Behavior of morphological associative memories with true-color image patterns

doi:10.1016/j.neucom.2009.09.004

Neurocomputing

Volume 73, Issues 1–3, December 2009, Pages 225-244

https://doi.org/10.1016/j.neucom.2009.09.004 Get rights and content

Abstract

Morphological associative memories (MAMs) are a special type of associative memory which exhibit optimal absolute storage capacity and one-step convergence. This associative model substitutes the additions and multiplications used by other models by computing maximums and minimums. This type of associative model has been applied to different pattern recognition problems including face localization and gray scale image restoration. Despite of his power, MAMs have not been applied in problems that involve true-color patterns. In this paper it is described how a MAM can be applied in problems involving true-color patterns. Furthermore, a complete study of the behavior of this associative model in the restoration of true-color images is performed using a benchmark of 14 400 images altered by different type of noises.

Introduction

The concept of associative memory (AM) emerges from psychological theories of human and animals learning. These memories store information by learning correlations among different stimuli. When a stimulus is presented as a memory cue, the other is retrieval as a consequence; this means that the two stimuli have become associated each other in the memory.

An AM can be seen as a particular type of neural network designed to recall output patterns in terms of input patterns that can appear altered by some kind of noise. Several associative models have been described in the last years (refer for example [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]). Most of these AMs have several constraints that limit their applicability in complex problems. Among these constraints we could mention their capacity of storage (limited), the type of patterns (only binary, bipolar, integer or real patterns), robustness to noise (additive, subtractive, mixed, Gaussian noise, etc.).

A first attempt in formulating useful morphological neural networks was proposed by Davidson et al. [12]. Since then, only a few papers involving morphological neural networks have appeared. Refer for example to [13], [14]. In 1998, Ritter et al. [8] proposed the concept of morphological associative memory (MAM) and the concept of morphological auto-associative memory (MAAM). Basically, the authors substituted the outer product by max and min operations. One year later, the authors introduced their morphological bidirectional associative memories [15]. Their properties, compared with Hopfield Associative model are completely different. For example, they exhibit optimal absolute storage capacity and one-step convergence in the auto-associative case.

This type of associative model has been applied to the reconstruction of gray scale images [9], [16], [17], [18], [19], [20]. Despite of his power, it has not been applied to problems involving true-color patterns; neither a deep study of this associative model under true-color image patterns has been reported.

In this paper it is described how a MAM can be applied in problems involving true-color patterns. Furthermore, a complete study of the behavior of this associative model in the restoration of true-color images is performed. For this a benchmark of 14 400 images altered by different type of noises is used. In addition, the potential of the described model is tested in two real scenarios: image categorization and image restoration.

Section snippets

Basics on morphological associative memories

The basic computations occurring in the morphological network proposed by Ritter et al. are based on the algebraic lattice structure $(R, \land, \lor, +)$ where the symbols $\land$ and $\lor$ denote the binary operations of minimum and maximum, respectively.

Let $x \in R^{n}$ and $y \in R^{m}$ an input and output pattern, respectively. An association between input pattern x and output pattern y is denoted as $(x^{ξ}, y^{ξ})$ , where ξ is the corresponding association. Associative memory W is represented by a matrix whose components $w_{ij}$ can be

Behavior of $W_{XY}$ under true-color noisy patterns

In this section a study of the behavior of $W_{XY}$ under true-color noisy patterns is presented. Two types of experiments will be performed. In the first case we study the auto-associative version of $W_{XY}$ , in the second case we study the hetero-associative version.

For the case of the auto-associative version, first to all, we verified if the MAAM $W_{XY}$ was capable to recall the complete set of associations. Then we verified the behavior of $W_{XY}$ using noisy versions of the images used to train the MAAM.

Behavior of $M_{XY}$ under true-color noisy patterns

In this section a study of the behavior of $M_{XY}$ under true-color noisy patterns is presented. Two type of experiments will be performed: in the first case we study the auto-associative version of $M_{XY}$ , and then the hetero-associative version.

