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.

Volumetric data arises in many scientific disciplines, describing the tissue in our body, the composition of the Earth, or the distribution of matter in the universe, to name a few. Such volume data is commonly visualized with direct methods such as ray marching, and with indirect methods such as isocontours. The combination of these approaches is fruitful to provide context and to enhance depth perception. A common challenge, however, is that three-dimensional data inherently leads to occlusions, for which visibility optimization techniques have been employed in the past. The joint visibility optimization of volumetric data (direct volume rendering) and surface geometry (indirect volume rendering) is challenging. With this paper, we provide an optimization approach in which explicit and implicit geometry, such as isocontours or closed context geometry, as well as the volume itself give way to reveal structures that have been identified as important by a user. We model the geometry implicitly as level set of a signed distance field, which is evolved under a normal flow to reduce the occlusion. The non-linear optimization involves gradient descent solvers, level set propagation, and multi-grid optimization. We compare our approach to previous visibility optimizations.

In recent years, grid-free Monte Carlo methods have gained increasing popularity for solving fundamental partial differential equations. For a given point in the domain, the Walk-on-Spheres method solves a boundary integral equation by integrating recursively over the largest possible sphere. When the walks approach boundaries with Dirichlet conditions, the number of path vertices increases considerably, since the step size becomes smaller with decreasing distance to the boundary. In practice, the walks are terminated once they reach an epsilon-shell around the boundary. This, however, introduces bias, leading to a trade-off between accuracy and performance. Instead of using spheres, we propose to utilize geometric primitives that share more than one point with the boundary to increase the likelihood of immediately terminating. Along the boundary of those new geometric primitives a sampling probability is needed, which corresponds to the exit probability of a Brownian motion. This is known as a first passage problem. Utilizing that Laplace equations are invariant under conformal maps, we transform exit points from unit circles to the exit points of our geometric primitives, for which we describe a suitable placement strategy. With this, we obtain a novel approach to solve the Laplace equation in two dimensions, which does not require an epsilon-shell, significantly reduces the number of path vertices, and reduces inaccuracies near Dirichlet boundaries.

Abstract A long‐standing challenge in volume visualization is the effective communication of relevant spatial structures that might be hidden due to occlusions. Given a scalar field that indicates the importance of every point in the domain, previous work synthesized volume visualizations by weighted averaging of samples along view rays or by optimizing a spatially‐varying extinction field through an energy minimization. This energy minimization, however, did not directly measure the contribution of an individual sample to the final pixel color. In this paper, we measure the visibility of relevant structures directly by incorporating the transmittance into a non‐linear energy minimization. For the first time, we not only perform a transmittance‐based extinction optimization, we concurrently optimize the camera position to find ideal viewpoints. We derive the partial derivatives for the gradient‐based optimization symbolically, which makes the application of automatic differentiation methods unnecessary. The transmittance‐based formulation gives a direct visibility measure that is communicated to the user in order to make aware of potentially overlooked relevant structures. Our approach is compatible with any measure of importance and its versatility is demonstrated in multiple data sets.