Marcio Luiz de Andrade Netto

Clustering of complex shaped data sets via Kohonen maps and mathematical morphology

Clustering is the process of discovering groups within the data, based on similarities, with a minimal, if any, knowledge of their structure. The self-organizing (or Kohonen) map (SOM) is one of the best known neural network algorithms. It has been widely studied as a software tool for visualization of high-dimensional data. Important features include information compression while preserving topological and metric relationship of the primary data items. Although Kohonen maps had been applied for clustering data, usually the researcher sets the number of neurons equal to the expected number of clusters, or manually segments a two-dimensional map using some a priori knowledge of the data. This paper proposes techniques for automatic partitioning and labeling SOM networks in clusters of neurons that may be used to represent the data clusters. Mathematical morphology operations, such as watershed, are performed on the U-matrix, which is a neuron-distance image. The direct application of watershed leads to an oversegmented image. It is used markers to identify significant clusters and homotopy modification to suppress the others. Markers are automatically found by performing a multi-level scan of connected regions of the U-matrix. Each cluster of neurons is a sub-graph that defines, in the input space, complex and nonparametric geometries which approximately describes the shape of the clusters. The process of map partitioning is extended recursively. Each cluster of neurons gives rise to a new map, which are trained with the subset of data that were classified to it. The algorithm produces dynamically a hierarchical tree of maps, which explains the clusters structure in levels of granularity. The distributed and multiple prototypes cluster representation enables the discoveries of clusters even in the case when we have two or more non-separable pattern classes.Clustering is the process of discovering groups within the data, based on similarities, with a minimal, if any, knowledge of their structure. The self-organizing (or Kohonen) map (SOM) is one of the best known neural network algorithms. It has been widely studied as a software tool for visualization of high-dimensional data. Important features include information compression while preserving topological and metric relationship of the primary data items. Although Kohonen maps had been applied for clustering data, usually the researcher sets the number of neurons equal to the expected number of clusters, or manually segments a two-dimensional map using some a-priori knowledge of the data. This paper proposes techniques for automatic partitioning and labeling SOM networks in clusters of neurons that may be used to represent the data clusters. Mathematical morphology operations, such as watershed, are performed on the U-matrix, which is a neuron-distance image. The direct application of watershed leads to an oversegmented image. It is used markers to identify significant clusters and homotopy modification to suppress the others. Markers are automatically found by performing a multilevel scan of connected regions of the U-matrix. Each cluster of neurons is a sub-graph that defines, in the input space, complex and non-parametric geometries which approximately describes the shape of the clusters. The process of map partitioning is extended recursively. Each cluster of neurons gives rise to a new map, which are trained with the subset of data that were classified to it. The algorithm produces dynamically a hierarchical tree of maps, which explains the clusters structure in levels of granularity. The distributed and multiple prototypes cluster representation enables the discoveries of clusters even in the case when we have two or more non-separable pattern classes.© (2001) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Estimating the Number of Clusters in Multivariate Data by Self-Organizing Maps

Determining the structure of data without prior knowledge of the number of clusters or any information about their composition is a problem of interest in many fields, such as image analysis, astrophysics, biology, etc. Partitioning a set of n patterns in a p-dimensional feature space must be done such that those in a given cluster are more similar to each other than the rest. As there are approximately Kn/K! possible ways of partitioning the patterns among K clusters, finding the best solution is very hard when n is large. The search space is increased when we have no a priori number of partitions. Although the self-organizing feature map (SOM) can be used to visualize clusters, the automation of knowledge discovery by SOM is a difficult task. This paper proposes region-based image processing methods to post-processing the U-matrix obtained after the unsupervised learning performed by SOM. Mathematical morphology is applied to identify regions of neurons that are similar. The number of regions and their labels are automatically found and they are related to the number of clusters in a multivariate data set. New data can be classified by labeling it according to the best match neuron. Simulations using data sets drawn from finite mixtures of p-variate normal densities are presented as well as related advantages and drawbacks of the method.

Hierarchical SOM applied to image compression

The increase of the need for image storage and transmission in computer systems has increased the importance of signal and image compression algorithms. The approach involving vector quantization (VQ) relies on the design of a finite set of codes which will substitute the original signal during transmission with a minimal of distortion, taking advantage of the spatial redundancy of image to compress them. Algorithms such as LBG and SOM work in an unsupervised way toward finding a good codebook for a given training data. However, the number of code vectors (N) needed for VQ increases with the vector dimension, and full-search algorithms such as LBG and SOM can lead to large training and coding times. An alternative for reducing the computational complexity is the use of a tree-structured vector quantization algorithm. This paper presents an application of a hierarchical SOM for image compression which reduces the search complexity from O(N) to O(log N), enabling a faster training and image coding. Results are given for conventional SOM, LBG and HSOM, showing the advantage of the proposed method.

Cell nuclei segmentation in noisy images using morphological watersheds

A major problem in image processing and analysis is the segmentation of its components. Many computer vision tasks process image regions after segmentation, and the minimization of errors is then crucial for a good automatic inspection system. This paper presents an applied work on automatic segmentation of cell nuclei in digital noisy images. One of the major problems when using morphological watersheds is oversegmentation. By using an efficient homotopy image modification module, we prevent oversegmentation. This module utilizes diverse operations, such as sequential filters, distance transforms, opening by reconstruction, top hat, etc., some in parallel, some in cascade form, leading to a new set of internal and external cell nuclei markers. Very good results have been obtained and the proposed technique should facilitate better analysis of visual perception of cell nuclei for human and computer vision. All steps are presented, as well as the associated images. Implementations wee done in the Khoros system using the MMach toolbox.

