Usman R. Alim

The Lattice-Boltzmann Method on Optimal Sampling Lattices

In this paper, we extend the single relaxation time lattice-Boltzmann method (LBM) to the 3D body-centered cubic (BCC) lattice. We show that the D3bQ15 lattice defined by a 15 neighborhood connectivity of the BCC lattice is not only capable of more accurately discretizing the velocity space of the continuous Boltzmann equation as compared to the D3Q15 Cartesian lattice, it also achieves a comparable spatial discretization with 30 percent less samples. We validate the accuracy of our proposed lattice by investigating its performance on the 3D lid-driven cavity flow problem and show that the D3bQ15 lattice offers significant cost savings while maintaining a comparable accuracy. We demonstrate the efficiency of our method and the impact on graphics and visualization techniques via the application of line-integral convolution on 2D slices as well as the extraction of streamlines of the 3D flow. We further study the benefits of our proposed lattice by applying it to the problem of simulating smoke and show that the D3bQ15 lattice yields more detail and turbulence at a reduced computational cost.

Gradient Estimation Revitalized

We investigate the use of a Fourier-domain derivative error kernel to quantify the error incurred while estimating the gradient of a function from scalar point samples on a regular lattice. We use the error kernel to show that gradient reconstruction quality is significantly enhanced merely by shifting the reconstruction kernel to the centers of the principal lattice directions. Additionally, we exploit the algebraic similarities between the scalar and derivative error kernels to design asymptotically optimal gradient estimation filters that can be factored into an infinite impulse response interpolation prefilter and a finite impulse response directional derivative filter. This leads to a significant performance gain both in terms of accuracy and computational efficiency. The interpolation prefilter provides an accurate scalar approximation and can be re-used to cheaply compute directional derivatives on-the-fly without the need to store gradients. We demonstrate the impact of our filters in the context of volume rendering of scalar data sampled on the Cartesian and Body-Centered Cubic lattices. Our results rival those obtained from other competitive gradient estimation methods while incurring no additional computational or storage overhead.

High-quality volumetric reconstruction on optimal lattices for computed tomography

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.

Decal-Maps: Real-Time Layering of Decals on Surfaces for Multivariate Visualization

We introduce the use of decals for multivariate visualization design. Decals are visual representations that are used for communication; for example, a pattern, a text, a glyph, or a symbol, transferred from a 2D-image to a surface upon contact. By creating what we define as decal-maps, we can design a set of images or patterns that represent one or more data attributes. We place decals on the surface considering the data pertaining to the locations we choose. We propose a (texture mapping) local parametrization that allows placing decals on arbitrary surfaces interactively, even when dealing with a high number of decals. Moreover, we extend the concept of layering to allow the co-visualization of an increased number of attributes on arbitrary surfaces. By combining decal-maps, color-maps and a layered visualization, we aim to facilitate and encourage the creative process of designing multivariate visualizations. Finally, we demonstrate the general applicability of our technique by providing examples of its use in a variety of contexts.

Compressive Volume Rendering

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.

CINAPACT-Splines: A Family of Infinitely Smooth, Accurate and Compactly Supported Splines

We introduce CINAPACT-splines, a class of \(C^\infty \), accurate and compactly supported splines. The integer translates of a CINAPACT-spline form a reconstruction space that can be tuned to achieve any order of accuracy. CINAPACT-splines resemble traditional B-splines in that higher orders of accuracy are achieved by successive convolutions with a B-spline of degree zero. Unlike B-splines however, the starting point for CINAPACT-splines is an infinitely smooth and compactly supported bump function that has been properly normalized so that it fulfills the partition of unity criterion. We use our construction to design two CINAPACT-splines, and explore their properties in the context of rendering volumetric data sampled on Cartesian grids. Our results show that CINAPACT-splines, while being infinitely smooth, are capable of providing similar reconstruction accuracy compared to some well-established filters of similar cost.

Compactly Supported Biorthogonal Wavelet Bases on the Body Centered Cubic Lattice

In this work, we present a family of compact, biorthogonal wavelet filter banks that are applicable to the Body Centered Cubic (BCC) lattice. While the BCC lattice has been shown to have superior approximation properties for volumetric data when compared to the Cartesian Cubic (CC) lattice, there has been little work in the way of designing wavelet filter banks that respect the geometry of the BCC lattice. Since wavelets have applications in signal de‐noising, compression, and sparse signal reconstruction, these filter banks are an important tool that addresses some of the scalability concerns presented by the BCC lattice. We use these filters in the context of volumetric data compression and reconstruction and qualitatively evaluate our results by rendering images of isosurfaces from compressed data.

A Quality-Preserving Cartesian to Body-Centered Cubic Downsampling Transform

The body-centered cubic lattice is the optimal sampling lattice in three dimensions. However, most volumetric datasets are acquired on the well-known Cartesian cubic lattice. In order to leverage the approximation capabilities of the body-centred cubic lattice, we propose a factor-of-four Cartesian to body-centered downsampling transform. We derive a Fourier domain post-aliasing error kernel and use it to optimize the cosine-weighted trilinear B-spline kernel. We demonstrate that our downsampling transform preserves fidelity when an oversampled function of interest is reconstructed with trilinear interpolation on the fine-scale Cartesian grid, and optimized cosine-weighted trilinear approximation on the coarse-scale body-centered cubic grid.

A closed PP form of box splines via Green’s function decomposition

Abstract 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.

Illustrative Multivariate Visualization for Geological Modelling

In this paper, we present a novel illustrative multivariate visualization for geological modelling to assist geologists and reservoir engineers in visualizing multivariate datasets in superimposed representations, in contrast to the single‐attribute visualizations supported by commercial software. Our approach extends the use of decals from a single surface to 3D irregular grids, using the layering concept to represent multiple attributes. We also build upon prior work to augment the design and implementation of different geological attributes (namely, rock type, porosity, and permeability). More specifically, we propose a new sampling strategy to generate decals for porosity on the geological grid, a hybrid visualization for permeability which combines 2D decals and 3D ellipsoid glyphs, and a perceptually‐based design that allows us to visualize additional attributes (e.g., oil saturation) while avoiding visual interference between layers. Furthermore, our visual design draws from traditional geological illustrations, facilitating the understanding and communication between interdisciplinary teams. An evaluation by domain experts highlights the potential of our approach for geological modelling and interpretation in this complex domain.