Abstract Recently, it has been shown that the quality of GPU‐based trilinear volume resampling can be significantly improved if the six additional trilinear samples evaluated for the gradient estimation also contribute to the reconstruction of the underlying function [Csé19]. Although this improvement increases the approximation order from two to three without any extra cost, the continuity order remains C0. In this paper, we go one step further showing that a C1continuous triquadratic B‐spline reconstruction and its analytic partial derivatives can be evaluated by taking only one more trilinear sample into account. Thus, our method is the first volume‐resampling technique that is nearly as fast as trilinear interpolation combined with on‐the‐fly central differencing, but provides a higher‐quality reconstruction together with a consistent analytic gradient calculation. Furthermore, we show that our fast evaluation scheme can also be adapted to the Mitchell‐Netravali [MN88] notch filter, for which a fast GPU implementation has not been known so far.

In volume-rendering applications, it is a de facto standard to reconstruct the underlying continuous function by using trilinear interpolation, and to estimate the gradients for the shading computations by calculating central differences on the fly. In a GPU implementation, this requires seven trilinear texture samples: one for the function reconstruction, and six for the gradient estimation. In this paper, for the first time, we show that the six additional samples can be used not just for gradient estimation, but for significantly improving the quality of the function reconstruction as well. As the additional arithmetic operations can be performed in the shadow of the texture fetches, we can achieve this quality improvement for free without reducing the rendering performance at all. Therefore, our method can completely replace the standard trilinear interpolation in the practice of GPU-accelerated volume rendering.

Abstract It is well known that cubic texture filtering can be efficiently implemented on the GPU by using a method published by Sigg and Hadwiger [ SH05 ], which simplifies the evaluation to a linear combination of linear texture fetches. However, their method cannot be directly applied if the filter kernel takes also negative values like the popular Catmull‐Rom spline, for example. In this paper, we propose a modified algorithm that is able to handle also the negative weights. Therefore, using our method, the Catmull‐Rom spline interpolation can also be evaluated in one, two, and three dimensions by taking two, four, and eight linear texture fetches, respectively.

Abstract Cosine‐Weighted B‐spline (CWB) interpolation [ Csé13 ] has been originally proposed for volumetric data sampled on the Body‐Centered Cubic (BCC) lattice. The BCC lattice is well known to be optimal for sampling isotropically band‐limited signals above the Nyquist limit. However, the Face‐Centered Cubic (FCC) lattice has been recently proven to be optimal for low‐rate sampling. The CWB interpolation is a state‐of‐the‐art technique on the BCC lattice, which outperforms, for example, the previously proposed box‐spline interpolation in terms of both efficiency and visual quality. In this paper, we show that CWB interpolation can be adapted to the FCC lattice as well, and results in similarly isotropic signal reconstructions as on the BCC lattice.

A new lower bound on the average reconstruction error variance of multidimensional sampling and reconstruction is presented. It applies to sampling on arbitrary lattices in arbitrary dimensions, assuming a stochastic process with constant, isotropically bandlimited spectrum and reconstruction by the best linear interpolator. The lower bound is exact for any lattice at sufficiently high and low sampling rates. The two threshold rates where the error variance deviates from the lower bound gives two optimality criteria for sampling lattices. It is proved that at low rates, near the first threshold, the optimal lattice is the dual of the best sphere-covering lattice, which for the first time establishes a rigorous relation between optimal sampling and optimal sphere covering. A previously known result is confirmed at high rates, near the second threshold, namely, that the optimal lattice is the dual of the best sphere-packing lattice. Numerical results quantify the performance of various lattices for sampling and support the theoretical optimality criteria.

Abstract 3D box splines are defined by convolving a 1D box function with itself along different directions. In volume visualization, box splines are mainly used as reconstruction kernels that are easy to adapt to various sampling lattices, such as the Cartesian Cubic (CC), Body‐Centered Cubic (BCC), and Face‐Centered Cubic (FCC) lattices. The usual way of tailoring a box spline to a specific lattice is to span the box spline by exactly those principal directions that span the lattice itself. However, in this case, the preferred directions of the box spline and the lattice are the same, amplifying the anisotropic effects of each other. This leads to an anisotropic volume representation with strongly preferred directions. Therefore, in this paper, we retailor box splines to lattices such that the sets of vectors that span the box spline and the lattice are disjoint sets. As the preferred directions of the box spline and the lattice compensate each other, a more isotropic volume representation can be achieved. We demonstrate this by comparing different combinations of box splines and lattices concerning their anisotropic behavior in tomographic reconstruction and volume visualization.

