Topology-preserving Mapping Research Articles

Topology-preserving image registration with novel multi-dimensional Beltrami regularization

Beltrami coefficient provides a proper measure to quantify local distortions for a topology-preserving mapping, a.k.a general quasi-conformal mapping, in 2D. Recent studies managed to extend Beltrami coefficient from 2D to nD, and introduced its corresponding Beltrami-like measures to encode local deformation information in challenging nD topology-preserving image registration tasks. However, all these nD measures are not consistent with the classical 2D Beltrami coefficient. We propose a novel nD measure for evaluating local distortions of an nD diffeomorphic mapping in this work. We show its key mathematical properties, and prove that the new measure coincides with the classical 2D Beltrami coefficient. We further introduce a new nD topology-preserving image registration model by means of regularizing such proposed Beltrami-consistent coefficient, and show the existence of its solution. Meanwhile, a generalized Gauss-Newton scheme is applied to establish a fast multi-level numerical solver. Extensive experiments over different 3D medical image registration tasks, including the challenging multi-modal medical image registration, illustrate that the proposed approach by regularizing the introduced Beltrami-consistent coefficient outperforms the other state-of-the-art topology-preserving methods in both efficiency and accuracy.

A Novel Ensemble Beta-Scale Invariant Map Algorithm

This research presents a novel topology preserving map (TPM) called Weighted Voting Supervision -Beta-Scale Invariant Map (WeVoS-Beta-SIM), based on the application of the Weighted Voting Supervision (WeVoS) meta-algorithm to a novel family of learning rules called Beta-Scale Invariant Map (Beta-SIM). The aim of the novel TPM presented is to improve the original models (SIM and Beta-SIM) in terms of stability and topology preservation and at the same time to preserve their original features, especially in the case of radial datasets, where they all are designed to perform their best. These scale invariant TPM have been proved with very satisfactory results in previous researches. This is done by generating accurate topology maps in an effectively and efficiently way. WeVoS meta-algorithm is based on the training of an ensemble of networks and the combination of them to obtain a single one that includes the best features of each one of the networks in the ensemble. WeVoS-Beta-SIM is thoroughly analyzed and successfully demonstrated in this study over 14 diverse real benchmark datasets with diverse number of samples and features, using three different well-known quality measures. In order to present a complete study of its capabilities, results are compared with other topology preserving models such as Self Organizing Maps, Scale Invariant Map, Maximum Likelihood Hebbian Learning-SIM, Visualization Induced SOM, Growing Neural Gas and Beta- Scale Invariant Map. The results obtained confirm that the novel algorithm improves the quality of the single Beta-SIM algorithm in terms of topology preservation and stability without losing performance (where this algorithm has proved to overcome other well-known algorithms). This improvement is more remarkable when complexity of the datasets increases, in terms of number of features and samples and especially in the case of radial datasets improving the Topographic Error.

SOINN+, a Self-Organizing Incremental Neural Network for Unsupervised Learning from Noisy Data Streams

The goal of continuous learning is to acquire and fine-tune knowledge incrementally without erasing already existing knowledge. How to mitigate this erasure, known as catastrophic forgetting, is a grand challenge for machine learning, specifically when systems are trained on evolving data streams. Self-organizing incremental neural networks (SOINN) are a class of neural networks that are designed for continuous learning from non-stationary data. Here, we propose a novel method, SOINN+, which is fundamentally different from the previous generations of SOINN with respect to how “forgetting” is modeled. To achieve a more graceful forgetting, we developed three new concepts: idle time, trustworthiness, and unutility of a node. SOINN+ learns a topology-preserving mapping of the input data to a network structure, which reveals clusters of arbitrary shapes in streams of noisy data. Our experiments with synthetic and real-world data sets showed that SOINN+ can maintain discovered structures even under sudden and recurring concept drifts.

Unsupervised stochastic segmentation of behaviour for learning by demonstration

A method for autonomous segmentation of motion primitives from continuous observation of goal-directed behaviours is proposed. In the proposed method, the iterative patterns of the observed behaviours, which are viewed as motion primitives, are segmented using a stochastic topology preserving map (TPM). The stochastic TPM, which is derived from a combination of self-organising map and Gaussian mixture, provides substantial capabilities for unsupervised learning and clustering of high-dimensional data sequences. The results of experiments conducted, in which the proposed method was applied to a dataset of daily activities captured using a Kinect, verify that the proposed method is reliable.

Visual Encoding of Dissimilarity Data via Topology-Preserving Map Deformation.

