Frame Permutation Quantization

Frame permutation quantization (FPQ) is a new vector quantization technique using finite frames. In FPQ, a vector is encoded using a permutation source code to quantize its frame expansion. This means that the encoding is a partial ordering of the frame expansion coefficients. Compared to ordinary permutation source coding, FPQ produces a greater number of possible quantization rates and a higher maximum rate. Various representations for the partitions induced by FPQ are presented, and reconstruction algorithms based on linear programming, quadratic programming, and recursive orthogonal projection are derived. Implementations of the linear and quadratic programming algorithms for uniform and Gaussian sources show performance improvements over entropy-constrained scalar quantization for certain combinations of vector dimension and coding rate. Monte Carlo evaluation of the recursive algorithm shows that mean-squared error (MSE) decays as 1/M^4 for an M-element frame, which is consistent with previous results on optimal decay of MSE. Reconstruction using the canonical dual frame is also studied, and several results relate properties of the analysis frame to whether linear reconstruction techniques provide consistent reconstructions.Comment: 29 pages, 5 figures; detailed added to proof of Theorem 4.3 and a few minor correction

Dilations of frames, operator valued measures and bounded linear maps

We will give an outline of the main results in our recent AMS Memoir, and include some new results, exposition and open problems. In that memoir we developed a general dilation theory for operator valued measures acting on Banach spaces where operator-valued measures (or maps) are not necessarily completely bounded. The main results state that any operator-valued measure, not necessarily completely bounded, always has a dilation to a projection-valued measure acting on a Banach space, and every bounded linear map, again not necessarily completely bounded, on a Banach algebra has a bounded homomorphism dilation acting on a Banach space. Here the dilation space often needs to be a Banach space even if the underlying space is a Hilbert space, and the projections are idempotents that are not necessarily self-adjoint. These results lead to some new connections between frame theory and operator algebras, and some of them can be considered as part of the investigation about "noncommutative" frame theory.Comment: Contemporary Mathematics, 21 pages. arXiv admin note: substantial text overlap with arXiv:1110.583

Lower bounds on volumes of hyperbolic Haken 3-manifolds

In this paper, we find lower bounds for volumes of hyperbolic 3-manifolds with various topological conditions. Let V_3 = 1.01494 denote the volume of a regular ideal simplex in hyperbolic 3-space. As a special case of the main theorem, if a hyperbolic manifold M contains an acylindrical surface S, then Vol(M)>= -2 V_3 chi(S). We also show that if beta_1(M)>= 2, then Vol(M)>= 4/5 V_3.Comment: 20 pages, 15 figure

Towards a High Quality Real-Time Graphics Pipeline

Modern graphics hardware pipelines create photorealistic images with high geometric complexity in real time. The quality is constantly improving and advanced techniques from feature film visual effects, such as high dynamic range images and support for higher-order surface primitives, have recently been adopted. Visual effect techniques have large computational costs and significant memory bandwidth usage. In this thesis, we identify three problem areas and propose new algorithms that increase the performance of a set of computer graphics techniques. Our main focus is on efficient algorithms for the real-time graphics pipeline, but parts of our research are equally applicable to offline rendering. Our first focus is texture compression, which is a technique to reduce the memory bandwidth usage. The core idea is to store images in small compressed blocks which are sent over the memory bus and are decompressed on-the-fly when accessed. We present compression algorithms for two types of texture formats. High dynamic range images capture environment lighting with luminance differences over a wide intensity range. Normal maps store perturbation vectors for local surface normals, and give the illusion of high geometric surface detail. Our compression formats are tailored to these texture types and have compression ratios of 6:1, high visual fidelity, and low-cost decompression logic. Our second focus is tessellation culling. Culling is a commonly used technique in computer graphics for removing work that does not contribute to the final image, such as completely hidden geometry. By discarding rendering primitives from further processing, substantial arithmetic computations and memory bandwidth can be saved. Modern graphics processing units include flexible tessellation stages, where rendering primitives are subdivided for increased geometric detail. Images with highly detailed models can be synthesized, but the incurred cost is significant. We have devised a simple remapping technique that allowsfor better tessellation distribution in screen space. Furthermore, we present programmable tessellation culling, where bounding volumes for displaced geometry are computed and used to conservatively test if a primitive can be discarded before tessellation. We introduce a general tessellation culling framework, and an optimized algorithm for rendering of displaced Bézier patches, which is expected to be a common use case for graphics hardware tessellation. Our third and final focus is forward-looking, and relates to efficient algorithms for stochastic rasterization, a rendering technique where camera effects such as depth of field and motion blur can be faithfully simulated. We extend a graphics pipeline with stochastic rasterization in spatio-temporal space and show that stochastic motion blur can be rendered with rather modest pipeline modifications. Furthermore, backface culling algorithms for motion blur and depth of field rendering are presented, which are directly applicable to stochastic rasterization. Hopefully, our work in this field brings us closer to high quality real-time stochastic rendering