Granulometry and size distribution
3.3.2. Average filter: The simplest average filter
operation, that is, the average of two pixel values x and y, is expressed by the minimum and the supremum, as follows: or as shown in Fig. 7. Fig. 6. Median expressed by the maximum and minimum. Fig. 7. Average expressed by the maximum and minimum.4. GRANULOMETRY AND TEXTURE ANALYSIS
Texture is an image composed by repetitive appearance of small structures, for example surfaces of textiles, microscopic images of ores, etc. Texture analysis is a fundamental application of mathematical morphology, since it was developed for the analysis of minerals in ores. In this section, the concept of size in the sense of mathematical morphology and the idea of granulometry for measuring granularity of image objects are explained. An example of texture analysis applying granulometry by the author is also presented.4.1. Granulometry and size distribution
Opening of image X with respect to structuring element B means residue of X obtained by removing smaller structures than B. It indicates that opening works as a filter to distinguish object structures by their sizes. Let 2B, 3B, . . . , be homothetic magnifications of the basic structuring element B. We then perform opening of X with respect to the homothetic structuring elements, and obtain the image sequence XB, X2B, X3B, . . . . In this sequence, XB is obtained by removing the regions smaller than B, X2B is obtained by removing the regions smaller than X2B, X3B is obtained by removing the regions smaller than 3B, . . . . If B is convex, it holds that X XB X2B X3B . . . . The size of rB is defined as r, and this sequence of opening is called granulometry [10]. We then calculate the ratio of the area for binary case or the sum of pixel values for gray scale case of XrB to that of the original X at each r. The area of an image is defined by the area occupied by an image object, i. e. the number of pixels composing an image object in the case of discrete images. The function from a size r to the corresponding ratio is monotonically decreasing, and unity when the size is zero. This function is called size distribution function. The size distribution function of size r indicates the area ratio of the regions whose sizes are greater than or equal to r. Information and Communication Technology Seminar, Vol. 1 No. 1, August 2005 ISSN 1858-1633 2005 ICTS 6 Fig. 8. Granulometry and size density function The r-times magnification of B, denoted rB, is usually defined in the context of mathematical morphology as follows: where {0} denotes a single dot at the origin. Let us consider a differentiation of the size distribution function. In the case of discrete sizes, it is equivalent to the area differences of the image pairs corresponding to adjacent sizes in X B , X 2B , X 3B , .... For example, the area difference between X 2B and X 3B corresponds to the part included inX 2B but excluded fromX 3B , that is, the part whose size is exactly 2. The sequence of the areas corresponding to each size exactly, derived as the above, is called pattern spectrum [13], and the sequence of the areas relative to the area of the original object is called size density function [14]. An example of granulometry and size density function is illustrated in Fig. 8. Size distribution function and size density function have similar properties to probability distribution function and probability density function, respectively, so that such names are given to these functions. Similarly to probability distributions, the average and the variance of size of objects in an image can be considered. Higher moments of a size distribution can be also defined, which are called granulometric moments, and image objects can be characterized using these moments [14–16] .4.2. Application to texture analysis
Parts
» INTRODUCTION ICTS2005 The Proceeding
» Opening Fundamental Operations of Mathematical Morphology
» Morphological filter Filter theorem
» Granulometry and size distribution
» PGPC texture model and estimation of the optimal structuring element: The PGPC
» CONCLUSIONS ICTS2005 The Proceeding
» Non-ergodicity parameters RESULTS AND DISCUSSIONS 1 Partial structure factors and
» SIMULATIONS CONCLUSION ICTS2005 The Proceeding
» IMAGE RECONSTRUCTION SYSTEM DESIGN
» RESULT CONCLUSION ICTS2005 The Proceeding
» MULTI-RESOLUTION HISTOGRAM TECHNIQUE DATA
» VALIDATION STRATEGY RESULTS AND DISCUSSION
» CONCLUSION ICTS2005 The Proceeding
» INTRODUCTION DISTILATION COLUMN AND ARTIFICIAL NEURAL NETWORK
» Using Temperature Correlation Using Flow Rate Correlation
» INTRODUCTION DETECTION OF SINGLE TREE FELLING WITH SOFT
» Supervised Fuzzy c-means Method
» Neural Network classification METHOD 1. Datasets
» Neural Network Classification Results
» Comparison of Classification Results
» DISCUSSIONS ICTS2005 The Proceeding
» CONCLUSION ACKNOWLEDGEMENT ICTS2005 The Proceeding
» Caching Access List BANDWIDTH MANAGEMENT IMPLEMENTATION
» Rate Limiting BANDWIDTH MANAGEMENT IMPLEMENTATION
» BANDWIDTH MANAGEMENT CONCEPTS RESULT
» The Architecture of UML Elements Model Element
» Diagram Element Editing SYSTEM ARCHITECTURE
» Server Application Architecture Undo
» INTRODUCTION IMPLEMENTATION TESTING ICTS2005 The Proceeding
» INTRODUCTION E-PURSE ICTS2005 The Proceeding
» Interfaces Verification Tool POS – Smart Card
» MULTI AGENT SYSTEM MAS A WEIGHTED-TREE SIMILARITY ALGORITHMS
» RESULTS ICTS2005 The Proceeding
» Facial Animation Morphing and Deformation Cross Dissolve
» Feature Morphing Mesh Morphing Text-to-Speech TTS Basic Block
» Text-to-Video Algorithm Text-To-Video Stake And Desain
» Suggestion CONCLUSION AND SUGGESTION 1 Conclusion
» The Concept SHARE-IT SYSTEM ARCHITECTURE
» SHARING SCENARIO CONCLUSION ICTS2005 The Proceeding
» The Bayesian Network Model and Modified Bayesian Optimization
» Designs and Implementation SCHEDULING MODEL AND IMPLEMENTATION
» Comparison Proposed Schedule with Real Schedule
» Face-to-Face Technique Long Distance Technique
» Scenario to motivate. Context_Selection Applikasi.