Abstract In the literature on optimal regular volume sampling, the Body‐Centered Cubic (BCC) lattice has been proven to be optimal for sampling spherically band‐limited signals above the Nyquist limit. On the other hand, if the sampling frequency is below the Nyquist limit, the Face‐Centered Cubic (FCC) lattice was demonstrated to be optimal in reducing the prealiasing effect. In this paper, we confirm that the FCC lattice is indeed optimal in this sense in a certain interval of the sampling frequency. By theoretically estimating the prealiasing error in a realistic range of the sampling frequency, we show that in other frequency intervals, the BCC lattice and even the traditional Cartesian Cubic (CC) lattice are expected to minimize the prealiasing. The BCC lattice is superior over the FCC lattice if the sampling frequency is not significantly below the Nyquist limit. Interestingly, if the original signal is drastically undersampled, the CC lattice is expected to provide the lowest prealiasing error. Additionally, we give a comprehensible clarification that the sampling efficiency of the FCC lattice is lower than that of the BCC lattice. Although this is a well‐known fact, the exact percentage has been erroneously reported in the literature. Furthermore, for the sake of an unbiased comparison, we propose to rotate the Marschner‐Lobb test signal such that an undue advantage is not given to either lattice.

In this paper, Cosine-Weighted B-spline (CWB) filters are proposed for interpolation on the optimal Body-Centered Cubic (BCC) lattice. We demonstrate that our CWB filters can well exploit the fast trilinear texture-fetching capability of modern GPUs, and outperform the state-of-the-art box-spline filters not just in terms of efficiency, but in terms of visual quality and numerical accuracy as well. Furthermore, we rigorously show that the CWB filters are better tailored to the BCC lattice than the previously proposed quasi-interpolating BCC B-spline filters, because they form a Riesz basis; exactly reproduce the original signal at the lattice points; but still provide the same approximation order.

In this paper, we demonstrate that quasi-interpolation of orders two and four can be efficiently implemented on the Body-Centered Cubic (BCC) lattice by using tensor-product B-splines combined with appropriate discrete prefilters. Unlike the nonseparable box-spline reconstruction previously proposed for the BCC lattice, the prefiltered B-spline reconstruction can utilize the fast trilinear texture-fetching capability of the recent graphics cards. Therefore, it can be applied for rendering BCC-sampled volumetric data interactively. Furthermore, we show that a separable B-spline filter can suppress the postaliasing effect much more isotropically than a nonseparable box-spline filter of the same approximation power. Although prefilters that make the B-splines interpolating on the BCC lattice do not exist, we demonstrate that quasi-interpolating prefiltered linear and cubic B-spline reconstructions can still provide similar or higher image quality than the interpolating linear box-spline and prefiltered quintic box-spline reconstructions, respectively.

In this paper prefiltered reconstruction techniques are evaluated for volume-rendering applications. All the analyzed methods perform a discrete prefiltering as a preprocessing of the input samples in order to improve the quality of the continuous reconstruction afterwards. Various prefiltering schemes have been proposed to fulfill either spatial-domain or frequency domain criteria. According to our best knowledge, however, their thorough comparative study has not been published yet. Therefore we derive the frequency responses of the different prefilteredreconstruction techniques to analyze their global behavior such as aliasing or smoothing. Furthermore, we introduce a novel mathematical basis to compare also their spatial-domain behavior in terms of the asymptotic local error effect. For the sake of fair comparison, we use the same linear and cubic B-splines as basis functions but combined with different discrete prefilters. Our goal with this analysis is to help the potential users to select the optimal prefiltering scheme for their specific applications.

Volume rendering techniques are conventionally classified as either direct or indirect methods. Indirect methods require to transform the initial volumetric model into an intermediate geometrical model in order to efficiently visualize it. In contrast, direct volume rendering (DVR) methods can directly process the volumetric data. Modern CT scanners usually provide data as a set of samples on a rectilinear grid, which is computed from the measured projections by discrete tomographic reconstruction. Therefore the rectilinear grid can already be considered as an intermediate volume representation. In this paper we introduce direct direct volume rendering (D2VR). D2VR does not require a rectilinear grid, since it is based on an immediate processing of the measured projections. Arbitrary samples for ray casting are reconstructed from the projections by using the Filtered Back-Projection algorithm. Our method removes a lossy resampling step from the classical volume rendering pipeline. It provides much higher accuracy than traditional grid-based resampling techniques do. Furthermore we also present a novel high-quality gradient estimation scheme, which is also based on the Filtered Back-Projection algorithm.

