Abstract Dynamic tomography reconstructs a time activity curve (TAC) for every voxel assuming that the algebraic form of the function is known a priori. The algebraic form derived from the analysis of compartmental models depends nonlinearly on the nonnegative parameters to be determined. Direct methods apply fitting in every iteration step. Because of the iterative nature of the maximum likelihood–expectation maximization (ML–EM) reconstruction, the fitting result of the previous step can serve as a good starting point in the current step; thus, after the first iteration we have a guess that is not far from the solution, which allows the use of gradient-based local optimization methods. However, finding good initial guesses for the first ML–EM iteration is a critical problem since gradient-based local optimization algorithms do not guarantee convergence to the global optimum if they are started at an inappropriate location. This paper examines the robust solution of the fitting problem both in the initial phase and during the ML–EM iteration. This solution is implemented on GPUs and is built into the 4D reconstruction module of the TeraTomo software.

Abstract Curved spaces are very un-intuitive to our eyes trained on Euclidean geometry. Games provide an interesting way to explore these strange worlds. Games are written with the help of modeling tools and game engines based on Euclidean geometry. This paper addresses the problem of adapting 3D game engines to the rules of curved spaces. We consider the conversion of Euclidean objects, geometric calculations, transformation pipeline, lighting and physical simulation. Finally, we identify where existing game engines should be modified.

Abstract Rendering inhomogeneous participating media requires a lot of volume samples since the extinction coefficient needs to be integrated along light paths. Ray marching makes small steps, which is time consuming and leads to biased algorithms. Woodcocklike approaches use analytic sampling and a random rejection scheme guaranteeing that the expectations will be the same as in the original model. These models and the application of control variates for the extinction have been successful to compute transmittance and single scattering but were not fully exploited in multiple scattering simulation. Our paper attacks the multiple scattering problem in heterogeneous media and modifies the light–medium interaction model to allow the use of simple analytic formulae while preserving the correct expected values. The model transformation reduces the variance of the estimates with the help of Rao‐Blackwellization and control variates applied both for the extinction coefficient and the incident radiance. Based on the transformed model, efficient Monte Carlo rendering algorithms are obtained.

This paper presents a new stochastic particle model for efficient and unbiased Monte Carlo rendering of heterogeneous participating media. We randomly add and remove material particles to obtain a density with which free flight sampling and transmittance estimation are simple, while material particle properties are simultaneously modified to maintain the true expectation of the radiance. We show that meeting this requirement may need the introduction of light particles with negative energy and materials with negative extinction, and provide an intuitive interpretation for such phenomena. Unlike previous unbiased methods, the proposed approach does not require a-priori knowledge of the maximum medium density that is typically difficult to obtain for procedural models. However, the method can benefit from an approximate knowledge of the density, which can usually be acquired on-the-fly at little extra cost and can greatly reduce the variance of the proposed estimators. The introduced mechanism can be integrated in participating media renderers where transmittance estimation and free flight sampling are building blocks. We demonstrate its application in a multiple scattering particle tracer, in transmittance computation, and in the estimation of the inhomogeneous air-light integral.

Abstract We reexamine in this paper the variance for the Multiple Importance Sampling (MIS) estimator for multi‐sample and one‐sample model. As a result of our analysis we can obtain the optimal estimator for the multi‐sample model for the case where the weights do not depend on the count of samples. We extend the analysis to include the cost of sampling. With these results in hand we find a better estimator than balance heuristic with equal count of samples. Further, we show that the variance for the one‐sample model is larger or equal than for the multi‐sample model, and that there are only two cases where the variance is the same. Finally, we study on four examples the difference of variances for equal count as used by Veach, our new estimator, and a recently introduced heuristic.

Abstract Stackless traversal algorithms for ray tracing acceleration structures require significantly less storage per ray than ordinary stack‐based ones. This advantage is important for massively parallel rendering methods, where there are many rays in flight. On SIMD architectures, a commonly used acceleration structure is the MBVH, which has multiple bounding boxes per node for improved parallelism. It scales to branching factors higher than two, for which, however, only stack‐based traversal methods have been proposed so far. In this paper, we introduce a novel stackless traversal algorithm for MBVHs with up to four‐way branching. Our approach replaces the stack with a small bitmask, supports dynamic ordered traversal, and has a low computation overhead. We also present efficient implementation techniques for recent CPU, MIC (Intel Xeon Phi) and GPU (NVIDIA Kepler) architectures.