For the case of the auto-associative version, first to all, we verified if the MAAM $M_{XY}$ was capable of recalling the complete set of associations. Then we verified the behavior of $M_{XY}$ using noisy versions of the images used to train the MAAM. After that, we

Robustness of MAMs under other image transformations

Until now, we have analyzed the behavior of the model when images are altered with noise. However, it is also important to analyze the model with other kind of noises. In this section we analyze the accuracy of the MAAM $W_{XY}$ with some simple images transformations: increasing and /or decreasing the value of all pixels from the image.

As in previous experiments, before we train the MAAM $W_{XY}$ with 40 associations, we transform each image into an image pattern, and then we proceed to evaluate the

Image categorization using MAMs

Image categorization is not trivial when pictures are taken from real life situations. This implies that categorization must be invariant to several image transformations such as translations, rotations, scale changes, illumination changes, orientation changes, noise, and so on [21].

In this section we describe how images can be categorized using the MAM already described and the methodology introduced in [21]. Suppose that we feed a MAM with a picture and we expect that it responds with

Image restoration using MAMs

In this section we describe how the associative model already exposed can be applied to the problem of image restoration. Particularly, we will show the applicability of the model in the restoration of several (k) famous paintings.

Each painting was first digitalized to a RGB format of size 250×250 pixels. Something important to remember is that for memory reasons so larger images cannot be stored in the described model. For this reason we propose to split each painting into c sub-images (refer

Conclusions

In this paper, a complete study of the behavior of the morphological associative memory in the reconstruction of true-color images using a benchmark of 14 400 images altered by different type of noises was presented.

Due to this associative model had been only applied to binary and gray level patterns; this paper is useful to really understand the power and limitations of this model. Two types of experiments were performed. In the first case we studied the auto-associative version of MAM, and

Acknowledgments

The authors give thanks to CIC-IPN and COFAA-IPN for the economical support. Authors also thank SIP-IPN grants 20071438, 20082948, 20091421 and the European Commission and CONACYT under grant FONCICYT 93829. We also greatly appreciate the reviewers’ comments for the improvement of this paper.

Roberto A. Vazquez, was born in Mexico City, Mexico, 1981. He received his B.Sc. degree from the School of Computer Sciences, National Polytechnic Institute (ESCOM-IPN), Mexico City, Mexico, 2003. He received his M.Sc. degree from the Center for Computing Research, National Polytechnic Institute (CIC-IPN),Mexico City, Mexico, 2005. He is now pursuing the Ph.D. degree at the Center for Computing Research, National Polytechnic Institute (CIC-IPN), Mexico City, Mexico. His main research interests

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There are more references available in the full text version of this article.

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Humberto Sossa, was born in Guadalajara, Jalisco, Mexico in 1956. He received a B.Sc. Degree in Electronics from the University of Guadalajara in 1981, a M.Sc. in Electrical Engineering from CINVESTAV-IPN in 1987 and a Ph.D. in Informatics from the National Polytechnic Institute of Grenoble, France in 1992. He is full time professor at the Center for Computing Research of the National Polytechnic Institute of Mexico. His main research interests are in Pattern Recognition, Associative Memories, Image Analysis, and Robot Control using Image Analysis.

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Neurocomputing

Behavior of morphological associative memories with true-color image patterns

Abstract

Introduction

Section snippets

Basics on morphological associative memories

Behavior of WXY under true-color noisy patterns

Behavior of MXY under true-color noisy patterns

Robustness of MAMs under other image transformations

Image categorization using MAMs

Image restoration using MAMs

Conclusions

Acknowledgments

A simple neural network generating an interactive memory

Math. Biosci.

Morphological bidirectional associative memories

Neural Networks

Associative morphological memories based on variations of the kernel and dual kernel methods

Neural Network

Die Lernmatrix

Kybernetik

Correlation matrix memories

IEEE Trans. Comput.

Neural networks and physical systems with emergent collective computational abilities

Proc. Natl. Acad. Sci.

Generalizing operations of binary auto-associative morphological memories using fuzzy set theory

J. Math. Imaging Vision

Reconstruction of patterns from noisy inputs using morphological associative memories

J. Math. Imaging Vision

Transforming fundamental set of patterns to a canonical form to improve pattern recall

Morphological associative memories

IEEE Trans. Neural Networks

Behavior of $W_{XY}$ under true-color noisy patterns

Behavior of $M_{XY}$ under true-color noisy patterns