Research

Research of the computer graphics group at RWTH Aachen focuses on geometry acquisition and processing, on interactive visualization, and on related areas such as computer vision, photo-realistic image synthesis, and ultra high speed multimedia data transmission.

In our projects we are cooperating with various industry companies (e.g. BMW, Siemens, Philips, ...) as well as with academic research groups around the world. Additional funding sources are the Deutsche Forschungsgemeinschaft, the Bundesministerium für Bildung und Forschung, the European Union, and the German Israelian Foundation.

This page presents some examples of our current research projects. If you are interested in one of the topics please contact us by email for further detail information or check out our publications page where you can find papers, software, and additional material for download.

Computer Vision and 3D Reconstruction

The faithful digitization and digital reproduction of three dimensional real world objects is one of the fundamental challenges in computer graphics and computer vision. Although established technologies such as laser scanning are able to produce high quality 3D reconstructions, they still lack flexibility with respect to material and lighting conditions, are relatively expensive, and are often applicable only to restricted types of objects.

In this research area we are focusing on new techniques to reconstruct 3D objects from simple photos or video. We investigate new solutions to involved problems such as camera calibration, structure from motion, and methods for volumetric and explicit surface reconstruction.

During the last years we have developed new techniques to efficiently solve two of the central questions in image-based reconstruction. Our highly efficient and illumination invariant photo-consistency measure for image-based, volumetric 3D reconstruction allows us to compute reliable probability estimates whether the desired object surface passes through a specific region in space or not. Furthermore, we presented new methods for surface extraction from such photo-consistency volumes, which allow us to generate triangle meshes that are faithful reproductions of the real 3D object surface solely from images.

We also showed that our solutions are applicable to a number of other, difficult 3D reconstruction problems such as unoriented point clouds.

Geometry Processing and Modeling

3D model repair

A common dilemma in todays CAM production environments are the different geometry representations that are employed by CAD systems on the one hand and downstream applications like computational fluid- or structure simulation, rapid prototyping, and numerically controlled machining on the other hand. The conversion between different representations creates artifacts like gaps, holes, intersections and overlaps which have to be removed in tedious and often manual post-processing steps.

We are focussing on the development of new algorithms to efficiently solve the model repair problem. Our hybrid approaches combine the advantages of traditional surface-oriented and volumetric algorithms. We exploit the topological simplicity of a voxel grid to reconstruct a cleaned up surface in the vicinity of intersections and cracks, but keep the input tessellation in regions that are away from these inconsistencies. We are thus able to preserve the characteristic structure of the input tessellation, close gaps up to a user-defined maximum diameter, resolve intersections, handle incompatible patch orientations and produce a feature-sensitive, manifold output that stays within a prescribed error-tolerance to the input model.

Dynamic simulation of physical behaviour

In many applications (prototyping and simulation, movies, games, ... ) the physically correct behavior of 3D objects is an improtant issue. This behavior should for example consider the dynamics of rigid bodies. Here objects can move through space, collide with each other and are liable to the laws of friction. Once one can simulate that, there is the possibility to add more degrees of freedom, e.g. object deformation by external and internal forces. With finite element methods these deformations can be handled in an efficient way. If we further increase the deformable character of the objects we may get materials like fluids or gases. Once all these phenomenas can be handled, a next step would be to simulate the interaction between all these different matters.

Our goal in this context one the one hand is to integrate as many of these physical effects as possible to gain maximum physical correctness and on the other hand we want do to this as fast as possible, i.e. in realtime. Clearly there is a tradeoff between these goals. Desirable are algorithms which cover the whole range of application whereas the tradeoff between efficiency and precision can be controlled by a simple parameter.

Multiresolution modeling

When modeling geometric objects it is important to be able to switch between different levels of resolution of the object: At one time one might want to create fine details, like the eyes of a character, another time the designer may want to change the overall shape of the object without losing these details. This problem requires a suitable internal representation of the object and algorithms that enable to switch between the various levels of detail. In fact a decomposition of the geometric shape into disjoint "frequency bands" is necessary (multiresolution representation). If the different levels of detail are defined relative to each other then we have a fully hierarchical representation of the underlying object.

Our research is focussing on algorithms that allow the designer to interactively edit a freeform object on arbitrary levels of resolution. Besides the underlying mathematical methods we are investigating new design metaphors to facilitate the handling of multiresolution surfaces within an interactive framework.

