Klaus Mueller

Rapid 3-D cone-beam reconstruction with the simultaneous algebraic reconstruction technique (SART) using 2-D texture mapping hardware

Algebraic reconstruction methods, such as the algebraic reconstruction technique (ART) and the related simultaneous ART (SART), reconstruct a two-dimensional (2-D) or three-dimensional (3-D) object from its X-ray projections. The algebraic methods have, in certain scenarios, many advantages over the more popular Filtered Backprojection approaches and have also recently been shown to perform well for 3-D cone-beam reconstruction. However, so far the slow speed of these iterative methods have prohibited their routine use in clinical applications. Here, the authors address this shortcoming and investigate the utility of widely available 2-D texture mapping graphics hardware for the purpose of accelerating the 3-D algebraic reconstruction. They find that this hardware allows 3-D cone-beam reconstructions to be obtained at almost interactive speeds, with speed-ups of over 50 with respect to implementations that only use general-purpose CPUs. However the authors also find that the reconstruction quality is rather sensitive to the resolution of the framebuffer, and to address this critical issue they propose a scheme that extends the precision of a given framebuffer by 4 bits, using the color channels. With this extension, a 12-bit framebuffer delivers useful reconstructions for 0.5% tissue contrast, while an 8-bit framebuffer requires 4%. Since graphics hardware generates an entire image for each volume projection, it is most appropriately used with an algebraic reconstruction method that performs volume correction at that granularity as well, such as SART or SIRT. The authors chose SART for its faster convergence properties.

Anti-aliased three-dimensional cone-beam reconstruction of low-contrast objects with algebraic methods

Examines the use of the algebraic reconstruction technique (ART) and related techniques to reconstruct 3-D objects from a relatively sparse set of cone-beam projections. Although ART has been widely used for cone-beam reconstruction of high-contrast objects, e.g., in computed angiography, the work presented here explores the more challenging low-contrast case which represents a little-investigated scenario for ART. Preliminary experiments indicate that for cone angles greater than 20/spl deg/, traditional ART produces reconstructions with strong aliasing artifacts. These artifacts are in addition to the usual off-midplane inaccuracies of cone-beam tomography with planar orbits. The authors find that the source of these artifacts is the nonuniform reconstruction grid sampling and correction by the cone-beam rays during the ART projection-backprojection procedure. A new method to compute the weights of the reconstruction matrix is devised, which replaces the usual constant-size interpolation filter by one whose size and amplitude is dependent on the source-voxel distance. This enables the generation of reconstructions free of cone-beam aliasing artifacts, at only little extra cost. An alternative analysis reveals that simultaneous ART (SART) also produces reconstructions without aliasing artifacts, however, at greater computational cost. Finally, the authors thoroughly investigate the influence of various ART parameters, such as volume initialization, relaxation coefficient /spl lambda/, correction scheme, number of iterations, and noise in the projection data on reconstruction quality. The authors find that ART typically requires only 3 iterations to render satisfactory reconstruction results.

Splatting without the blur

Splatting is a volume rendering algorithm that combines efficient volume projection with a sparse data representation. Only voxels that have values inside the iso-range need to be considered, and these voxels can be projected via efficient rasterization schemes. In splatting, each projected voxel is represented as a radially symmetric interpolation kernel, equivalent to a fuzzy ball. Projecting such a basis function leaves a fuzzy impression, called a footprint or splat, on the screen. Splatting traditionally classifies and shades the voxels prior to projection, and thus each voxel footprint is weighted by the assigned voxel color and opacity. Projecting these fuzzy color balls provides a uniform screen image for homogeneous object regions, but leads to a blurry appearance of object edges. The latter is clearly undesirable, especially when the view is zoomed on the object. In this work, we manipulate the rendering pipeline of splatting by performing the classification and shading process after the voxels have been projected onto the screen. In this way volume contributions outside the iso-range never affect the image. Since shading requires gradients, we not only splat the density volume, using regular splats, but we also project the gradient volume, using gradient splats. However alternative to gradient splats, we can also compute the gradients on the projection plane using central differencing. This latter scheme cuts the number of footprint rasterization by a factor of four since only the voxel densities have to be projected.

High-quality splatting on rectilinear grids with efficient culling of occluded voxels

Splatting is a popular volume rendering algorithm that pairs good image quality with an efficient volume projection scheme. The current axis-aligned sheet-buffer approach, however, bears certain inaccuracies. The effect of these is less noticeable in still images, but clearly revealed in animated viewing, where disturbing popping of object brightness occurs at certain view angle transitions. In previous work, we presented a new variant of sheet-buffered splatting in which the compositing sheets are oriented parallel to the image plane. This scheme not only eliminates the condition for popping, but also produces images of higher quality. In this paper, we summarize this new paradigm and extend it in a number of ways. We devise a new solution to render rectilinear grids of equivalent cost to the traditional approach that treats the anisotropic volume as being warped into a cubic grid. This enables us to use the usual radially symmetric kernels, which can be projected without inaccuracies. Next, current splatting approaches necessitate the projection of all voxels in the iso-interval(s), although only a subset of these voxels may eventually be visible in the final image. To eliminate these wasteful computations we propose a novel front-to-back approach that employs an occlusion map to determine if a splat contributes to the image before it is projected, thus skipping occluded splats. Additional measures are presented for further speedups. In addition, we present an efficient list-based volume traversal scheme that facilitates the quick modification of transfer functions and iso-values.

