Image segmentation is a critical step in computer vision tasks constituting an essential issue for pattern recognition and visual interpretation. In this paper, we study the behavior of entropy in digital images through an iterative algorithm of mean shift filtering. The order of a digital image in gray levels is defined. The behavior of Shannon entropy is analyzed and then compared, taking into account the number of iterations of our algorithm, with the maximum entropy that could be achieved under the same order. The use of equivalence classes it induced, which allow us to interpret entropy as a hyper-surface in real m dimensional space. The difference of the maximum entropy of order n and the entropy of the image is used to group the the iterations, in order to caractrizes the performance of the algorithm.
A COMPARATIVE STUDY ON DISTANCE MEASURING APPROACHES FOR CLUSTERINGIJORCS
Clustering plays a vital role in the various areas of research like Data Mining, Image Retrieval, Bio-computing and many a lot. Distance measure plays an important role in clustering data points. Choosing the right distance measure for a given dataset is a biggest challenge. In this paper, we study various distance measures and their effect on different clustering. This paper surveys existing distance measures for clustering and present a comparison between them based on application domain, efficiency, benefits and drawbacks. This comparison helps the researchers to take quick decision about which distance measure to use for clustering. We conclude this work by identifying trends and challenges of research and development towards clustering.
This document presents a scalable method for image classification using sparse coding and dictionary learning. It proposes parallelizing the computation of image similarity for faster recognition. Specifically, it distributes the task of measuring similarity between images among multiple cores in a cluster. Experimental results on a face recognition dataset show nearly linear speedup when balancing the dataset size and number of nodes. Reconstruction errors are used as a similarity measure, with dictionaries learned using K-SVD for each image. The proposed parallel method distributes this similarity computation process to achieve faster image classification.
The document proposes a novel Spatial Fuzzy C-Means (PET-SFCM) clustering algorithm to segment PET scan images of patients with neurodegenerative disorders like Alzheimer's disease. The algorithm incorporates spatial neighborhood information into the traditional Fuzzy C-Means algorithm. It was tested on real patient data sets and showed satisfactory results compared to conventional FCM and K-Means clustering algorithms. The PET-SFCM algorithm provides an effective way to segment PET images and analyze brain changes related to neurological conditions.
Online Multi-Person Tracking Using Variance Magnitude of Image colors and Sol...Pourya Jafarzadeh
The document describes a multi-object tracking method that formulates tracking as a Short Minimum Clique Problem (SMCP). It uses three consecutive frames divided into three clusters, where each clique between clusters represents a tracklet (partial trajectory) of a person. Edges between clusters are weighted based on color histogram similarity and eigenvalue similarity of bounding boxes. Occlusion handling is performed by saving color histograms of occluded people in a buffer and comparing them to newly detected people. The method was evaluated on challenging datasets and shown to achieve promising results compared to state-of-the-art methods.
This document presents a new approach for automatic fuzzy clustering of magnetic resonance images. The approach combines multi-degree immersion and entropy algorithms (multi-degree entropy algorithm) to determine the optimal number of clusters in an image without human input. Multi-degree immersion first segments the image into multiple levels based on intensity. Entropy is then used to merge regions to arrive at the final cluster number based on a validity function. The method is tested on simulated and real MRI data and shown to produce accurate results, outperforming other validity indices. The approach provides an automatic way to determine the appropriate number of clusters for segmenting medical images.
Trust Region Algorithm - Bachelor DissertationChristian Adom
The document summarizes the trust region algorithm for solving unconstrained optimization problems. It begins by introducing trust region methods and comparing them to line search algorithms. The basic trust region algorithm is then outlined, which approximates the objective function within a region using a quadratic model at each iteration. It discusses solving the trust region subproblem to find a step that minimizes the model within the trust region. Finally, it introduces the Cauchy point and double dogleg step as methods for solving the subproblem.
The aim of this research is to find accurate solution for the Troesch’s problem by using high performance technique based on parallel processing implementation.
Design/methodology/approach – Feed forward neural network is designed to solve important type of differential equations that arises in many applied sciences and engineering applications. The suitable designed based on choosing suitable learning rate, transfer function, and training algorithm. The authors used back propagation with new implement of Levenberg - Marquardt training algorithm. Also, the authors depend new idea for choosing the weights. The effectiveness of the suggested design for the network is shown by using it for solving Troesch problem in many cases.
Findings – New idea for choosing the weights of the neural network, new implement of Levenberg - Marquardt training algorithm which assist to speeding the convergence and the implementation of the suggested design demonstrates the usefulness in finding exact solutions.
This document summarizes recent convergence results for the fuzzy c-means clustering algorithm (FCM). It discusses both numerical convergence, referring to how well the algorithm attains the minima of an objective function, and stochastic convergence, referring to how accurately the minima represent the actual cluster structure in data. For numerical convergence, the document outlines global and local convergence theorems, showing FCM converges to minima or saddle points globally and linearly to local minima. For stochastic convergence, it discusses a consistency result showing the minima accurately represent cluster structure under certain statistical assumptions.
A COMPARATIVE STUDY ON DISTANCE MEASURING APPROACHES FOR CLUSTERINGIJORCS
Clustering plays a vital role in the various areas of research like Data Mining, Image Retrieval, Bio-computing and many a lot. Distance measure plays an important role in clustering data points. Choosing the right distance measure for a given dataset is a biggest challenge. In this paper, we study various distance measures and their effect on different clustering. This paper surveys existing distance measures for clustering and present a comparison between them based on application domain, efficiency, benefits and drawbacks. This comparison helps the researchers to take quick decision about which distance measure to use for clustering. We conclude this work by identifying trends and challenges of research and development towards clustering.
This document presents a scalable method for image classification using sparse coding and dictionary learning. It proposes parallelizing the computation of image similarity for faster recognition. Specifically, it distributes the task of measuring similarity between images among multiple cores in a cluster. Experimental results on a face recognition dataset show nearly linear speedup when balancing the dataset size and number of nodes. Reconstruction errors are used as a similarity measure, with dictionaries learned using K-SVD for each image. The proposed parallel method distributes this similarity computation process to achieve faster image classification.
The document proposes a novel Spatial Fuzzy C-Means (PET-SFCM) clustering algorithm to segment PET scan images of patients with neurodegenerative disorders like Alzheimer's disease. The algorithm incorporates spatial neighborhood information into the traditional Fuzzy C-Means algorithm. It was tested on real patient data sets and showed satisfactory results compared to conventional FCM and K-Means clustering algorithms. The PET-SFCM algorithm provides an effective way to segment PET images and analyze brain changes related to neurological conditions.
Online Multi-Person Tracking Using Variance Magnitude of Image colors and Sol...Pourya Jafarzadeh
The document describes a multi-object tracking method that formulates tracking as a Short Minimum Clique Problem (SMCP). It uses three consecutive frames divided into three clusters, where each clique between clusters represents a tracklet (partial trajectory) of a person. Edges between clusters are weighted based on color histogram similarity and eigenvalue similarity of bounding boxes. Occlusion handling is performed by saving color histograms of occluded people in a buffer and comparing them to newly detected people. The method was evaluated on challenging datasets and shown to achieve promising results compared to state-of-the-art methods.
This document presents a new approach for automatic fuzzy clustering of magnetic resonance images. The approach combines multi-degree immersion and entropy algorithms (multi-degree entropy algorithm) to determine the optimal number of clusters in an image without human input. Multi-degree immersion first segments the image into multiple levels based on intensity. Entropy is then used to merge regions to arrive at the final cluster number based on a validity function. The method is tested on simulated and real MRI data and shown to produce accurate results, outperforming other validity indices. The approach provides an automatic way to determine the appropriate number of clusters for segmenting medical images.
Trust Region Algorithm - Bachelor DissertationChristian Adom
The document summarizes the trust region algorithm for solving unconstrained optimization problems. It begins by introducing trust region methods and comparing them to line search algorithms. The basic trust region algorithm is then outlined, which approximates the objective function within a region using a quadratic model at each iteration. It discusses solving the trust region subproblem to find a step that minimizes the model within the trust region. Finally, it introduces the Cauchy point and double dogleg step as methods for solving the subproblem.
The aim of this research is to find accurate solution for the Troesch’s problem by using high performance technique based on parallel processing implementation.
Design/methodology/approach – Feed forward neural network is designed to solve important type of differential equations that arises in many applied sciences and engineering applications. The suitable designed based on choosing suitable learning rate, transfer function, and training algorithm. The authors used back propagation with new implement of Levenberg - Marquardt training algorithm. Also, the authors depend new idea for choosing the weights. The effectiveness of the suggested design for the network is shown by using it for solving Troesch problem in many cases.
Findings – New idea for choosing the weights of the neural network, new implement of Levenberg - Marquardt training algorithm which assist to speeding the convergence and the implementation of the suggested design demonstrates the usefulness in finding exact solutions.
This document summarizes recent convergence results for the fuzzy c-means clustering algorithm (FCM). It discusses both numerical convergence, referring to how well the algorithm attains the minima of an objective function, and stochastic convergence, referring to how accurately the minima represent the actual cluster structure in data. For numerical convergence, the document outlines global and local convergence theorems, showing FCM converges to minima or saddle points globally and linearly to local minima. For stochastic convergence, it discusses a consistency result showing the minima accurately represent cluster structure under certain statistical assumptions.
