide a necessary and sufficient condition on the sequence $(\mathcal{M}_N)_{N\in \N}$ that ensures weak convergence of the projections $(\mu^*_N)_{N\in \N}$ to $\pi$.We then provide a numerical algorithm to solve a discretized version of the problem and show several illustrations related to computer-assisted synthesis of artistic paintings/drawings

Structure-aware halftoning

our result faithfully preserves the texture details as well as the local tone. All images have the same resolution of 445×377. This paper presents an optimization-based halftoning technique that preserves the structure and tone similarities between the original and the halftone images. By optimizing an objective function consisting of both the structure and the tone metrics, the generated halftone images preserve visually sensitive texture details as well as the local tone. It possesses the blue-noise property and does not introduce annoying patterns. Unlike the existing edge-enhancement halftoning, the proposed method does not suffer from the deficiencies of edge detector. Our method is tested on various types of images. In multiple experiments and the user study, our method consistently obtains the best scores among all tested methods.

Variational blue noise sampling

Blue noise point sampling is one of the core algorithms in computer graphics. In this paper, we present a new and versatile variational framework for generating point distributions with high-quality blue noise characteristics while precisely adapting to given density functions. Different from previous approaches based on discrete settings of capacity-constrained Voronoi tessellation, we cast the blue noise sampling generation as a variational problem with continuous settings. Based on an accurate evaluation of the gradient of an energy function, an efficient optimization is developed which delivers significantly faster performance than the previous optimization-based methods. Our framework can easily be extended to generating blue noise point samples on manifold surfaces and for multi-class sampling. The optimization formulation also allows us to naturally deal with dynamic domains, such as deformable surfaces, and to yield blue noise samplings with temporal coherence. We present experimental results to validate the efficacy of our variational framework. Finally, we show a variety of applications of the proposed methods, including nonphotorealistic image stippling, color stippling, and blue noise sampling on deformable surfaces. © 1995-2012 IEEE.published_or_final_versio

Scalable multi-class sampling via filtered sliced optimal transport

We propose a multi-class point optimization formulation based on continuous Wasserstein barycenters. Our formulation is designed to handle hundreds to thousands of optimization objectives and comes with a practical optimization scheme. We demonstrate the effectiveness of our framework on various sampling applications like stippling, object placement, and Monte-Carlo integration. We a derive multi-class error bound for perceptual rendering error which can be minimized using our optimization. We provide source code at https://github.com/iribis/filtered-sliced-optimal-transport.Comment: 15 pages, 17 figures, ACM Trans. Graph., Vol. 41, No. 6, Article 261. Publication date: December 202