On various eigen fuzzy sets and their application to image reconstruction

In this study, we formulate and solve a problem of image reconstruction using eigen fuzzy sets. Treating images as fuzzy relations, we propose two algorithms of generating eigen fuzzy sets that are used in the reconstruction process. The first one corresponds to a convex combination of eigen fuzzy set equations, i.e., fuzzy relational equations involving convex combination of max–min and min–max compositions. In the case of the first algorithm, various eigen fuzzy sets can be generated by changing the parameter controlling the convex combination of the corresponding equations. The second algorithm generates various eigen fuzzy sets with respect to the original fuzzy relation using a permutation matrix. A thorough comparison of the proposed algorithms and a conventional algorithm which reconstructs an image using the greatest and smallest eigen fuzzy sets is presented as well. In the experiments, 10,000 artificial images of size 5 × 5 pixels. The approximation error in the case of the first/second algorithm is decreased to 68.2%/97.9% of that of the conventional algorithm, respectively. Furthermore, through the experimentation using real images extracted from Standard Image DataBAse (SIDBA), it is confirmed that the approximation error of the first algorithm is decreased to 41.5% of that of the conventional one.