We propose a novel approach to represent maps between two discrete surfaces of the same genus and to minimize intrinsic mapping distortion. Our maps are well-defined at every surface point and are guaranteed to be continuous bijections (surface homeomorphisms). As a key feature of our approach, only the images of vertices need to be represented explicitly, since the images of all other points (on edges or in faces) are properly defined implicitly. This definition is via unique geodesics in metrics of constant Gaussian curvature. Our method is built upon the fact that such metrics exist on surfaces of arbitrary topology, without the need for any cuts or cones (as asserted by the uniformization theorem). Depending on the surfaces' genus, these metrics exhibit one of the three classical geometries: Euclidean, spherical or hyperbolic. Our formulation handles constructions in all three geometries in a unified way. In addition, by considering not only the vertex images but also the discrete metric as degrees of freedom, our formulation enables us to simultaneously optimize the images of these vertices and images of all other points.

The problem of discrete surface parametrization, i.e. mapping a mesh to a planar domain, has been investigated extensively. We address the more general problem of mapping between surfaces. In particular, we provide a formulation that yields a map between two disk-topology meshes, which is continuous and injective by construction and which locally minimizes intrinsic distortion. A common approach is to express such a map as the composition of two maps via a simple intermediate domain such as the plane, and to independently optimize the individual maps. However, even if both individual maps are of minimal distortion, there is potentially high distortion in the composed map. In contrast to many previous works, we minimize distortion in an end-to-end manner, directly optimizing the quality of the composed map. This setting poses additional challenges due to the discrete nature of both the source and the target domain. We propose a formulation that, despite the combinatorial aspects of the problem, allows for a purely continuous optimization. Further, our approach addresses the non-smooth nature of discrete distortion measures in this context which hinders straightforward application of off-the-shelf optimization techniques. We demonstrate that, despite the challenges inherent to the more involved setting, discrete surface-to-surface maps can be optimized effectively.

Non‐linear optimization is essential to many areas of geometry processing research. However, when experimenting with different problem formulations or when prototyping new algorithms, a major practical obstacle is the need to figure out derivatives of objective functions, especially when second‐order derivatives are required. Deriving and manually implementing gradients and Hessians is both time‐consuming and error‐prone. Automatic differentiation techniques address this problem, but can introduce a diverse set of obstacles themselves, e.g. limiting the set of supported language features, imposing restrictions on a program's control flow, incurring a significant run time overhead, or making it hard to exploit sparsity patterns common in geometry processing. We show that for many geometric problems, in particular on meshes, the simplest form of forward‐mode automatic differentiation is not only the most flexible, but also actually the most efficient choice. We introduce TinyAD: a lightweight C++ library that automatically computes gradients and Hessians, in particular of sparse problems, by differentiating small (tiny) sub‐problems. Its simplicity enables easy integration; no restrictions on, e.g., looping and branching are imposed. TinyAD provides the basic ingredients to quickly implement first and second order Newton‐style solvers, allowing for flexible adjustment of both problem formulations and solver details. By showcasing compact implementations of methods from parametrization, deformation, and direction field design, we demonstrate how TinyAD lowers the barrier to exploring non‐linear optimization techniques. This enables not only fast prototyping of new research ideas, but also improves replicability of existing algorithms in geometry processing. TinyAD is available to the community as an open source library.

Figure 1: We embed a given layout connectivity (left, visualized as a coarse mesh for illustration) into a target surface with prescribed landmark positions (center left) by successively embedding edges as shortest paths in an optimized order. Previous methods choose a greedy sequence of locally optimal decisions, which can lead to severe topological artifacts (center right) due to accidental blocking of subsequently inserted paths. Our branch-and-bound method globally optimizes over all possible edge insertion sequences to find an embedding of shortest total path length, which yields the expected homotopy class (right).

We present an algorithm for realtime rendering of large-scale city models with procedurally generated facades. By using highly detailed assets like windows, doors, and decoration such city models can provide an extremely high geometric level of detail but on the downside they also consist of billions of polygons which makes it infeasible to even store them as explicit polygonal meshes. Moreover, when rendering urban scenes usually only a very small fraction of the city is actually visible which calls for effective culling mechanisms. For procedural textures there are efficient screen space techniques that evaluate, e.g., a split grammar on a per-pixel basis in the fragment shader and thus render a textured facade in a view dependent manner. We take this idea further by introducing 3D geometric detail in addition to flat textures. Our approach is a two-pass procedure that first renders a flat procedural facade. During rasterization the fragment shader triggers the instantiation of a detailed asset whenever a geometric facade element is potentially visible. The set of instantiated detail models are then rendered in a second pass. The major challenges arise from the fact that geometric details belonging to a facade can be visible even if the base polygon of the facade itself is not visible. Hence we propose measures to conservatively estimate visibility without introducing excessive redundancy. We further extend our technique by a simple level of detail mechanism that switches to baked textures (of the assets) depending on the distance to the camera. We demonstrate that our technique achieves realtime frame rates for large-scale city models with massive detail on current commodity graphics hardware.

A homeomorphism between two surfaces not only defines a (continuous and bijective) geometric correspondence of points but also (by implication) an identification of topological features, i.e. handles and tunnels, and how the map twists around them. However, in practice, surface maps are often encoded via sparse correspondences or fuzzy representations that merely approximate a homeomorphism and are therefore inherently ambiguous about map topology. In this work, we show a way to infer topological information from an imperfect input map between two shapes. In particular, we compute a homology map, a linear map that transports homology classes of cycles from one surface to the other, subject to a global consistency constraint. Our inference robustly handles imperfect (e.g., partial, sparse, fuzzy, noisy, outlier‐ridden, non‐injective) input maps and is guaranteed to produce homology maps that are compatible with true homeomorphisms between the input shapes. Homology maps inferred by our method can be directly used to transfer homological information between shapes, or serve as foundation for the construction of a proper homeomorphism guided by the input map, e.g., via compatible surface decomposition.