1 Functional Representations

Scalable functional representations have been studied in many research areas in order to interpolate, approximate, and abstract data. Many disciplines no longer explore raw data, but functionally derived and processed information from the raw data. Examples include CT and MRI images in medical applications, and the derived high-level products from remote sensing data. Common techniques for functional representation include radial basis functions, wavelets, and spherical harmonics. The nature of this functional representation is to find functions hierarchically, therefore, it is possible to represent data from abstract to detail levels according to compression ratio and level of detail. The abstract level of the representation allows us to visualize large data interactively, whereas, the detail level requires more computational power for the visualization. Moreover it is a unified representation regardless of data formats. Details of the functional representations are presented as follows.

1.1 Radial Basis Functions

Previously, most radial basis function (RBF) encoding work has concentrated on surface fitting. In the early 1970s, RBFs were used to interpolate geography surface data using multiquadric RBFs [13] and results showed that RBFs are a good interpolation basis function for smooth surface data sets. Hardy [14] presented 20  years of discovery in the theory and applications of multiquadric RBFs and surveys RBF work from 1968 to 1988. Franke [10] showed scattered data interpolation and tests using several methods, such as the inverse distance weighted method, the rectangle based blending method, and the triangle based blending method. He compared these methods and showed that Hardy’s multiquadric approach is best. Since Franke’s work, multiquadrics have been considered the best basis function in most surface fitting research. After Franke’s survey, Franke and Nielson [12] collected more work on surface fitting and presented their research by surveying and comparing several techniques. For better interpolation, the least squares approach were used by Franke and Hagen [11]. For the approximation of surface fitting, knot (center) selection [23] was introduced using thin plate splines by Dirichlet tessellation. Through knot selection, encoded data can be reduced and a small number of basis function can represent the whole data set.

Although RBFs have been used to reconstruct surfaces by approximating scattered data sets, they were primarily used for mesh reduction of surface representations [4, 25, 32, 39]. In more recent work on surface fitting, Carr et al. [4] showed surface fitting as an approximation using multiquadric RBFs. They iteratively added basis functions using a greedy algorithm by computing fitting errors, where basis functions were added at larger error points. In their work, the zero level set implicit surface of the distance function was fit and energy (error) was minimized for the smoothest interpolant. Ohtake et al. [31] also showed the fitting of implicit surfaces. They selected centers based on the density of data points. More basis functions were added in higher density areas. By linking the RBF approximation and the partition of unity method [30], Ohtake el al. presented a robust approximation for noisy data.

Volume fitting using RBFs was introduced by Nielson et al. [28, 29], where they extended surface fitting methods to volume fitting. Their approaches showed good approximation of volume data. In more recent work on volume fitting, Co et al. [5] showed a hierarchical representation of volumetric data sets based on clusters computed by Principal Component Analysis (PCA). A level of detail representation was extracted by either the hierarchical level or the error. Jang et al. [16, 17] and Weiler et al. [41] proposed a functional representation approach for interactive volume rendering. Their approaches were designed for any scattered datasets and directly volume rendered the basis functions without resampling. Moreover, using ellipsoidal basis functions, they improved the functional representation statistically and visually [16]. Recently, Ledergerber et al. [19] applied a moving least square to interpolate the volumetric data and Vuçini et al. [40] reconstructed non-uniform volumetric data by B-splines.

1.2 Wavelets

As a hierarchical data representation for comparison, there is an approach, wavelet [3, 6, 15, 26, 27, 35–37]. In the area of volume encoding, most work [15, 26, 27] has been performed for regular grids and shows effective 3D compressed volumes and rendering. Recently, this research has been extended to irregular grids organized using polygonal meshes [6, 36, 37]. The wavelet encoding of irregular grids is usually performed using irregular sampling and adaptive subdivision. However, the wavelet for the irregular data set is based on polygonal meshes. Therefore, it is not easily extended to arbitrary scattered volume data. Sohn et al. [35] presented a compression scheme for encoding time-varying isosurface and volume features. They encoded only the significant blocks using a block-based wavelet transform.

1.3 Spherical Harmonics

Most of the work in spherical harmonics has been done for surface fitting, especially, 3D object modeling and molecular surface modeling. For 3D object modeling [7, 18, 38], the datasets were decomposed into high and low frequency components and represented by the properties of the spherical harmonic basis functions. Also this modeling was used for shape deformation [8, 9]. In molecular surface modeling [22], spherical harmonics give a sequence of smooth approximations to the molecular surface since the shapes of the spherical harmonics are very similar to the shapes of the molecules. For volume fitting, Misner [24] showed spherical harmonic decomposition, however, the volume fitting is based on a rectangular grid.

1.4 Time Series Data Representations

The large volume of time-varying data makes visualization a challenging problem. Many techniques for volume rendering of time-varying data have been proposed and these techniques enable the visualization of large amount of time-varying datasets. One approach is to use data coherency between consecutive time steps to speed up volume rendering [1, 2, 33, 34]. Another approach is to encode and compress the time-varying data appropriately for the volume rendering [20, 21, 35, 42]. Shen and Johnson [34] proposed an algorithm that exploits data coherency between time steps and extracts the differential information for biomedical and computational fluid dynamics datasets. Shen et al. [33] showed the time-space partitioning (TSP) tree and this structure improves the rendering speed and reduces the amount of volumetric data I/O. For temporal compression approach, Westermann [42] proposed a memory minimizing algorithm based on multi-resolution representations of both the spatial distribution and the time evolutions. Ma and Shen [21] presented quantization and octree encoding of time-varying datasets and reduced the rendering speed over time. Lum et al. [20] presented temporal encoding using discrete cosine transform (DCT) and accelerated the rendering speed using the graphics hardware.