Textural Feature Extraction of Natural Objects for Image ClassificationCSCJournals
The field of digital image processing has been growing in scope in the recent years. A digital image is represented as a two-dimensional array of pixels, where each pixel has the intensity and location information. Analysis of digital images involves extraction of meaningful information from them, based on certain requirements. Digital Image Analysis requires the extraction of features, transforms the data in the high-dimensional space to a space of fewer dimensions. Feature vectors are n-dimensional vectors of numerical features used to represent an object. We have used Haralick features to classify various images using different classification algorithms like Support Vector Machines (SVM), Logistic Classifier, Random Forests Multi Layer Perception and Naïve Bayes Classifier. Then we used cross validation to assess how well a classifier works for a generalized data set, as compared to the classifications obtained during training.
1. The document presents a new approach for steganography detection using a combination of Fisher's linear discriminant function (FLD) and radial basis function neural network (RBF).
2. In the training phase, FLD is used to project high-dimensional image data onto a lower dimensional space, then an RBF network is trained to classify images as containing hidden data or not.
3. Experiments show the combined FLD-RBF method provides promising results for steganography detection compared to existing supervised methods, though extracting the hidden information remains challenging.
This paper proposes a new fuzzy similarity measure called Fuzzy Monotonic Inclusion (FMI) to measure similarity between images for image retrieval systems. The FMI approach segments images into regions, extracts features for each region, and maps the features into a fuzzy similarity model based on fuzzy inclusion. Experimental results on the Label Me image dataset show the FMI approach achieves higher precision than other methods like Unified Feature Matching and Fuzzy Histogram in identifying images by semantic class.
Satellite image Compression reduces redundancy in data representation in order to achieve saving in the
cost of storage and transmission image compression compensates for the limited on-board resources, in
terms of mass memory and downlink bandwidth and thus it provides a solution to the (bandwidth vs. data
volume) dilemma of modern spacecraft Thus compression is very important feature in payload image
processing units of many satellites, In this paper, an improvement of the quantization step of the input
vectors has been proposed. The k-nearest neighbour (KNN) algorithm was used on each axis. The three
classifications considered as three independent sources of information, are combined in the framework of
the evidence theory the best code vector is then selected. After Huffman schemes is applied for encoding
and decoding.
OTSU Thresholding Method for Flower Image Segmentationijceronline
Segmentation is basic process in image processing. It always produces an effective result for next process. In this paper, we proposed the flower image segmentation. Oxford flower collection is used for segmentation.Different segmentation techniques are available. Different techniques and algorithm are developed to describe the segmentation.We proposed a OTSU thresholding technique for flower image segmentation in this paper. which gives good result as compared with the other methods and simple also.Segmentation subdivide the image into different parts.firstly, segmentation techniques and then otsu thresholding method described in this paper.CIE L*a*b color space is used in thresholding for better results.Thresholding apply seperatly on each L, a and b component. accordingly the features can be extracted like shape, color, texture etc. finally, results with the flower images are shown.
A COMPREHENSIVE ANALYSIS OF QUANTUM CLUSTERING : FINDING ALL THE POTENTIAL MI...IJDKP
Quantum clustering (QC), is a data clustering algorithm based on quantum mechanics which is
accomplished by substituting each point in a given dataset with a Gaussian. The width of the Gaussian is a
σ value, a hyper-parameter which can be manually defined and manipulated to suit the application.
Numerical methods are used to find all the minima of the quantum potential as they correspond to cluster
centers. Herein, we investigate the mathematical task of expressing and finding all the roots of the
exponential polynomial corresponding to the minima of a two-dimensional quantum potential. This is an
outstanding task because normally such expressions are impossible to solve analytically. However, we
prove that if the points are all included in a square region of size σ, there is only one minimum. This bound
is not only useful in the number of solutions to look for, by numerical means, it allows to to propose a new
numerical approach “per block”. This technique decreases the number of particles by approximating some
groups of particles to weighted particles. These findings are not only useful to the quantum clustering
problem but also for the exponential polynomials encountered in quantum chemistry, Solid-state Physics
and other applications.
Blind Image Seperation Using Forward Difference Method (FDM)sipij
In this paper, blind image separation is performed, exploiting the property of sparseness to represent images. A new sparse representation called forward difference method is proposed. It is known that most of the independent component analysis (ICA) basis functions, extracted from images are sparse and gives unreliable sparseness measure. In the proposed method, the image mixture is first transformed to sparse images. These images are divided into blocks and for each block the sparseness measure ε0 norm is applied. The block having the most sparseness is considered to determine the separation matrix. The efficiency of the proposed method is compared with other sparse representation functions.
Sensing Method for Two-Target Detection in Time-Constrained Vector Poisson Ch...sipij
It is an experimental design problem in which there are two Poisson sources with two possible and known rates, and one counter. Through a switch, the counter can observe the sources individually or the counts can be combined so that the counter observes the sum of the two. The sensor scheduling problem is to determine an optimal proportion of the available time to be allocated toward individual and joint sensing, under a total time
constraint. Two different metrics are used for optimization: mutual information between the sources and the observed counts, and probability of detection for the associated source detection problem. Our results, which are primarily computational, indicate similar but not identical results under the two cost functions.
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
MARGINAL PERCEPTRON FOR NON-LINEAR AND MULTI CLASS CLASSIFICATION ijscai
Generalization error of classifier can be reduced by larger margin of separating hyperplane. The proposed classification algorithm implements margin in classical perceptron algorithm, to reduce generalized errors by maximizing margin of separating hyperplane. Algorithm uses the same updation rule with the perceptron, to converge in a finite number of updates to solutions, possessing any desirable fraction of the margin. This solution is again optimized to get maximum possible margin. The algorithm can process linear, non-linear and multi class problems. Experimental results place the proposed classifier equivalent to the support vector machine and even better in some cases. Some preliminary experimental results are briefly discussed.
This document discusses various clustering techniques for image segmentation. It begins by defining clustering and image segmentation. It then describes four main clustering techniques - exclusive clustering (e.g. k-means), overlapping clustering (e.g. fuzzy c-means), hierarchical clustering, and probabilistic-D clustering. For each technique, it provides details on the clustering algorithm and steps. It concludes that fuzzy c-means is superior to other approaches for image segmentation efficiency but has high computational time, while probabilistic-D clustering aims to reduce this time.
MULTI-OBJECTIVE ENERGY EFFICIENT OPTIMIZATION ALGORITHM FOR COVERAGE CONTROL ...ijcseit
Many studies have been done in the area of Wireless Sensor Networks (WSNs) in recent years. In this kind of networks, some of the key objectives that need to be satisfied are area coverage, number of active sensors and energy consumed by nodes. In this paper, we propose a NSGA-II based multi-objective algorithm for optimizing all of these objectives simultaneously. The efficiency of our algorithm is demonstrated in the simulation results. This efficiency can be shown as finding the optimal balance point among the maximum coverage rate, the least energy consumption, and the minimum number of active nodes while maintaining the connectivity of the network
Improving search time for contentment based image retrieval via, LSH, MTRee, ...IOSR Journals
This document proposes a new index structure called LSH-LUBMTree to improve search time for content-based image retrieval using the Earth Mover's Distance metric. LSH-LUBMTree combines Locality Sensitive Hashing (LSH) and the LUBMTree index. Images hashed to the same bucket via LSH are then stored in the LUBMTree to reduce false positives and accelerate search time. Experimental results show LSH-LUBMTree performs better than standard LSH in terms of search time by leveraging advantages of both LSH and LUBMTree indexing.
Research Inventy : International Journal of Engineering and Scienceinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Spectroscopy or hyperspectral imaging consists in the acquisition, analysis, and extraction of the spectral information measured on a specific region or object using an airborne or satellite device. Hyperspectral imaging has become an active field of research recently. One way of analysing such data is through clustering. However, due to the high dimensionality of the data and the small distance between the different material signatures, clustering such a data is a challenging task.In this paper, we empirically compared five clustering techniques in different hyperspectral data sets. The considered clustering techniques are K-means, K-medoids, fuzzy Cmeans, hierarchical, and density-based spatial clustering of applications with noise. Four data sets are used to achieve this purpose which is Botswana, Kennedy space centre, Pavia, and Pavia University. Beside the accuracy, we adopted four more similarity measures: Rand statistics, Jaccard coefficient, Fowlkes-Mallows index, and Hubert index. According to accuracy, we found that fuzzy C-means clustering is doing better on Botswana and Pavia data sets, K-means and K-medoids are giving better results on Kennedy space centre data set, and for Pavia University the hierarchical clustering is better
One of the important steps in routing is to find a feasible path based on the state information. In order to support real-time multimedia applications, the feasible path that satisfies one or more constraints has to be computed within a very short time. Therefore, the paper presents a genetic algorithm to solve the paths tree problem subject to cost constraints. The objective of the algorithm is to find the set of edges connecting all nodes such that the sum of the edge costs from the source (root) to each node is minimized. I.e. the path from the root to each node must be a minimum cost path connecting them. The algorithm has been applied on two sample networks, the first network with eight nodes, and the last one with eleven nodes to illustrate its efficiency.