Subdivision surfaces

Subdivision schemes have become increasingly popular in recent years because they provide a simple and efficient construction of smooth curves and surfaces. In contrast to plain piecewise polynomial representations like Bezier patches and NURBS, subdivision schemes can easily represent smooth surfaces of arbitrary topology.

At our institute we are investigating the analysis of subdivision surfaces as well as their applications in geometric modeling and other computer graphics areas. We develop new subdivision schemes that accomodate specific requirements and carry over classical NURBS tools to subdivision surfaces.

Quad-dominant remeshing

Remeshing algorithms are fundamental for the generation of high-quality CAD models in rapid prototyping, reverse engineering and conceptual design. Traditionally, remeshing algorithms where focused on improving the local mesh characteristics, i.e., on generating mesh elements with optimal shape and regularity (nearly quadratic). Recently the focus has widened to also incorporate structural considerations, i.e. the alignment of the elements to mesh features like edges and corners. We propose a two-step approach: In a first step the model is segmented into a set of (almost) planar regions that capture the overall structure of the model. In the second step, these regions are remeshed while taking consisteny constraints along their abutting boundaries into account. This way we obtain a high-quality, quad-dominant remesh that properly reflects the features of the input model.

Tool-path generation

In 3-axis CNC milling, excess material is removed slice by slice by a tool head from a solid block of material. A number of different strategies have been explored which generally try to optimize the tool-paths to achieve high throughput at low machine wear-off. Classical approaches for computing the tool-paths are plagued by various numerical problems that result in poor performance and complex program architectures.

We adopt an alternative approach that exploits the computing power of modern GPUs to achieve fast and robust algorithms. Our idea is to compute the signed distance field to a given contour and extract the tool-paths as isocurves of the field. This approach avoids the handling of special cases and leads to unprecedented speed when implemented in parallel on a graphics processor.

Currently we are working on a generalization of these techniques to 5-axis machining.

Global Illumination

Photorealistic image synthesis

Using global illumination techniques, we are able to generate photorealistic images and movie sequences. This is important in areas of visualization, architecture, advertisement, movies and video games. Both offline and real-time rendering are of interest, and we in particular focus on real-time global illumination which can be used to create even more convincing video games and a stunning virtual reality for the user.

Radio wave propagation

Global illumination can also be applied to compute the propagation of radio waves, e.g. in buildings or urban environments. This is for example important in the context of wireless networks or mobile phone networks. Predicting reception quality of mobile nodes and simulation of mobile networks is a hot topic in the network computing community. With the knowledge gained from the image synthesis aspects of global illumination, we are able to create accurate and physics based solutions to the wave propagation problem.

Image Processing

Image completion / texture synthesis

If the processing of digital images is supposed to go beyond mere filtering, one of the most basic operations is to add or remove objects from a given image. While adding an object to the foreground of an image is comparably easy (if we ignore relighting issues), removing an object is usually quite difficult because the now-visible background has to be reconstructed properly in order to hide the modification to the viewer.

Besides image completion approaches that are based on filtering or texture-synthesis, the fragment-based approaches have been shown to lead to the most convincing results. In these approaches the image regions, where foreground objects have been removed, are filled by iteratively copying small patches (source fragments) of the input image to the boundary of the unknown region (target fragment) until they completely cover the unknown region.

We have developed a new approach for computing fragment-based image completion which enables its integration into an interactive inpainting tool.

Character animation from 2D pictures and 3D motion data

In recent years, research on two-dimensional image manipulation has received a huge amount of interest. Very powerful solutions for problems such as matting, image completion, texture synthesis, or rigid image manipulation have been presented. Based on these and similar methods it has now become possible to explore interesting new ideas to re-animate still pictures.

In this project we want to take the idea of creating animations directly in image space one step further by making photographed persons move. We developed a purely image based approach to combine real images with real 3D motion data in order to generate visually convincing animations of 2D characters from only a single image.