This paper proposes the application of multiple importance sampling in fully 3-D positron emission tomography to speed up the iterative reconstruction process. The proposed method combines the results of lines of responses (LOR) driven and voxel driven projections keeping their advantages, like importance sampling, performance and parallel execution on graphics processing units. Voxel driven methods can focus on point like features while LOR driven approaches are efficient in reconstructing homogeneous regions. The theoretical basis of the combination is the application of the mixture of the samples generated by the individual importance sampling methods, emphasizing a particular method where it is better than others. The proposed algorithms are built into the Tera-tomo system.

Iterative positron emission tomography (PET) reconstruction computes projections between the voxel space and the lines of response (LOR) space, which are mathematically equivalent to the evaluation of multi-dimensional integrals. The dimension of the integration domain can be very high if scattering needs to be compensated. Monte Carlo (MC) quadrature is a straightforward method to approximate high-dimensional integrals. As the numbers of voxels and LORs can be in the order of hundred millions and the projection also depends on the measured object, the quadratures cannot be precomputed, but Monte Carlo simulation should take place on-the-fly during the iterative reconstruction process. This paper presents modifications of the maximum likelihood, expectation maximization (ML-EM) iteration scheme to reduce the reconstruction error due to the on-the-fly MC approximations of forward and back projections. If the MC sample locations are the same in every iteration step of the ML-EM scheme, then the approximation error will lead to a modified reconstruction result. However, when random estimates are statistically independent in different iteration steps, then the iteration may either diverge or fluctuate around the solution. Our goal is to increase the accuracy and the stability of the iterative solution while keeping the number of random samples and therefore the reconstruction time low. We first analyze the error behavior of ML-EM iteration with on-the-fly MC projections, then propose two solutions: averaging iteration and Metropolis iteration. Averaging iteration averages forward projection estimates during the iteration sequence. Metropolis iteration rejects those forward projection estimates that would compromise the reconstruction and also guarantees the unbiasedness of the tracer density estimate. We demonstrate that these techniques allow a significant reduction of the required number of samples and thus the reconstruction time. The proposed methods are built into the Teratomo system.

Abstract This paper presents efficient algorithms for free path sampling in heterogeneous participating media defined either by high‐resolution voxel arrays or generated procedurally. The method is based on the concept of mixing ‘virtual’ material or particles to the medium, augmenting the extinction coefficient to a function for which the free path can be sampled in a straightforward way. The virtual material is selected such that it modifies the volume density but does not alter the radiance. We define the total extinction coefficient of the real and virtual particles by a low‐resolution grid of super‐voxels that are much larger than the real voxels defining the medium. The computational complexity of the proposed method depends just on the resolution of the super‐voxel grid and does not grow with the resolution above the scale of super‐voxels. The method is particularly efficient to render large, low‐density, heterogeneous volumes, which should otherwise be defined by enormously high resolution voxel grids and where the average free path length would cross many voxels.

Abstract This paper presents a hatching algorithm which – while also allows for an implementation in real‐time – is integrated into the production pipeline of computer generated motion picture. Motion picture production pipelines impose special functional and quality requirements. From the functional point of view, the stages of modeling, 3D rendering, and compositing form a pipeline without feed‐back, and frames are rendered independently, possibly on different computers. Thus, no temporal data can be shared between them while flicker free animation needs to be generated. Quality requirements can be grasped as that of dual consistency: the generated hatching must consistently follow object movement and deformation, and, at the same time, it should have a consistent pattern and density in image‐space to provide the hand‐crafted look. In order to meet both requirements, we apply a particle based method and develop an image‐space density control mechanism using rejection sampling and low‐discrepancy sequences. We also discuss the decomposition of rendering tasks according to the main stages of the production pipeline and demonstrate how the artist can define the illustration style in a convenient way.

