Abstract To support the design and subsequent generation of terrestrial planets for use in the creative media, we propose a solution that employs a generative model trained on satellite data from planetary bodies with a defined solid surface, such as the Earth and Mars. A user sketches coarse elevation, landcover, temperature, and precipitation directly onto a globe. Our model then infers high‐resolution heightmap and surface appearance layers at planetary scales, with sufficient detail to enable animated flyovers within the exosphere at a distance of a few thousand kilometers from the planet surface. We address the issue of distortion in the mapping from atlas to globe using a quadsphere representation, and the consistency of large‐scale geomorphological features by extracting a global river network from the sketch inputs and providing this as conditioning to the diffusion. As our results demonstrate, our generative model provides a balance between: authoring control through a multi‐layer painting interface with a satellite image pre‐visualization; computation times proportional to the surface area being generated; landscape diversity, displaying, without repetition artefacts, the full range of elevation and landcover features drawn from multiple source planets, and geomorphological plausibility through the provision of a consistent uninterrupted exorheic global river network, where the input sketches allow.

Abstract Our goal is to reconstruct scenes with stochastic, incoherent motion such as leaves moving in the wind, that can be particularly challenging because of small objects with similar appearance that move independently. Previous dynamic 3D Gaussian Splatting solutions either represent motion implicitly with neural networks achieving good quality but lower framerate, or explicitly with a function, often with higher training times and lower quality. To overcome these limitations, we propose an explicit method that introduces adaptive space‐time densification and smoother optimization. We introduce a new densification approach based on error moments that are used to guide primitive splitting, and we adaptively refine the number of keyframes used based on the variance of error. We observe that dynamic reconstruction from monocular video is hard for standard optimization pipelines. To counter this, we introduce a weighted Adam approach that improves results based on primitive visibility. Finally, to handle the hard case of independent motion of similar‐looking objects, we introduce an image‐driven as‐rigid‐as‐possible regularization. Our method has higher quality than previous explicit solutions, and has significantly higher framerate for rendering.

A common process in authoring digital scenes for games, films, and virtual environments is for artists to construct 3D geometry that matches a 2D perspective reference image. In the case of the bare-earth terrain, this is typically a manual process since, unlike for trees and buildings, few inverse reconstruction methods currently exist. To address this, we introduce a method for automatically inferring a detailed, consistent, and complete terrain heightmap from a single photographic image. Our initial phase involves extracting camera parameters and a 3D pointmap from the input image, which is then transformed into a heightmap. However, this only recovers the unoccluded portions of the terrain visible from the perspective of the image. The next phase thus entails the generation of plausible occluded regions using a diffusion model trained on terrain elevation data. The entire process is guided by three consistency principles: geomorphological consistency (the features of the occluded terrain resemble the visible portions), hydrological consistency (the river network is uninterrupted and flows reliably), and view consistency (the shape of the rendered terrain accurately matches the input image). We demonstrate that our method obeys these principles, reliably generates terrains across various scales, and integrates with scene authoring workflows.

Mountainous terrains evolve over geological timescales through erosion processes driven by the complex interplay of transported quantities such as water, sediment, and rockfall. A key challenge in erosion modeling is the simultaneous simulation of transport and erosive processes, which differ in temporal scales by several orders of magnitude. We address this challenge with a novel, parallel, stochastic particle-based method capable of simulating transport over geological timescales. Our approach relaxes the strong assumptions on velocity required by prior works ( e.g. , based on the Stream Power Law), enabling a new erosion model grounded in a more general form of momentum conservation. We demonstrate that our scheme accurately solves the underlying conservation laws and avoids artifacts common in previous works. Furthermore, we show that our new erosion model captures multiscale geomorphological features, producing coherent basin structures and dynamic phenomena such as braided rivers, meanders, and deltas.

We introduce Arenite, a novel physics-based approach for modeling sandstone structures. The key insight of our work is that simulating a combination of stress and multi-factor erosion enables the generation of a wide variety of sandstone structures observed in nature. We isolate the key shape-forming phenomena: multi-physics fabric interlocking, wind and fluvial erosion, and particle-based deposition processes. Complex 3D structures such as arches, alcoves, hoodoos, or buttes can be achieved by creating simple 3D structures with user-painted erodable areas and vegetation and running the simulation. We demonstrate the algorithm on a wide variety of structures, and our GPU-based implementation achieves the simulation in less than 5 minutes on a desktop computer for our most complex example.

