Let C be a nonempty closed and convex subset of a uniformly smooth and uniformly convex real Banach space E with dual space E * . A novel hybrid method for finding a solution of an equilibrium problem and a common element of fixed points for a family of a general class of nonlinear nonexpansive maps is constructed. The sequence of the method is proved to converge strongly to a common element of the family and a solution of the equilibrium problem. Finally, an application of our theorem complements, generalizes and extends some recent important results (Takahashi et al., Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces,
LetEbe a uniformly convex and uniformly smooth real Banach space, and letE* be its dual. LetA : E→ 2E*be a bounded maximal monotone map. Assume thatA−1(0) ≠ Ø. A new iterative sequence is constructed which convergesstronglyto an element ofA−1(0). The theorem proved complements results obtained on strong convergence ofthe proximal point algorithmfor approximating an element ofA−1(0) (assuming existence) and also resolves an important open question. Furthermore, this result is applied to convex optimization problems and to variational inequality problems. These results are achieved by combining a theorem of Reich on the strong convergence of the resolvent of maximal monotone mappings in a uniformly smooth real Banach space and new geometric properties of uniformly convex and uniformly smooth real Banach spaces introduced by Alber, with a technique of proof which is also of independent interest.
In this work, we focused on 3D image segmentation where the segmented surface is reconstructed by the use of 3D digital image information and information from thresholded 3D image in a local neighbourhood. To this end, we applied a mathematical model based on the level set formulation and numerical method which is based on the so-called reduced diamond cell approach. The segmentation approach is based on surface evolution governed by a nonlinear PDE, the modified subjective surface equation. This is done by defining the input to the edge detector function as the weighted sum of norm of presmoothed 3D image and norm of presmoothed thresholded 3D image in a local neighbourhood. For the numerical discretization, we used a semi-implicit finite volume scheme. The method was applied to real data representing 3D microscopy images of cell nuclei within the zebrafish pectoral fin.
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