Emil Praun

Spherical parametrization and remeshing

Recently, Gu et al. [2002] introduced geometry images, in which geometry is resampled into a completely regular 2D grid. The process involves cutting the surface into a disk using a network of cut paths, and then mapping the boundary of this disk to a square. Both geometry and other signals are stored as 2D grids, with grid samples in implicit correspondence, obviating the need to store a parametrization. Also, the boundary parametrization makes both geometry and textures seamless. The traditional approach for parametrizing a surface involves cutting it into charts and mapping these piecewise onto a planar domain. We introduce a robust technique for directly parametrizing a genus-zero surface onto a spherical domain. A key ingredient for making such a parametrization practical is the minimization of a stretch-based measure, to reduce scaledistortion and thereby prevent undersampling. Our second contribution is a scheme for sampling the spherical domain using uniformly subdivided polyhedral domains, namely the tetrahedron, octahedron, and cube. We show that these particular semiregular samplings can be conveniently represented as completely regular 2D grids, i.e. geometry images. Moreover, these images have simple boundary extension rules that aid many processing operations. Applications include geometry remeshing, level-ofdetail, morphing, compression, and smooth surface subdivision. In all three approaches, the surface is first cut into one or more disk-like charts using a network of cut paths, and a parametrization is formed piecewise on each chart. The a priori construction of the chart boundaries or cut paths is heuristic, and constrains the quality of the attainable parametrization. In texture atlases, both the number of charts and their surface extents are selected heuristically to minimize parametric distortion onto planar polygons, while also maintaining good packing efficiency. In semi-regular remeshing, surface charts are selected to have low-distortion maps onto regular domain faces, and to have approximately the same size. Finally, in geometry images, the surface is heuristically cut into a disk that hopefully maps well onto a square.

Lapped textures

We present for creating texture over an surface mesh using an example 2D texture. The approach is to identify interesting regions (texture patches) in the 2D example, and to repeatedly paste them onto the surface until it is completely covered. We call such a collection of overlapping patches a lapped texture. It is rendered using compositing operations, either into a traditional global texture map during a preprocess, or directly with the surface at runtime. The runtime compositing approach avoids resampling artifacts and drastically reduces texture memory requirements. Through a simple interface, the user specifies a tangential vector field over the surface, providing local control over the texture scale, and for anisotropic textures, the orientation. To paste a texture patch onto the surface, a surface patch is grown and parametrized over texture space. Specifically, we optimize the parametrization of each surface patch such that the tangential vector field aligns everywhere with the standard frame of the texture patch. We show that this optimization is solved efficiently as a sparse linear system.

Robust mesh watermarking

We describe a robust method for watermarking triangle meshes. Watermarking provides a mechanism for copyright protection of digital media by embedding information identifying the owner in the data. The bulk of the research on digital watermarks has focused on media such as images, video, audio, and text. Robust watermarks must be able to survive a variety of “attacks”, including resizing, cropping, and filtering. For resilience to such attacks, recent watermarking schemes employ a “spread-spectrum” approach – they transform the document to the frequency domain and perturb the coefficients of the perceptually most significant basis functions. We extend this spread-spectrum approach to work for the robust watermarking of arbitrary triangle meshes. Generalizing spread spectrum techniques to surfaces presents two major challenges. First, arbitrary surfaces lack a natural parametrization for frequency-based decomposition. Our solution is to construct a set of scalar basis function over the mesh vertices using multiresolution analysis. The watermark perturbs vertices along the direction of the surface normal, weighted by the basis functions. The second challenge is that simplification and other attacks may modify the connectivity of the mesh. We use an optimization technique to resample an attacked mesh using the original mesh connectivity. Results show that our watermarks are resistant to common mesh operations such as translation, rotation, scaling, cropping, smoothing, simplification, and resampling, as well as malicious attacks such as the insertion of noise, modification of low-order bits, or even insertion of other watermarks. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Surface Representations.

Real-time hatching

Drawing surfaces using hatching strokes simultaneously conveys material, tone, and form. We present a real-time system for non-photorealistic rendering of hatching strokes over arbitrary surfaces. During an automatic preprocess, we construct a sequence of mipmapped hatch images corresponding to different tones, collectively called a tonal art map. Strokes within the hatch images are scaled to attain appropriate stroke size and density at all resolutions, and are organized to maintain coherence across scales and tones. At runtime, hardware multitexturing blends the hatch images over the rendered faces to locally vary tone while maintaining both spatial and temporal coherence. To render strokes over arbitrary surfaces, we build a lapped texture parametrization where the overlapping patches align to a curvature-based direction field. We demonstrate hatching strokes over complex surfaces in a variety of styles.