Analysis of mass based and density based clustering techniques on numerical d...Alexander Decker
This document compares and analyzes mass-based and density-based clustering techniques. It summarizes DBSCAN and OPTICS, two popular density-based clustering algorithms, and introduces DEMassDBSCAN, a mass-based clustering algorithm. The document tests the algorithms on several datasets and finds that DEMassDBSCAN has better runtime than DBSCAN, especially on larger datasets, while producing fewer unassigned clusters.
This document presents an improved multi-SOM clustering algorithm that uses the Davies-Bouldin index to determine the optimal number of clusters. The multi-SOM algorithm iteratively clusters an initial self-organizing map (SOM) grid using the DB index at each level until the index reaches its minimum value, indicating the best number of clusters. Experimental results on five datasets show the proposed algorithm performs as well as or better than k-means, BIRCH, and a previous multi-SOM algorithm in determining the correct number of clusters.
A New Method Based on MDA to Enhance the Face Recognition PerformanceCSCJournals
A novel tensor based method is prepared to solve the supervised dimensionality reduction problem. In this paper a multilinear principal component analysis(MPCA) is utilized to reduce the tensor object dimension then a multilinear discriminant analysis(MDA), is applied to find the best subspaces. Because the number of possible subspace dimensions for any kind of tensor objects is extremely high, so testing all of them for finding the best one is not feasible. So this paper also presented a method to solve that problem, The main criterion of algorithm is not similar to Sequential mode truncation(SMT) and full projection is used to initialize the iterative solution and find the best dimension for MDA. This paper is saving the extra times that we should spend to find the best dimension. So the execution time will be decreasing so much. It should be noted that both of the algorithms work with tensor objects with the same order so the structure of the objects has been never broken. Therefore the performance of this method is getting better. The advantage of these algorithms is avoiding the curse of dimensionality and having a better performance in the cases with small sample sizes. Finally, some experiments on ORL and CMPU-PIE databases is provided.
Soft computing is likely to play aprogressively important role in many applications including image enhancement. The paradigm for soft computing is the human mind. The soft computing critique has been particularly strong with fuzzy logic. The fuzzy logic is facts representationas a
rule for management of uncertainty. Inthis paperthe Multi-Dimensional optimized problem is addressed by discussing the optimal thresholding usingfuzzyentropyfor Image enhancement. This technique is compared with bi-level and multi-level thresholding and obtained optimal
thresholding values for different levels of speckle noisy and low contrasted images. The fuzzy entropy method has produced better results compared to bi-level and multi-level thresholding techniques.
BEHAVIOR STUDY OF ENTROPY IN A DIGITAL IMAGE THROUGH AN ITERATIVE ALGORITHM O...ijscmcj
Image segmentation is a critical step in computer vision tasks constituting an essential issue for pattern
recognition and visual interpretation. In this paper, we study the behavior of entropy in digital images
through an iterative algorithm of mean shift filtering. The order of a digital image in gray levels is defined.
The behavior of Shannon entropy is analyzed and then compared, taking into account the number of
iterations of our algorithm, with the maximum entropy that could be achieved under the same order. The
use of equivalence classes it induced, which allow us to interpret entropy as a hyper-surface in real m-
dimensional space. The difference of the maximum entropy of order n and the entropy of the image is used
to group the the iterations, in order to caractrizes the performance of the algorithm
The document describes a statistical property-based blind source separation algorithm for separating mixed signals without additional information. It discusses how previous algorithms have relied on assumptions about linear mixing and independence of sources. The proposed algorithm separates signals based on their statistical properties, such as sources having fewer gradients than mixtures. It estimates mixing parameters by correlating gradients between mixtures and reconstructs sources by optimizing a loss function. The algorithm is tested on its ability to separate mixtures with varying levels of texture, illumination, identical motions between sources, and real-world mixtures. The results indicate it works best when sources have sufficient texture differences and non-identical motions but can fail when mixtures have too little texture or identical source motions.
Textural Feature Extraction of Natural Objects for Image ClassificationCSCJournals
The field of digital image processing has been growing in scope in the recent years. A digital image is represented as a two-dimensional array of pixels, where each pixel has the intensity and location information. Analysis of digital images involves extraction of meaningful information from them, based on certain requirements. Digital Image Analysis requires the extraction of features, transforms the data in the high-dimensional space to a space of fewer dimensions. Feature vectors are n-dimensional vectors of numerical features used to represent an object. We have used Haralick features to classify various images using different classification algorithms like Support Vector Machines (SVM), Logistic Classifier, Random Forests Multi Layer Perception and Naïve Bayes Classifier. Then we used cross validation to assess how well a classifier works for a generalized data set, as compared to the classifications obtained during training.
1. The document presents a new approach for steganography detection using a combination of Fisher's linear discriminant function (FLD) and radial basis function neural network (RBF).
2. In the training phase, FLD is used to project high-dimensional image data onto a lower dimensional space, then an RBF network is trained to classify images as containing hidden data or not.
3. Experiments show the combined FLD-RBF method provides promising results for steganography detection compared to existing supervised methods, though extracting the hidden information remains challenging.
This paper proposes a new fuzzy similarity measure called Fuzzy Monotonic Inclusion (FMI) to measure similarity between images for image retrieval systems. The FMI approach segments images into regions, extracts features for each region, and maps the features into a fuzzy similarity model based on fuzzy inclusion. Experimental results on the Label Me image dataset show the FMI approach achieves higher precision than other methods like Unified Feature Matching and Fuzzy Histogram in identifying images by semantic class.
Satellite image Compression reduces redundancy in data representation in order to achieve saving in the
cost of storage and transmission image compression compensates for the limited on-board resources, in
terms of mass memory and downlink bandwidth and thus it provides a solution to the (bandwidth vs. data
volume) dilemma of modern spacecraft Thus compression is very important feature in payload image
processing units of many satellites, In this paper, an improvement of the quantization step of the input
vectors has been proposed. The k-nearest neighbour (KNN) algorithm was used on each axis. The three
classifications considered as three independent sources of information, are combined in the framework of
the evidence theory the best code vector is then selected. After Huffman schemes is applied for encoding
and decoding.
OTSU Thresholding Method for Flower Image Segmentationijceronline
Segmentation is basic process in image processing. It always produces an effective result for next process. In this paper, we proposed the flower image segmentation. Oxford flower collection is used for segmentation.Different segmentation techniques are available. Different techniques and algorithm are developed to describe the segmentation.We proposed a OTSU thresholding technique for flower image segmentation in this paper. which gives good result as compared with the other methods and simple also.Segmentation subdivide the image into different parts.firstly, segmentation techniques and then otsu thresholding method described in this paper.CIE L*a*b color space is used in thresholding for better results.Thresholding apply seperatly on each L, a and b component. accordingly the features can be extracted like shape, color, texture etc. finally, results with the flower images are shown.
A COMPREHENSIVE ANALYSIS OF QUANTUM CLUSTERING : FINDING ALL THE POTENTIAL MI...IJDKP
Quantum clustering (QC), is a data clustering algorithm based on quantum mechanics which is
accomplished by substituting each point in a given dataset with a Gaussian. The width of the Gaussian is a
σ value, a hyper-parameter which can be manually defined and manipulated to suit the application.
Numerical methods are used to find all the minima of the quantum potential as they correspond to cluster
centers. Herein, we investigate the mathematical task of expressing and finding all the roots of the
exponential polynomial corresponding to the minima of a two-dimensional quantum potential. This is an
outstanding task because normally such expressions are impossible to solve analytically. However, we
prove that if the points are all included in a square region of size σ, there is only one minimum. This bound
is not only useful in the number of solutions to look for, by numerical means, it allows to to propose a new
numerical approach “per block”. This technique decreases the number of particles by approximating some
groups of particles to weighted particles. These findings are not only useful to the quantum clustering
problem but also for the exponential polynomials encountered in quantum chemistry, Solid-state Physics
and other applications.
Blind Image Seperation Using Forward Difference Method (FDM)sipij
In this paper, blind image separation is performed, exploiting the property of sparseness to represent images. A new sparse representation called forward difference method is proposed. It is known that most of the independent component analysis (ICA) basis functions, extracted from images are sparse and gives unreliable sparseness measure. In the proposed method, the image mixture is first transformed to sparse images. These images are divided into blocks and for each block the sparseness measure ε0 norm is applied. The block having the most sparseness is considered to determine the separation matrix. The efficiency of the proposed method is compared with other sparse representation functions.
Sensing Method for Two-Target Detection in Time-Constrained Vector Poisson Ch...sipij
It is an experimental design problem in which there are two Poisson sources with two possible and known rates, and one counter. Through a switch, the counter can observe the sources individually or the counts can be combined so that the counter observes the sum of the two. The sensor scheduling problem is to determine an optimal proportion of the available time to be allocated toward individual and joint sensing, under a total time
constraint. Two different metrics are used for optimization: mutual information between the sources and the observed counts, and probability of detection for the associated source detection problem. Our results, which are primarily computational, indicate similar but not identical results under the two cost functions.
