text: Chains are vector-valued signals sampling a curve. They are important to motion signal processing and to many scientific applications including location sensors. We propose a novel measure of smoothness for chains curves by generalizing the scalar-valued concept of monotonicity. Monotonicity can be defined by the connectedness of the inverse image of balls. This definition is coordinate-invariant and can be computed efficiently over chains. Monotone curves can be discontinuous, but continuous monotone curves are differentiable a.e. Over chains, a simple sphere-preserving filter shown to never decrease the degree of monotonicity. It outperforms moving average filters over a synthetic data set. Applications include Time Series Segmentation, chain reconstruction from unordered data points, Optical Character Recognition, and Pattern Matching.
  @cite_16: Smooth motion generation is an important issue in the computer animation and virtual reality areas. The motion of a rigid body consists of translation and orientation. The former is described by a space curve in 3-dimensional Euclidean space, while the latter is represented by a curve in the unit quaternion space. Although there exist well-known techniques for smoothing the translation data, smoothing the orientation data is yet to be explored due to the nonlinearity of the unit quaternion space. This paper presents a wavelet-based algorithm for smoothing noise-embedded motion data and the experiment shows the effectiveness of the proposed algorithm.
  @cite_2: Multiresolution motion analysis has gained considerable research interest as a unified framework to facilitate a variety of motion editing tasks. Within this framework, motion data are represented as a collection of coefficients that form a coarse-to-fine hierarchy. The coefficients at the coarsest level describe the global pattern of a motion signal, while those at fine levels provide details at successively finer resolutions. Due to the inherent nonlinearity of the orientation space, the challenge is to generalize multiresolution representations for motion data that contain orientations as well as positions. Our goal is to develop a multiresolution analysis method that guarantees coordinate-invariance without singularity. To do so, we employ two novel ideas: hierarchical displacement mapping and motion filtering. Hierarchical displacement mapping provides an elegant formulation to describe positions and orientations in a coherent manner. Motion filtering enables us to separate motion details level-by-level to build a multiresolution representation in a coordinate-invariant way. Our representation facilitates multiresolution motion editing through level-wise coefficient manipulation that uniformly addresses issues raised by motion modification, blending, and stitching.
summary: A motion signal is comprised of two components: orientation and translation. The orientation vector indicates where the object is facing, whereas the translation component determines the object's location. Recent work has focused on smoothing the orientation vectors @cite_2 @cite_16 , whereas the results of the present paper apply equally well to orientation vectors (points on the surface of a unit sphere) as to arbitrary translation signals.