Fuzzy roughness via ideals

In this paper, we join the notion of fuzzy ideal to the notion of fuzzy approximation space to define the notion of fuzzy ideal approximation spaces. We introduce the fuzzy ideal approximation interior operator int Φ λ and the fuzzy ideal approximation closure operator cl Φ λ , and moreover, we define the fuzzy ideal approximation preinterior operator p int Φ λ and the fuzzy ideal approximation preclosure operator p cl Φ λ with respect to that fuzzy ideal defined on the fuzzy approximation space (X, R) associated with some fuzzy set λ ∈ IX. Also, we define fuzzy separation axioms, fuzzy connectedness and fuzzy compactness in fuzzy approximation spaces and in fuzzy ideal approximation spaces as well, and prove the implications in between.

An Improved Fuzzy Connectedness Method for Automatic Three-Dimensional Liver Vessel Segmentation in CT Images

In this paper, an improved fuzzy connectedness (FC) method was proposed for automatic three-dimensional (3D) liver vessel segmentation in computed tomography (CT) images. The vessel-enhanced image (i.e., vesselness image) was incorporated into the fuzzy affinity function of FC, rather than the intensity image used by traditional FC. An improved vesselness filter was proposed by incorporating adaptive sigmoid filtering and a background-suppressing item. The fuzzy scene of FC was automatically initialized by using the Otsu segmentation algorithm and one single seed generated adaptively, while traditional FC required multiple seeds. The improved FC method was evaluated on 40 cases of clinical CT volumetric images from the 3Dircadb (n=20) and Sliver07 (n=20) datasets. Experimental results showed that the proposed liver vessel segmentation strategy could achieve better segmentation performance than traditional FC, region growing, and threshold level set. Average accuracy, sensitivity, specificity, and Dice coefficient of the improved FC method were, respectively, (96.4 ± 1.1)%, (73.7 ± 7.6)%, (97.4 ± 1.3)%, and (67.3 ± 5.7)% for the 3Dircadb dataset and (96.8 ± 0.6)%, (89.1 ± 6.8)%, (97.6 ± 1.1)%, and (71.4 ± 7.6)% for the Sliver07 dataset. It was concluded that the improved FC may be used as a new method for automatic 3D segmentation of liver vessel from CT images.