We present an efficient technique for topology-preserving map deformation and apply it to the visualization of dissimilarity data in a geographic context. Map deformation techniques such as value-by-area cartograms are well studied. However, using deformation to highlight (dis)similarity between locations on a map in terms of their underlying data attributes is novel. We also identify an alternative way to represent dissimilarities on a map through the use of visual overlays. These overlays are complementary to deformation techniques and enable us to assess the quality of the deformation as well as to explore the design space of blending the two methods. Finally, we demonstrate how these techniques can be useful in several-quite different-applied contexts: travel-time visualization, social demographics research and understanding energy flowing in a wide-area power-grid.

Detection of Brain Tumor using Hybrid Topology Preserving Map

Segmentation of an image is the division or separation of the image into disjoint regions of similar feature. In clinical practice, Magnetic Resonance Imaging (MRI) is used to distinguish pathologic tissue from normal tissues, especially for the detection of brain tumors. Among the thousands of segmentation methods, clustering algorithm is one of the powerful tools in segmenting medical images. In this paper we proposed a new approach based on Hybrid Topology Preserving Map (HTPM) and Adaptive Mean Shift (AMS) algorithm which combines the advantages of both mean shift based clustering algorithm and Hierarchical Topology Preserving Map. And also the performance of the proposed method is compared with the traditional method Adaptive Mean Shift (AMS) algorithm.

Mapping Patterns of Multiple Deprivation and Well-Being using Self-Organizing Maps: An Application to Swiss Household Panel Data

The aim of this paper is to propose multidimensional measures of deprivation and wellbeing in contemporary Switzerland, in order to overcome the limitations of standard approaches. More precisely, we have developed self organising maps (SOM) using data drawn from the 2009 Swiss Household Panel wave, in order to identify highly homogeneous clusters of individuals characterized by distinct profiles across 44 indicators of deprivation and well-being. SOM is a vector quantiser that performs a topology-preserving mapping of the k-dimensional input data to a two-dimensional, rectangular grid of output units, preserving as much as possible the information contained in the original input data. “Topology-preserving” means that, when an SOM is properly developed, units that are close in the output space are also close in the input space. Our results suggest that the SOM approach could improve our understanding of complex and multidimensional phenomena, like those of well-being, deprivation, vulnerability, that show only a partial overlapping with standard income poverty measures.

WeVoS-ViSOM: An ensemble summarization algorithm for enhanced data visualization

This study presents a novel version of the Visualization Induced Self-Organizing Map based on the application of a new fusion algorithm for summarizing the results of an ensemble of topology-preserving mapping models. The algorithm is referred to as Weighted Voting Superposition (WeVoS). Its main feature is the preservation of the topology of the map, in order to obtain the most accurate possible visualization of the data sets under study. To do so, a weighted voting process between the units of the maps in the ensemble takes place, in order to determine the characteristics of the units of the resulting map. Several different quality measures are applied to this novel neural architecture known as WeVoS-ViSOM and the results are analyzed, so as to present a thorough study of its capabilities. To complete the study, it has also been compared with the well-know SOM and its fusion version, with the WeVoS-SOM and with two other previously devised fusion Fusion by Euclidean Distance and Fusion by Voronoi Polygon Similarity—based on the analysis of the same quality measures in order to present a complete analysis of its capabilities. All three summarization methods were applied to three widely used data sets from the UCI Repository. A rigorous performance analysis clearly demonstrates that the novel fusion algorithm outperforms the other single and summarization methods in terms of data sets visualization.

Self-Organizing Map with Generating and Moving Neurons in Visible Space

Kohonen’s Self-Organizing feature Map (SOM) is used to obtain topology-preserving mapping from high-dimensional feature space to visible space of two or fewer dimensions. The SOM algorithm uses a fixed structure of neurons in visible space and learns a dataset by updating reference points in feature space. The mapping result depends on mapping parameters fixed, which are the number and visible positions of neurons, and parameters of learning, which are the learning rate, total iteration, and the setting of neighboring radii. To obtain a satisfactory result, the user usually must try many combinations of parameters. It is wasteful, however, to set up every possible combination of parameters and to repeatedly run the algorithm from the beginning because the computation cost for learning is large, especially for a large-scale dataset. These problems arise due to the fixing of two types of mapping parameters, i.e., the number and visible positions of neurons. The high computation cost is mainly in the calculation of distances from each sample to all reference points. At the beginning of learning, reference points should be adjusted globally to preserve the topology well because they are initially set far from optimal positions in feature space, e.g. randomly. Such many reference points subdivides feature space into unnecessarily fine Voronoi regions. To avoid this computational waste, it is natural to start learning with a small number of neurons and increase the number of neurons during learning. We propose a new SOM method that varies the number and visible positions of neurons, and thus is applicable also to visible torus and sphere spaces. We apply our proposal to spherical visible space. We use central Voronoi tessellation to move visible positions for two reasons: to tessellate visible space evenly for easy visualization and to level the number of neighboring neurons and better preserve topology. We demonstrate the effect of generating neurons to reduce computation cost and of moving visible positions in visualization and topology preservation.