A System of Knowledge Representation Based in Formulae of Predicate Calculus whose Variables are Annotated by Expressions of A FuzzyTerminological Logic

In this paper one presents a system for introducing assertions in a knowledge base (kb). These assertions are represented as formulae of Predicate Calculus (PC) whose variables are annotated by concepts of Terminological Logic (TL). The system answers questions by using methods of inference both from PC and TL. The terminological language used is characterized by a fuzzy treatment of concepts. The main contribuitions are: a unification algorithm which closes the semantic gap between PC and TL; introduction of uncertainty in subsumption; use of subsumption to simplify the tracing of the proof; use of Partial Evaluation to link assertional and terminological reasoning.

Segmentação de mapas auto-organizáveis com espaço de saída 3-D

The self-organizing map (SOM) has been widely used as a software tool for visualization of high-dimensional data. Important SOM features include information compression while trying to preserve topological and metric relationship of the primary data items. Similar data in the input space would be mapped to the same neuron or in a nearby unit. The clustering properties of a trained SOM 2-D can be visualized by the U-matrix, which is a neurons neighborhood distance based image. This assumption of topological preservation is not true for many SOM mappings involving dimension reduction. With the automation of cluster detection in SOM network higher output dimensions can be used in problems involving discovery of classes in multidimensional data. Results of topological errors are shown in a simple 2-D clustering in a 1-D output grid SOM. This paper presents the U-array as an extension of the U-matrix for 3-D output grids. The advantage of the method relies in working with higher dimensions in the output space, which can lead to a better topological preservation in data analysis. Examples of automatic class discovery using U-arrays are also presented.

Land-Cover Classification Using Self-Organizing Maps Clustered with Spectral and Spatial Information

Digital classification methods of remotely sensed images have acquired a growing importance in the automatic recognition of the land cover patterns. The enormous quantity of images that are being generated from an increasing number of highly sophisticated sensor systems require the development of innovative classification methodologies, which allow an automatic and efficient detection of the great volume of data available in the images and at the same time makes the mapping process of terrestrial surfaces less subjective and with greater potential for reuse in subsequent situations. Particularly, unsupervised classification methods have traditionally been considered as an important approach for the interpretation of remotely sensed images. This approach of classification plays an especially significant role when very little a priori information about image data is available, and for that reason continues to be a popular choice for analysts without ample field knowledge or for those wanting to avoid introduced bias in classification analysis (Duda & Canty, 2002; Kelly et al., 2004). Unsupervised classification is frequently performed through clustering methods. These methods examine the unknown pixels in an image and incorporate them into a set of classes defined through the natural clusters of the gray levels of the pixels. Cluster analysis provides a practical method for organizing a large set of data so that the retrieval of information may be made more efficiently. However, although there is a large quantity of different clustering methods in the pattern recognition area (Xu & Wunsch II, 2005), only a limited quantity of them can be used in remote sensing applications. As pointed out in Tran et. al. (2005), several problems are encountered when clustering remotely sensed images, but above all the image size and the feature dimension problems are those that often make a method inappropriate due to computation time and computer memory. The most common clustering method applied to remotely sensed data is partitional. The majority of software or computational systems meant for the digital processing of remotely

A fractal Fuzzy approach to clustering tendency analysis

A hybrid system was implemented with the combination of Fractal Dimension Theory and Fuzzy Approximate Reasoning, in order to analyze datasets. In this paper, we describe its application in the initial phase of clustering methodology: the clustering tendency analysis. The Box-Counting Algorithm is carried out on a dataset, and with its resultant curve one obtains numeric indications related to the features of the dataset. Then, a fuzzy inference system acts upon these indications and produces information which enable the analysis mentioned above.

SABIO: an automatic portuguese text summarizer through artificial neural networks in a more biologically plausible model

An implementation of a computational tool to generate new summaries from new source texts in Portuguese language, by means of connectionist approach (artificial neural networks) is presented. Among other contributions that this work intends to bring to natural language processing research, the employment of more biologically plausible connectionist architecture and training for automatic summarization is emphasized. The choice relies on the expectation that it may lead to an increase in computational efficiency when compared to the so-called biologically implausible algorithms.

U-array: Extensão da U-matrix para Mapas Auto-Organizáveis com Espaço de Saída 3-D

The self-organizing map (SOM) has been widely used as a software tool for visualization of highdimensional data. Important SOM features include information compression while trying to preserve topological and metric relationship of the primary data items. As the SOM networks are used extensively for data display, usually the output grid is 2-D. The assumption of topological preservation in SOM is not true for many data mappings involving dimension reduction. With the automation of cluster detection in SOM, higher output dimensions can be used in problems involving discovery of classes in multidimensional data. This paper presents the U-array as an extension of the U-matrix for 3-D SOM output grids. The algorithm uses the watershed transform as well as a heuristic approach for determining the image markers to perform volume segmentation. Examples of automatic class discovery using U-arrays are also presented.