Obradovic and Parberry showed that any n-input k-ary function can be computed by a depth 4 unit-weight k-ary threshold circuit of size O(nkⁿ). They also showed that any n-input k-ary symmetric function can be computed by a depth 6 unit-weight k-ary threshold circuit of size O(n^k+1). In this paper, we improve upon and expand their results. The k-ary threshold circuits of nonunit weight and unit weight are considered. We show that any n-input k-ary function can be computed by a depth 2 k-ary threshold circuit of size O(k^n-1). This means that depth 2 is optimal for computing some k-ary functions (e.g., a PARITY function). We also show that any n-input k-ary function can be computed by a depth 3 unit-weight k-ary threshold circuit of size O(kⁿ). Next, we show that any n-input k-ary symmetric function can be computed by a depth 3 k-ary threshold circuit of size O(n^k-1), and can be computed by a depth 3 unit-weight k-ary threshold circuit of size O(kn^k-1). Finally, we show that if the weights of the circuit are polynomially bounded, some k-ary symmetric functions cannot be computed by any depth 2 k-ary threshold circuit of polynomial-size.

Introducing a mathematical model of noise in stereo images, we propose a new criterion for intelligent statistical inference about the scene we are viewing by using the geometric information criterion (geometric AIC). Using synthetic and real-image experiments, we demonstrate that a robot can test whether or not the object is located very far away or the object is a planar surface without using any knowledge about the noise magnitude or any empirically adjustable thresholds.

High-speed display of 3-D objects in virtual reality environments is one of the currently important subjects. Shape simplification is considered an efficient method. This paper presents a method of hierarchical cube-based segmentation for shape simplification and multiresolution model construction. The relations among shape simplification, resolution and visual distance are derived firstly. The first level model is generated from scattered range data by cube-base segmentation with the first level cube size. Multiresolution models are then generated by re-sampling polygonal patch vertices of each former level model with hierarchical cube-based segmentation structure. The results show that the algorithm is efficient for constructing multiresolution models of free-form shape 3-D objects from scattered range data and high compression ratio can be obtained with little noticeable difference during the visualization.

A lot of optimum filters have been proposed for an image restoration problem. Parametric filter, such as Parametric Wiener Filter, Parametric Projection Filter, or Parametric Partial Projection Filter, is often used because it requires to calculate a generalized inverse of one operator. These optimum filters are formed by a degradation operator, a covariance operator of noise, and one of original images. In practice, these operators are estimated based on empirical knowledge. Unfortunately, it happens that such operators differ from the true ones. In this paper, we show the unified formulae of inducing them to clarify their common properties. Moreover, we investigate their properties for perturbation of a degradation operator, a covariance operator of noise, and one of original images. Some numerical examples follow to confirm that our description is valid.

In this paper, a new method for measuring three-dimensional (3D) moving facial shapes is introduced. This method uses two light sources and a slit pattern projector. First, the normal vectors at points on a face are computed by the photometric stereo method with two light sources and a conventional video camera. Next, multiple light stripes are projected onto the face with a slit pattern projector. The 3D coordinates of the points on the stripes are measured using the stereo vision algorithm. The normal vectors are then integrated within 2D finite intervals around the measured points on the stripes. The 3D curved segment within each finite interval is computed by the integration. Finally, all the curved segments are blended into the complete facial shape using a family of exponential functions. By switching the light rays at high speed, the time required for sampling data can be reduced, and the 3D shape of a moving human face at each instant can be measured.

In this paper, we prove that for a class of nonsymmetric neural networks with connection matrices T having nonnegative off-diagonal entries, -T is an M-matrix is a necessary and sufficient condition for absolute exponential stability of the network belonging to this class. While this result extends the existing one of absolute stability in Forti, et al., its proof given in this paper is simpler, which is completed by an approach different from one used in Forti, et al. The most significant consequence is that the class of nonsymmetric neural networks with connection matrices T satisfying -T is an M-matrix is the largest class of nonsymmetric neural networks that can be employed for embedding and solving optimization problem with global exponential rate of convergence to the optimal solution and without the risk of spurious responses. An illustrating numerical example is also given.

Efforts have been cumulated to measure tooth mobility, in order to accurately characterize the mechanical features of periodontal tissues. This paper provides a totally new technique for accomplishing the task of measuring tooth displacement in 6 degrees of freedom, using a range finder. Its intraoral equipment comprises two elements, a moving polyhedron and a referential device, both of which are secured to a subject tooth and several other teeth splinted together. The polyhedron has 6 planar surfaces, each oriented in a distinctly different direction, with each plane facing an opposing range finder mounted on the referential part. If the sensor geometry is provided, the position and orientation of the movable part, vis-a-vis the reference, can be determined theoretically from the distances between all the range finders and their opposing surfaces. This computation was mathematically formulated as a non-linear optimization problem, the numerical solution of which can be obtained iteratively. Its error-propagation formula was also provided as a linear approximation.