Pose space deformation generalizes and improves upon both shape interpolation and common skeleton-driven deformation techniques. This deformation approach proceeds from the observation that several types of deformation can be uniformly represented as mappings from a pose space, defined by either an underlying skeleton or a more abstract system of parameters, to displacements in the object local coordinate frames. Once this uniform representation is identified, previously disparate deformation types can be accomplished within a single unified approach. The advantages of this algorithm include improved expressive power and direct manipulation of the desired shapes yet the performance associated with traditional shape interpolation is achievable. Appropriate applications include animation of facial and body deformation for entertainment, telepresence, computer gaming, and other applications where direct sculpting of deformations is desired or where real-time synthesis of a deforming model...

This is a survey of techniques for the interpolation of scattered data in three or more independent variables. It covers schemes that can be used for any number of variables as well as schemes specifically designed for three variables. Emphasis is on breadth rather than depth, but there are explicit illustrations of different techniques used in the solution of multivariate interpolation problems. List of Contents 1. Introduction 2. Rendering of Trivariate Functions 3. Tensor Product Schemes 4. Point Schemes 4.1 Shepard's Methods 4.2 Radial Interpolants 4.2.1 Hardy Multiquadrics 4.2.2 Duchon Thin Plate Splines 5. Natural Neighbor Interpolation 6. k-dimensional Triangulations 7. Tetrahedral Schemes 7.1 Polynomial Schemes 7.2 Rational Schemes 8. Simplicial Schemes 8.1 Polynomial Schemes 8.2 Rational Schemes 8.3 A Transfinite Scheme 9. Multivariate Splines 10. Transfinite Hypercubal Methods 11. Derivative Generation 12. Interpolation on the sphere and other surfa...

This paper explores a memory-based approach to robot learning, using memorybased neural networks to learn models of the task to be performed. Steinbuch and Taylor presented neural network designs to explicitly store training data and do nearest neighbor lookup in the early 1960s. In this paper their nearest neighbor network is augmented with a local model network, which fits a local model to a set of nearest neighbors. This network design is equivalent to a statistical approach known as locally weighted regression, in which a local model is formed to answer each query, using a weighted regression in which nearby points (similar experiences) are weighted more than distant points (less relevant experiences). We illustrate this approach by describing how it has been used to enable a robot to learn a difficult juggling task. Keywords: memory-based, robot learning, locally weighted regression, nearest neighbor, local models. 1 Introduction An important problem in motor learning is approxim...

Recent efforts in visualization have concentrated on high volume data sets from numerical simulations and medical imaging. There is another large class of data, characterized by their spatial sparsity with noisy and possibly missing data points, that also need to be visualized. Two places where these type of data sets can be found are in oceanographic and atmospheric science studies. In such cases, it is not uncommon to have on the order of one percent of sampled data available within a space volume. Techniques that attempt to deal with the problem of filling-in-the-holes range in complexity from simple linear interpolation to more sophisticated multiquadric and optimal interpolation techniques. These techniques will generally produce results that do not fully agree with each other. To avoid misleading the users, it is important to highlight these differences and make sure the users are aware of the idiosyncrasies of the different methods. This paper compares some of these interpolatio...

The classical approaches in fuzzy control (Zadeh and Mamdani) deal with dense rule bases. When this is not the case, i.e. in sparse rule bases, one has to choose another method. Fuzzy rule interpolation (proposed first by Koczy and Hirota [15]) offers a possibility to construct fuzzy controllers (KH controllers) under such conditions. The main result of this paper shows that the KH interpolation method is stable. It also contributes to the application oriented use of Balazs-Shepard interpolation operators investigated extensively by researchers in approximation theory. The numerical analysis aspect of the result contributes to the well-known problem of finding a stable interpolation method in the following sense.

