Search

Interpolation is a foundational concept in scientific computing and is at the heart of many scientific visualization techniques. There is usually a tradeoff between the approximation capabilities of an interpolation scheme and its evaluation efficiency. For many applications, it is important for a user to navigate their data in real time. In practice, evaluation efficiency outweighs any incremental improvements in reconstruction fidelity. We first analyze, from a general standpoint, the use of compact piece-wise polynomial basis functions to efficiently interpolate data that is sampled on a lattice. We then detail our automatic code-generation framework on both CPU and GPU architectures. Specifically, we propose a general framework that can produce a fast evaluation scheme by analyzing the algebro-geometric structure of the convolution sum for a given lattice and basis function combination. We demonstrate the utility and generality of our framework by providing fast implementations of various box splines on the Body Centered and Face Centered Cubic lattices, as well as some non-separable box splines on the Cartesian lattice. We also provide fast implementations for certain Voronoi-splines that have not yet appeared in the literature. Finally, we demonstrate that this framework may also be used for non-Cartesian lattices in 4D. [ABSTRACT FROM AUTHOR]

For the class of non-degenerate box splines, we present a set construction scheme that separably decomposes the Green’s function of a box spline, yielding its explicit piecewise polynomial form. While it is possible to use the well known recursive formulation to obtain these polynomial pieces, that procedure is quite expensive. We prove that, under certain conditions, our decomposition procedure is asymptotically orders of magnitude lower than the recursive procedure. This allows us to evaluate box splines with more direction vectors than what would be feasible under the recursive scheme. Finally, using the explicit polynomials in each region of the box spline, we show how to create fast evaluation schemes using this explicit characterization and a spatial data structure. [ABSTRACT FROM AUTHOR]

Compressive rendering refers to the process of reconstructing a full image from a small subset of the rendered pixels, thereby expediting the rendering task. In this paper, we empirically investigate three image order techniques for compressive rendering that are suitable for direct volume rendering. The first technique is based on the theory of compressed sensing and leverages the sparsity of the image gradient in the Fourier domain. The latter techniques exploit smoothness properties of the rendered image; the second technique recovers the missing pixels via a total variation minimization procedure while the third technique incorporates a smoothness prior in a variational reconstruction framework employing interpolating cubic B-splines. We compare and contrast the three techniques in terms of quality, efficiency and sensitivity to the distribution of pixels. Our results show that smoothness-based techniques significantly outperform techniques that are based on compressed sensing and are also robust in the presence of highly incomplete information. We achieve high quality recovery with as little as 20% of the pixels distributed uniformly in screen space. [ABSTRACT FROM AUTHOR]

Within the context of emission tomography, we study volumetric reconstruction methods based on the Expectation Maximization (EM) algorithm. We show, for the first time, the equivalence of the standard implementation of the EM-based reconstruction with an implementation based on hardware-accelerated volume rendering for nearest-neighbor (NN) interpolation. This equivalence suggests that higher-order kernels should be used with caution and do not necessarily lead to better performance. We also show that the EM algorithm can easily be adapted for different lattices, the body-centered cubic (BCC) one in particular. For validation purposes, we use the 3D version of the Shepp-Logan synthetic phantom, for which we derive closed-form analytical expressions of the projection data. The experimental results show the theoretically-predicted optimality of NN interpolation in combination with the EM algorithm, for both the noiseless and the noisy case. Moreover, reconstruction on the BCC lattice leads to superior accuracy, more compact data representation, and better noise reduction compared to the Cartesian one. Finally, we show the usefulness of the proposed method for optical projection tomography of a mouse embryo. [ABSTRACT FROM AUTHOR]

According to the circle-packing theorem, the packing efficiency of a hexagonal lattice is higher than an equivalent square tessellation. Consequently, in several contexts, hexagonally sampled images compared to their Cartesian counterparts are better at preserving information content. In this paper, novel mapping techniques alongside the wavelet compression scheme are presented for hexagonal images. Specifically, we introduce two tree-based coding schemes, referred to as SBHex (spirally-mapped branch-coding for hexagonal images) and BBHex (breadth-first block-coding for hexagonal images). Both of these coding schemes respect the geometry of the hexagonal lattice and yield better compression results. Our empirical results show that the proposed algorithms for hexagonal images produce better reconstruction quality at low bits per pixel representations compared to the tree-based coding counterparts for the Cartesian grid. • Tree-based hexagonal wavelet sub-band coding scheme for hexagonally sampled images. • Spiral wavelet tree rearrangement that preserves spatial coherence of coefficients. • Traversal scheme and parent-to-children relationships that respect hexagonal geometry. • Compressed files that are upto 2.5 times smaller compared to similar Cartesian schemes. • Improved quality at low bit rates compared to similar Cartesian schemes. [ABSTRACT FROM AUTHOR]