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
MARGINAL PERCEPTRON FOR NON-LINEAR AND MULTI CLASS CLASSIFICATION ijscai
Generalization error of classifier can be reduced by larger margin of separating hyperplane. The proposed classification algorithm implements margin in classical perceptron algorithm, to reduce generalized errors by maximizing margin of separating hyperplane. Algorithm uses the same updation rule with the perceptron, to converge in a finite number of updates to solutions, possessing any desirable fraction of the margin. This solution is again optimized to get maximum possible margin. The algorithm can process linear, non-linear and multi class problems. Experimental results place the proposed classifier equivalent to the support vector machine and even better in some cases. Some preliminary experimental results are briefly discussed.
This document discusses various clustering techniques for image segmentation. It begins by defining clustering and image segmentation. It then describes four main clustering techniques - exclusive clustering (e.g. k-means), overlapping clustering (e.g. fuzzy c-means), hierarchical clustering, and probabilistic-D clustering. For each technique, it provides details on the clustering algorithm and steps. It concludes that fuzzy c-means is superior to other approaches for image segmentation efficiency but has high computational time, while probabilistic-D clustering aims to reduce this time.
MULTI-OBJECTIVE ENERGY EFFICIENT OPTIMIZATION ALGORITHM FOR COVERAGE CONTROL ...ijcseit
Many studies have been done in the area of Wireless Sensor Networks (WSNs) in recent years. In this kind of networks, some of the key objectives that need to be satisfied are area coverage, number of active sensors and energy consumed by nodes. In this paper, we propose a NSGA-II based multi-objective algorithm for optimizing all of these objectives simultaneously. The efficiency of our algorithm is demonstrated in the simulation results. This efficiency can be shown as finding the optimal balance point among the maximum coverage rate, the least energy consumption, and the minimum number of active nodes while maintaining the connectivity of the network
Improving search time for contentment based image retrieval via, LSH, MTRee, ...IOSR Journals
This document proposes a new index structure called LSH-LUBMTree to improve search time for content-based image retrieval using the Earth Mover's Distance metric. LSH-LUBMTree combines Locality Sensitive Hashing (LSH) and the LUBMTree index. Images hashed to the same bucket via LSH are then stored in the LUBMTree to reduce false positives and accelerate search time. Experimental results show LSH-LUBMTree performs better than standard LSH in terms of search time by leveraging advantages of both LSH and LUBMTree indexing.
Research Inventy : International Journal of Engineering and Scienceinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
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Analysis of mass based and density based clustering techniques on numerical d...Alexander Decker
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BEHAVIOR STUDY OF ENTROPY IN A DIGITAL IMAGE THROUGH AN ITERATIVE ALGORITHM O...ijscmcj
Image segmentation is a critical step in computer vision tasks constituting an essential issue for pattern
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The behavior of Shannon entropy is analyzed and then compared, taking into account the number of
iterations of our algorithm, with the maximum entropy that could be achieved under the same order. The
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The document describes a statistical property-based blind source separation algorithm for separating mixed signals without additional information. It discusses how previous algorithms have relied on assumptions about linear mixing and independence of sources. The proposed algorithm separates signals based on their statistical properties, such as sources having fewer gradients than mixtures. It estimates mixing parameters by correlating gradients between mixtures and reconstructs sources by optimizing a loss function. The algorithm is tested on its ability to separate mixtures with varying levels of texture, illumination, identical motions between sources, and real-world mixtures. The results indicate it works best when sources have sufficient texture differences and non-identical motions but can fail when mixtures have too little texture or identical source motions.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
DOMAIN SPECIFIC CBIR FOR HIGHLY TEXTURED IMAGEScseij
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Colour-Texture Image Segmentation using Hypercomplex Gabor Analysissipij
Texture analysis such as segmentation and classification plays a vital role in computer vision and pattern recognition and is widely applied to many areas such as industrial automation, bio-medical image processing and remote sensing. In this paper, we first extend the well-known Gabor filters to color images using a specific form of hypercomplex numbers known as quaternions. These filters are
constructed as windowed basis functions of the quaternion Fourier transform also known as hypercomplex Fourier transform. Based on this extension this paper presents the use of these new
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Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Particle Swarm Optimization for Nano-Particles Extraction from Supporting Mat...CSCJournals
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AUTOMATIC THRESHOLDING TECHNIQUES FOR OPTICAL IMAGESsipij
Image segmentation is one of the important tasks in computer vision and image processing. Thresholding is
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and shape measures. Otsu technique splits the object from the background by increasing the separability
factor between the classes. Our aim form this work is (1) making a comparison among five thresholding
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and intensity contrast technique, and variance discrepancy technique)on different applications. (2)
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experimental results ensure that every thresholding technique has shown a superior level of performance
on specific type of bimodal images.
Texture classification based on overlapped texton co occurrence matrix (otcom...eSAT Journals
Abstract
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Keywords: co-occurrence matrix, texton, Texture Classification
Decision trees have been widely used in machine learning. However, due to some reasons, data collecting
in real world contains a fuzzy and uncertain form. The decision tree should be able to handle such fuzzy
data. This paper presents a method to construct fuzzy decision tree. It proposes a fuzzy decision tree
induction method in iris flower data set, obtaining the entropy from the distance between an average value
and a particular value. It also presents an experiment result that shows the accuracy compared to former
ID3.
This paper proposes a new facial recognition method that combines fuzzy theory and Shannon entropy. It calculates the entropy ratios between facial features and determines the fuzzy membership degree to quantify similarity. This approach is simpler than other methods as it only requires two data points for training and is unaffected by size differences between images. The method was able to achieve high accuracy by focusing on stable features that do not usually change over a person's life.
OBIA on Coastal Landform Based on Structure Tensor csandit
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landforms. That is, develop Hessian matrix by Gabor filtering and calculate multiscale structure
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watershed segment of the image. Then, develop texons and create texton histogram. Finally,
obtain the final results by means of maximum likelihood classification with KL divergence as
the similarity measurement. The study findings show that structure tensor could obtain
multiscale and all-direction information with small data redundancy. Moreover, the method
described in the current paper has high classification accuracy
REMOVING OCCLUSION IN IMAGES USING SPARSE PROCESSING AND TEXTURE SYNTHESISIJCSEA Journal
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Computer apparition plays the most important role in human perception, which is limited to only the visual band of the electromagnetic spectrum. The need for Radar imaging systems, to recover some sources that
are not within human visual band, is raised. This paper present new algorithm for Synthetic Aperture Radar (SAR) images segmentation based on thresholding technique. Entropy based image thresholding has
received sustainable interest in recent years. It is an important concept in the area of image processing.
Pal (1996) proposed a cross entropy thresholding method based on Gaussian distribution for bi-modal images. Our method is derived from Pal method that segment images using cross entropy thresholding based on Gamma distribution and can handle bi-modal and multimodal images. Our method is tested using
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GRAY SCALE IMAGE SEGMENTATION USING OTSU THRESHOLDING OPTIMAL APPROACHJournal For Research
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A Thresholding Method to Estimate Quantities of Each ClassWaqas Tariq
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Behavior study of entropy in a digital image through an iterative algorithm
1. International Journal of Soft Computing, Mathematics and Control (IJSCMC),Vol. 4, No. 3, August 2015
DOI : 10.14810/ijscmc.2015.4301 1
BEHAVIOR STUDY OF ENTROPY IN A DIGITAL IMAGE
THROUGH AN ITERATIVE ALGORITHM OF THE
MEAN SHIFT FILTERING
Esley Torres, Yasel Garces, Osvaldo Pereira and Roberto Rodriguez
Digital Signal Processing Group,
Institute of Cybernetics, Mathematics and Physics (ICIMAF), Havana, Cuba
ABSTRACT
Image segmentation is a critical step in computer vision tasks constituting an essential issue for pattern
recognition and visual interpretation. In this paper, we study the behavior of entropy in digital images
through an iterative algorithm of mean shift filtering. The order of a digital image in gray levels is defined.
The behavior of Shannon entropy is analyzed and then compared, taking into account the number of
iterations of our algorithm, with the maximum entropy that could be achieved under the same order. The
use of equivalence classes it induced, which allow us to interpret entropy as a hyper-surface in real m-
dimensional space. The difference of the maximum entropy of order n and the entropy of the image is used
to group the the iterations, in order to caractrizes the performance of the algorithm.
KEYWORDS
Maximum entropy of order n, equivalence classes, image segmentation, mean shift, relative entropy
1. INTRODUCTION
Image segmentation, that is, classification of the image gray-level values into homogeneous areas
is recognized to be one of the most important steps in any image analysis system. This allows one
to analyze and to interpret the relevant zones according to the aims of observer. Although, most
of the time, the final result of segmentation depends largely on the interest of the observer; it is
possible to develop unsupervised algorithms, which reach the expected results of the interpreter.
The creation of image segmentation algorithms with a fewer adjustment parameters is a task that
has been addressed in the last decade which makes the manipulation of these algorithms easier
and less complicated.
Mean shift (MSH) is a robust technique which has been applied in many computer vision tasks,
for example: image segmentation, visual tracking, etc. [13]. MSH technique was proposed by
Fukunaga and Hostetler [5] and largely forgotten until Cheng’s paper [1] rekindled interest in it.