In this paper a novel high-quality reconstruction scheme is presented. Although our method is mainly proposed to reconstruct volumetric data sampled on an optimal body-centered cubic (BCC) grid, it can be easily adapted lo the conventional regular rectilinear grid as well. The reconstruction process is decomposed into two steps. The first step, which is considered to be a preprocessing, is a discrete Gaussian deconvolution performed only once in the frequency domain. Afterwards, the second step is a spatial-domain convolution with a truncated Gaussian kernel, which is used to interpolate arbitrary samples for ray casting. Since the preprocessing is actually a discrete prefiltering, we call our technique prefiltered Gaussian reconstruction (PGR). It is shown that the impulse response of PGR well approximates the ideal reconstruction kernel. Therefore the quality of PGR is much higher than that of previous reconstruction techniques proposed for optimally sampled data, which are based on linear and cubic box splines adapted to the BCC grid. Concerning the performance, PGR is slower than linear box-spline reconstruction but significantly faster than cubic box-spline reconstruction.

In this paper a novel volume-rendering technique based on Monte Carlo integration is presented. As a result of a preprocessing, a point cloud of random samples is generated using a normalized continuous reconstruction of the volume as a probability density function. This point cloud is projected onto the image plane, and to each pixel an intensity value is assigned which is proportional to the number of samples projected onto the corresponding pixel area. In such a way a simulated X-ray image of the volume can be obtained. Theoretically, for a fixed image resolution, there exists an M number of samples such that the average standard deviation of the estimated pixel intensities us under the level of quantization error regardless of the number of voxels. Therefore Monte Carlo Volume Rendering (MCVR) is mainly proposed to efficiently visualize large volume data sets. Furthermore, network applications are also supported, since the trade-off between image quality and interactivity can be adapted to the bandwidth of the client/server connection by using progressive refinement.

We report on using computed tomography (CT) as a model acquisition tool for complex objects in computer graphics. Unlike other modeling and scanning techniques the complexity of the object is irrelevant in CT, which naturally enables to model objects with, for example, concavities, holes, twists or fine surface details. Once the data is scanned, one can apply post-processing techniques for data enhancement, modification or presentation. For demonstration purposes we chose to scan a Christmas tree which exhibits high complexity which is difficult or even impossible to handle with other techniques. However, care has to be taken to achieve good scanning results with CT. Further, we illustrate post-processing by means of data segmentation and photorealistic as well as non-photorealistic surface and volume rendering techniques.

In this paper we present a fast visualization technique for volumetric data, which is based on a recent non-photorealistic rendering technique. Our new approach enables alternative insights into 3D data sets (compared to traditional approaches such as direct volume rendering or iso-surface rendering). Object contours, which usually are characterized by locally high gradient values, are visualized regardless of their density values. Cumbersome tuning of transfer functions, as usually needed for setting up DVR views is avoided. Instead, a small number of parameters is available to adjust the non-photorealistic display. Based on the magnitude of local gradient information as well as on the angle between viewing direction and gradient vector, data values are mapped to visual properties (color, opacity), which then are combined to form the rendered image (MIP is proposed as the default compositing stragtegy here). Due to the fast implementation of this alternative rendering approach, it is possible to interactively investigate the 3D data, and quickly learn about internal structures. Several further extensions of our new approach, such as level lines are also presented in this paper.

In this paper a new gradient estimation method is presented which is based on linear regression. Previous contextual shading techniques try to fit an approximate function to a set of surface points in the neighborhood of a given voxel. Therefore a system of linear equations has to be solved using the computationally expensive Gaussian elimination. In contrast, our method approximates the density function itself in a local neighborhood with a 3D regression hyperplane. This approach also leads to a system of linear equations but we will show that it can be solved with an efficient convolution. Our method provides at each voxel location the normal vector and the translation of the regression hyperplane which are considered as a gradient and a filtered density value respectively. Therefore this technique can be used for surface smoothing and gradient estimation at the same time.

This paper presents an enhanced transillumination radiosity method that can provide accurate solutions at relatively low computational cost. The proposed algorithm breaks down the double integral of the gathered power to an area integral that is computed analytically and to a directional integral that is evaluated by quasi‐Monte Carlo techniques. Since the analytical integration results in a continuous function of finite variation, the quasi‐Monte Carlo integration that follows the analytical integration will be efficient and its error can be bounded by the Koksma‐Hlawka inequality. The paper also analyses the requirements of the convergence, presents theoretical error bounds and proposes error reduction techniques. The theoretical bounds are compared with simulation results.