Abstract space

Abstract: Introduction, 89. — I. The idea of abstract space and economic analysis, 91; distinction between geonomic (banal) space and economic spaces, 92; space defined by a plan, 95; space defined as a field of forces, 95; space defined as a homogeneous aggregate, 96. — II. Some applications of the distinction between the three types of economic space, 97: monetary space, 97; national space, 99; the European economy, 102.

Abstract: A common motion interpolation technique for realistic human animation is to blend similar motion samples with weighting functions whose parameters are embedded in an abstract space. Existing methods, however, are insensitive to statistical properties, such as correlations between motions. In addition, they lack the capability to quantitatively evaluate the reliability of synthesized motions. This paper proposes a method that treats motion interpolations as statistical predictions of missing data in an arbitrarily definable parametric space. A practical technique of geostatistics, called universal kriging, is then introduced for statistically estimating the correlations between the dissimilarity of motions and the distance in the parametric space. Our method statistically optimizes interpolation kernels for given parameters at each frame, using a pose distance metric to efficiently analyze the correlation. Motions are accurately predicted for the spatial constraints represented in the parametric space, and they therefore have few undesirable artifacts, if any. This property alleviates the problem of spatial inconsistencies, such as foot-sliding, that are associated with many existing methods. Moreover, numerical estimates for the reliability of predictions enable motions to be adaptively sampled. Since the interpolation kernels are computed with a linear system in real-time, motions can be interactively edited using various spatial controls.

Abstract: : A major source of inefficiency in automated problem solvers is their inability to decompose problems and work on the more difficult parts first. This issue can be addressed by employing a hierarchy of abstract problem spaces to focus the search. Instead of solving a problem in the original problem space, a problem is first solved in an abstract space, and the abstract solution is then refined at successive levels in the hierarchy. While this use of abstraction can significantly reduce search, it is often difficult to find good abstractions, and the abstractions must be manually engineered by the designer of a problem domain.