MSH is a versatile nonparametric density analysis tool and can provide reliable solutions in many
applications [3], [2]. In essence, MSH is an iterative mode detection algorithm in the density
distribution space. The MSH procedure moves to a kernel-weighted average of the observations
within a smoothing window. This computation is repeated until convergence is obtained at a local
density mode. This way the density modes can be located without explicitly estimating the
density. An elegant relation between the MSH and other techniques can be found in [13].
The Mean Shift iterative algorithm (ܵܯℎ݅) that is used in this paper is based on the mean shift
and was previously introduced and applied in several works [4], [7], [9], [10], [11]. The proposed
2. International Journal of Soft Computing, Mathematics and Control (IJSCMC),Vol. 4, No. 3, August 2015
2
algorithm uses entropy as a stopping criterion. As result of applying this algorithm the segmented
image is obtained without loss of quality of segmentation.
The aim of this work is to study the function of entropy (E) in real digital images, with the
purpose of analyzing the behavior of this entropy regarding the probabilities of occurrence of the
gray levels, while the proposed algorithm of the mean shift is running. For more details about
entropy function see [14]. In order to achieve the results of this study it is necessary to define the
associated equivalent classes with the gray levels and the maximum entropy of n order.
The work continues as follows. In section 2, the related theoretical aspects with the largest value
reached by the entropy in images with the same quantity of gray levels will be discussed. Section
3 will present the obtained experimental results according to the comparison of entropy of the
images with respect to its maximum value. Also, an analysis of these experimental results is
carried out. In section 4, the most important conclusions are given.
This document describes, and is written to conform to, author guidelines for the journals of WSP
series. It is prepared in Microsoft Word as a .doc document. Although other means of
preparation are acceptable, final, camera-ready versions must conform to this layout. Microsoft
Word terminology is used where appropriate in this document. Although formatting instructions
may often appear daunting, the simplest approach is to use this template and insert headings and
text into it as appropriate.
2. THEORETICAL ASPECTS
In this section the important theoretical aspects corresponding to this study will be exposed, with
the aim that one can understand with more clarity the analysis that will be carried out in the
transformations of the images in the application of the iterative algorithm of the mean shift [9],
[11].
2.1. Mean Shift
We first review the basic concepts of the MSh algorithm [5]. One of the most popular
nonparametric density estimators is kernel density estimation. Given ݊ data points ݔ ,
݅ = 1, 2, 3,··· , ݊, in a neighborhood of radius ℎ, drawn from a population with density
function ݂(,)ݔ ݔ ∈ ℝௗ
the estimated general multivariate kernel density at ݔ is defined
by:
݂(ݔ) =
1
݊ℎௗ
ܭ
ୀଵ
ቀ
ݔ − ݔ
ℎ
ቁ, (1)
By the use of theory and profile notation given in [2], the Mean Shift vector is given by
ܵܯℎ,ீ()ݔ =
∑ ݔ݃ ቀቛ
ݔ − ݔ
ℎ
ቛቁ
ୀଵ
∑ ݃ ቀቛ
ݔ − ݔ
ℎ
ቛቁ
ୀଵ
− )2( ݔ
where g is the profile of G and K is a shadow kernel of G. The length of the Mean Shift vector
give a measure of how much it is close a local maximum from the point x. For more details about
this topic see [1], [2], [3].
3. International Journal of Soft Computing, Mathematics and Control (IJSCMC),Vol. 4, No. 3, August 2015
3
In [13] was proved that the segmentation algorithm, by recursively applying mean shift, guarantee
the convergence. By simplicity, in this paper, we only approach the case d = 1 corresponding to
gray level images. For more details about this algorithm see [4], [9], [10], [11].
Therefore, if the individual mean shift procedure is guaranteed to converge, a recursively
procedure of the mean shift also converges. For more details about this algorithm see [5], [19],
[20], [21], [22].
2.2. Entropy
The entropy of a digital image is a statistical measure that expresses the randomness of gray
levels (colors) and it is defined as:
Definition 1: Comparable images
ܧଵ = − ݈݃ଶ
ଶಳିଵ
ୀ
,
where ܤ is the total quantity of bits of the digitized image and by agreement ݈)0(2݃ = 0, also
=
ଵ ∙ଶ
, is the probability of occurrence of color ݅,
݇, is the frequency of occurrence of color ݅ in the image,
݉ଵ, is number of rows of the image,
݉ଶ, is number of columns of the image,
∈ [0; 1] and must satisfy that ∑ = 1
ୀଵ , as a condition of probability .
One can observe in the first expression 1 that the entropy of a digital image is a sum of terms that
depend on the probability of occurrence of gray levels of pixels, in this way its value will not only
depend on pi, but also on the quantity of gray levels present in the image. This characteristic of
entropy denotes that it is a non-trivial function to analyze its values, when several images are
compared that do not have the same quantity of gray levels. Basically, we are interested in
knowing the maximum of the function of entropy under the conditions in which the gray levels
have a fixed number.
2.3. Relative Entropy
The Relative Entropy or Kullback Leibler Distance is a nonsymetric mesure of the distance of
two distributions [14]. This function is defined as follows:
Definition 2 (Relative Entropy):
(ܦ ∥ )ݍ =
௫∈॒
݈݃
ݍ
, (2)
where and ݍ are two probability distributions of occurrence of variable ݔ at the set ॒.
2.4. Classes of equivalence Abstract
Definition 3 (Relation of Equivalence)
Let K be a given non-empty set and ܴ a defined binary relationship over .ܭ It is said that ܴ is a
relation of equivalence if this satisfies the following properties [8]:
4. International Journal of Soft Computing, Mathematics and Control (IJSCMC),Vol. 4, No. 3, August 2015
4
1) Reflexivity: ∀ݔ ∈ ܭ ⟹ .ݔܴݔ
2) Symmetry: ∀,ݔ ݕ ∈ ,ܭ ݕܴݔ ⟹ .ݔܴݕ
3) Transitivity: ∀,ݔ ,ݕ ݖ ∈ ,ܭ ,ݕܴݔ ݖܴݕ ⟹ .ݖܴݔ
A relation of equivalence ܴ over a ܭ set can be denoted as the ordered pair (,ܭ ~ ). The relation
of equivalence denoted by the symbol ~, defines disjoint sets in ܭ called classes of equivalence,
i.e., given an element ݔ ∈ ܭ , the given set by all the related elements with ݔ , i.e., ܥ =
ሼݕ ∈ ݔܴݕ |ܭሽ, is called the class of equivalence associated to element .ݔ The element ݔ is called a
representative of the class.
Definition 4 (Order of Relation of Equivalence)
The order of relation of equivalence is defined as the number of classes that generates a relation,
it is denoted by the letter ݊, and the number of elements of the classes ܥ௫ is denoted by the
symbol ݈௫. The concept of a class of equivalence is very important for digital image processing.
Indeed, given a set of objects or abstract entities, relations of equivalence based on some criterion
can be created, where the resulting classes are the “types” in which one can classify the entire
range of objects. Later, we will build classes of equivalence by using ܵܯℎ݅ and we will create a
link to the relation of equivalence defined in this section.
2.5. Class of equivalence of gray levels
Definition 5 (Relation of equivalence among pixels)
Let ܩ be a digital image in gray levels. Let ݕ,ݔ be the pixels and let ܿ,)ݔ(݈ ܿ)ݕ(݈ be their
respective gray levels, the symbolized relation on ܩ is such that ݕ ~ݔ if ܿ)ݔ(݈ = ܿ)ݕ(݈,
∀݅ = 0, 1, 2, ⋯ , 2
− 1 (see section 2.4). By using the notation of classes of equivalence, this can
be written as ܥ௫ = ሼݕ ∈ )ݔ(݈ܿ|ܩ = ܿ)ݕ(݈ሽ, which represents all pixels in image whose gray
level coincides with x. Due to the fact that the pixels in the images are visually represented by
gray levels, it is not convenient to refer them by classes, but by the gray levels.
Suppose that col(x) = i, then it follows that ܥ௫ = ሼݕ ∈ )ݔ(݈ܿ|ܩ = ܿ)ݕ(݈ሽ, provided that
col(x) = i. Taking in consideration the above-mentioned, it is convenient to establish the classes
of equivalence as ܥ = ሼܾ ∈ )ܾ(݈ܿ|ܩ = ݅ሽ ∀݅ = 0, 1, 2, ⋯ , 2
− 1. This class represents the set
of pixels having gray level ݅, so the order or the size of the class ܥ coincides with what is known
as ݇, i.e., the frequency of gray level ݅ in the image.
In a digital image, it may occurs that a certain gray level has = 0, this means that physically
this gray level is not visually present for the observer. For this reason, the agreement log2(0) = 0
is assumed, and therefore these terms do not affect the domain of the function of entropy. On the
other hand, the trivial case = 1 may occurs, which means that all pixels in the image have the
same colour, i.e., the region is quite homogeneous, and its entropy is equal to zero. Therefore, we
are not interested in these cases because entropy reaches the minimum value. In this way, the
values = 0 and = 1 are ignored. This means that the image to analyze has a certain number
of gray levels (such as minimum 2).
