The success of advanced Background Subtraction (BGS) algorithms for dynamic backgrounds is mostly in land scenes such as those in CDNet benchmarks; few handle underwater scenes, since existing underwater video datasets are either in low resolution or with only static backgrounds. Consequently, the lack of reliable BGS support makes supervised Moving-Objects Segmentation (MOS) algorithms much harder to adapt to unknown underwater scenes because of the diversities of the aquatic environments. For example, those trained by the latest underwater image dataset, SUIM, are ineffective in the underwater videos of our experiments.The underwater waving objects (e.g., plants) often render existing BGS algorithms inaccurate due to three types of errors: (a) incompletely identified MOs (Moving Objects), (b) missing MOs, and (c) falsely identified MOs. In this paper, we propose a novel Clustering-Based Multi-State Background Representation (CBMSBR) model to learn and represent the repeatable patterns of waving movements in k background states (i.e., color ranges) per pixel, and thus accurately subtract the background waving objects to reduce these errors. In addition, we further develop a CBMSBR+ model to remove the more challenging background objects in unusually large magnitudes of wavings. Both models come from a basic observation: the video pixels in the waving zones repeatedly switch among multiple background states; e.g., a pixel switches among water state, plant 1 state, and plant 2 state. To test our proposed models, we create experiments using three types of challenging scenarios that each often covers at least two error types, i.e., the scattered MOs scenario covering (b) and (c), the crowded MOs scenario covering (a) - (c), and the slow MOs scenario covering (a) and (c). Experiments on these scenarios demonstrate the accuracy, effectiveness, and efficiency of our models and their applications in MOS improvements.

In the author list, "Ferry Sansoto" should be Ferry Susanto.• To reflect more accurately the contribution of the article, the title should be changed to "A permutation and equinumerosity based polynomial-time algorithm for simple undirected graph isomorphism."• In the abstract, the "Pythagorean Triples Theorem" should be removed.• In the abstract, "squared sums of elements" should be "nth power sums."• In Section 2.2, "and the sum of the individual squared elements.By checking two sums," should be ", the sum of the individual squared elements and until the sum of the nth power of the nth element in the array.By checking these sums,"• In Section 2.2, "For both vertex and edge arrays of row/column sum based on the vertex and edge adjacency matrices, if and only if one array is a permutation of another one, the corresponding two graphs are isomorphic."should be "For both the vertex and edge arrays of row/column sum based on the vertex and edge adjacency matrices, if and only if one array is a permutation of another one and the corresponding edge and vertex's adjacent relationship has been preserved, the corresponding two graphs are isomorphic."

Summary The graph isomorphism problem is to determine two finite graphs that are isomorphic which is not known with a polynomial‐time solution. This paper solves the simple undirected graph isomorphism problem with an algorithmic approach as NP=P and proposes a polynomial‐time solution to check if two simple undirected graphs are isomorphic or not. Three new representation methods of a graph as vertex/edge adjacency matrix and triple tuple are proposed. A duality of edge and vertex and a reflexivity between vertex adjacency matrix and edge adjacency matrix were first introduced to present the core idea. Beyond this, the mathematical approval is based on an equivalence between permutation and bijection. Because only addition and multiplication operations satisfy the commutative law, we propose a permutation theorem to check fast whether one of two sets of arrays is a permutation of another or not. The permutation theorem was mathematically approved by Integer Factorization Theory, Pythagorean Triples Theorem, and Fundamental Theorem of Arithmetic. For each of two n ‐ary arrays, the linear and squared sums of elements were respectively calculated to produce the results.

The topological distance is to measure the structural difference between two graphs in a metric space. Graphs are ubiquitous, and topological measurements over graphs arise in diverse areas, including, e.g. COVID-19 structural analysis, DNA/RNA alignment, discovering the Isomers, checking the code plagiarism. Unfortunately, popular distance scores used in these applications, that scale over large graphs, are not metrics, and the computation usually becomes NP-hard. While, fuzzy measurement is an uncertain representation to apply for a polynomial-time solution for undirected multigraph isomorphism. But the graph isomorphism problem is to determine two finite graphs that are isomorphic, which is not known with a polynomial-time solution. This paper solves the undirected multigraph isomorphism problem with an algorithmic approach as NP=P and proposes a polynomial-time solution to check if two undirected multigraphs are isomorphic or not. Based on the solution, we define a new fuzzy measurement based on graph isomorphism for topological distance/structural similarity between two graphs. Thus, this paper proposed a fuzzy measure of the topological distance between two undirected multigraphs. If two graphs are isomorphic, the topological distance is 0; if not, we will calculate the Euclidean distance among eight extracted features and provide the fuzzy distance. The fuzzy measurement executes more efficiently and accurately than the current methods.

Abstract The problem of risk classification and prediction, an essential research direction, aiming to identify and predict risks for various applications, has been researched in this paper. To identify and predict risks, numerous researchers build models on discovering hidden information of a label (positive credit or negative credit). Fuzzy logic is robust in dealing with ambiguous data and, thus, benefits the problem of classification and prediction. However, the way to apply fuzzy logic optimally depends on the characteristics of the data and the objectives, and it is extraordinarily tricky to find such a way. This paper, therefore, proposes a general membership function model for fuzzy sets (GMFMFS) in the fuzzy decision tree and extend it to the fuzzy random forest method. The proposed methods can be applied to identify and predict the credit risks with almost optimal fuzzy sets. In addition, we analyze the feasibility of our GMFMFS and prove our GMFMFS‐based linear membership function can be extended to a nonlinear membership function without a significant increase in computing complex. Our GMFMFS‐based fuzzy decision tree is tested with a real dataset of US credit, Susy dataset of UCI, and synthetic datasets of big data. The results of experiments further demonstrate the effectiveness and potential of our GMFMFS‐based fuzzy decision tree with linear membership function and nonlinear membership function.

With the advance of computing and electronic technology, quantitative data, for example, continuous data (i.e., sequences of floating point numbers), become vital and have wide applications, such as for analysis of sensor data streams and financial data streams. However, existing association rule mining generally discover association rules from discrete variables, such as boolean data (`O' and `l') and categorical data (`sunny', `cloudy', `rainy', etc.) but very few deal with quantitative data. In this paper, a novel optimized fuzzy association rule mining (OFARM) method is proposed to mine association rules from quantitative data. The advantages of the proposed algorithm are in three folds: 1) propose a novel method to add the smoothness and flexibility of membership function for fuzzy sets; 2) optimize the fuzzy sets and their partition points with multiple objective functions after categorizing the quantitative data; and 3) design a two-level iteration to filter frequent-item-sets and fuzzy association-rules. The new method is verified by three different data sets, and the results have demonstrated the effectiveness and potentials of the developed scheme.