For animated Videos and the corresponding paper, see also:
Character Animation from 2D Pictures and 3D Motion Data

Medical Applications

Shape extraction with control over geometry and topology

Extracting isosurfaces from volumetric datasets is an essential step for indirect volume rendering algorithms. For physically measured data like it is used, e.g. in medical imaging applications one often introduces topological errors such as small handles that stem from measurement inaccuracy and cavities that are generated by tight folds of an organ. During isosurface extraction these measurement errors result in a surface whose genus is much higher than that of the actual surface. In many cases however, the topological type of the object under consideration is known beforehand, e.g., the cortex of a human brain is always homeomorphic to a sphere.

We investigate algorithms for shape extraction that allow the user to have control over the topology of the reconstructed surface and at the same time guarantee high geometric fidelity. We do not only consider binary volumes but also look for generalizations to multi-valued volume datasets.

High quality and realtime volume visualization

Direct volume rendering algorithms often suffer from occlusion artefacts, e.g. when a certain region of interest like the cerebral cortex is occluded by other regions like the cranial bone. However, their advantage is that they allow to compute visualizations of the 3D image in a flexible and efficient way. On the other hand, surfaces which are extracted to be used for indirect visualization methods have a well defined geometry and therefore can be used for generating lighting and shadow computations. However, isosurfaces might be made of multiple connected components, too, and thus occlusions might also occur in this case.

An alternative approach is based on a re-labeling of the input volume's set of isosurfaces which allows the user to peel off the outer layers and to distinguish unconnected voxel components which happen to have the same voxel values. We use these new labels to mask out certain components and render the original data directly. The masking process is implemented on the GPU which enables interactive frame rates even when the selection of the user changes dynamically. The integration of lighting and shadows is also possible and clearly enhances the 3D impression of the image.

Virtual reality and motion tracking

The digitization of 3D human motion data plays a central role in a variety of applications, such as immersive interfaces for virtual environments, motion analysis in medical contexts, or character animation in movies or games. The process of capturing and processing motion data is a computationally complex task which usually requires a significant amount of manual pre- and post-processing even for established techniques such as optical motion capture systems. This necessity for expert knowledge makes current motion capturing techniques too inflexible in many potential application scenarios.

In this project we develop techniques to support the use of motion capture as a general, flexible input device without the above mentioned restrictions. We implement a self-calibrating framework for optical motion capture, enabling the reconstruction and tracking of arbitrary articulated objects in real-time. Our method automatically estimates relevant model parameters on-the-fly without any information on the initial tracking setup or the distribution of optical markers, and computes the geometry and topology of multiple tracked skeletons.

Multimedia Data Processing

Efficient spectral watermarking of 3D meshes

Digital watermarking techniques as they are known from classical media like images, videos and sound become more and more important in geometric models to enable copyright protection and ownership verification. In particular, the widely used spread-spectrum methods can be generalized to 3D datasets but are often far too slow to cope with very large meshes due to their complex computations.

We explore alternative spectral watermarking schemes that are based on orthogonalized radial basis functions. Our algorithm achieves high resistance against various real-world attacks but even more importantly, it runs faster by two orders of magnitude and thus can efficiently watermark even very large models.

Point Based Graphics

High-quality progressive surface splatting

Because of their conceptual simplicity and superior flexibility, point-based geometries evolved into a valuable alternative to surface representations based on polygonal meshes. In particular elliptical surface splats were shown to allow for high-quality anti-aliased rendering by sophisticated EWA filtering. Our research focuses on improving the efficiency and visual quality of these algorithms in two ways: First, we leverage as much as possible of the involved computations to the GPU which allows us to generate high-quality, anti-aliased and dynamically shaded visualizations at a rate of of more than 20M elliptical splats per second. Second, we impose a hierarchical structure on the set of input points that allows us to adapt the object complexity to the requirements and features of the target hardware in a multi-resolution approach.