This paper presents a fast parallel method to solve the radiative transport equation in inhomogeneous participating media. We apply a novel approximation scheme to find a good initial guess for both the direct and scattered components. Then, the initial approximation is used to bootstrap an iterative multiple scattering solver, i.e., we let the iteration concentrate just on the residual problem. This kind of bootstrapping makes the volumetric source approximation more uniform, thus it helps to reduce the discretization artifacts and improves the efficiency of the parallel implementation. The iterative refinement is executed on a face-centered cubic grid. The implementation is based on CUDA and runs on the GPU. For large volumes that do not fit into the GPU memory, we also consider the implementation on a GPU cluster, where the volume is decomposed to blocks according to the available GPU nodes. We show how the communication bottleneck can be avoided in the cluster implementation by not exchanging the boundary conditions in every iteration step. In addition to light photons, we also discuss the generalization of the method to γ-photons that are relevant in medical simulation.

Abstract This paper proposes a deterministic importance sampling algorithm that is based on the recognition that delta‐sigma modulation is equivalent to importance sampling. We propose a generalization for delta‐sigma modulation in arbitrary dimensions, taking care of the curse of dimensionality as well. Unlike previous sampling techniques that transform low‐discrepancy and highly stratified samples in the unit cube to the integration domain, our error diffusion sampler ensures the proper distribution and stratification directly in the integration domain. We also present applications, including environment mapping and global illumination rendering with virtual point sources.

Abstract This survey reviews algorithms that can render specular, i.e. mirror reflections, refractions, and caustics on the GPU. We establish a taxonomy of methods based on the three main different ways of representing the scene and computing ray intersections with the aid of the GPU, including ray tracing in the original geometry, ray tracing in the sampled geometry, and geometry transformation. Having discussed the possibilities of implementing ray tracing, we consider the generation of single reflections/refractions, interobject multiple reflections/refractions, and the general case which also includes self‐reflections or refractions. Moving the focus from the eye to the light sources, caustic effect generation approaches are also examined.

Abstract Monte Carlo approaches use random lines to distribute the light power in the scene but the cost of creating a set of random single lines is very costly. In this paper, we present several software and hardware techniques in order to reduce the computational cost of the generation of random single lines by using bundles of parallel lines. The bundle of parallel lines is simulated with a general purpose polygon filling algorithm, like the painter's algorithm. We also present two graphics hardware implementations. The first approach uses two depth buffers in order to represent stochastically a bundle of parallel global lines. The second one uses multiple depth buffers and the aim is to exploit coherence between projection planes for each iteration. All algorithms were implemented with the multipath method.

Abstract This paper reviews the latest developments of displacement mapping algorithms implemented on the vertex, geometry, and fragment shaders of graphics cards. Displacement mapping algorithms are classified as per‐vertex and per‐pixel methods. Per‐pixel approaches are further categorized as safe algorithms that aim at correct solutions in all cases, to unsafe techniques that may fail in extreme cases but are usually much faster than safe algorithms, and to combined methods that exploit the robustness of safe and the speed of unsafe techniques. We discuss the possible roles of vertex, geometry and fragment shaders to implement these algorithms. Then the particular GPU‐based bump, parallax, relief, sphere, horizon mapping, cone stepping, local ray tracing, pyramidal and view‐dependent displacement mapping methods, as well as their numerous variations are reviewed providing also implementation details of the shader programs. We present these methods using uniform notations and also point out when different authors called similar concepts differently. In addition to basic displacement mapping, self‐shadowing and silhouette processing are also reviewed. Based on our experiences gained having reimplemented these methods, their performance and quality are compared, and the advantages and disadvantages are fairly presented.

Abstract Light source animation is a particularly hard field of real‐time global illumination algorithms since moving light sources result in drastic illumination changes and make coherence techniques less effective. However, the animation of small (point‐like) light sources represents a special but practically very important case, for which the reuse of the results of other frames is possible. This paper presents a fast light source animation algorithm based on the virtual light sources illumination method. The speed up is close to the length of the animation, and is due to reusing paths in all frames and not only in the frame where they were obtained. The possible applications of this algorithm are the lighting design and systems to convey shape and features with relighting.