Let ܫ = ሼ݅ ∈ [0, 2
− 1] |݊ ≠ 0ሽ be the set of gray levels whose classes of equivalence are non-
zero. It is not difficult to see that the order of the relation of equivalence coincides with the
quantity of gray levels that can be appreciated in .ܩ In this way, one can say that the image has
5. International Journal of Soft Computing, Mathematics and Control (IJSCMC),Vol. 4, No. 3, August 2015
5
order ݊. Therefore, it is possible to give a more precise definition of the order of an image in gray
levels.
Definition 6 (Order of an image in gray levels):
Let ܩ be a digital image in gray levels, the order of ܩ is defined as ݊, as the quantity of elements
of the set ሼ݅ ݃| ݈݁ݒ݈݁ ݕܽݎ ≠ 0ሽ.
Figure 1 represents a region of a digital image of size 5x10 (50 pixels), where one can see the
gray levels corresponding to the values 15 , 16 , 49 and 159 . Note that the order of the
equivalence relation is ݊ = 4, such that the equivalence classes presented on the region are
ܥଵହ, ܥଵ, ܥସଽ, ܥଵହଽ being respectively, ݈ଵହ = 42, ݈ଵ = 6, ݈ସଽ = 1, ݈ଵହଽ = 1 and therefore, ଵହ =
0.84, ଵ = 0.32, ସଽ = 0.02, ଵହଽ = 0.02. It is not casual that the value of the order of the
relation of equivalence coincides with the value of the quantity of gray levels in the image. In fact,
we are referring to the same physical entity under different names. Therefore, starting from now,
we will refer to the number of gray levels in the image with the letter ݊ (order of the relation of
equivalence), and we will indistinctly use the terms: “order of the relation of equivalence”, “order
of image” and “number of gray levels in the image”. Therefore, taking into consideration Figure 1,
the set of probabilities for a given image in gray levels can be expressed in the following way:
ܫ = ሼ݅ ∈ [0, 2
− 1] |݇ ≠ 0ሽ. (3)
Figure 1. Classes of equivalence of gray levels with respect to classes of equivalence.
Observe that the difference in the expression 3 with regarding of Definition 1, it is that we now
only take into account those gray levels whose probabilities are not zero. In other words, here, we
do not consider the gray levels that are not present in the image.
2.6. Other considerations about notation of the entropy formula
It is known from information theory that the given data by a symbol ݅ of an object is supplied by
−݈݃ଶ ݍ , where ݍ is the relative frequency of symbol ݅ [12]. Taking the mathematical
expectation of the given information by n symbols, we obtain the entropy formula
ܧ = − ݍ݈݃ଶݍ
ୀ
,
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where −ݍ݈݃ଶ ݍ, it is the average contribution of each symbol ݅. For such a reason, entropy is a
measure of the information average contained from each symbol of the object that is being
analyzed. Digital images in gray levels are a particular case of the information theory in which
images are objects and each gray levels represent the symbols.
Suppose one has a gray level image (assuming ݊ > 1), and starting from now, we assume that the
݊ gray levels in the image always will be present. This supposition permits that the entropy
formula can be written in the following way:
ܧ = − ݍ݈݃ଶݍ
∈ூ
, (4)
where ܫ = ሼ݅ ∈ [0, 2
− 1] |݇ ≠ 0ሽ, (see Section 2.4). Without loss of generality and to obtain a
better manipulation of the previous formula, the same can be expressed as,
ܧ = − ݍ݈݃ଶݍ
ୀ
, (5)
Starting from now, the expression 5 will be widely used and we shall call entropy of order ݊,
which is interpreted as that entropy that is composed of the sum of n terms and the associated
image has ݊ gray levels, which are visually present to the observer.
2.7. Maximum entropy of an image of order
It is known from information theory that entropy is seen as a measure of information [12], this
reaches its maximum value when all symbols are equally probable (equal probability of
occurrence). The classical formula for the entropy for a digital image in gray levels is given by
expression 1 and it is defined in (2
− 1)-dimensional real space. However, in most of the cases
it is possible to reduce the quantity of variables of the space by using expression 5, especially in
images where a good level of segmentation is attained. In this section, we present a theorem about
the maximum value reached by entropy when expression 5 is applied.
Theorem 1 (Maximum entropy of an image of order ݊)
Let ܩ be an arbitrary digital image of order ݊, then has maximal entropy of order n, if =
ଵ
,
∀݅ ∈ ,ܫ where ܫ = ሼ݅ ∈ [0, 2
− 1]|݇ ≠ 0ሽ.
The proof of this theorem appears on appendix.
Lemma 1 (Maximum entropy of an image of order ݊)
Let ܩ be an arbitrary digital image in gray levels of order ݊, then if ܩ has maximum entropy of
order n, the entropy value ܧ is given by ݈݃ଶ݊.
Proof:
If ܩ is a digital image in gray levels of order ݊ and this has maximum entropy order ݊, then the
probability of occurrence of gray levels are =
ଵ
, ∀݅ ∈ .ܫ Calculating the value of the entropy,
one has that:
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ܧଵ = − ݈݃ଶ
ଶಳିଵ
ୀ
= −
1
݊
݈݃ଶ
1
݊
ଶಳିଵ
ୀ
= −݊ ൬
1
݊
݈݃ଶ
1
݊
൰ = −݈݃ଶ
1
݊
= ݈݃ଶ݊. ∎
The result obtaining from the previous lemma is not new. In what we are interested, it is in the
expression of the formula and their later use in the following sections.
Given the previous lemma it follows that any gray level image of order ݊, its entropy cannot be
greater than ݈݃ଶ݊. Images that have the property of =
ଵ
, ∀݅ ∈ ܫ will be called “ideal images
of order n” and its entropy by “maximum entropy order ݊”. The last term will be denoted by .݉ܧ
In Figure 2(a) is shown a digital image of order 8, (݊ = 8) and 88ݔ dimension, in which the
equivalence classes can be appreciated; namely, ܥ, ܥଷଶ, ܥସ, ܥଽ, ܥଵଶ଼, ܥଵ, ܥଵଽଶ, ܥଶଶସ and their
corresponding frequencies are ݈ = 7, ݈ଷଶ = 4, ݈ସ = 8, ݈ଽ = 9 , ݈ଵଶ଼ = 13, ݈ଵ = 12, ݈ଵଽଶ = 3,
݈ଶଶସ = 8. In Figure 2(b), 2(c), 2(d) different regions are shown whose entropies are maxima of
order 8. These regions have the gray levels and relative frequencies in common. Observe that
Figure 2(b) and 2(c) have the same dimensions as the image shown in Figure 2(a), but differ in
the spatial distribution of gray levels, while in Figure 2(d) the region has a smaller dimension.
However, the entropy in Figure 2(b) and 2(c) is the same. In the same way that in Figure 2(d) it is
possible to find a region with a much bigger dimension than in Figure 2(a), for example of
dimension 64 ,46 ݔ and to have a maximum entropy of order 8. This fact shows that for any
image in gray levels with entropy of order n it is always possible to obtain an image with
maximum entropy of order n and that it presents the same gray levels.
(a) (b)
(a) (b)
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Figure. 2. Digital images of order 8. 2(a) ܧ = 2.8740. 2(b) Digital image of order 8, size 8x8, maximum
entropy of order 8, ݉ܧ = 3. 2(c) Image digital order 8, size 8x8, maximum entropy of order 8, ݉ܧ = 3.
2(d) digital image of order 8, size 4x2, maximum entropy of order 8, ݉ܧ = 3.
2.8. Applications
The previously obtained result becomes important when an image is compared with another after
running the iterativealgorithm of the mean shift. This will be seen in section 3.
2.8.1 Entropy and its relation with the classes of equivalence
One can observe in expression (5) that in the entropy of a digital image in gray levels, each term
of the same is associated to one and only a gray level. If this way, a one-one relation among the
terms of entropy is established; that is, gray levels and classes of equivalence. With the aim of
revealing other properties in the entropy of image, in the segmentation process this link will be
used.
2.8.2 Entropy as hypersurface
As was seen in the proof of the Theorem 1, when an image is of order ݊, its entropy depends on (݊ − 1)-
variables related by the formula ∑
ିଵ
ୀଵ + ൫1 − ∑
ିଵ
ୀଵ ൯ = 1 , with 0 < < 1 . The domain of
definition of the variables , with ݅ = 1, ݊ − 1തതതതതതതതതത, is 0 < ∑ ݅
݊−1
݅=1 < 1, which is convex and open set of
ℝିଵ
, and its topological properties could be employed for obtaining information about the
evolution of the entropy during the segmentation process. Of this way, entropy can be seen as a
hypersurface in (݊ − 1)-dimensional space and it can be applied tools of the theory of differential
geometry.
Note that if ݊ > 3 it is not possible to obtain the graph of the entropy function for all pi due to
the inability of representing in ܴଷ
sets with dimensions greater than 3. However, in spite of this, it
is possible to plot the entropy function upon the domain by pairs of pi which are of interest to
analyze. For example, those , that have not been annulled during all the segmentation process.
