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Instability of the eikonal equation and shape from shading

Ian Barnes, Kewei Zhang (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In the shape from shading problem of computer vision one attempts to recover the three-dimensional shape of an object or landscape from the shading on a single image. Under the assumptions that the surface is dusty, distant, and illuminated only from above, the problem reduces to that of solving the eikonal equation |Du|=f on a domain in ℝ 2 . Despite various existence and uniqueness theorems for smooth solutions, we show that this problem is unstable, which is catastrophic for general numerical algorithms. ...

Integrated region-based segmentation using color components and texture features with prior shape knowledge

Mehryar Emambakhsh, Hossein Ebrahimnezhad, Mohammad Hossein Sedaaghi (2010)

International Journal of Applied Mathematics and Computer Science

Segmentation is the art of partitioning an image into different regions where each one has some degree of uniformity in its feature space. A number of methods have been proposed and blind segmentation is one of them. It uses intrinsic image features, such as pixel intensity, color components and texture. However, some virtues, like poor contrast, noise and occlusion, can weaken the procedure. To overcome them, prior knowledge of the object of interest has to be incorporated in a top-down procedure...

Intelligent decision-making system for autonomous robots

Zdzisław Kowalczuk, Michał Czubenko (2011)

International Journal of Applied Mathematics and Computer Science

The paper gives an account of research results concerning a project on creating a fully autonomous robotic decisionmaking system, able to interact with its environment and based on a mathematical model of human cognitive-behavioural psychology, with some key elements of personality psychology included. The principal idea of the paper is focused on the concept of needs, with a certain instrumental role of emotions.

Interpretability of linguistic variables: a formal account

Ulrich Bodenhofer, Peter Bauer (2005)

Kybernetika

This contribution is concerned with the interpretability of fuzzy rule-based systems. While this property is widely considered to be a crucial one in fuzzy rule-based modeling, a more detailed formal investigation of what “interpretability” actually means is not available. So far, interpretability has most often been associated with rather heuristic assumptions about shape and mutual overlapping of fuzzy membership functions. In this paper, we attempt to approach this problem from a more general...

Interpretable random forest model for identification of edge 3-uncolorable cubic graphs

Adam Dudáš, Bianka Modrovičová (2023)

Kybernetika

Random forest is an ensemble method of machine learning that reaches a high level of accuracy in decision-making but is difficult to understand from the point of view of interpreting local or global decisions. In the article, we use this method as a means to analyze the edge 3-colorability of cubic graphs and to find the properties of the graphs that affect it most strongly. The main contributions of the presented research are four original datasets suitable for machine learning methods, a random...

Interpretation and optimization of the k -means algorithm

Kristian Sabo, Rudolf Scitovski (2014)

Applications of Mathematics

The paper gives a new interpretation and a possible optimization of the well-known k -means algorithm for searching for a locally optimal partition of the set 𝒜 = { a i ∈ ℝ n : i = 1 , ⋯ , m } which consists of k disjoint nonempty subsets π 1 , ⋯ , π k , 1 ≤ k ≤ m . For this purpose, a new divided k -means algorithm was constructed as a limit case of the known smoothed k -means algorithm. It is shown that the algorithm constructed in this way coincides with the k -means algorithm if during the iterative procedure no data points appear in the Voronoi diagram....

Interpretation of pattern classification results, obtained from a test set

Edgard Nyssen (1998)

Kybernetika

The present paper presents and discusses a methodology for interpreting the results, obtained from the application of a pattern classifier to an independent test set. It addresses the problem of testing the random classification null hypothesis in the multiclass case, by introducing an exact probability technique. The discussion of this technique includes the presentation of an interval estimation technique for the probability of correct classification, which is slightly more accurate than the ones...

Interval analysis for certified numerical solution of problems in robotics

Jean-Pierre Merlet (2009)

International Journal of Applied Mathematics and Computer Science

Interval analysis is a relatively new mathematical tool that allows one to deal with problems that may have to be solved numerically with a computer. Examples of such problems are system solving and global optimization, but numerous other problems may be addressed as well. This approach has the following general advantages: (a) it allows to find solutions of a problem only within some finite domain which make sense as soon as the unknowns in the problem are physical parameters; (b) numerical computer...

Intrinsic dimensionality and small sample properties of classifiers

Šarūnas Raudys (1998)

Kybernetika

Small learning-set properties of the Euclidean distance, the Parzen window, the minimum empirical error and the nonlinear single layer perceptron classifiers depend on an “intrinsic dimensionality” of the data, however the Fisher linear discriminant function is sensitive to all dimensions. There is no unique definition of the “intrinsic dimensionality”. The dimensionality of the subspace where the data points are situated is not a sufficient definition of the “intrinsic dimensionality”. An exact...

Isomorphisms of Direct Products of Cyclic Groups of Prime Power Order

Hiroshi Yamazaki, Hiroyuki Okazaki, Kazuhisa Nakasho, Yasunari Shidama (2013)

Formalized Mathematics

In this paper we formalized some theorems concerning the cyclic groups of prime power order. We formalize that every commutative cyclic group of prime power order is isomorphic to a direct product of family of cyclic groups [1], [18].

Isomorphisms of Direct Products of Finite Commutative Groups

Hiroyuki Okazaki, Hiroshi Yamazaki, Yasunari Shidama (2013)

Formalized Mathematics

We have been working on the formalization of groups. In [1], we encoded some theorems concerning the product of cyclic groups. In this article, we present the generalized formalization of [1]. First, we show that every finite commutative group which order is composite number is isomorphic to a direct product of finite commutative groups which orders are relatively prime. Next, we describe finite direct products of finite commutative groups

Isomorphisms of Direct Products of Finite Cyclic Groups

Kenichi Arai, Hiroyuki Okazaki, Yasunari Shidama (2012)

Formalized Mathematics

In this article, we formalize that every finite cyclic group is isomorphic to a direct product of finite cyclic groups which orders are relative prime. This theorem is closely related to the Chinese Remainder theorem ([18]) and is a useful lemma to prove the basis theorem for finite abelian groups and the fundamental theorem of finite abelian groups. Moreover, we formalize some facts about the product of a finite sequence of abelian groups.

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