Basic Idea

The basic idea of our new approach is visualized in the right picture of Fig. 2.1, whereas in the left drawing the original shear-warp algorithm (cf. Sec.1) is depicted. First, as we have discussed, in the original method rays are considered to be perpendicular to the sheared slices and the intermediate image. At the intersections positions of the rays and the sheared slices values are bi-linearly reconstructed using the data from the lo- cal neighborhood. Note that the coefficients used for bi-linear reconstruction has to be computed once per slice only, which are then applied to the local neighborhood of four data values within a slice to reconstruct a value between them. Thus, after shearing a slice, i.e. computing the position (projection) of that slice in the intermediate image and after determining the interpolation weights, space-leaping within the considered voxel scan-lines of a slice synchronously with early-ray termination by using the correspond- ing scan-lines of the intermediate image is applied. The bi-linearly interpolated values are composited in front-to-back manner into the appropriate pixels of the intermediate image. Although this approach is quite fast it discriminates the quality of the resulting images. However, by applying the more time-consuming intermediate slices approach as discussed in Sec. 1 it is possible to generate high-quality images. On the other hand some questions remain open. How many intermediate slices have to be chosen to obtain high-quality visual results? Does this depend on the underlying data? Since this ap- proach applies a trilinear model (namely the intermediate slices) for the reconstruction

Figure 2.1: Simplified two-dimensional view of the new shear-warp factorization for ortho- graphic projections. The red lines denote the final image, whereas the cyan lines denote the intermediate image. The black grid is associated with the volume and the cross points (black dots) are the data values. Left: The original shear-warp algorithm. Rays are considered perpen- dicular to the slices and the intermediate image. At the intersections positions of the rays and the slices (blue dots) values are bi-linearly interpolated (reconstructed). Right: In the new shear-warp technique rays may vary up to 45◦ according to the main viewing axis. The intersections posi- tions of the rays and the grid planes (blue dots) are pre-computed, the values at these positions and anywhere else along a ray can be reconstructed by using several reconstruction models for the data. Since a parallel projection matrix is used all intersection position of the rays and the grid may be represented by shifting the column template (yellow highlighted), which stores the pre-computed positions.

of the volume data, other questions arise immediately. Is that model accurate enough? Does it represent the data appropriately? What about different reconstruction models, i.e. can we use piecewise quadratic or cubic spline models as well? Can they be easily introduced into the shear-warp method? What about a more accurate approximation of the volume rendering integral? Then, another goal is to remove the threefold redun- dancy of run-length encoded data sets used in the shear-warp approach, because more and more data sets obtained from imaging systems like CT, MRI, etc. grow to several gigabytes. This can be achieved only by developing some new data structures as well. On that score in our new shear-warp approach (cf. left drawing in Fig. 2.1) we follow a somewhat different technique. Instead of shearing the slices we let the rays vary up to 45◦ according to the main viewing direction1, both is determined from the model matrix

M of Equ. (1.2). If the angle between the current ray direction and the current main

viewing direction exceeds 45◦ then the shear-warp factorization automatically accounts for this and selects another main viewing axis, such that the current ray direction will always stay below this threshold value. Thus, only the processing order of the data set is different. First of all, the intersection positions (blue dots in Fig. 2.1) of the rays and the local grid planes are pre-computed and we further allow different data reconstruction models to obtain data values between the grid points (i.e. piecewise linear-, quadratic-, and cubic models defined on type-0 and type-6 partitions of the volume). This setting

1_{A main viewing direction is always the direction parallel to one of the three main viewing axis.}

CHAPTER 2. COMBINATION OF RAY-CASTING AND SHEAR-WARP

permits us to easily reconstruct values at these intersection positions and anywhere else along the rays (but of course not only on the rays). Then, on each interval (i.e. between two consecutive intersection positions) along different rays r_ν,μ2 we approximate the volume rendering integral, compute the iso-surface or the maximum intensity projection and composite the result in front-to-back order into the appropriate pixels ν, μ of the intermediate image. We can observe from the right drawing of Fig. 2.1, that only one such called column template is necessary to represent the intersection positions of all rays with the local grid planes. During rendering this template is shifted through the volume grid to quickly find the intersection positions and the other information necessary for visualization of the volume data.