ABSTRACT
Computing and visualizing features in fluid flow often depends on the observer, or reference frame, relative to which the input velocity field is given. A desired property of feature detectors is therefore that they are objective, meaning independent of the input reference frame. However, the standard definition of objectivity is only given for Euclidean domains and cannot be applied in curved spaces. We build on methods from mathematical physics and Riemannian geometry to generalize objectivity to curved spaces, using the powerful notion of symmetry groups as the basis for definition. From this, we develop a general mathematical framework for the objective computation of observer fields for curved spaces, relative to which other computed measures become objective. An important property of our framework is that it works intrinsically in 2D, instead of in the 3D ambient space. This enables a direct generalization of the 2D computation via optimization of observer fields in flat space to curved domains, without having to perform optimization in 3D. We specifically develop the case of unsteady 2D geophysical flows given on spheres, such as the Earth. Our observer fields in curved spaces then enable objective feature computation as well as the visualization of the time evolution of scalar and vector fields, such that the automatically computed reference frames follow moving structures like vortices in a way that makes them appear to be steady.
ABSTRACT
Flow fields are usually visualized relative to a global observer, i.e., a single frame of reference. However, often no global frame can depict all flow features equally well. Likewise, objective criteria for detecting features such as vortices often use either a global reference frame, or compute a separate frame for each point in space and time. We propose the first general framework that enables choosing a smooth trade-off between these two extremes. Using global optimization to minimize specific differential geometric properties, we compute a time-dependent observer velocity field that describes the motion of a continuous field of observers adapted to the input flow. This requires developing the novel notion of an observed time derivative. While individual observers are restricted to rigid motions, overall we compute an approximate Killing field, corresponding to almost-rigid motion. This enables continuous transitions between different observers. Instead of focusing only on flow features, we furthermore develop a novel general notion of visualizing how all observers jointly perceive the input field. This in fact requires introducing the concept of an observation time, with respect to which a visualization is computed. We develop the corresponding notions of observed stream, path, streak, and time lines. For efficiency, these characteristic curves can be computed using standard approaches, by first transforming the input field accordingly. Finally, we prove that the input flow perceived by the observer field is objective. This makes derived flow features, such as vortices, objective as well.
ABSTRACT
Dynamic contrast-enhanced image data (perfusion data) are used to characterize regional tissue perfusion. Perfusion data consist of a sequence of images, acquired after a contrast agent bolus is applied. Perfusion data are used for diagnostic purposes in oncology, ischemic stroke assessment or myocardial ischemia. The diagnostic evaluation of perfusion data is challenging, since the data is complex and exhibits various artifacts, e.g., motion artifacts. We provide an overview on existing methods to analyze, and visualize CT and MR perfusion data. The integrated visualization of several 2D parameter maps, the 3D visualization of parameter volumes and exploration techniques are discussed. An essential aspect in the diagnosis of perfusion data is the correlation between perfusion data and derived time-intensity curves as well as with other image data, in particular with high resolution morphologic image data. We discuss visualization support with respect to the three major application areas: ischemic stroke diagnosis, breast tumor diagnosis and the diagnosis of coronary heart disease.