Although, in literature various results can be found claiming that fuzzy rule-based systems (FRBSs) possess the universal approximation property, to reach arbitrary accuracy the necessary number of rules are unbounded. Therefore, the inherent property of FRBSs in the original sense of Zadeh, namely that they can be characterized by a semantic relying on linguistic terms is lost. If we restrict the number of rules, universal approximation is not valid anymore as it was shown for, including others, Sugeno and TSK type models [10,19]. Due to this theoretic bound there is recently a great demand among researchers on finding trade-off techniques between a required accuracy and the number of rules, and as such, they attempt to determine the (optimal) number of rules as a function of accuracy. Naturally, to obtain such results one has to restrict somehow the set of continuous functions, usually requiring some smoothness conditions on the approximated function. In terms of approximation theory this is the so-called saturation problem, the determination of optimal order and class of approximation. Hitherto, saturation classes and orders have not been determined for FRBSs and neural networks. In this paper we solve the saturation problem for a special type of fuzzy controller, for the generalized KH-interpolator, being a suitable inference method in sparse rule bases.

Fuzzy control is the most successfull application area of fuzzy theory. The advantage of fuzzy controllers against conventionel ones is that, they can by used for modelling systems with complicated (non linearizable) or unknown behaviour by means of linguistic variables anf fuzzy If-then rules. Later on, the first approximating model can be tuned to obtain appropriate result. However, an essential problem of these algorithms is that their time complexity grows exponentially with the number of input variables. Fuzzy rule interpolation methods are one of the technique developed to reduce the complexity of fuzzy reasoning approaches. Purposes of this work are the following: -- Modification of the first published Kóczy--Hirota (KH) interpolation method to alleviate the so-called abnormal conclusion while maintaining its advantageous complexity behaviour. -- Investigation of the mathematical stability of the KH-method. -- Examination of the universal approximation property of certain fuzzy controllers. A modification of the original KH approach was proposed, whose main idea is the following. The consequent fuzzy sets are transformed by a proper coordinate transformation to such a space where the convexity of these consequents excludes abnormality of the conclusion. After the conclusion is calculated in this space, the inverse of the aforementioned transformation is used to obtain the corresponding conclusion in the original output space. The proposed method is closed for convex and normal fuzzy sets (Theorem 2.1). The new interpolation method was compared with the KH-approach one in several aspects. It was investigated how the proposed method differs form linear between characteristic points, and finally a comparison among the main interpolation techniques is given with respect to the relation of the observation's and conclusion's fuzziness. It was proven that the input-output function of the KH interpolation converges uniformly to the arbitrary approximated continuous function if the measurement points are uniformly distributed on the domain. A generalization of this theorem is also given for a wider class of interpolatory operators. It was pointed out that the stability of the well-known Shepard-interpolation (investigated extensively by approximation theorists) is can be derived from the one of the KH interpolation. The third main statement characterizes a set of certain type fuzzy controllers with bounded number of rules concerning the universal approximation property. As a generalization of Moser's result, it was shown that this property does not hold for the set of T-controllers (which includes Sugeno, Takagi--Sugeno, Takagi--Sugeno--Kang inference methods) if the number of rules is prerestricted, although that is a considerable practical limitation. It contradicts to those statements which state that fuzzy controllers are universal approximators, i.e., they lie dense in the space of continuous functions.

This paper presents an approach for the global optimization of constrained nonlinear programming problems in which some of the constraints are non-analytical (non-factorable), defined by a computational model for which no explicit analytical representation is available.

This paper investigates the approximation behaviour of the alpha-cut based fuzzy interpolators. First, it is shown that the so-called KH fuzzy interpolator is a fuzzy generalization of a well-known and thoroughly investigated parameterized interpolatory operator from approximation theory, the Shepard operator. Exploiting the aforementioned relationship, we establish analog results on the approximation rate of KH controllers. The optimal order and class of approximation (saturation problem) are determined for certain values of the parameter $\lambda$. Corresponding results on the MACI method, being a modification of the KH interpolator, are also provided.