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Home Research Projects UROP Who is who Address Teaching Publications Jobs Theses Activities	Research Research of the computer graphics group at RWTH Aachen focuses on geometry acquisition and processing, on interactive visualization, and on related areas such as computer vision, photo-realistic image synthesis, and ultra high speed multimedia data transmission. In our projects we are cooperating with various industry companies (e.g. BMW, Siemens, Philips, ...) as well as with academic research groups around the world. Additional funding sources are the Deutsche Forschungsgemeinschaft, the Bundesministerium für Bildung und Forschung, the European Union, and the German Israelian Foundation. This page presents some examples of our current research projects. If you are interested in one of the topics please contact us by email for further detail information or check out our publications page where you can find papers, software, and additional material for download. Computer Vision and 3D Reconstruction The faithful digitization and digital reproduction of three dimensional real world objects is one of the fundamental challenges in computer graphics and computer vision. Although established technologies such as laser scanning are able to produce high quality 3D reconstructions, they still lack flexibility with respect to material and lighting conditions, are relatively expensive, and are often applicable only to restricted types of objects. In this research area we are focusing on new techniques to reconstruct 3D objects from simple photos or video. We investigate new solutions to involved problems such as camera calibration, structure from motion, and methods for volumetric and explicit surface reconstruction. During the last years we have developed new techniques to efficiently solve two of the central questions in image-based reconstruction. Our highly efficient and illumination invariant photo-consistency measure for image-based, volumetric 3D reconstruction allows us to compute reliable probability estimates whether the desired object surface passes through a specific region in space or not. Furthermore, we presented new methods for surface extraction from such photo-consistency volumes, which allow us to generate triangle meshes that are faithful reproductions of the real 3D object surface solely from images. We also showed that our solutions are applicable to a number of other, difficult 3D reconstruction problems such as unoriented point clouds. See also: Iterative Multi-View Plane Fitting Hierarchical Volumetric Multi-view Stereo Reconstruction of Manifold Surfaces based on Dual Graph Embedding Robust and Efficient Photo-Consistency Estimation for Volumetric 3D Reconstruction Robust Reconstruction of Watertight 3D Models from Non-uniformly Sampled Point Clouds Without Normal Information Geometry Processing and Modeling 3D model repair A common dilemma in todays CAM production environments are the different geometry representations that are employed by CAD systems on the one hand and downstream applications like computational fluid- or structure simulation, rapid prototyping, and numerically controlled machining on the other hand. The conversion between different representations creates artifacts like gaps, holes, intersections and overlaps which have to be removed in tedious and often manual post-processing steps. We are focussing on the development of new algorithms to efficiently solve the model repair problem. Our hybrid approaches combine the advantages of traditional surface-oriented and volumetric algorithms. We exploit the topological simplicity of a voxel grid to reconstruct a cleaned up surface in the vicinity of intersections and cracks, but keep the input tessellation in regions that are away from these inconsistencies. We are thus able to preserve the characteristic structure of the input tessellation, close gaps up to a user-defined maximum diameter, resolve intersections, handle incompatible patch orientations and produce a feature-sensitive, manifold output that stays within a prescribed error-tolerance to the input model. See also: Structure Preserving CAD Model Repair Automatic Restoration of Polygon Models Dynamic simulation of physical behaviour In many applications (prototyping and simulation, movies, games, ... ) the physically correct behavior of 3D objects is an improtant issue. This behavior should for example consider the dynamics of rigid bodies. Here objects can move through space, collide with each other and are liable to the laws of friction. Once one can simulate that, there is the possibility to add more degrees of freedom, e.g. object deformation by external and internal forces. With finite element methods these deformations can be handled in an efficient way. If we further increase the deformable character of the objects we may get materials like fluids or gases. Once all these phenomenas can be handled, a next step would be to simulate the interaction between all these different matters. Our goal in this context one the one hand is to integrate as many of these physical effects as possible to gain maximum physical correctness and on the other hand we want do to this as fast as possible, i.e. in realtime. Clearly there is a tradeoff between these goals. Desirable are algorithms which cover the whole range of application whereas the tradeoff between efficiency and precision can be controlled by a simple parameter. Multiresolution modeling When modeling geometric objects it is important to be able to switch between different levels of resolution of the object: At one time one might want to create fine details, like the eyes of a character, another time the designer may want to change the overall shape of the object without losing these details. This problem requires a suitable internal representation of the object and algorithms that enable to switch between the various levels of detail. In fact a decomposition of the geometric shape into disjoint "frequency bands" is necessary (multiresolution