Parallel Learning Model and Topological Measurement for Self-Organizing Maps

The parallel learning model we propose for self-organizing maps (SOMs) uses reference vectors for simultaneously given multiple input and introduces a topological measurement of ordering for reference vectors in a multidimensional array. We term the parallel SOM algorithm parallel learning and the criterion of ordering the twist index. Parallel learning simultaneously updates reference vectors corresponding to individual input when multiple input is prepared. The twist index is the criterion for evaluating multidimensional ordering of the topological array for reference vectors. When the parallel degree changes for a SOM, the topology preserving map (TPM) for post-learning is evaluated using the twist index. Although adaptation generally yields different results for single and multiple input given at each step in neural networks, parallel learning in SOMs produces results almost the same as sequential learning. Discussing the formation rate of TPMs and average distortion, we examine the effectiveness of our approach through numerical experiments.

An adaptive non-linear map and its application

A novel adaptive non-linear mapping (ANLM), integrating an adaptive mapping error (AME) with a chaosgenetic algorithm algorithm (CGA) including chaotic variable, was proposed to overcome the deficiencies of non-linear mapping (NLM). The value of AME weight factor is determined according to the relative deviation square of distance between the two mapping points and the corresponding original objects distance. The larger the relative deviation square between two distances is, the larger the value of the corresponding weight factor is. Due to chaotic mapping operator, the evolutional process of CGA makes the individuals of subgenerations distributed ergodically in the defined space and circumvents the premature of the individuals of subgenerations. The comparison results demonstrated that the whole performance of CGA is better than that of traditional genetic algorithm. Furthermore, a typical example of mapping eight-dimensional olive oil samples onto two-dimensional plane was employed to verify the effectiveness of ANLM. The results showed that the topology-preserving map obtained by ANLM can well represent the classification of original objects and is much better than that obtained by NLM.

Self-Organizing Topological Tree for Online Vector Quantization and Data Clustering

The self-organizing Maps (SOM) introduced by Kohonen implement two important operations: vector quantization (VQ) and a topology-preserving mapping. In this paper, an online self-organizing topological tree (SOTT) with faster learning is proposed. A new learning rule delivers the efficiency and topology preservation, which is superior of other structures of SOMs. The computational complexity of the proposed SOTT is O(log N) rather than O(N) as for the basic SOM. The experimental results demonstrate that the reconstruction performance of SOTT is comparable to the full-search SOM and its computation time is much shorter than the full-search SOM and other vector quantizers. In addition, SOTT delivers the hierarchical mapping of codevectors and the progressive transmission and decoding property, which are rarely supported by other vector quantizers at the same time. To circumvent the shortcomings of clustering performance of classical partition clustering algorithms, a hybrid clustering algorithm that fully exploit the online learning and multiresolution characteristics of SOTT is devised. A new linkage metric is proposed which can be updated online to accelerate the time consuming agglomerative hierarchical clustering stage. Besides the enhanced clustering performance, due to the online learning capability, the memory requirement of the proposed SOTT hybrid clustering algorithm is independent of the size of the data set, making it attractive for large database.

Multiple outlier detection in multivariate data using self-organizing maps title

The problem of detection of multidimensional outliers is a fundamental and important problem in applied statistics. The unreliability of multivariate outlier detection techniques such as Mahalanobis distance and hat matrix leverage has led to development of techniques which have been known in the statistical community for well over a decade. The literature on this subject is vast and growing. In this paper, we propose to use the artificial intelligence technique ofself-organizing map (SOM) for detecting multiple outliers in multidimensional datasets. SOM, which produces a topology-preserving mapping of the multidimensional data cloud onto lower dimensional visualizable plane, provides an easy way of detection of multidimensional outliers in the data, at respective levels of leverage. The proposed SOM based method for outlier detection not only identifies the multidimensional outliers, it actually provides information about the entire outlier neighbourhood. Being an artificial intelligence technique, SOM based outlier detection technique is non-parametric and can be used to detect outliers from very large multidimensional datasets. The method is applied to detect outliers from varied types of simulated multivariate datasets, a benchmark dataset and also to real life cheque processing dataset. The results show that SOM can effectively be used as a useful technique for multidimensional outlier detection.

IBR retrieval method based on topology preserving mappings

Case-based reasoning systems, in general, and instance-based reasoning systems, in particular, are used more and more in industrial applications nowadays. During the last few years, researchers have been working in the development of techniques to automate the reasoning stages identified in this methodology. This paper presents a method for automating the retrieval stage and indexation of instance-based reasoning systems. This method is based on a modification of a new type of topology preserving map that can be used for scale invariant classification. The scale invariant map is an implementation of the negative feedback network to form a topology-preserving mapping. Maximum/minimum likelihood learning is applied in this paper to the scale invariant map and its possibilities are explored. This method automates the organization of cases and the retrieval stage of case-based reasoning systems. The proposed methodology groups instances with similar structure, identifying clusters automatically in a data set in an unsupervised mode. The method has been successfully used to completely automate the reasoning process of an oceanographic forecasting system and to improve its performance.

