Alireza Entezari

Practical Box Splines for Reconstruction on the Body Centered Cubic Lattice

We introduce a family of box splines for efficient, accurate, and smooth reconstruction of volumetric data sampled on the body-centered cubic (BCC) lattice, which is the favorable volumetric sampling pattern due to its optimal spectral sphere packing property. First, we construct a box spline based on the four principal directions of the BCC lattice that allows for a linear C0 reconstruction. Then, the design is extended for higher degrees of continuity. We derive the explicit piecewise polynomial representations of the C0 and C2 box splines that are useful for practical reconstruction applications. We further demonstrate that approximation in the shift-invariant space - generated by BCC-lattice shifts of these box splines - is twice as efficient as using the tensor-product B-spline solutions on the Cartesian lattice (with comparable smoothness and approximation order and with the same sampling density). Practical evidence is provided demonstrating that the BCC lattice not only is generally a more accurate sampling pattern, but also allows for extremely efficient reconstructions that outperform tensor-product Cartesian reconstructions.

Linear and Cubic Box Splines for the Body Centered Cubic Lattice

We derive piecewise linear and piecewise cubic box spline reconstruction filters for data sampled on the body centered cubic (BCC) lattice. We analytically derive a time domain representation of these reconstruction filters and using the Fourier slice-projection theorem we derive their frequency responses. The quality of these filters, when used in reconstructing BCC sampled volumetric data, is discussed and is demonstrated with a raycaster. Moreover, to demonstrate the superiority of the BCC sampling, the resulting reconstructions are compared with those produced from similar filters applied to data sampled on the Cartesian lattice.

Efficient volume rendering on the body centered cubic lattice using box splines

We demonstrate that non-separable box splines deployed on body centered cubic lattices (BCC) are suitable for fast evaluation on present graphics hardware. Therefore, we develop the linear and quintic box splines using a piecewise polynomial (pp)-form as opposed to their currently known basis (B)-form. The pp-form lends itself to efficient evaluation methods such as de Boors algorithm for splines in box splines basis. Further on, we offer a comparison of quintic box splines with the only other interactive rendering available on BCC lattices that is based on separable kernels for interleaved Cartesian cubic (CC) lattices. While quintic box splines result in superior quality, interleaved CC lattices are still faster, since they can take advantage of the highly optimized circuitry for CC lattices, as it is the case in graphics hardware nowadays. This result is valid with and without prefiltering. Experimental results are shown for both a synthetic phantom and data from optical projection tomography. We provide shader code to ease the adaptation of box splines for the practitioner.

Box Spline Reconstruction On The Face-Centered Cubic Lattice

We introduce and analyze an efficient reconstruction algorithm for FCC-sampled data. The reconstruction is based on the 6-direction box spline that is naturally associated with the FCC lattice and shares the continuity and approximation order of the triquadratic B-spline. We observe less aliasing for generic level sets and derive special techniques to attain the higher evaluation efficiency promised by the lower degree and smaller stencil-size of the C1 6-direction box spline over the triquadratic B-spline.

A Geometric Construction of Multivariate Sinc Functions

We present a geometric framework for explicit derivation of multivariate sampling functions (sinc) on multidimensional lattices. The approach leads to a generalization of the link between sinc functions and the Lagrange interpolation in the multivariate setting. Our geometric approach also provides a frequency partition of the spectrum that leads to a nonseparable extension of the 1-D Shannon (sinc) wavelets to the multivariate setting. Moreover, we propose a generalization of the Lanczos window function that provides a practical and unbiased approach for signal reconstruction on sampling lattices. While this framework is general for lattices of any dimension, we specifically characterize all 2-D and 3-D lattices and show the detailed derivations for 2-D hexagonal body-centered cubic (BCC) and face-centered cubic (FCC) lattices. Both visual and numerical comparisons validate the theoretical expectations about superiority of the BCC and FCC lattices over the commonly used Cartesian lattice.

Shading for Fourier volume rendering

The work presented here describes two methods to incorporate viable illumination models into Fourier Volume Rendering (FVR). The lack of adequate illumination has been one of the impediments for the wide spread acceptance of FVR. Our first method adapts the Gamma Corrected Hemispherical Shading (GCHS) proposed by Scoggins et al. (2001) for FVR. We achieve interactive rendering for constant diffusive light sources. Our second method operates on data transformed by spherical harmonic functions. This latter approach allows for illumination under varying light directions. It should be noted that we only consider diffuse lighting in this paper. We demonstrate and compare the effect of these two new models on the rendered image and document speed and accuracy improvements.

Voronoi Splines

We introduce a framework for construction of non-separable multivariate splines that are geometrically tailored for general sampling lattices. Voronoi splines are B-spline-like elements that inherit the geometry of a sampling lattice from its Voronoi cell and generate a lattice-shift-invariant spline space for approximation in Rd. The spline spaces associated with Voronoi splines have guaranteed approximation order and degree of continuity. By exploiting the geometric properties of Voronoi polytopes and zonotopes, we establish the relationship between Voronoi splines and box splines which are used for a closed-form characterization of the former. For Cartesian lattices, Voronoi splines coincide with tensor-product B-splines and for the 2-D hexagonal lattice, the proposed approach offers a reformulation of hex-splines in terms of multi-box splines. While the construction is for general multidimensional lattices, we particularly characterize bivariate and trivariate Voronoi splines for all 2-D and 3-D lattices and specifically study them for body centered cubic and face centered cubic lattices.

On visual quality of optimal 3D sampling and reconstruction

This paper presents a user study of the visual quality of an imaging pipeline employing the optimal body-centered cubic (BCC) sampling lattice. We provide perceptual evidence supporting the theoretical expectation that sampling and reconstruction on the BCC lattice offer superior imaging quality over the traditionally popular Cartesian cubic (CC) sampling lattice. We asked 12 participants to choose the better of two images: one image rendered from data sampled on the CC lattice and one image that is rendered from data sampled on the BCC lattice. We used both synthetic and CT volumetric data, and confirm that the theoretical advantages of BCC sampling carry over to the perceived quality of rendered images. Using 25% to 35% fewer samples, BCC sampled data result in images that exhibit comparable visual quality to their CC counterparts.

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.

A granular three dimensional multiresolution transform

We propose a three dimensional multi-resolution scheme to represent volumetric data in resolutions which are powers of two, resolving the rigidity of the commonly used separable Cartesian multi-resolution schemes in 3D that only allow for change of resolution by a power of eight. Through in-depth comparisons with the counterpart resampling solutions on the Cartesian lattice, we demonstrate the superiority of our subsampling scheme. We derive and document the Fourier domain analysis of this representation. Using such an analysis one can obtain ideal and discrete multidimensional filters for this multi-resolution scheme.