representation). If the different levels of detail are defined relative to each other then we have a fully hierarchical representation of the underlying object. Our research is focussing on algorithms that allow the designer to interactively edit a freeform object on arbitrary levels of resolution. Besides the underlying mathematical methods we are investigating new design metaphors to facilitate the handling of multiresolution surfaces within an interactive framework. See also: An Intuitive Framework for Real-Time Freeform Modeling Interactive Multi-Resolution Modeling on Arbitrary Meshes GPU-Based Multiresolution Deformation Using Approximate Normal Field Reconstruction PriMo: Coupled Prisms for Intuitive Surface Modeling Subdivision surfaces Subdivision schemes have become increasingly popular in recent years because they provide a simple and efficient construction of smooth curves and surfaces. In contrast to plain piecewise polynomial representations like Bezier patches and NURBS, subdivision schemes can easily represent smooth surfaces of arbitrary topology. At our institute we are investigating the analysis of subdivision surfaces as well as their applications in geometric modeling and other computer graphics areas. We develop new subdivision schemes that accomodate specific requirements and carry over classical NURBS tools to subdivision surfaces. See also: Interpolatory Subdivision for Open quadrilateral meshes Sqrt(3) subdivision Quad-dominant remeshing Remeshing algorithms are fundamental for the generation of high-quality CAD models in rapid prototyping, reverse engineering and conceptual design. Traditionally, remeshing algorithms where focused on improving the local mesh characteristics, i.e., on generating mesh elements with optimal shape and regularity (nearly quadratic). Recently the focus has widened to also incorporate structural considerations, i.e. the alignment of the elements to mesh features like edges and corners. We propose a two-step approach: In a first step the model is segmented into a set of (almost) planar regions that capture the overall structure of the model. In the second step, these regions are remeshed while taking consisteny constraints along their abutting boundaries into account. This way we obtain a high-quality, quad-dominant remesh that properly reflects the features of the input model. See also: Direct Anisotropic Quad-Dominant Remeshing A Robust Two-Step Procedure for Quad-Dominant Remeshing Tool-path generation In 3-axis CNC milling, excess material is removed slice by slice by a tool head from a solid block of material. A number of different strategies have been explored which generally try to optimize the tool-paths to achieve high throughput at low machine wear-off. Classical approaches for computing the tool-paths are plagued by various numerical problems that result in poor performance and complex program architectures. We adopt an alternative approach that exploits the computing power of modern GPUs to achieve fast and robust algorithms. Our idea is to compute the signed distance field to a given contour and extract the tool-paths as isocurves of the field. This approach avoids the handling of special cases and leads to unprecedented speed when implemented in parallel on a graphics processor. Currently we are working on a generalization of these techniques to 5-axis machining. Global Illumination Photorealistic image synthesis Using global illumination techniques, we are able to generate photorealistic images and movie sequences. This is important in areas of visualization, architecture, advertisement, movies and video games. Both offline and real-time rendering are of interest, and we in particular focus on real-time global illumination which can be used to create even more convincing video games and a stunning virtual reality for the user. Radio wave propagation Global illumination can also be applied to compute the propagation of radio waves, e.g. in buildings or urban environments. This is for example important in the context of wireless networks or mobile phone networks. Predicting reception quality of mobile nodes and simulation of mobile networks is a hot topic in the network computing community. With the knowledge gained from the image synthesis aspects of global illumination, we are able to create accurate and physics based solutions to the wave propagation problem. See also: Wave Propagation Using the Photon Path Map The Effect of the Radio Wave Propagation Model in Mobile Ad Hoc Networks Image Processing Image completion / texture synthesis If the processing of digital images is supposed to go beyond mere filtering, one of the most basic operations is to add or remove objects from a given image. While adding an object to the foreground of an image is comparably easy (if we ignore relighting issues), removing an object is usually quite difficult because the now-visible background has to be reconstructed properly in order to hide the modification to the viewer. Besides image completion approaches that are based on filtering or texture-synthesis, the fragment-based approaches have been shown to lead to the most convincing results. In these approaches the image regions, where foreground objects have been removed, are filled by iteratively copying small patches (source fragments) of the input image to the boundary of the unknown region (target fragment) until they completely cover the unknown region. We have developed a new approach for computing fragment-based image completion which enables its integration into an interactive inpainting tool. See also: Image Completion with Perspective Correction Character animation from 2D pictures and 3D motion data In recent years, research on two-dimensional image manipulation has received a huge amount of interest. Very powerful solutions for problems such as matting, image completion, texture synthesis, or rigid image manipulation have been presented. Based on these and similar methods it has now become possible to explore interesting new ideas to re-animate still pictures. In