A comparison of normal estimation schemes

The task of reconstructing the derivative of a discrete function is essential for its shading and rendering as well as being widely used in image processing and analysis. We survey the possible methods for normal estimation in volume rendering and divide them into two classes based on the delivered numerical accuracy. The three members of the first class determine the normal in two steps by employing both interpolation and derivative filters. Among these is a new method which has never been realized. The members of the first class are all equally accurate. The second class has only one member and employs a continuous derivative filter obtained through the analytic derivation of an interpolation filter. We use the new method to analytically compare the accuracy of the first class with that of the second. As a result of our analysis we show that even inexpensive schemes can in fact be more accurate than high order methods. We describe the theoretical computational cost of applying the schemes in a volume rendering application and provide guidelines for helping one choose a scheme for estimating derivatives. In particular we find that the new method can be very inexpensive and can compete with the normal estimations which pre-shade and pre-classify the volume (M. Levoy, 1988).

Fast implementations of algebraic methods for three-dimensional reconstruction from cone-beam data

The prime motivation of this work is to devise techniques that make the algebraic reconstruction technique (ART) and related methods more efficient for routine clinical use, while not compromising their accuracy. Since most of the computational effort of ART is spent for projection/backprojection operations, the authors first seek to optimize the projection algorithm. Existing projection algorithms are surveyed and it is found that these algorithms either lack accuracy or speed, or are not suitable for cone-beam reconstruction. The authors hence devise a new and more accurate extension to the splatting algorithm, a well-known voxel-driven projection method. They also describe a new three-dimensional (3-D) ray-driven projector that is considerably faster than the voxel-driven projector and, at the same time, more accurate and perfectly suited for the demands of cone beam. The authors then devise caching schemes for both ART and simultaneous ART (SART), which minimize the number of redundant computations for projection and backprojection and, at the same time, are very memory conscious. They find that with caching, the cost for an ART projection/backprojection operation can be reduced to the equivalent cost of 1.12 projections. They also find that SART, due to its image-based volume correction scheme, is considerably harder to accelerate with caching. Implementations of the algorithms yield runtime ratios T/sub SART//T/sub ART/ between 1.5 and 1.15, depending on the amount of caching used.

Design of accurate and smooth filters for function and derivative reconstruction

The correct choice of function and derivative reconstruction filters is paramount to obtaining highly accurate renderings. Most filter choices are limited to a set of commonly used functions, and the visualization practitioner has so far no way to state his preferences in a convenient fashion. Much work has been done towards the design and specification of filters using frequency based methods. However for visualization algorithms it is more natural to specify a filter in terms of the smoothness of the resulting reconstructed function and the spatial reconstruction error. Hence, the authors present a methodology for designing filters based on spatial smoothness and accuracy criteria. They first state their design criteria and then provide an example of a filter design exercise. They also use the filters so designed for volume rendering of sampled data sets and a synthetic test function. They demonstrate that their results compare favorably with existing methods.

Eliminating popping artifacts in sheet buffer-based splatting

Splatting is a fast volume rendering algorithm which achieves its speed by projecting voxels in the form of pre-integrated interpolation kernels, or splats. Presently, two main variants of the splatting algorithm exist: (i) the original method, in which all splats are composited back-to-front, and (ii) the sheet-buffer method, in which the splats are added in cache-sheets, aligned with the volume face most parallel to the image plane, which are subsequently composited back-to-front. The former method is prone to cause bleeding artifacts from hidden objects, while the latter method reduces bleeding, but causes very visible color popping artifacts when the orientation of the compositing sheets changes suddenly as the image screen becomes more parallel to another volume face. We present a new variant of the splatting algorithm in which the compositing sheets are always parallel to the image plane, eliminating the condition for popping, while maintaining the insensitivity to color bleeding. This enables pleasing animated viewing of volumetric objects without temporal color and lighting discontinuities. The method uses a hierarchy of partial splats and employs an efficient list-based volume traversal scheme for fast splat access. It also offers more accuracy for perspective splatting as the decomposition of the individual splats facilitates a better approximation to the diverging nature of the rays that traverse the splatting kernels.

Fast perspective volume rendering with splatting by utilizing a ray-driven approach

Volume ray casting is based on sampling the data along sight rays. In this technique, reconstruction is achieved by a convolution, which collects the contribution of multiple voxels to one sample point. Splatting, on the other hand, is based on projecting data points on to the screen, and reconstruction is implemented by an inverted convolution, where the contribution of one data element is distributed to many sample points (i.e. pixels). Splatting produces images of a quality comparable to ray casting but at greater speeds. This is achieved by pre-computing the projection footprint that the interpolation kernel leaves on the image plane. However, while fast incremental schemes can be utilized for orthographic projection, the perspective projection complicates the mapping of the footprints and is therefore rather slow. In this paper, we merge the technique of splatting with the principles of ray casting to yield a ray-driven splatting approach. We imagine splats as being suspended in object space, a splat at every voxel. Rays are then spawned to traverse the space and intersect the splats. An efficient and accurate way of intersecting and addressing the splats is described. Not only is ray-driven splatting inherently insensitive to the complexity of the perspective viewing transform, it also offers acceleration methods such as early ray termination and bounding volumes, which are methods that traditional voxel-driven splatting cannot benefit from. This results in competitive or superior performance for parallel projection, and superior performance for perspective projection.

Splatting errors and antialiasing

The paper describes three new results for volume rendering algorithms utilizing splatting. First, an antialiasing extension to the basic splatting algorithm is introduced that mitigates the spatial aliasing for high resolution volumes. Aliasing can be severe for high resolution volumes or volumes where a high depth of field leads to converging samples along the perspective axis. Next, an analysis of the common approximation errors in the splatting process for perspective viewing is presented. In this context, we give different implementations, distinguished by efficiency and accuracy, for adding the splat contributions to the image plane. We then present new results in controlling the splatting errors and also show their behavior in the framework of our new antialiasing technique. Finally, current work in progress on extensions to splatting for temporal antialiasing is demonstrated. We present a simple but highly effective scheme for adding motion blur to fast moving volumes.