Abstract Terrain analysis plays an important role in computer graphics, hydrology and geomorphology. In particular, analyzing the path of material flow over a terrain with consideration of local depressions is a precursor to many further tasks in erosion, river formation, and plant ecosystem simulation. For example, fluvial erosion simulation used in terrain modeling computes water discharge to repeatedly locate erosion channels for soil removal and transport. Despite its significance, traditional methods face performance constraints, limiting their broader applicability. In this paper, we propose a novel GPU flow routing algorithm that computes the water discharge in 𝒪(log n) iterations for a terrain with n vertices (assuming n processors). We also provide a depression routing algorithm to route the water out of local minima formed by depressions in the terrain, which converges in 𝒪(log2 n) iterations. Our implementation of these algorithms leads to a 5× speedup for flow routing and 34 × to 52 × speedup for depression routing compared to previous work on a 10242 terrain, enabling interactive control of terrain simulation.

Abstract Explosive volcanic eruptions rank among the most terrifying natural phenomena, and are thus frequently depicted in films, games, and other media, usually with a bespoke once‐off solution. In this paper, we introduce the first general‐purpose model for bi‐directional interaction between the atmosphere and a volcano plume. In line with recent interactive volcano models, we approximate the plume dynamics with Lagrangian disks and spheres and the atmosphere with sparse layers of 2D Eulerian grids, enabling us to focus on the transfer of physical quantities such as temperature, ash, moisture, and wind velocity between these sub‐models. We subsequently generate volumetric animations by noise‐based procedural upsampling keyed to aspects of advection, convection, moisture, and ash content to generate a fully‐realized volcanic skyscape. Our model captures most of the visually salient features emerging from volcano‐sky interaction, such as windswept plumes, enmeshed cap, bell and skirt clouds, shockwave effects, ash rain, and sheathes of lightning visible in the dark.

Abstract Terrain generation methods have long been divided between procedural and physically‐based. Procedural methods build upon the fast evaluation of a mathematical function but suffer from a lack of geological consistency, while physically‐based simulation enforces this consistency at the cost of thousands of iterations unraveling the history of the landscape. In particular, the simulation of the competition between tectonic uplift and fluvial erosion expressed by the stream power law raised recent interest in computer graphics as this allows the generation and control of consistent large‐scale mountain ranges, albeit at the cost of a lengthy simulation. In this paper, we explore the analytical solutions of the stream power law and propose a method that is both physically‐based and procedural, allowing fast and consistent large‐scale terrain generation. In our approach, time is no longer the stopping criterion of an iterative process but acts as the parameter of a mathematical function, a slider that controls the aging of the input terrain from a subtle erosion to the complete replacement by a fully formed mountain range. While analytical solutions have been proposed by the geomorphology community for the 1D case, extending them to a 2D heightmap proves challenging. We propose an efficient implementation of the analytical solutions with a multigrid accelerated iterative process and solutions to incorporate landslides and hillslope processes – two erosion factors that complement the stream power law.

Abstract While the past of terrain cannot be known precisely because an effect can result from many different causes, exploring these possible pasts opens the way to numerous applications ranging from movies and games to paleogeography. We introduce unerosion, an attempt to recover plausible past topographies from an input terrain represented as a height field. Our solution relies on novel algorithms for the backward simulation of different processes: fluvial erosion, sedimentation, and thermal erosion. This is achieved by re‐formulating the equations of erosion and sedimentation so that they can be simulated back in time. These algorithms can be combined to account for a succession of climate changes backward in time, while the possible ambiguities provide editing options to the user. Results show that our solution can approximately reverse different types of erosion while enabling users to explore a variety of alternative pasts. Using a chronology of climatic periods to inform us about the main erosion phenomena, we also went back in time using real measured terrain data. We checked the consistency with geological findings, namely the height of river beds hundreds of thousands of years ago.

Erosion simulation is a common approach used for generating and authoring mountainous terrains. While water is considered the primary erosion factor, its simulation fails to capture steep slopes near the ridges. In these low-drainage areas, erosion is often approximated with slope-reducing erosion, which yields unrealistically uniform slopes. However, geomorphology observed that another process dominates the low-drainage areas: erosion by debris flow, which is a mixture of mud and rocks triggered by strong climatic events. We propose a new method to capture the interactions between debris flow and fluvial erosion thanks to a new mathematical formulation for debris flow erosion derived from geomorphology and a unified GPU algorithm for erosion and deposition. In particular, we observe that sediment and debris deposition tend to intersect river paths, which motivates the design of a new, approximate flow routing algorithm on the GPU to estimate the water path out of these newly formed depressions. We demonstrate that debris flow carves distinct patterns in the form of erosive scars on steep slopes and cones of deposited debris competing with fluvial erosion downstream.

