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.
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.
@cite_16 @cite_2
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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. The problem considered involves the use of a sequence of noisy monocular images of a three-dimensional moving object to estimate both its structure and kinematics. The object is assumed to be rigid, and its motion is assumed to be smooth. A set of object match points is assumed to be available, consisting of fixed features on the object, the image plane coordinates of which have been extracted from successive images in the sequence. Structure is defined as the 3-D positions of these object feature points, relative to each other. Rotational motion occurs about the origin of an object-centered coordinate system, while translational motion is that of the origin of this coordinate system. In this work, which is a continuation of the research done by the authors and reported previously (ibid., vol.PAMI-8, p.90-9, Jan. 1986), results of an experiment with real imagery are presented, involving estimation of 28 unknown translational, rotational, and structural parameters, based on 12 images with seven feature points. > The error sensitivity in the estimation of the 3-D motion and the normal of a planar surface from an instantaneous motion field is studied. The statistical theory or the Cramer-Rao lower bound is used for error covariance in the estimated motion and structure parameters. This enables the derivation of results valid for any unbiased estimator under the assumption of Gaussian noise in the motion field. The obtained lower-bound-matrix is studied analytically with respect to measurement noise, size of the field of view, and the motion-geometry configuration. The coupling between translation and rotation is exacerbated if the field of view and the slant of the plane become smaller, and the deviation of the translation from the viewing direction becomes larger. The relationships of the uncertainty bounds for every unknown motion parameter to the angle between translation and the plane-normal, the size of the field of view, and the distance from the perceived plane and the translation magnitude are discussed. > Inspired by Weber's Law, this paper proposes a simple, yet very powerful and robust local descriptor, called the Weber Local Descriptor (WLD). It is based on the fact that human perception of a pattern depends not only on the change of a stimulus (such as sound, lighting) but also on the original intensity of the stimulus. Specifically, WLD consists of two components: differential excitation and orientation. The differential excitation component is a function of the ratio between two terms: One is the relative intensity differences of a current pixel against its neighbors, the other is the intensity of the current pixel. The orientation component is the gradient orientation of the current pixel. For a given image, we use the two components to construct a concatenated WLD histogram. Experimental results on the Brodatz and KTH-TIPS2-a texture databases show that WLD impressively outperforms the other widely used descriptors (e.g., Gabor and SIFT). In addition, experimental results on human face detection also show a promising performance comparable to the best known results on the MIT+CMU frontal face test set, the AR face data set, and the CMU profile test set.