An icon-based data image construction method for production data visualization

Visualization takes advantage of human's rapid and flexible pattern recognition capacity to provide a powerful information-processing environment. It is especially suitable for identifying patterns that are difficult, or impossible, to be defined by a mathematical or computer model. This paper presents an icon-based method for constructing images to visualize the abstract, multidimensional production data stored in tables. The method employs a simple, topology-preserving mapping to convert numbers in a table to icons. This mapping strategy not only makes the meaning of the resulted data image easy to understand, but also makes the maximum number of dimensions and data items that could be displayed be virtually unlimited. Visual patterns in the data image are formed by adjacent icons that have similar grey scales. The area sampling techniques could be applied to reduce the size of the data image while still retaining its visual patterns. The proposed visualization method has been applied to several production planning and control activities, including analysing WIP patterns, production indices, and the machine idle cost performance of production plans. These industrial case studies demonstrate that the ability to quickly and easily examine large amount of manufacturing data through visualization is critical for manufacturing companies to understand their performance, discover possible problems and take necessary actions so that they could remain competitive in the market.

Shape indexing using self-organizing maps

In this paper, we propose a novel approach to generate the topology-preserving mapping of structural shapes using self-organizing maps (SOMs). The structural information of the geometrical shapes is captured by relational attribute vectors. These vectors are quantised using an SOM. Using this SOM, a histogram is generated for every shape. These histograms are treated as inputs to train another SOM which yields a topology-preserving mapping of the geometric shapes. By appropriately choosing the relational vectors, it is possible to generate a mapping that is invariant to some chosen transformations, such as rotation, translation, scale, affine, or perspective transformations. Experimental results using trademark objects are presented to demonstrate the performance of the proposed methodology.

Modelling topology preservation in a cortex by controlled deformation of its energy landscape

Soft topology-preserving mapping with fixed lateral interactions has been applied for the formation of retinotopy and ocular dominance. The learning is based on controlled deformation of the energy landscape modelled by deterministic annealing, which can be more realistic metaphor than changing the lateral interaction range.

Self-Organising Maps as a Relevance Feedback Technique in Content-Based Image Retrieval

Self-Organising Maps (SOMs) can be used in implementing a powerful relevance feedback mechanism for Content-Based Image Retrieval (CBIR). This paper introduces the PicSOM CBIR system, and describes the use of SOMs as a relevance feedback technique in it. The technique is based on the SOM’s inherent property of topology-preserving mapping from a high-dimensional feature space to a two-dimensional grid of artificial neurons. On this grid similar images are mapped in nearby locations. As image similarity must, in unannotated databases, be based on low-level visual features, the similarity of images is dependent on the feature extraction scheme used. Therefore, in PicSOM there exists a separate tree-structured SOM for each different feature type. The incorporation of the relevance feedback and the combination of the outputs from the SOMs are performed as two successive processing steps. The proposed relevance feedback technique is described, analysed qualitatively, and visualised in the paper. Also, its performance is compared with a reference method.

Kohonen's SOM with cache

The Kohonen self-organizing map (SOM),is a topology-preserving map that maps data from higher dimensions onto a (typically) two-dimensional grid of lattice points [1]. The aim of self-organization is to generate a topology-preserving mapping, where the neighborhood relations in the input space are preserved as well as possible, in the neighborhood relations of the units of the map [2]. One of the most time consuming steps during the training of the SOM is the sub-problem of locating the winner node for a given sample. A winner node is the best matching unit for each input vector.

A self-organizing neural tree for large-set pattern classification

Neural networks have been successfully applied to various pattern classification problems in terms of their learning ability, high discrimination power, and excellent generalization ability. However, for the case of classifying large-set and complex patterns, the greater part of conventional neural networks suffer from several difficulties such as the determination of the structure and size of the network, the computational complexity, and so on. In this paper, to cope with these difficulties, we propose a structurally adaptive intelligent neural tree (SAINT). The basic idea is to partition hierarchically input pattern space using a tree-structured network which is composed of subnetworks with topology-preserving mapping ability. The main advantage of SAINT is that it attempts to find automatically a network structure and size suitable for the classification of large-set and complex patterns through structure adaptation. Experimental results reveal that SAINT is very effective for the classification of large-set real world handwritten characters with high variations, as well as multilingual, multifont, and multisize large-set characters.