The real world exhibits rich structure and detail across many scales of observation. It is difficult, however, to capture and represent a broad spectrum of scales using ordinary images. We devise a novel paradigm for learning a representation that captures an orders-of-magnitude variety of scales from an unstructured collection of ordinary images. We treat this collection as a distribution of scale-space slices to be learned using adversarial training, and additionally enforce coherency across slices. Our approach relies on a multiscale generator with carefully injected procedural frequency content, which allows to interactively explore the emerging continuous scale space. Training across vastly different scales poses challenges regarding stability, which we tackle using a supervision scheme that involves careful sampling of scales. We show that our generator can be used as a multiscale generative model, and for reconstructions of scale spaces from unstructured patches. Significantly outperforming the state of the art, we demonstrate zoom-in factors of up to 256x at high quality and scale consistency.

Abstract Recent advances in Neural Radiance Fields enable the capture of scenes with motion. However, editing the motion is hard; no existing method allows editing beyond the space of motion existing in the original video, nor editing based on physics. We present the first approach that allows physically‐based editing of motion in a scene captured with a single hand‐held video camera, containing vibrating or periodic motion. We first introduce a Lagrangian representation, representing motion as the displacement of particles, which is learned while training a radiance field. We use these particles to create a continuous representation of motion over the sequence, which is then used to perform a modal analysis of the motion thanks to a Fourier transform on the particle displacement over time. The resulting extracted modes allow motion synthesis, and easy editing of the motion, while inheriting the ability for free‐viewpoint synthesis in the captured 3D scene from the radiance field. We demonstrate our new method on synthetic and real captured scenes.

Abstract The design of car shapes requires a delicate balance between aesthetic and performance. While fluid simulation provides the means to evaluate the aerodynamic performance of a given shape, its computational cost hinders its usage during the early explorative phases of design, when aesthetic is decided upon. We present an interactive system to assist designers in creating aerodynamic car profiles. Our system relies on a neural surrogate model to predict fluid flow around car shapes, providing fluid visualization and shape optimization feedback to designers as soon as they sketch a car profile. Compared to prior work that focused on time‐averaged fluid flows, we describe how to train our model on instantaneous, synchronized observations extracted from multiple pre‐computed simulations, such that we can visualize and optimize for dynamic flow features, such as vortices. Furthermore, we architectured our model to support gradient‐based shape optimization within a learned latent space of car profiles. In addition to regularizing the optimization process, this latent space and an associated encoder‐decoder allows us to input and output car profiles in a bitmap form, without any explicit parameterization of the car boundary. Finally, we designed our model to support pointwise queries of fluid properties around car shapes, allowing us to adapt computational cost to application needs. As an illustration, we only query our model along streamlines for flow visualization, we query it in the vicinity of the car for drag optimization, and we query it behind the car for vortex attenuation.

We introduce the first solution for simulating the formation and evolution of glaciers, together with their attendant erosive effects, for periods covering the combination of glacial and inter-glacial cycles. Our efficient solution includes both a fast yet accurate deep learning-based estimation of highorder ice flows and a new, multi-scale advection scheme enabling us to account for the distinct time scales at which glaciers reach equilibrium compared to eroding the terrain. We combine the resulting glacial erosion model with finer-scale erosive phenomena to account for the transport of debris flowing from cliffs. This enables us to model the formation of terrain shapes not previously adequately modeled in Computer Graphics, ranging from U-shaped and hanging valleys to fjords and glacial lakes.

Solving linear system of equations stemming from Laplacian operators is at the heart of a wide range of applications. Due to the sparsity of the linear systems, iterative solvers such as Conjugate Gradient and Multigrid are usually employed when the solution has a large number of degrees of freedom. These iterative solvers can be seen as sparse approximations of the Green’s function for the Laplacian operator. In this paper we propose a machine learning approach that regresses a Green’s function from boundary conditions. This is enabled by a Green’s function that can be effectively represented in a multi-scale fashion, drastically reducing the cost associated with a dense matrix representation. Additionally, since the Green’s function is solely dependent on boundary conditions, training the proposed neural network does not require sampling the right-hand side of the linear system. We show results that our method outperforms state of the art Conjugate Gradient and Multigrid methods.