Figure. 3. Graphs of entropy as a function of two variables, ݊ = 3 (rotated for better visualization)
In space R3 the entropy is seen as a surface (see Figure 3) and its graphical view gives an overview of
treatment for the extension to larger spaces. In Figure 3, the graphs of the entropy function for the case that
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9
presents three gray levels are shown, the domain of definition is 0 < ଵ + ଶ < 1, which is represented by
the shaded region by contour lines in the ܻܺ plane. One can observe that this set is open and convex. It is
also shown the maximum point in the coordinates ൫1
3ൗ , 1
3ൗ ൯ and the maximum value of entropy function
is 1.5850.
2.9 Relative entropy in images
As it was point out in Definition 2 the Relative Entropy express by:
(ܦ ∥ )ݍ =
௫∈॒
݈݃
ݍ
= −
௫∈॒
݈ݍ݃ − ൭−
௫∈॒
݈݃൱ =
−
௫∈॒
݈ݍ݃ − (ܧ), (6)
Supose that probability ݑ is the uniform probability of occurrence of colors ݅ ∈ ,ܫ i.e., ݍ =
1
݊
∀݅ =
1, ݊തതതതത, then:
)ݑ(ܧ = −
௫∈॒
݈ݍ݃ = −
௫∈॒
݈݃
1
݊
= − ൬݈݃
1
݊
൰
௫∈॒
= −݈݃
1
݊
= ݈,݊݃
which it is an entropy of ݊ order. Combining Definition 2 and the concept of maximum entropy
of ݊ we obtain the following concept:
Definition 7 (Relative entropy in gray level images)
Let ܩ a gray level image with ݊ gray levels, then the relative entropy in image ܩ is defined by
ܴ)ܩ(ܧ = ݈݊݃ − ܧ൫݅
൯, (7)
where is the probability of occurrence of color ݅.
Note that Definition 7 is the difference between the maximum entropy of ݊ order and the entropy
of the given image ,ܩ for that reason it will denoted by ݉ܧ − .ܧ
Expression 7 will let us characterize the ݅ܵܪܯ algorithm as it is treated in the following section
related to the experimental results.
3. EXPERIMENTS AND ANALYSIS OF RESULTS
With theoretical results it is possible to carry out a study of entropy while the ܵܯℎ݅ algorithm is
running. The used images in this study are represented in Figure 4.
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(a) Barbara (b) Cameraman (c) Bird (d) Baboon (e) Montage
Figure. 4. Images used in the experimentation.
The images and graphics that are shown, have been obtained using the system MATLAB. The
images have 256x256 pixels, the used stopping threshold is 0.000001. The parameters chosen
were hs = 3 and hr = 5. For more details on our algorithm, one can refer to [11], [10]. In Figure
5(a) first result is presented.
When comparing the images shown in Figure 5, one can observe that some details were lost (see
arrows). In Figure 5(b) one can see some more homogeneous areas, in which a certain
segmentation level has been achieved.
(a) Original image (b) Segmented image
Figure. 5. Homogeneous regions
The graph in Figure 6 shows that the entropy gradually decreases (black points), while the
maximum entropy of order ݊ has 6 disjoint intervals remaining constant for the iterations
1 − 2, 6 − 7, 9 − 10, 13 − 15, 23 − 26, 29 − 95 (squares). Throughout this section, the sets of
points upon the graphs formed by these intervals will be called groups of iterations, and will be
identified as the first group of iterations, the second group of iterations, last group of iterations.
Figure 6. Graphs of entropy. Em vs. Number of iterations (Barbara)
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The variation of the order of the image is associated with the appearance or disappearance of gray
levels. However, due to the homogenization that produces the iterative algorithm, certain gray
levels are replaced by others that were not present, in this way it is achieved, that the order of the
image remains constant. The search and analysis of the pi that are annulled in the mentioned
iterations led us to the following conclusion: appearance and disappearance of the gray levels
only in the fourth and fifth groups of iterations, with iterations 13 − 15 and 23 − 26 respectively.
These results appear in Figure 7.
Figure 7 shows the values of for iterations from 13 to 15. At iterations 13 and 14 the gray level
216 is absent (light gray bar, discontinuous edges) and this appears at iteration 15, while the gray
level 217 decreases until it disappears (dark gray bar, continuous edges). This justifies why gray
levels remains constant. Another way of seeing this issue is analyzing the quantity of “null ” per
iteration, which is always the same; i.e., by iteration only one of the is annulled. In a simple
graphical way, it can be seen that the number of columns per iteration (corresponding to no “null
”) is always the same. Similar analysis is carried out with the group of fifth iterations.
(a) 13 − 14 iterations (b) 23 − 26 iterations
Figure. 7. Graph of pi vs. iterations.
In Figure 7(b) we observed that most of the takes the value 1.526 ∗ 10ିହ
, which is not a casual
number. As we pointed out in the beginning of this section all images are of dimension 256x256,
this makes that the total number of pixels is 65536. If one calculates the minimum probability
(no null) of occurrence of a gray level in the image gives a value of 0. 0000152587890625,
which one can express as 1.526 ∗ 10ିହ
. This indicates that when seeing this value in the graphics,
the gray level that corresponds can be only expressed in a pixel of the 65536. In other words, one
can consider this gray level as noise, and if its value did not vary in the last group of iterations can
be eliminated, which will contribute to improve the homogenization in the segmented image. On
the other hand, this gray level will not affect the value of the entropy since its contribution is
minimum. Similarly, one can check that the value 0.3052 ∗ 10ିହ
corresponds to the probability
that its gray level appears only in two pixels in the whole image.
With regard to the last group of iterations it is important to point out that it is quite high (69 %
bigger than the rest), which shows that the number of gray levels tends to be stabilized when
increasing the iterations.
The appearance or disappearance of gray levels in the image determines the variation of the value
of the maximum entropy of order n, since this depends only on the quantity of gray levels present
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in the image. Taking the difference between the maximum entropy of order n with the entropy of
the image, difference which will be denoted by ݉ܧ − ,ܧ an idea of entropy away from its upper
bound can be given (see Figure 8).
The graph in Figure 8, shows that starting from the iteration 29 the entropy of the image tends to
move away from the maximum entropy of order n, without varying the order. The abrupt change
in the graph shows clearly that it varied the quantity of gray levels. Iterations which not belong to
groups of iterations are marked in black color triangles. Observe that several of these iterations
have the same order; for example, the iterations 12, 16, 18, 20 and 22. The difference ݉ܧ − ܧ
corresponding to the iteration 1 it is bigger than the final one (iteration 95).
Figure. 8. Graph of ݉ܧ − ܧ vs. number of iterations (Barbara)
Figure 9 was obtained from Figure 8 by removing black points. The connected symbols by dotted
lines belong to consecutive iterations (groups of iterations), and those with the same shape are
associated to iterations of the same order, a “type of separation by levels”, it is being well-defined.
Figure 9. Graph ݉ܧ − ܧ vs number of iterations.
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It is highlighted with different geometrical figures and the iterations in which the image has the
same order (Barbara).
The points in the first group of iterations have been differentiated with squares from the points of
the last groups of iterations with points. One can appreciate that as the number of iterations
increase the number of groups of iterations with same gray level diminishes until one. In
consecutive the iterations that have the same quantity of gray levels, the values of the difference
݉ܧ − ܧ tend to increase, and the slope of the curve tends to diminish its inclination. On the other
hand, the iterations of the first group have higher values in the difference ݉ܧ − ܧ than the last
group.
Figure 10 shows another example using the image of Cameraman. In Figure 10(b) the arrows
indicate two regions that have been homogenized.
(a) Original image (b) Segmented image
Figure. 10. Homogenous regions.
Figure 11 shows the plot of ݉ܧ (squares) and entropy of the image (points) versus the number of
iterations. Observe that after iteration 10, the number of gray levels did not change, it is constant.
The consecutive iterations that did not change were 1-2, 6-7, 8-13, 14-78, of which the second
group of iterations presented the appearance and disappearance of gray levels. The last group of
iterations (14-78) was 82 % higher than the rest of the iterations in the segmentation process.
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Figure. 11. Graphics Entropy and Em vs. number of iterations (Cameraman)
Figure 12 shows the values of pi for the iterations 6 and 7. One can note that there is only one
column per iteration, and these have the same height but different colors. One can interpret that
the 244 gray level in the iteration 6 was replaced completely by the 245 gray level, maintaining
the order of the image. The value 0.4578 ∗ 10ିହ
corresponds to the probability that its respective
gray level appears only in three pixels of the entire image.
Figure. 12. Graph of vs pi 6-7 iteration (Cameraman).
In Figure 13 it is possible to observe that the iterations previous to iteration 14 do not go through
as many changes of gray levels in comparison to the image of Barbara. This iteration has been
marked with number 14, which shows the instant in that the last group of iterations begins. The
difference ݉ܧ − ܧ for the iteration 1 is smaller than the final iteration (iteration 78), contrary to
the Barbara image.
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Figure. 13. Graphics ݉ܧ − ܧ vs number of iterations (Cameran).
Figure 14 was obtained from Figure 13 by removing only 3 points. In this way, four groups of
iterations are obtained. Abrupt changes are not observed in the groups in which in which the order
of the image remains constant as was in the Barbara image. It can be observed that the value of
݉ܧ − ܧ tends to increase inside the groups when the quantity of iterations increases. The last
group of iterations (black points) presents a bigger number of points in comparison with the rest.