Abstract This paper presents a general variance reduction method that is a quasi‐optimal combination of correlated and importance sampling. The weights of the combination are selected automatically in order to keep the merits of both importance and correlated sampling. The proposed sampling method is used for efficient direct light source computation of large area sources and for the calculation of the reflected illumination of environment maps. Importance sampling would be good in these cases if the sources are hidden, while correlated sampling is efficient if the sources are fully visible. The proposed method automatically detects the particular case and provides results that inherit the advantages of both techniques.

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.

This paper presents a new mutation strategy for the Metropolis light transport algorithm, which works in the unit cube of pseudo-random numbers instead of mutating in the path space. This transformation makes the integrand have lower variation and thus increases the acceptance probability of the mutated samples. Higher acceptance ratio, in turn, reduces the correlation of the samples, which increases the speed of convergence. We use both local mutations that choose a new random sample in the neighborhood of the previous one, and global mutations that make ``large steps'', and find the samples independently. Local mutations smooth out the result, while global mutations guarantee the ergodicity of the process. Due to the fact that samples are generated independently in large steps, this method can also be considered as a combination of the Metropolis algorithm with a classical random walk. If we use multiple importance sampling for this combination, the combined method will be as good at bright regions as the Metropolis algorithm and at dark regions as random walks. The resulting scheme is robust, efficient, but most importantly, is easy to implement and to combine with an arbitrary random-walk algorithm.

The paper introduces a global illumination method that combines continuous and finite-element approaches, pre-serving the speed of finite-element based iteration and the accuracy of continuous random walks. The basic idea is to decompose the radiance function to a finite-element component that is only a rough estimate and to a difference component that is obtained by Monte-Carlo techniques. Iteration and random walk are handled uniformly in the framework of stochastic iteration. This uniform treatment allows the finite-element component to be built up adap-tively aiming at minimizing the Monte-Carlo component. The method is also suited for interactive walkthrough animation in glossy scenes since when the viewpoint changes, only the small Monte-Carlo component needs to be recomputed.

The paper presents simple, physically plausible, but not physically based reflectance models for metals and other specular materials. So far there has been no metallic BRDF model that is easy to compute, suitable for fast importance sampling and is physically plausible. This gap is filled by appropriate modifications of the Phong, Blinn and the Ward models. The Phong and the Blinn models are known not to have metallic characteristics. On the other hand, this paper also shows that the Cook‐Torrance and the Ward models are not physically plausible, because of their behavior at grazing angles. We also compare the previous and the newly proposed models. Finally, the generated images demonstrate how the metallic impression can be provided by the new models.

We introduce a physically plausible mathematical model for a large class of BRDFs. The new model is as simple as the well‐known Phong model, but eliminates its disadvantages. It gives a good visual approximation for many practical materials: coated metals, plastics, ceramics, retro‐reflective paints, anisotropic and retro‐reflective materials, etc. Because of its illustrative properties it can be used easily in most commercial software and because of its low computational cost it is practical for virtual reality. The model is based on a special basic BRDF definition, which meets the requirements of reciprocity and of energy conservation. Then a class of BRDFs is constructed from this basic BRDF with different weight functions. The definition of such weight functions requires the user to specify the profile of the highlights, from which the weight function is obtained by derivation. It is also demonstrated how importance sampling can be used with the new BRDFs.

This paper presents a single‐pass, view‐dependent method to solve the rendering equation, using a stochastic iterational scheme where the transport operator is selected randomly in each iteration. The requirements of convergence are given for the general case. To demonstrate the basic idea, a very simple,continuous random transport operator is examined, which gives back the light tracing algorithm incorporating Russian roulette. Then, a new mixed continuous and finite‐element based iteration method is proposed, which uses ray‐bundles to transfer the radiance in a single random direction. The resulting algorithm is fast, it provides initial results in seconds and accurate solutions in minutes and does not suffer from the error accumulation problem and the high memory demand of other finite‐element and hierarchical approaches.

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.

Abstract Using the two ray coherence theorems of Ohta and Maekawa the computation time of ray tracing algorithms for scenes of spheres and convex polyhedra can be reduced considerably. This paper presents further theorems which, together with the first two, may enable further reduction in computation time.