Abstract Creative processes of artists often start with hand‐drawn sketches illustrating an object. Pre‐visualizing these keyframes is especially challenging when applied to volumetric materials such as smoke. The authored 3D density volumes must capture realistic flow details and turbulent structures, which is highly non‐trivial and remains a manual and time‐consuming process. We therefore present a method to compute a 3D smoke density field directly from 2D artist sketches, bridging the gap between early‐stage prototyping of smoke keyframes and pre‐visualization. From the sketch inputs, we compute an initial volume estimate and optimize the density iteratively with an updater CNN. Our differentiable sketcher is embedded into the end‐to‐end training, which results in robust reconstructions. Our training data set and sketch augmentation strategy are designed such that it enables general applicability. We evaluate the method on synthetic inputs and sketches from artists depicting both realistic smoke volumes and highly non‐physical smoke shapes. The high computational performance and robustness of our method at test time allows interactive authoring sessions of volumetric density fields for rapid prototyping of ideas by novice users.

Abstract Fluid control often uses optimization of control forces that are added to a simulation at each time step, such that the final animation matches a single or multiple target density keyframes provided by an artist. The optimization problem is strongly under‐constrained with a high‐dimensional parameter space, and finding optimal solutions is challenging, especially for higher resolution simulations. In this paper, we propose two novel ideas that jointly tackle the lack of constraints and high dimensionality of the parameter space. We first consider the fact that optimized forces are allowed to have divergent modes during the optimization process. These divergent modes are not entirely projected out by the pressure solver step, manifesting as unphysical smoke sources that are explored by the optimizer to match a desired target. Thus, we reduce the space of the possible forces to the family of strictly divergence‐free velocity fields, by optimizing directly for a vector potential. We synergistically combine this with a smoothness regularization based on a spectral decomposition of control force fields. Our method enforces lower frequencies of the force fields to be optimized first by filtering force frequencies in the Fourier domain. The mask‐growing strategy is inspired by Kolmogorov's theory about scales of turbulence. We demonstrate improved results for 2D and 3D fluid control especially in higher‐resolution settings, while eliminating the need for manual parameter tuning. We showcase various applications of our method, where the user effectively creates or edits smoke simulations.

We present a novel method for authoring landscapes with flora and fauna while considering their mutual interactions. Our algorithm outputs a steady-state ecosystem in the form of density maps for each species, their daily circuits, and a modified terrain with eroded trails from a terrain, climatic conditions, and species with related biological information. We introduce the Resource Access Graph, a new data structure that encodes both interactions between food chain levels and animals traveling between resources over the terrain. A novel competition algorithm operating on this data progressively computes a steady-state solution up the food chain, from plants to carnivores. The user can explore the resulting landscape, where plants and animals are instantiated on the fly, and interactively edit it by over-painting the maps. Our results show that our system enables the authoring of consistent landscapes where the impact of wildlife is visible through animated animals, clearings in the vegetation, and eroded trails. We provide quantitative validation with existing ecosystems and a user-study with expert paleontologist end-users, showing that our system enables them to author and compare different ecosystems illustrating climate changes over the same terrain while enabling relevant visual immersion into consistent landscapes.

Abstract Although an important component of natural scenes, the representation of skyscapes is often relatively simplistic. This can be largely attributed to the complexity of the thermodynamics underpinning cloud evolution and wind dynamics, which make interactive simulation challenging. We address this problem by introducing a novel layered model that encompasses both terrain and atmosphere, and supports efficient meteorological simulations. The vertical and horizontal layer resolutions can be tuned independently, while maintaining crucial inter‐layer thermodynamics, such as convective circulation and land‐air transfers of heat and moisture. In addition, we introduce a cloud‐form taxonomy for clustering, classifying and upsampling simulation cells to enable visually plausible, finely‐sampled volumetric rendering. As our results demonstrate, this pipeline allows interactive simulation followed by up‐sampled rendering of extensive skyscapes with dynamic clouds driven by consistent wind patterns. We validate our method by reproducing characteristic phenomena such as diurnal shore breezes, convective cells that contribute to cumulus cloud formation, and orographic effects from moist air driven upslope.

Abstract Terrains are a crucial component of three‐dimensional scenes and are present in many Computer Graphics applications. Terrain modeling methods focus on capturing landforms in all their intricate detail, including eroded valleys arising from the interplay of varied phenomena, dendritic mountain ranges, and complex river networks. Set against this visual complexity is the need for user control over terrain features, without which designers are unable to adequately express their artistic intent. This article provides an overview of current terrain modeling and authoring techniques, organized according to three categories: procedural modeling, physically‐based simulation of erosion and land formation processes, and example‐based methods driven by scanned terrain data. We compare and contrast these techniques according to several criteria, specifically: the variety of achievable landforms; realism from both a perceptual and geomorphological perspective; issues of scale in terms of terrain extent and sampling precision; the different interaction metaphors and attendant forms of user‐control, and computation and memory performance. We conclude with an in‐depth discussion of possible research directions and outstanding technical and scientific challenges.