this project we want to take the idea of creating animations directly in image space one step further by making photographed persons move. We developed a purely image based approach to combine real images with real 3D motion data in order to generate visually convincing animations of 2D characters from only a single image. For animated Videos and the corresponding paper, see also: Character Animation from 2D Pictures and 3D Motion Data Medical Applications Shape extraction with control over geometry and topology Extracting isosurfaces from volumetric datasets is an essential step for indirect volume rendering algorithms. For physically measured data like it is used, e.g. in medical imaging applications one often introduces topological errors such as small handles that stem from measurement inaccuracy and cavities that are generated by tight folds of an organ. During isosurface extraction these measurement errors result in a surface whose genus is much higher than that of the actual surface. In many cases however, the topological type of the object under consideration is known beforehand, e.g., the cortex of a human brain is always homeomorphic to a sphere. We investigate algorithms for shape extraction that allow the user to have control over the topology of the reconstructed surface and at the same time guarantee high geometric fidelity. We do not only consider binary volumes but also look for generalizations to multi-valued volume datasets. See also: Extracting consistent and manifold interfaces from multi-valued volume data sets Sub-Voxel Topology Control for Level Set Surfaces High quality and realtime volume visualization Direct volume rendering algorithms often suffer from occlusion artefacts, e.g. when a certain region of interest like the cerebral cortex is occluded by other regions like the cranial bone. However, their advantage is that they allow to compute visualizations of the 3D image in a flexible and efficient way. On the other hand, surfaces which are extracted to be used for indirect visualization methods have a well defined geometry and therefore can be used for generating lighting and shadow computations. However, isosurfaces might be made of multiple connected components, too, and thus occlusions might also occur in this case. An alternative approach is based on a re-labeling of the input volume's set of isosurfaces which allows the user to peel off the outer layers and to distinguish unconnected voxel components which happen to have the same voxel values. We use these new labels to mask out certain components and render the original data directly. The masking process is implemented on the GPU which enables interactive frame rates even when the selection of the user changes dynamically. The integration of lighting and shadows is also possible and clearly enhances the 3D impression of the image. Virtual reality and motion tracking The digitization of 3D human motion data plays a central role in a variety of applications, such as immersive interfaces for virtual environments, motion analysis in medical contexts, or character animation in movies or games. The process of capturing and processing motion data is a computationally complex task which usually requires a significant amount of manual pre- and post-processing even for established techniques such as optical motion capture systems. This necessity for expert knowledge makes current motion capturing techniques too inflexible in many potential application scenarios. In this project we develop techniques to support the use of motion capture as a general, flexible input device without the above mentioned restrictions. We implement a self-calibrating framework for optical motion capture, enabling the reconstruction and tracking of arbitrary articulated objects in real-time. Our method automatically estimates relevant model parameters on-the-fly without any information on the initial tracking setup or the distribution of optical markers, and computes the geometry and topology of multiple tracked skeletons. See also: Self-Calibrating Optical Motion Tracking for Articulated Bodies Multimedia Data Processing Efficient spectral watermarking of 3D meshes Digital watermarking techniques as they are known from classical media like images, videos and sound become more and more important in geometric models to enable copyright protection and ownership verification. In particular, the widely used spread-spectrum methods can be generalized to 3D datasets but are often far too slow to cope with very large meshes due to their complex computations. We explore alternative spectral watermarking schemes that are based on orthogonalized radial basis functions. Our algorithm achieves high resistance against various real-world attacks but even more importantly, it runs faster by two orders of magnitude and thus can efficiently watermark even very large models. See also: Efficient Spectral Watermarking of Large Meshes with Orthogonal Basis Functions Point Based Graphics High-quality progressive surface splatting Because of their conceptual simplicity and superior flexibility, point-based geometries evolved into a valuable alternative to surface representations based on polygonal meshes. In particular elliptical surface splats were shown to allow for high-quality anti-aliased rendering by sophisticated EWA filtering. Our research focuses on improving the efficiency and visual quality of these algorithms in two ways: First, we leverage as much as possible of the involved computations to the GPU which allows us to generate high-quality, anti-aliased and dynamically shaded visualizations at a rate of of more than 20M elliptical splats per second. Second, we impose a hierarchical structure on the set of input points that allows us to adapt the object complexity to the requirements and features of the target hardware in a multi-resolution approach. See also: High-Quality Surface Splatting on Today's GPUs Phong Splatting Efficient high quality rendering of point sampled geometry
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