Consistent mesh parameterizations

A basic element of Digital Geometry Processing algorithms is the establishment of a smooth parameterization for a given model. In this paper we propose an algorithm which establishes parameterizations for a set of models. The parameterizations are called consistent because they share the same base domain and respect features. They give immediate correspondences between models and allow remeshes with the same connectivity. Such remeshes form the basis for a large class of algorithms, including principal component analysis, wavelet transforms, detail and texture transfer between models, and n-way shape blending. We demonstrate the versatility of our algorithm with a number of examples.

Inter-surface mapping

We consider the problem of creating a map between two arbitrary triangle meshes. Whereas previous approaches compose parametrizations over a simpler intermediate domain, we directly create and optimize a continuous map between the meshes. Map distortion is measured with a new symmetric metric, and is minimized during interleaved coarse-to-fine refinement of both meshes. By explicitly favoring low inter-surface distortion, we obtain maps that naturally align corresponding shape elements. Typically, the user need only specify a handful of feature correspondences for initial registration, and even these constraints can be removed during optimization. Our method robustly satisfies hard constraints if desired. Inter-surface mapping is shown using geometric and attribute morphs. Our general framework can also be applied to parametrize surfaces onto simplicial domains, such as coarse meshes (for semi-regular remeshing), and octahedron and toroidal domains (for geometry image remeshing). In these settings, we obtain better parametrizations than with previous specialized techniques, thanks to our fine-grain optimization.

Building and using a scalable display wall system

Princetons scalable display wall project explores building and using a large-format display with commodity components. The prototype system has been operational since March 1998. Our goal is to construct a collaborative space that fully exploits a large-format display system with immersive sound and natural user interfaces. Our prototype system is built with low-cost commodity components: a cluster of PCs, PC graphics accelerators, consumer video and sound equipment, and portable presentation projectors. This approach has the advantages of low cost and of tracking technology well, as high-volume commodity components typically have better price-performance ratios and improve at faster rates than special-purpose hardware. We report our early experiences in building and using the display wall system. In particular, we describe our approach to research challenges in several specific research areas, including seamless tiling, parallel rendering, parallel data visualization, parallel MPEG decoding, layered multiresolution video input, multichannel immersive sound, user interfaces, application tools, and content creation.

Real-time fur over arbitrary surfaces

We introduce a method for real-time rendering of fur on surfaces of arbitrary topology. As a pre-process, we simulate virtual hair with a particle system, and sample it into a volume texture. Next, we parameterize the texture over a surface of arbitrary topology using “lapped textures” — an approach for applying a sample texture to a surface by repeatedly pasting patches of the texture until the surface is covered. The use of lapped textures permits specifying a global direction field for the fur over the surface. At runtime, the patches of volume textures are rendered as a series of concentric shells of semi-transparent medium. To improve the visual quality of the fur near silhouettes, we place “fins” normal to the surface and render these using conventional 2D texture maps sampled from the volume texture in the direction of hair growth. The method generates convincing imagery of fur at interactive rates for models of moderate complexity. Furthermore, the scheme allows real-time modification of viewing and lighting conditions, as well as local control over hair color, length, and direction. Additional

Gaussian transfer functions for multi-field volume visualization

Volume rendering is a flexible technique for visualizing dense 3D volumetric datasets. A central element of volume rendering is the conversion between data values and observable quantities such as color and opacity. This process is usually realized through the use of transfer functions that are precomputed and stored in lookup tables. For multidimensional transfer functions applied to multivariate data, these lookup tables become prohibitively large. We propose the direct evaluation of a particular type of transfer functions based on a sum of Gaussians. Because of their simple form (in terms of number of parameters), these functions and their analytic integrals along line segments can be evaluated efficiently on current graphics hardware, obviating the need for precomputed lookup tables. We have adopted these transfer functions because they are well suited for classification based on a unique combination of multiple data values that localize features in the transfer function domain. We apply this technique to the visualization of several multivariate datasets (CT, cryosection) that are difficult to classify and render accurately at interactive rates using traditional approaches.

Shape Compression using Spherical Geometry Images

We recently introduced an algorithm for spherical parametrization and remeshing, which allows resampling of a genus-zero surface onto a regular 2D grid, a spherical geometry image. These geometry images offer several advantages for shape compression. First, simple extension rules extend the square image domain to cover the infinite plane, thereby providing a globally smooth surface parametrization. The 2D grid structure permits use of ordinary image wavelets, including higher-order wavelets with polynomial precision. The coarsest wavelets span the entire surface and thus encode the lowest frequencies of the shape. Finally, the compression and decompression algorithms operate on ordinary 2D arrays, and are thus ideally suited for hardware acceleration. In this paper, we detail two wavelet-based approaches for shape compression using spherical geometry images, and provide comparisons with previous compression schemes.