We develop the first approach for interactive volume visualization based on a sophisticated rendering method of shear-warp type, wavelet data encoding techniques, and a trivariate spline model, which has been introduced [24] recently. As a first step of our algorithm, we apply standard wavelet expansions [6, 31] to represent and decimate the given gridded three-dimensional data. Based on this data encoding, we give a sophisticated version of the shearwarp based volume rendering method [13]. Our new algorithm visits each voxel only once taking advantage of the particular data organization of octrees. In addition, the hierarchies of the data guide the local (re)construction of the quadratic super-spline models, which we apply as a pure visualization tool. The low total degree of the polynomial pieces allows to numerically approximate the volume rendering integral efficiently. Since the coefficients of the splines are almost immediately available from the given data, Bernstein-B?ezier techniques can be fully employed in our algorithms. In this way, we demonstrate that these models can be successfully applied to full volume rendering of hierarchically organized data. Our computational results show that (even when hierarchical approximations are used) the new approach leads to almost artifact-free visualizations of high quality for complicated and noise-contaminated volume data sets, while the computational effort is considerable low, i.e. our current implementation yields 1-2 frames per second for parallel perspective rendering a 256^3 volume data set (using simple opacity transfer functions) in a 512^2 view-jor directions in interactive volume rendering are the classical image based approaches like ray-casting [14, 25, 34], object based methods like splatting [16, 34, 36, 37], and intermediate algorithms based on the shear-warp transform [13, 29]. The latter have shown to be very efficient due to their ability to optimally use the cache architecture found in modern computers. Artifacts arising in the original implementation of [13] led to modifications and essential further developments of the shear-warp approach. In [30] a sophisticated improvement using intermediate slices was introduced to reduce the overall sampling distance, which is often considered as a major source for artifacts visible on the screen. A similar approach using Hermite curves was given in [27]. More recently, [28] included pre-integrated volume rendering in the shear-warp algorithm for parallel projection to further reduce potential problems connected with the classification step in the rendering. The approach presented below is orthogonal to pre-integrated volume rendering since [28] uses a linear univariate model of the opacity and color function defined within a cubic cell whereas we consider trivariate piecewise quadratic polynomials on tetrahedral partitions. Therefore, we can easily extend this method using pre-integrated rendering as well. In addition, the three-fold overhead for coding the volume data has been solved by reducing the coding to two and one coded volume(s) [27, 30]. Furthermore, one basically distinguishes between the above full volume rendering methods, where the integral equation of a physical emission-transport model has to be numerically solved along rays and the less expensive visualization of isosurfaces extracted from the volume (cf. [15, 23], and the references therein), where the first intersection point of an isosurface along rays has to be found. A common property of these approaches is that the non-discrete model used as a tool for the visualization are trilinear splines, i.e. splines with tensor-product structure, which are linear in each of the three space directions. Hence, for full volume rendering the integral equation has to be numerically approximated for the piecewise cubic models (total degree of the polynomial pieces is 3). Moreover, in the standard approach the models involve no smoothness conditions and therefore the evaluation of the necessary gradient field requires a separate model. This is one of the reasons why different models of the discrete data have been proposed [4, 15, 18, 19, 20, 32, 33] as tools for the visualization with improved visual quality. Here, the models are often local approximations with tensor-product splines of higher degrees (for instance triquadratic and tricubic splines) involving smoothness conditions. The simplicity of these (re)constructions allows to keep the computational costs relatively low, while simultaneously the visualization of the isosurfaces often lead to satisfying results. On the other hand, according to the relatively high total polynomial degree of these models (which is 6 and 9 for the triquadratic and tricubic models, respectively) the evaluation of its values and gradients may sometimes become inexact and the computational costs are higher as in the approaches based on trilinear splines. As a compromise between visual quality and computational efficiency issues, a different model for the local approximation with splines has been proposed, recently. The approach [24] is based on trivariate splines involving smoothness conditions on a uniform tetrahedral partition of the volumetric domain, where the polynomial pieces are quadratic, i.e. the total degree is 2. It has been shown that the piecewise Bernstein-B?ezier form of these splines allows to apply well-established tools from Computer Aided Geometric Design (CAGD) to compute its values and gradients, which leads to high-quality visualizations of iso-surfaces from gridded volume data. Note that recently the spline model proposed in [24] has been used [26] for the analysis and visualization of vector fields. For further information on the field of volume visualization, we refer the interested reader to recent books [3, 5], the surveys [5, 21, 32, 35], and the references therein.