The trend in science and engineering applications has been to produce larger data sets, since computers and imaging technology are getting faster and storage space is increasing. Large amounts of data are difficult to visualize, and it is impossible to directly visualize them on inexpensive computers. Many visualization techniques exist that visualize certain types of large data. However, a general solution does not exist. A hierarchical method provides the foundation for a solution. Linear and quadratic decomposition elements can be used to form an approximation hierarchy representing large data; a user can then visualize this hierarchy on low-end machines. A hierarchical approximation method is described that uses linear, quadratic, and curved-quadratic decompositional elements. Linear element approximation and visualization has been studied extensively in the past. Higher-order element approximation and visualization is not nearly as developed as that for linear elements, thus, more research is needed. Fundamental visualization techniques--such as isosurfacing, ray casting, and cutting planes--for quadratic elements are described.

We present a method for the construction of hierarchies of single-valued functions in one, two, and three variables. The input to our method is a coarse decomposition of the compact domain of a function in the form of an interval (univariate case), triangles (bivariate case), or tetrahedra (trivariate case). We compute the best linear spline approximations, understood in an integral least squares sense, for functions defined over such triangulations and refine triangulations using repeated bisection. This requires the identification of the interval (triangle, tetrahedron) with largest error and splitting it into two intervals (triangles, tetrahedra). Each bisection step requires the re-computation of all spline coefficients due to the global nature of the best approximation problem. Nevertheless, this can be done efficiently by bisecting multiple intervals (triangles, tetrahedra) in one step and by reducing the bandwidths of the matrices resulting from the normal equations.

We present a method for the hierarchical approximation of functions in one, two, or three variables based on the finite element method (Ritz approximation). Starting with a set of data sites with associated function, we first determine a smooth (scattered-data) interpolant. Next, we construct an initial triangulation by triangulating the region bounded by the minimal subset of data sites defining the convex hull of all sites. We insert only original data sites, thus reducing storage requirements. For each triangulation we solve a minimization problem: computing the best linear spline approximation of the interpolant of all data, based on a functional involving function values and first derivatives. The error of a best linear spline approximation is computed in a Sobolev-like norm, leading to element-specific error values. We use these interval-/triangle-/tetrahedron-specific values to identify the element to subdivide next. The subdivision of an element with largest error value requires the re-computation of all spline coefficients due to the global nature of the problem. We improve efficiency by (i) subdividing multiple elements simultaneously and (ii) by using a sparse-matrix representation and system solver.

We present a method for the construction of multiple levels of tetrahedral meshes approximating a trivariate function at different levels of detail. Starting with an initial, high-resolution triangulation of a three-dimensional region, we construct coarser representation levels by collapsing tetrahedra. Each triangulation defines a linear spline function, where the function values associated with the vertices are the spline coefficients. Based on predicted errors, we collapse tetrahedron in the grid that do not cause the maximum error to exceed a use-specified threshold. Bounds are stored for individual tetrahedra and are updated as the mesh is simplified. We continue the simplification process until a certain error is reached. The result is a hierarchical data description suited for the efficient visualization of large data sets at varying levels of detail.

Nursing influences growth rate and overall health of mammals; however, the behavior is difficult to study in wild cetaceans because it occurs below the surface and can thus be misidentified from surface observations. Nursing has been observed in humpback whales on the breeding and calving grounds, but the behavior remains unstudied on the feeding grounds. We instrumented three dependent calves (four total deployments) with combined video and 3D-accelerometer data loggers (CATS) on two United States feeding grounds to document nursing events. Two associated mothers were also tagged to determine if behavior diagnostic of nursing was evident in the mothers movement. Animal-borne video was manually analyzed and the average duration of successful nursing events was 23 s (±7 sd, n = 11). Nursing occurred at depths between 4.1-64.4 m (along the seafloor) and in close temporal proximity to foraging events by the mothers, but could not be predicted solely by relative positions of mother and calf. When combining all calf deployments, successful nursing was documented eleven times; totaling only 0.3% of 21.0 hours of video. During nursing events, calves had higher overall dynamic body acceleration (ODBA) and increased fluke-stroke rate (FSR) compared to non-nursing segments (Mixed effect models, ODBA: F1,107 = 13.57756, p = 0.0004, FSR: F1,107 = 32.31018, p < 0.0001). In contrast, mothers had lower ODBA and reduced FSR during nursing events compared to non-nursing segments. These data provide the first characterization of accelerometer data of humpback whale nursing confirmed by animal-borne video tags and the first analysis of nursing events on feeding grounds. This is an important step in understanding the energetic consequences of lactation while foraging.

We discuss spline refinement methods that approximate multi-valued data defined over one, two, and three dimensions. The input to our method is a coarse decomposition of the compact domain of the function to be approximated consisting of intervals (univariate case), triangles (bivariate case), and tetrahedra (trivariate case). We first describe a best linear spline approximation scheme, understood in a least squares sense, and refine on initial mesh using repeated bisection of simplices (intervals, triangles, or tetrahedra) of maximal error. We discuss three enhancements that improve the performance and quality of our basic bisection ap- proach. The enhancements we discuss are: (i) using a -nite element approach that only considers original data sites during subdivision, (ii) including -rst-derivative information in the error functional and spline-coe+-cient computations, and (iii) using quadratic (deformed, \curved'') rather than linear simplices to better ap- proximate bivariate and trivariate data. We improve e+-ciency of our re-nement algorithms by subdividing multiple simplices simultaneously and by using a sparse- matrix representation and system solver.

We have combined methods from volume visualization and data analysis to support better diagnosis and treatment of human retinal diseases. Many diseases can be identified by abnormalities in the thicknesses of various retinal layers captured using optical coherence tomography (OCT). We used a support vector machine (SVM) to perform semi-automatic segmentation of retinal layers for subsequent analysis including a comparison of layer thicknesses to known healthy parameters. We have extended and generalized an older SVM approach to support better performance in a clinical setting through performance enhancements and graceful handling of inherent noise in OCT data by considering statistical characteristics at multiple levels of resolution. The addition of the multi-resolution hierarchy extends the SVM to have "global awareness." A feature, such as a retinal layer, can therefore be modeled.

Suction-cup-attached biologging tags have led to major advances in our understanding of large whale behaviour. Getting close enough to a whale at sea to safely attach a tag is a major limiting factor when deploying these systems. Here we present an uncrewed aerial system (UAS)-based tagging technique for free-swimming large whales and provide data on efficacy from field testing on blue (Balaenoptera musculus) and fin (B. physalus) whales. Rapid transit speed and the birds-eye view of the animal during UAS tagging contributed to the techniques success. During 8 days of field testing, we had 29 occasions when a focal animal was identified for attempted tagging and tags were successfully attached 21 times. The technique was efficient, with mean flight time of 2 min 45 s from launch to deployment and a mean distance of 490 m from the launch vessel to tagged animal, reducing potential adverse effects resulting from close approaches for tagging. These data indicate that UAS are capable of attaching biologging tags to free-swimming large whales quickly and from large distances, potentially increasing success rates, decreasing attempt times, and reducing animal disruption during tagging.