# Unsupervised Image Classification Based on Fully Fuzzy Voronoi Tessellation
## 1 Introduction
It extended the algorithm to multidimensional image processing. In addition, [PERSON] et al. also proposed a novel FLICM with non-local information and back projection (FLICMLNLI) [28]. FLICMLNLI combined local and non-local information to balance the robustness and the retention of details. To further improve the noise resistance, [PERSON] et al. proposed a patch-based fuzzy local similarity C-means (PFLSCM) [29]. It used the structural similarity to model the neighborhood distance, and the local patches form to model the clustering distance. FLICMLNLI and PFLSCM are visually almost equivalent to KWFLICM. [PERSON] and [PERSON] also proposed a local feature-driven FLICM with kernel metric (LTPKW-FLICM) [30]. It embedded edge information by the local ternary model and improved the boundary classification effect. However, its visual effect is not as good as KWFLICM. In addition, the above algorithms only utilize one level of neighborhood information. To explore deeper neighborhood effects, [PERSON] and [PERSON] [31] proposed the master-slave hierarchy local information driven FCM (MSHLICM). MSHLICM proposed the master-slave neighborhood window to effectively describe the inherent relationship between the current pixel and its neighborhood pixels and incorporated it into FLICM. MSHLICM has stronger noise resistance and robustness. The above spatial neighborhood effects are all modeled based on the pixel level, and their ability to overcome noise is limited. [PERSON] et al. [32] developed a new style and proposed the Voronoi tessellation and hidden Markov random field FCM (VT-HMRF-FCM). It constructed a hierarchical membership model based on the Voronoi tessellation technique. Pixels are first assigned to Voronoi polygons based on distance minimization from seed points. Then, all pixels within a polygon are considered as a whole. They are forced to share the same degree of membership. Taking the Voronoi polygons as the basic processing unit, the classification is executed by optimizing the Voronoi polygons and clustering centers based on the HMRF-FCM framework [33]. Compared with traditional region-based classification algorithms, VT-HMRF-FCM achieves the ability to optimize the region during the iterative process. On the one hand, it can avoid getting stuck in local optima to better adapt to complex scenes. On the other hand, its shape is relatively regular and compact. With the help of this idea, [PERSON] et al. further proposed adaptive distance-weighted Voronoi tessellation technology to enhance polygon flexibility [34]. However, it is worth noting that the Voronoi polygon is an implicit variable that cannot be solved analytically. Thus, the spatial random sampling method is used to find the optimal Voronoi tessellations. It only selects one polygon in each iteration, and randomly selects only one seed point in the polygon. The objective function is recalculated based on the new seed points set. If the value decreases, accept the new seed point, otherwise reject it. Although VT-HMRF-FCM has good noise resistance, the accuracy of the boundaries often depends on a large number of polygons and iterative optimization. In order to perform classification faster and better, the unsupervised image classification based on fully fuzzy Voronoi tessellation algorithm (FVT-FCM) is proposed in this paper. It constructs a hierarchical fuzzy membership model, i.e., pixels fuzzily belong to Voronoi polygons and polygons fuzzily belong to clusters. In the first level, the objective function of fuzzy Voronoi tessellation is only modeled by spatial distance without spectra. It implements an explicit model representation of Voronoi tessellation, allowing it to obtain analytical solutions through the objective function. Additionally, the fuzzy Voronoi tessellation operation also greatly maintains the flexibility of pixels, making it advantageous for boundary classification. In the second level, the objective function of clustering is modeled by considering the transitivity of fuzzy based on the fuzzy Voronoi polygons.