Convex Images Research Articles

3-D Convex Outputs for the Blind - Image Quality Analysis

One of the means by which the blind communicate with the environment is text information in the Braille system and convex graphical images printed on the paper. Printing and reading 3-D images is of a special significance in the process of educating the blind. It is also used in any contact of the blind with the literature, in every-day life (e.g. convex guidebooks) and professional life (letters, documents, etc.).Because of a high diversification of the perception abilities of the blind as well as widespread application of the convex printings, the role of evaluating their quality becomes more significant. Quantitative measurements of the image quality of the convex printouts may be also used for evaluation of printing and copying devices.In the paper one analysed the main techniques of making 3-D printouts and copies for the blind. In the paper one proposed parameters for evaluating the quality of convex images and methodology of measuring them. One provided also preliminary results of measurements of these parameters. Results of the works carried out can be a basis for developing design of devices for printing in the Braille system and for convex copying.

Continuous Selections for Multivalued Mappings with Closed Convex Images and Applications

Some existence theorems of continuous selections for multivalued mappings with closed convex images are proved. As some simple applications, we give some results on fixed points and differential inclusions. The results presented in this paper generalize some recent results.

Regular Castaing Representations of Multifunctions with Applications to Stochastic Programming

We consider set-valued mappings defined on a topological space with closed convex images in $\mathbb R^n$. The measurability of a multifunction is characterized by the existence of a Castaing representation for it: a countable set of measurable selections that pointwise fills up the graph of the multifunction. Our aim is to construct a Castaing representation which inherits the regularity properties of the multifunction. The construction uses Steiner points. A notion of a generalized Steiner point is introduced. A Castaing representation called regular is defined by using generalized Steiner selections. All selections are measurable, continuous, resp., Hölder-continuous, or directionally differentiable, if the multifunction has the corresponding properties. The results are applied to various multifunctions arising in stochastic programming. In particular, statements about the asymptotic behavior of measurable selections of solution sets via the delta-method are obtained.

Inventing the Formula of the Trees: A Solution of the Representation of Self Similar Objects

Iterated Function Systems (IFS) seem to be used best to represent objects in the nature, because many of them are self similar. An IFS is a set of affine and contractive transformations. The union (so-called collage) of the subimages generated by transforming the whole image produces the image again - the self similar attractor of these transformations, which can be described by a binary image. For a fast and compact representation of those images, it would be desirable to calculate the transformations (the IFS-Codes) directly from the image that means to solve the inverse IFS-Problem. The solution presented in this paper will directly use the features of the self similar image. Subsets of the entire image and the subimage to be calculated are identified by the computation of the set difference between the pixels of the original and a rotated copy. The rotation and the scale factor of the transformation can be computed by the mapping of this two subsets onto each other, if the translation part - the fixed point - is predefined. The calculation of the transformation has to be repeated for each subimage. It will be proved, that with this method the IFS-Codes can be calculated for not convex, undistorted, and self similar images as long as the fixed point is known. An efficient algorithm for the identification of these fixed points within the image is introduced. Different ways to achieve this solutions are presented. In the conclusion the class of images, which can be coded by this method is defined, the results are pointed out, the advantages resp. the disadvantages of the method are evaluated, and possible ways to extend the method are discussed.

Differential inclusions with stable subinclusions

wherexER”,tE[tO,T]andF:R”x[tO,T] G. R is a set-valued mapping with compact and convex images. An absolutely continuous function x(m) is a solution of (1) if it satisfies (1) for a.e. t E [to, T]. Thus, a differential inclusion generalizes an ordinary differential equation by permitting a set-valued right-hand side. Differential equations containing either control/ uncertain parameters or discontinuity in the right-hand side can be transformed into the framework of (1) [l, 21. A well-known and very useful result due to Filippov [3] estimates the uniform distance from an arbitrary absolutely continuous function x( -) to the set of solutions of (1) starting from x(t,) by means of constant times the “deviation”

A remark on the lower semi-continuity of the set-valued metric projection

where E is a normed linear space and G C E a proximial linear subspace. In this context, lower and upper semi-continuity and continuous selections are three of the most important notions: Let F: E -+ 2G be a set-valued mapping (E, G Hausdorff-spaces.) F is called lower semi-continuous (l.s.c.), iff, for each x E E and each open subset UC G with U r F(x) # o , there exists a neighborhood V C E of x such that U n F(y) f o for each y E V. F is called upper semi-continuous (u.s.c.), iff, for each x E E and each open subset UC G with F(x) C U, there exists a neighborhood V C E of x with F(y) C U for each y E V. A single-valued mapping S: E + G is called a selection for F, iff s(x) E F(x) for each x E E. A well-known theorem of Michael [4, Theorem 3.2”] states that an 1.s.c. set-valued mapping from a paracompact Hausdorff-space into a Banachspace with closed convex images always admits a continuous selection. In his paper [5], Ntirnberger introduces the following notion: Let E be a normed linear space and G C E a proximinal linear subspace. A selection S: E + G for PG is said to have the “Nulleigenschaft,” iff S(X) = 0 for each x E E with 0 E PC(x). Using this notion, the lower semi-continuity of the set-valued metric projection can be characterized as follows: 83 0021-9045/80/010083-04$02.00/0