Figure. 14. Graph of the difference in ݉ܧ − ܧ vs number of iterations.
The iterations in which the image has the same order (Cameran) are highlighted with different
geometrical figures.
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Figure 15 shows the results of the image of Bird. In Figure 15(b) the arrows indicate two regions
that have been homogenized.
(a) original image (b) segmented image
Figure. 15. (a) original image. (b) segmented image.
The graph in Figure 16 shows the entropy and maximum entropy of order ݊ versus the number of
iterations of the segmented Bird image. It can be appreciated that the iterations whose resulting
images present the same order are separated into disjoint intervals, which elevate their longitude
when increasing the quantity of iterations. Inside the groups of iterations there is no appearance or
disappearance of gray levels. The last group of iterations is 63 % bigger than the rest of the
iterations in the segmentation process.
Figure 16. Graphics of Entropy and Em vs number of iterations (Bird)
In Figure 17 the graph of the value of the difference ݉ܧ − ܧ versus the quantity of iterations is
shown, where a triangle marks the iteration that does not belong to the groups of iterations. The
behavior of this image does not suffer very marked oscillations as in the case of Barbara image.
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Starting from the iteration 29 the quantity of gray levels is stabilized. The value of ݉ܧ − ܧ initial
is bigger than the final.
Figure 17. Graph ݉ܧ − ܧ vs. number of iterations (Bird)
The graph of Figure 18 presents 6 groups of consecutive iterations that have the same order. The
slopes of the groups of iterations are highlighted with different geometrical figures and one can
see these tend to decrease when the number of iterations increases. This behavior is common for
the previously analyzed images. However, in this example more clarity is evident.
Figure. 18. Graph ݉ܧ − ܧ vs. number of iterations.
The iterations in which the image has the same order (Bird) are highlighted with different
geometrical figures.
The range in that the values of the difference ݉ܧ − ܧ oscillate in the last group of iterations is
very specific to each image. It is shown clearly in Figure 19. However, these present the common
characteristic such as a slow growth of the value of the difference of ݉ܧ − ,ܧ where this slow
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growth is more accentuated with the increase of the number of iterations. The graph includes the
results of the images of Baboon and Montage.
Figure. 19. Graph ݉ܧ − ܧ vs number of iterations for the last group of iterations in different images.
4. CONCLUSIONS
In the graphs corresponding to the Entropy vs. the number of iterations can be appreciated that, as
the segmentation was reached, the entropy diminishes from one iteration to another. The order of
the image decreases as the number of iterations increases, this fact can be interpreted that when
the image is moving away from its ideal image of order ݊, it becomes more homogeneous.
However, it was possible to verify that, starting from an instant, the order of the image remains
constant until the final segmentation is reached, according to the selected stopping threshold. The
differences of the values of ݉ܧ − ܧ between the maximum entropy of order n and the entropy of
the image are significant and these differences increase in the groups of iterations that present the
same quantity of gray levels. In the graphs of bars (see Figures 12 and 13), one can check that
there are gray levels that disappeared from the image as the algorithm runs, contributing to more
homogenization of the image. It is possible to see that in the last stages of segmentation most
images present gray levels with “no null” probability of occurrence. This issue will be the subject
of future research as well as other aspects that arose in this study.
APPENDIX
MAXIMUM VALUE REACH BY ENTROPY
Definition 7: (Negative Definite Function)
Let ܣ ∈ ܯ(ℝ) be, such that ܣ = ܣ௧
, it is said to be negative definite it satisfies that ∀ݕ ≠ 0,
ݕ ∈ ℝ
⟹ ݕ௧
ݕܣ < 0, where ܯ(ℝ), is the set of square matrices of ݊ order with coefficients in
ℝ [6].
Definition 8: (Maximum Strict)
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Suppose that the function ݂()ݔ is defined over the set ܺ ⊂ ℝ
. The point ݔ()
∈ ܺ is called strict
maximum point if a neighborhood ܸ (ݔ()
) of the point ݔ()
that is, ∀ݔ ∈ V൫ݔ()
൯⋂ܺ, ݔ ≠ ݔ()
,
such that the inequality ݂ ()ݔ < ݂൫ݔ()
൯ is checked [6].
Theorem 3 (Condition of Maximum Entropy of Order ):
Let ܩ be an arbitrary digital image of order n, then has maximal entropy of order ݊, if =
ଵ
,
where ܫ = ሼ݅ ∈ [0, 2
− 1] |݇ ≠ 0ሽ.
Proof:
By hypothesis, since ∑
ୀଵ = 1, we construct the vector = (ଵ, ⋯ , ). Without loss of
generality, we will assume that = 1 − ∑
ୀଵ , substituting the expression (3) in the entropy
formula, obtain
ܧ = − ∑ ݈݃ଶ
ିଵ
ୀଵ − ൫1 − ∑ ݅
݊
݅=1 ൯݈݃ଶ൫1 − ∑ ݅
݊
݅=1 ൯ (ܧ depends on ݊ − 1 variables).
The partial derivatives of first order are,
߲ܧ
߲
= − ൬݈݃ଶ +
1
ln 2
൰ + ൭݈݃ଶ ൭1 −
ିଵ
ୀଵ
൱ +
1
ln 2
൱.
Making
డா
డ
= 0 with ݅ = 1, ݊തതതതത, one arrives to the expressions
݈݃ଶ = ݈݃ଶ(1 − ∑
ିଵ
ୀଵ ) and ݈݃ଶ = ݈݃ଶ, ∀݅ ≠ ݊, ݅ = 1, ݊തതതതത.
Because the logarithm function is strictly monotonous, this implies that = ∀݅ ≠ ݊ .
Therefore, we have arrived to that a candidate to strict extreme is obtained when = ∀݅ ≠ ݊,
i.e., when all the probabilities of occurrence of gray levels are equals, so
∑ ݊
ୀଵ = 1 ⟹ ݊݊
= 1 ⟹ ݊
=
1
݊
.
Since all have the same value, then =
ଵ
∀݅ i∀ , the obtained point = ቀ
ଵ
, ⋯ ,
ଵ
ቁ ∈ ℝ
.
We shall now determine the type of local extreme (maximum or minimum).
Analyzing the second partial derivatives of the matrix of the entropy function ,ܧ one has that
൬
డమா
డడೕ
൰ ∈ ܯିଵ(ℝ), i.e., the entropy being a function of (݊ − 1) variables, then the second
differential of the entropy function will depend only on (݊ − 1) variables and it will be a square
matrix of size (݊ − 1)x(݊ − 1). The elements of this matrix are:
డమா
డడೕ
= ൞
−
ଵ
୪୬ ଶ
൬
ଵ
ೕ
+
ଵ
൫1−∑ ݅
݊−1
݅=1 ൯
൰ ∀݅ = ݆
−
ଵ
୪୬ ଶ
൬
ଵ
൫1−∑ ݅
݊−1
݅=1 ൯
൰ ∀݅ ≠ ݆
.
Therefore, the matrix of second partial derivatives has the form
20. International Journal of Soft Computing, Mathematics and Control (IJSCMC),Vol. 4, No. 3, August 2015
20
+
+
+
=
babb
bbab
bbba
A
mL
MOLM
L
L
2
1
,
where ܽ =
ଵ
i∀ and ܾ =
1
൫ଵି∑
షభ
సభ ൯
.
Now, we shall prove that the matrix ൬
డమா
డడೕ
൰ is negative definite by Definition 7. Let ܣ =
൬
డమா
డడೕ
൰ and ݕ ∈ ℝ be, ݕ௧
ቀ−
ଵ
୪୬ ଶ
ܣቁ ݕ = −
ଵ
୪୬ ଶ
(ݕ௧.)ݕܣ Working with the expression ݕ௧,ݕܣ one
has that:
( ) ( )
2
11
2
1
2
1
1
+=
+
+
+
= ∑∑ ==
m
j
j
m
j
jj
m
m
m
t
ybay
y
y
babb
bbab
bbba
yyAyy M
L
MOLM
L
L
L .
Since 0, >bai , ni ,1= , then 0>Ayyt m
y ℜ∈∀ , ni ,1= and therefore
ݕ௧
൬−
1
ln 2
ܣ൰ ݕ = −
1
ln 2
(ݕ௧
)ݕܣ < 0
This last expression says that matrix ܣ is negative definite and of this way the entropy of order ݊
has at least a strict maximum ∀݅; that is, =
ଵ
∀݅. Now, we shall prove that this point is the
global maximum. Suppose that a value of entropy of order ݊ that is, which is bigger or similar to
the maximum than was previously found, but since the point = ቀ
ଵ
, ⋯ ,
ଵ
ቁ , it is a strict
maximum in a small neighborhood ܸ , the entropy values decreases from this point over
neighborhood ܸ ; therefore, there exists a value greater or equal to the entropy in another point,
then a saddle point or an absolute minimum, at least, must exist. However, this is not possible
because the previous analysis gives a negative definite matrix ൬
డమா
డడೕ
൰() for all =
(ଵ, ⋯ , ିଵ) that belongs to the domain of the entropy function. Therefore, the non-existence
of saddle or strict minimum point implies non-existence of other strict maximums, which leads to
only a global maximum.
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