Wavelet-Based Hierarchical Time-Varying Volume Representation With 4th-Root-Of-2 Subdivision

Due to the improvements in the performance of computing power and storage capacity achieved over the last decade, today's data-intensive scientific applications are capable of quickly generating and storing huge amounts of data. Down sampling can be used to reduce the data to a manageable amount. The reduced data can be examined by scientists to spot regions of interest, for which more detailed examinations can be performed. Today, visualization applications have to deal with large-scale data in the spatial as well as temporal dimensions and their representation at multiple levels of detail.
Multi-resolution methods for representing data at multiple levels of detail are widely used for large-scale two- and three-dimensional data sets. Furthermore, for time-varying data sets techniques have been developed that make use of temporal coherence of, for example, numerically simulated data. We developed a four-dimensional multiresolution approach, where time is treated as fourth dimension. We deal with large scales in spatial and temporal dimensions in a single hierarchical framework.
For large-scale volume representation, one should use regular rather than irregular data formats, since grid connectivity should be implicit and data should be easily and quickly accessible. To overcome regular data structures' disadvantage of coarse granularity, we have developed a 4th-root-of-2-subdivision scheme. Every nth-root-of-2-subdivision step only doubles the number of vertices, which is a factor of nth-root-of-2 in each of the n dimensions. The figure below shows four subdivision steps of the 4throot-of-2-subdivision scheme, starting with one hypercube (illustration stretched in temporal direction, not depicting the temporal connections) and leading to 16 hypercubes.
Another drawback of regular data structures is that down sampling is based purely on grid structure and without considering data values. Therefore, some scientifically interesting details in a data set can get lost and be overseen for further examinations. To avoid this, we use a linear B-spline wavelet scheme: The data value at a vertex is updated when changing the level of detail, i.e., the value varies with varying level of detail. On a coarse level, the value represents the value at the vertex itself as well as an average value of a certain region around the vertex. This approach leads to better approximations on coarser levels.
Quadrilinear B-spline wavelets have the property that the computation of the wavelet coefficient at a vertex p is not only based on the neighbors of p but also on vertices that are farther away in the spatial and temporal dimensions. Thus, when using out-of-core techniques to operate on or visualize large-scale data, substantial amounts of data must be loaded from external memory, with low I/O-performance. We developed lifting schemes with narrow filters to overcome this problem.