Schwinn,C., Hildenbrand,D., Stock,F., Koch,A.
Abstract:
In recent years, Geometric Algebra (GA) has become more and more popular in fields of science and engineering due to its potential for compact algorithms. However, the execution of GA algorithms and the related need for high computational power is still the limiting factor for these algorithms to be used in practice. Therefore, it would be desirable to automatically detect parts that can be calculated in parallel by a software tool. In this paper, we present Gaalop 2.0, a Geometric Algebra Algorithm Compiler, which takes as input the description of a GA algorithm, symbolically optimizes the output multivectors and compiles the optimized code into a target language source file like C++, for instance. For each output multivector the code for non-zero coefficients is generated, which is finally adjusted to contain only basic arithmetic operations instead. This allows the optimized output to be compiled for different parallel computing platforms like FPGAs, for instance.
Panozzo,D., Puppo,E., Rocca,L.
Abstract:
We consider the problem of multi-scale estimation of principal curvatures and crease lines on a surface represented with a mesh of triangles and affected by noise. We show that curvature at different scales can be efficiently and accurately estimated by modifying a fitting technique and applying it to neighborhoods of various size, depending on scale: we discard bending portions of surfaces during fitting; and we apply Monte-Carlo sampling to speed up computation. Next we propose a new technique for extracting crease lines and we show how such lines can summarize shape features at the various scales. This is a first step towards building a scale-space of surface features.
Gunn,C.
Abstract:
We describe a visualization system in which the two classical noneuclidean spaces -- elliptic and hyperbolic -- are integrated as equal citizens along with euclidean space. Such software we call metric-neutral. After surveying previous work in this direction, we review the mathematical foundations, particularly the projective models for these spaces. We give an overview of the issues involved in converting euclidean visualization software to be metric-neutral, beginning with non-interactive issues before turning to interaction, and finally, to immersive environments. We describe how the metric-neutral visualization system under discussion solves these challenges, highlighting a number of innovative features, including metric-neutral tubing, metric-neutral realtime shading, and metric-neutral tracking.
Rosner,J., Fassold,H., Bailer,W., Schallauer,P.
Abstract:
Frame interpolation (the insertion of artificially generated images in a film sequence) is often used in post production to change the temporal duration of a sequence, e.g. to achieve a slow-motion effect. Most frame interpolation algorithms first calculate the motion field between two neighboring images and scale it appropriately. Afterwards, the images are warped (mapped) with the scaled motion field, and regions to which no source pixel was mapped are filled up (image inpainting). In this paper, we will focus on the latter two steps, the warping of the images and the image inpainting. We present simple and fast algorithms for image warping and inpainting, and discuss their efficient implementation to GPUs, using the NVIDIA CUDA technology. We compare the CPU and corresponding GPU routines and notice a speedup of approximately 6 - 10 for image warping and image inpainting. Significantly higher speedups can be expected for the latest NVIDIA GPU generation codenamed Fermi due to several architectonical improvements (faster atomic operations, L1/L2 cache). When comparing the result images of the CPU and GPU routine visually, practically no difference can be seen.
Bin,L., Goel,V., Peters,J.
Abstract:
We map reyes-rendering to the GPU by leveraging new features of modern GPUs exposed by the Microsoft DirectX11 API. That is, we show how to accelerate reyes-quality rendering for realistic real-time graphics. In detail, we discuss the implementation of the split-and-dice phase via the Geometry Shader, including bounds on displacement mapping; micropolygon rendering without standard rasterization; and the incorporation of motion blur and camera defocus as one pass rather than a multi-pass. Our implementation achieves interactive rendering rates on the Radeon 5000 series.
Hitzer,E.
Abstract:
We first review the definition of the angle between subspaces and how it is computed using matrix algebra. Then we introduce the Grassmann and Clifford algebra description of subspaces. The geometric product of two subspaces yields the full relative angular information in an explicit manner. We explain and interpret the result of the geometric product of subspaces gaining thus full access to the relative orientation information.
Charrier,P., Hildenbrand,D.
Abstract:
The focus of the this work is on the better integration of algorithms expressed in Conformal Geometric Algebra (CGA) in modern high level computer languages, namely C++ and NVIDIA's Compute Unified Device Architecture (CUDA). A high runtime performance in terms of CGA is achieved using symbolic optimizing through the invocation of Gaalop.
Goerlitz,A., Siebert,H., Hildenbrand,D.
Abstract:
Geometric Algebra (GA) is a mathematical framework that allows a compact and geometrically intuitive description of geometric relationships and algorithms. In this paper a translation, rotation and scale invariant algorithm for registration of color images and other multichannel data is introduced. The use of Geometric Algebra allows to generalize the well known Fourier Transform which is widely used for the registration of scalar fields. In contrast to the original algorithm our algorithm allows to handle vector valued data in an appropriate way. As a proof of concept the registration results for artificial, as well as for real world data, are discussed.
Kovacic,M., Guggeri,F., Marras,S., Scateni.,R.
Abstract:
In this paper we propose an optimization of the Shape Diameter Function (SDF) that we call Accelerated SDF (ASDF). We discuss in detail the advantages and disadvantages of the original SDF definition, proposing theoretical and practical approaches for speedup and approximation. Using Poisson-based interpolation we compute the SDF value for a small subset of randomly distributed faces and propagate the values over the mesh. We show the results obtained with ASDF versus SDF in terms of timings and error.
Comic,L., De Floriani,L., Iuricic,F.
Abstract:
Ascending and descending Morse complexes, defined by the critical points and integral lines of a scalar field $f$ defined on a manifold $M$, induce a subdivision of $M$ into regions of uniform gradient flow, and thus provide a compact description of the morphology of $f$ on $M$. We propose a dual representation for the ascending and descending Morse complexes of $f$ in arbitrary dimensions in terms of an incidence graph. We describe atomic simplification and refinement operators on the Morse complexes and we investigate the effect of those operators on the graph-based representation of the two complexes. Simplification and refinement operators form a basis for a hierarchical multi-resolution representation of
Morse complexes, from which it will be possible to dynamically extract representations of the morphology of the scalar field $f$ over $M$, at both uniform and variable resolutions.
Rockwood,A.
Abstract:
A method is developed that simultaneously interpolates scattered points and parametric curves in space. The curves do not have an obvious topological structure; the curves might be disconnected, self intersecting and randomly distributed. The method involves minimizing an energy functional based on distance to simple curve and point pre-images in parameter space. The minimization yields an analytic surface definition, which can be shown to generalize Shepard's formula to vector-valued parametric curves. T he method also allows derivative information to be included; thus slopes and curvatures along the curves could be part of the input .
Benger,W.; Ritter,M.
Abstract:
The differential geometry of curves is described via the means of the Frenet-Serret formulas, which cast first, second and third order derivatives into curvature and torsion. While in usual vector calculus these quantities are usually considered to be scalar values, formulating the Frenet-Serret equations exhibits that they are best described by a bivector for the curvature and a trivector for the torsion. The bi-vector curvature field is directly suitable for visualization of integration curves for vector fields such as stemming from computational fluid dynamics (CFD). These are known as Frenet Ribbons which are much richer in their visual expressiveness than lines. We review the Frenet-Serret formulas in the context of Geometric Algebra and discuss the set of quantities derived allows to study numerical pitfalls for computing Frenet Ribbons. and how to address them. The applicability of the technique is demonstrated upon a complex numerical CFD data set.