José Alfredo Ferreira Costa

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.

An efficient approach to the travelling salesman problem using self-organizing maps.

This paper presents an approach to the well-known Travelling Salesman Problem (TSP) using Self-Organizing Maps (SOM). The SOM algorithm has interesting topological information about its neurons configuration on cartesian space, which can be used to solve optimization problems. Aspects of initialization, parameters adaptation, and complexity analysis of the proposed SOM based algorithm are discussed. The results show an average deviation of 3.7% from the optimal tour length for a set of 12 TSP instances.

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.

Cluster analysis using self-organizing maps and image processing techniques

Cluster analysis methods are used to classify R unlabeled objects in a P-dimensional space into groups based on their similarities. This paper focuses on the use of self organising maps (SOM) as a clustering tool and some of the additional procedures required to enable a meaningful clusters interpretation in the trained map. Topics discussed here include the usage of mathematical morphology segmentation method watershed to segment the neurons distance image (u-matrix). Finding good watershed markers and the modification of the u-matrix homotopy are discussed. The algorithm automatically produces labeled sets of neurons that are related to the clusters in the P-dimensional space. An example of non-spherical, complex shaped and nonlinearly separable clusters illustrate the capabilities of the method.

A new tree-structured self-organizing map for data analysis

This paper presents a new algorithm for dynamical generation of a hierarchical structure of self-organizing maps (SOM) with applications to data analysis. Different from other tree-structured SOM approaches, in this case the tree nodes are actually maps. From top to down, maps are automatically segmented by using the U-matrix information, which presents relations between neighboring neurons. The automatic map partitioning algorithm is based on mathematical morphology segmentation and it is applied to each map in each level of the hierarchy. Clusters of neurons are automatically identified and labeled and generate new sub-maps. Data are partitioned accordingly the label of its best match unit in each level of the tree. The algorithm may be seen as a recursive partition clustering method with multiple prototypes cluster representation, which enables the discoveries of clusters in a variety of geometrical shapes.

Data Clustering using Self-Organizing Maps segmented by Mathematic Morphology and Simplified Cluster Validity Indexes: an application in remotely sensed images

This paper presents a cluster analysis method which automatically finds the number of clusters as well as the partitioning of a data set without any type of interaction with the user. The data clustering is made using the self-organizing (or Kohonen) map (SOM). Different partitions of the trained SOM are obtained from different segmentations of the U-matrix (a neuron-distance image) that are generated by means of mathematical morphology techniques. The different partitions of the trained SOM produce different partitions for the data set which are evaluated by cluster validity indexes. To reduce the computational cost of the cluster analysis process this work also proposes the simplification of cluster validity indexes using the statistical properties of the SOM. The proposed methodology is applied in the cluster analysis of remotely sensed images.

Automatic data classification by a hierarchy of self-organizing maps

Clustering is the process by which discrete objects are assigned to groups that have similar characteristics. Self-organizing maps (SOM) have been widely used as a data visualization tool. Some of their advantages include information compression and density estimation while trying to preserve the topological and metric relationships of the primary data items. For using SOM as a clustering tool additional procedures are required to interpret the mapping obtained through unsupervised learning. Costa and Netto (1999) described the usage of image analysis and mathematical morphology to find automatically regions of similar neurons and their borders. The purpose of this paper is to enhance the clustering process in order to detail the underlying structure obtained in a first trial. Groups of neurons associated to clusters are further subdivided in new sub-networks, generating a tree-like structure of SOMs. Differently to other hierarchical SOM approaches, the number of sub-nets for a given SOM in a given height of the tree is not specified in advance. The process can be seen as a dynamic strategy for cluster discovery.

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.