Texture segmentation through eigen-analysis of the Pseudo-Wigner distribution

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Abstract

In this paper we propose a new method for texture segmentation based on the use of texture feature detectors derived from a decorrelation procedure of a modified version of a Pseudo-Wigner distribution (PWD). The decorrelation procedure is accomplished by a cascade recursive least squared (CRLS) principal component (PC) neural network. The goal is to obtain a more efficient analysis of images by combining the advantages of using a high-resolution joint representation given by the PWD with an effective adaptive principal component analysis (PCA) through the use of feedforward neural networks.

Introduction

It is well known that most of the statistical-based signal processing methods are based on hypothesis of linearity and stationarity. However, there exist situations in which these assumptions fail to be true. One way to solve this problem is to combine time-frequency analysis with other techniques such as neural networks. Neural networks-based methods provide an input-output mapping that can be considered as a non-parametric estimation of the data, i.e. the knowledge of the underlying probability distribution is not required. Most situations in signal processing filtering are based to create a sparse sampling of the space/spatial-frequency domain. However, there exist some applications, e.g. shape from texture, that require a more dense sampling as provided by the PWD or the spectrogram (Krumm, 1993; Malik and Rosenholtz, 1994). One of the main trade-offs of the use of this type of joint representation is the high dimensionality of the data to be processed. The PWD of a 2D image will increase the amount of redundant information and therefore data compression will be required. The main motivation of this paper is to obtain a more efficient analysis of images by combining the advantages of using a high-resolution joint representation given by the PWD with an effective adaptive PCA extraction through the use of feedforward neural nets. A precedent of the present method can be found in (Gorecki, 1991) in which an optical-digital approach for image classification by taking a Karhunen–Loève (KLT) expansion of Fourier spectra was proposed. The present work can be considered as an extension of the modular learning strategy to 2D signals described by Haykin (1996). What we are using here is a common detecting strategy used in pattern recognition tasks which consists in performing a space/spatial-frequency analysis first, followed by the usual feature extraction and classification steps. Many other examples can be found in the literature (Abeyskera and Boashash, 1991; Haykin and Bhattacharya, 1994). The main idea is to consider a local transform, i.e. the PWD, as an N-dimensional feature detector followed by a neural-based PCA as a fast and adaptive spectral decorrelator. In this way, we produce a substantial reduction in the volume of data due to the fact that after the PCA extraction we still have a local (but optimally reduced) representation.

In Section 2, a brief review of the discrete PWD is presented through the use of a new 2D analytical image for eliminating aliasing and reducing cross-terms. In Section 3, the Sanger's rule for training unsupervised feedforward neural networks is briefly outlined and its convergence analyzed by using variable learning rate rules. However, a CRLS method is used here for PCA of the PWD samples that constitutes a faster and more efficient adaptive network (Cichocki et al., 1996). Comparisons with other PCA extraction algorithms are also provided. Some simulation results of texture classification by K-means clustering are presented. Finally, conclusions are drawn in Section 4.

Section snippets

The 2D Pseudo-Wigner distribution

The WD was introduced in (Wigner, 1932) as a phase space representation in Quantum Mechanics. It gives a simultaneous representation of a signal in space and spatial-frequency variables. The WD presents a set of desirable mathematical properties as was formulated in (Claasen and Mecklenbrauker, 1980) including high-signal concentration (maximum auto-component concentration in space/spatial frequency). However, the presence of cross-terms has limited use in some practical applications.

Unsupervised learning strategies

Many methods have been proposed for finding the eigenvectors of a data set. The generalized Hebbian learning (GHA), also known as the Sanger's rule, has been proposed for one-layer feedforward unsupervised learning. For M-output networks, it has been proven that it extracts the M principal components (PCs) from the input distribution, which are formally equivalent to the KLT (Oja, 1982; Sanger, 1989), although other alternative approaches can be considered (Fritzke, 1993, Scheweizer et al., 1991

Conclusions

We have proposed a PWD–PCA texture segmentation method based on the use of a new analytic image with high resolution and frequency support and a CRLS network for PWD's PCA extraction. The PWD is computed through a new 2D analytic image method that outperforms traditional ones that use low-pass filtering for reducing aliasing and by removing cross-terms between higher and lower spatial-frequency regions without auto-term broadening. In this way, a substantial reduction in the volume of data is

Acknowledgements

This work has been supported in part by the following Grants: NATO Collaborative Grants Program, and EU INCO-DC AMOVIP 961646 Project. J. Hormigo was supported by a Spanish Ministry of Education and Culture fellowship. We thank Prof. Leon Cohen for a number of fruitful discussions about this paper and Nicolas Roa for providing results included in Fig. 2. Matlab programs are available at http://www.iv.optica.csic.es/projects/links.html.

References (30)

  • Cohen, L., 1995. Time-Frequency Analysis. Prentice-Hall, Englewood Cliffs,...
  • Diamantaras, K.I., Kung, S.Y., 1996. Principal Component Neural Networks: Theory and Applications. Wiley, New...
  • Fritzke, B., 1993. Vector quantization with a growing and splitting elastic net. In: Proceedings of the International...
  • D. Gabor

    Theory of Communication

    J. Inst. Electr. Eng.

    (1946)
  • C. Gorecki

    Surface classification by an optoelectronic implementation of the Karhunen–Loève expansion

    Appl. Optics

    (1991)
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