Abstract We introduce a novel method for interactive generation of visually consistent, snow‐covered landscapes and provide control of their dynamic evolution over time. Our main contribution is the real‐time phenomenological simulation of avalanches and other user‐guided events, such as tracks left by Nordic skiing, which can be applied to interactively sculpt the landscape. The terrain is modeled as a height field with additional layers for stable, compacted, unstable, and powdery snow, which behave in combination as a semi‐viscous fluid. We incorporate the impact of several phenomena, including sunlight, temperature, prevailing wind direction, and skiing activities. The snow evolution includes snow‐melt and snow‐drift, which affect stability of the snow mass and the probability of avalanches. A user can shape landscapes and their evolution either with a variety of interactive brushes, or by prescribing events along a winter season time‐line. Our optimized GPU‐implementation allows interactive updates of snow type and depth across a large (10 × 10km) terrain, including real‐time avalanches, making this suitable for visual assets in computer games. We evaluate our method through perceptual comparison against exiting methods and real snow‐depth data.

Most mountain ranges are formed by the compression and folding of colliding tectonic plates. Subduction of one plate causes large-scale asymmetry while their layered composition (or stratigraphy) explains the multi-scale folded strata observed on real terrains. We introduce a novel interactive modeling technique to generate visually plausible, large scale terrains that capture these phenomena. Our method draws on both geological knowledge for consistency and on sculpting systems for user interaction. The user is provided hands-on control on the shape and motion of tectonic plates, represented using a new geologically-inspired model for the Earth crust. The model captures their volume preserving and complex folding behaviors under collision, causing mountains to grow. It generates a volumetric uplift map representing the growth rate of subsurface layers. Erosion and uplift movement are jointly simulated to generate the terrain. The stratigraphy allows us to render folded strata on eroded cliffs. We validated the usability of our sculpting interface through a user study, and compare the visual consistency of the earth crust model with geological simulation results and real terrains.

One challenge in portraying large-scale natural scenes in virtual environments is specifying the attributes of plants, such as species, size and placement, in a way that respects the features of natural ecosystems, while remaining computationally tractable and allowing user design. To address this, we combine ecosystem simulation with a distribution analysis of the resulting plant attributes to create biome-specific databases, indexed by terrain conditions, such as temperature, rainfall, sunlight and slope. For a specific terrain, interpolated entries are drawn from this database and used to interactively synthesize a full ecosystem, while retaining the fidelity of the original simulations. A painting interface supplies users with semantic brushes for locally adjusting ecosystem age, plant density and variability, as well as optionally picking from a palette of precomputed distributions. Since these brushes are keyed to the underlying terrain properties a balance between user control and real-world consistency is maintained. Our system can be be used to interactively design ecosystems up to 5 × 5 km2 in extent, or to automatically generate even larger ecosystems in a fraction of the time of a full simulation, while demonstrating known properties from plant ecology such as succession, self-thinning, and underbrush, across a variety of biomes.

We introduce a novel framework for interactive landscape authoring that supports bi-directional feedback between erosion and vegetation simulation. Vegetation and terrain erosion have strong mutual impact and their interplay influences the overall realism of virtual scenes. Despite their importance, these complex interactions have been neglected in computer graphics. Our framework overcomes this by simulating the effect of a variety of geomorphological agents and the mutual interaction between different material and vegetation layers, including rock, sand, humus, grass, shrubs, and trees. Users are able to exploit these interactions with an authoring interface that consistently shapes the terrain and populates it with details. Our method, validated through side-by-side comparison with real terrains, can be used not only to generate realistic static landscapes, but also to follow the temporal evolution of a landscape over a few centuries.

Abstract At large scale, landscapes result from the combination of two major processes: tectonics which generate the main relief through crust uplift, and weather which accounts for erosion. This paper presents the first method in computer graphics that combines uplift and hydraulic erosion to generate visually plausible terrains. Given a user‐painted uplift map, we generate a stream graph over the entire domain embedding elevation information and stream flow. Our approach relies on the stream power equation introduced in geology for hydraulic erosion. By combining crust uplift and stream power erosion we generate large realistic terrains at a low computational cost. Finally, we convert this graph into a digital elevation model by blending landform feature kernels whose parameters are derived from the information in the graph. Our method gives high‐level control over the large scale dendritic structures of the resulting river networks, watersheds, and mountains ridges.