» INTRODUCTION ARCHITECTURE. CONCLUSION. ICTS2005 The Proceeding
» SUGGESTION ICTS2005 The Proceeding
» Data Flow Database Structure
» EXPERIMENTAL RESULT ICTS2005 The Proceeding
» Investment Stock Prototyping System Design
» Database Model Stock Valuation
» INTRODUCTION METHODOLOGY ICTS2005 The Proceeding
» Buffer Overrun Cryptography Random Numbers
» Anti-Tampering Error Handling Injection Flaws
» Encapsulate Field Restructuring Arrays
» Generating Secure Random Number Storing Deleting Passwords
» Smart Serialization Message Digest
» Convert Message with Private Key to Public Key
» INTRODUCTION CURRENT STATUS ICTS2005 The Proceeding
» INTRODUCTION PROPOSED SIMULATION MODEL
» PARALLELIZATION STRATEGY ICTS2005 The Proceeding
» EXPERIMENTS AND DISCUSSION CONCLUSION
» INTRODUCTION RESULTS AND DISCUSSION
» EXPERIMENTAL ICTS2005 The Proceeding
» RESULT AND DISCUSSION ICTS2005 The Proceeding
» Color segmentation SYSTEM CONFIGURATION
» FEATURE CHARACTERISTICS AND GENERAL RULE
» EXPERIMENTAL RESULT CONCLUSION ICTS2005 The Proceeding
» INTRODUCTION REVIEW OF LITERATURE
» Social Economics Impact. Restructuring Impact
» Manager Application Mobile Agent Generator MAG Mobile Agents MAs
» SNMP Table Polling SNMP Table Filtering
» BREAST CARCINOMA TUMOR ICTS2005 The Proceeding
» WATERSHED ALGORITHM METHODS ICTS2005 The Proceeding
» RESULT AND DISCUSION ICTS2005 The Proceeding
» FADED INFORMATION FIELD ARCHITECTURE
» ALGORITHMS TO CHOOSE NODES TO CREATE THE FADED
» SYSTEM SIMULATIONS ICTS2005 The Proceeding
» Model and Teory MODEL, TEORY, DESIGN, IMPLEMENTATION AND
» INTRODUCTION ANALYSIS AND RESULT
» INTRODUCTION A SIMPLE MODEL OF THE QUEUING SYSTEM
» SIMULATION RESULTS DISCUSSION ICTS2005 The Proceeding
» CONCLUSION INTRODUCTION ICTS2005 The Proceeding
» Dialog Processing ADDING NONVERBAL BEHAVIOUR
» Emotion Expression Experiment ADDING NONVERBAL BEHAVIOUR
» NATURAL LANGUAGE PROCESSING EMOTION REASONING
» Fuzzy Logic Control FLC System Planning
» Digital To Analog Converter DAC Motor Driver Position Sensor Display Unit
» INTRODUCTION CONCLUSION ICTS2005 The Proceeding
» Variable-Centered Rule Structure VARIABLE-CENTERED INTELLIGENT RULE SYSTEM
» Knowledge Refinement VARIABLE-CENTERED INTELLIGENT RULE SYSTEM
» Knowledge Building VARIABLE-CENTERED INTELLIGENT RULE SYSTEM
» Knowledge Inferencing VARIABLE-CENTERED INTELLIGENT RULE SYSTEM
» INTRODUCTION BASIC CONCEPTS OF FUZZY SETS
» Calculation of the Fitness Degree
» ESTIMATING MULTIPLE NULL VALUES IN RELATIONAL
» Chen’s [6] Result This Improving Method’s Result
» The Fuzzy Set HISTOGRAM THRESHOLDING
» Fuzzy Set Similarity HISTOGRAM THRESHOLDING
» EXPERIMENTAL RESULTS ICTS2005 The Proceeding
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