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 lowresolution 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.

Figure 1: Transmittance (left), single-scatter of a point light source (middle), and multiple-scatter (right) calculation for 12-octave Perlinnoise medium and comparison to Woodcock tracking with equal number of medium fetches. AbstractThis 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.

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In positron emISSion tomography (PET), photon scattering inside the body causes significant blurring and quan tification error in the reconstructed images. To solve this problem we have developed Monte Carlo (MC) based 3D PET recon struction algorithms implemented on the Graphics Processing Unit (GPU). Our implementation takes mUltiple Compton scat tering into account without any significant additional cost. The performance of the scatter correction is evaluated using GAT E simulation as well as by comparing reconstruction results of Tera-Torno to the reference reconstruction implementation of the Philips Gemini TOF PET which applies attenuation correction and single scatter simulation (SSS) for scatter correction. The comparative reconstruction results are based on the NEMA NU2-2007 image quality phantom. I. INTRODUCT IONIn positron emission tomography (PET) annihilating gamma photon pairs are detected in order to calculate the spatial distribution of the positron-emitting isotope injected into the patient's body. The events detected in coincidence define a line of response (LOR), along which the originating event of positron-electron annihilation is searched. However, gamma photons may interact with the patient's body which can result in single or even multiple scattering. As a result, the LOR is not a straight line any more, but a combination of line segments. If scattering is not addressed in the reconstruction algorithm, then the result may involve significant blurring and may lead to inaccurate quantification [3]. The scattering phenomenon can be handled by many approximating meth ods, such as the SSS algorithm [2], which approximates the expected distribution of scattered events by an iterative scheme along with a reconstruction operation that does not involve any scatter modeling.In the Tera-Torno project, MC based 3D reconstruction approaches have been implemented [6] on the GPU. We focused on the precise yet efficient, on-the-fty calculation of system matrix (SM) elements, that is, the path of the gamma photons are calculated and the relevant physical effects are taken into account, including the detector response [7], [5], the Compton scattering, and the absorption inside the body as well as positron range modeling. One of our approaches applies quasi-Monte Carlo techniques for evaluating the re lated high-dimensional integral of SM. This so-called adjoint solver is capable to model multiple-scattering effects inside the body without any significant additional cost compared to the geometry-only case. This enables scatter correction to be applied in a whole-body PET examination and results in a scatter-corrected reconstructed volume in a few minutes right after the end of the acquisition. II. METHODSThe employed reconstruction scheme is the total variation (TV) regularized expectation maximization (EM) method, where the SM elements of forward and back projections are calculated on-the-fty [6].Regarding the projectors of the adjoint solver the calcu lations are organized by LORs in the forward projection...