Fuzzy optimization issues have been discussed by many authors in recent articles, inspired by the concepts of generalized Hukuhara difference and generalized Hukuhara DIFFERENTIABILITY for one-dimensional functions from R to E, and have made considerable progress. In this paper, the concept of total generalized DIFFERENTIABILITY is introduced by using the concept of generalized difference from R^n to E for multi-dimensional mappings. Furthermore, the total generalized DIFFERENTIABILITY of the above is also examined and then the concept of DIRECTIONAL generalized DIFFERENTIABILITY and partial generalized DIFFERENTIABILITY for fuzzy multi-dimensional mappings is defined and discussed in detail. In addition, DIRECTIONAL generalized DIFFERENTIABILITY and partial generalized DIFFERENTIABILITY are expressed in terms of level-wise generalized DIFFERENTIABILITY. Also, the properties and the relationship between them are discussed. Finally, the relationships between total generalized DIFFERENTIABILITY, DIRECTIONAL generalized DIFFERENTIABILITY and partial generalized DIFFERENTIABILITY are shown to illustrate the power and ability of relationships between them by mentioning some examples.

For the eigenvalue function on symmetric matrices, we have gathered a number of it’s properties. We show that this map has the properties of continuity, strict continuity, DIRECTIONAL DIFFERENTIABILITY, Frechet DIFFERENTIABILITY, continuous DIFFERENTIABILITY. Eigenvalue function will be extended to a larger set of matrices and then the listed properties will prove again.

A novel method of parametric active contours with geometric shape prior is presented in this paper. The main idea of the method consists in minimizing an energy function that includes additional information on a shape reference called a prototype (mean shape). Shape prior introduced through a similarity measurement between evolving contour and Procrustes mean shape of desired object. This similarity measurement is Procrustes distance between these two contours that is invariant with respect to similarity transformations (translation, scaling, and rotation). Procrustes mean shape is extracted from a set of sample shapes. We apply k-mean clustering to sample shapes for overcoming non Gaussian distribution of them e.g., multiple views of a 3D object. In this case the final energy function is invariant with respect to changing viewpoint of 3D object. This extra shape knowledge enhances the model robustness to noise, occlusion and complex background. We also use gradient direction information in addition to its magnitude for more robustness to complex background and noise. We obtain promising results for 3D object tracking and segmentation showing the robustness of our method against noise, complex background, similarity transformations, occlusion, and changing viewpoint of 3D object.

In this paper, we study first order linear fuzzy differential equations with fuzzy coefficient and initial value. We use the generalized DIFFERENTIABILITY concept and apply the exponent matrix to present the general form of their solutions. Finally, one example is given to illustrate our results.