image recognition using simplified fuzzy artmap ... - Semantic Scholar
IMAGE RECOGNITION USING SIMPLIFIED FUZZY ARTMAP AUGMENTED WITH A MOMENT BASED FEATURE EXTRACTOR S. RAJASEKARAN∗ and G. A. VIJAYALAKSHMI PAI† ∗ Department of Civil Engineering of Mathematics & Computer Applications PSG College of Technology, Coimbatore 641 004, India ∗ E-mail : [email protected] † E-mail : [email protected]
† Department
The capability of Kasuba’s Simplified Fuzzy ARTMAP (SFAM) to behave as a Pattern Recognizer/Classifier of images both noisy and noise free has been investigated in this paper. This calls for augmenting the original Neuro–Fuzzy model with a modified moment-based RST invariant feature extractor. The potential of the SFAM based Pattern Recognizer to recognize patterns — monochrome and color, noisy and noise free — has been studied on two experimental problems. The first experiment which concerns monochrome images, pertains to recognition of satellite images, a problem discussed by Wang et al. The second experiment, which concerns color images, deals with the recognition of some sample test colored patterns. The results of the computer simulation have also been presented. Keywords: Simplified fuzzy ARTMAP; moments; feature extraction; pattern recognition.
1. INTRODUCTION Pattern Recognition (PR) which is a science that deals with the description or classification (recognition) of measurements has turned out to be an important component of a dominant technology such as Machine Intelligence. Various approaches to pattern recognition include statistical (or decision theoretic), syntactic (or structural) and neural approaches. 1 Of the three major approaches, neural technology is emerging quickly as a powerful means to solve PR problems. Fuzzy Logic (FL), which has turned out to be an excellent computational methodology has significantly contributed to the solution of PR problems.2 – 6 The fusion of Neural Networks and Fuzzy Systems,7 termed Neuro–Fuzzy Systems have also been applied for the solution of various PR applications. ARTMAP is a class of Neural Network architectures that perform incremental supervised learning of recognition categories and multidimensional maps in response to input vectors presented in arbitrary order.8 The first ARTMAP9 system classified input patterns represented as binary values. Carpenter et al.4 refined the system to a general one by incorporating Fuzzy ART dynamics and termed it Fuzzy ARTMAP. Fuzzy ARTMAP responds to both analog and fuzzy patterns. Kasuba6 propounded the Simplified Fuzzy ARTMAP (SFAM) system which is a vast simplification of Fuzzy ARTMAP. The network is a step ahead of Fuzzy ARTMAP in reducing the computational overhead and architectural redundancy of Fuzzy ARTMAP. 1081 International Journal of Pattern Recognition and Artificial Intelligence, Vol. 14, No. 8 (2000) 1081–1095 c World Scientific Publishing Company
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However, Kasuba’s SFAM despite its simplicity, does not display the characteristic of tolerance to pattern perturbations or noise while processing images. 9,10 The aim of the investigation is therefore to enhance the pattern recognition capability of SFAM by retaining its simple architecture but by augmenting it with a moment-based feature extractor to enable it display tolerance to pattern perturbations and noise. The feature extractor extracts features from patterns, which are “preprocessed” inputs to SFAM. Digital approximations of moment-based invariants1 have been employed. However, these approximations which are invariant to translation of patterns are not strictly invariant to rotation and scaling changes. The authors therefore, in a parallel investigation11 have mathematically modified the properties before putting it to use for SFAM. The SFAM augmented with the above-said modified feature extractor, has been investigated for its recognition of monochrome and color images. In this paper, the architecture of SFAM has been presented in Sec. 2. The moment-based feature extractor has been reviewed in Sec. 3. The recognition of monochrome and color images by SFAM has been presented in Secs. 4 and 5, respectively. Results of the computer simulation have also been presented in the sections. 2. SFAM — A REVIEW SFAM comprises of two layers viz. an input and an output layer [see Fig. 1]. The input to the network flows through the complement coder where the input string is stretched to double the size by adding its complement as well. The complement coded input then flows into the input layer. Weights (W ) from each of the output category nodes reach down to the input layer. The category layer is just an area to hold the names of the M number of categories that the network has to learn. The other mechanisms of the network architecture are primarily for network training. ρ is the vigilance parameter, which can range from 0 to 1. It controls the granularity
Fig. 1.
Architecture of SFAM network.
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of the output node encoding. Thus, high vigilance values make the output node much fussier when deciding how to encode input patterns whereas low vigilance allows much more relaxed matching criteria for the output node. The “match tracking” portion of the network lets itself adjust its vigilance during learning from some base level, in response to errors in classification during the training phase. It is through match tracking that the network adjusts its own learning parameter to decide when to sprout new output nodes or reshape its decision regions. During training, match tracking is evoked when the selected output node does not represent the same output category corresponding to the input vectors given. 2.1. Input Normalization Complement coding is used for input normalization and it represents the presence of a particular feature in the input pattern and its absence. For example, if a is the given input pattern vector of d features, the complement coded vector a represents the absence of each feature, where a is defined as a = 1 − a .
(1)
The previous equation is valid since, just as in fuzzy logic, all SFAM input values must be within the range 0 to 1. Therefore, the complement coded input vector I internal to SFAM is given by the two-dimensional vector: I = (a, a ) = (a1 , a2 , . . . ad , a1 , a2 . . . ad ) .
(2)
An interesting side-effect of the complement coding is the automatic normalization of input vectors such that |I| = |(a, a )| =
d
ai + (d −
i=1
d
ai )
(3)
i=1
where the norm || is defined as |p| =
d
pi .
(4)
i=1
2.2. Output Node Activation When SFAM is presented an input pattern whose complement coded representation is I, all output nodes become active to some degree. This output activation is denoted by Tj for the jth output node, where Wj is the corresponding weight vector given by: Tj (I) =
|I ∧ Wj | . α + |Wj |
(5)
Here, α is kept as a small value close to 0 usually about 0.0000001. The winning output node is the node with the highest activation Winner = max(Tj ) .
(6)
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If more than one Tj is maximal, the output node j with the smallest index is arbitrarily chosen to break the tie. The category associated with the winning output node is thus, the networks classification of the current input pattern. The match function given below helps determine if learning should occur. |I ∧ Wj | . (7) |I| When used in conjunction with the vigilance parameter, the match function value states whether the current input is a good enough match to a particular output node to be encoded by that output node, or instead, whether a new output node should be formed to encode the input pattern. If the match function value is greater than the vigilance parameter the network is said to be in a state of resonance. Resonance means that output node j is good enough to encode the input I, provided that output node j represents the same category as input I. A network state called “mismatch reset” occurs if the match function is less than vigilance. This state indicates that the current output node does not meet the encoding granularity represented by the vigilance parameter and therefore cannot update its weights even if the input patterns’ category is equal to the category of the winning output node. Once a winning output node j has been selected to learn a particular input pattern I, the top–down weight vector Wj from the output node is updated according to the equation. Wjnew = β(I ∧ Wjold ) + (1 − β)Wjold
(8)
where 0 < β ≤ 1. Once SFAM has been trained, the equivalent of a “feed forward” pass for an unknown pattern classification consists of passing the input pattern through the complement coder and into the input layer. The output node activation function is evaluated and the winner is the one with the highest value. The category of the input pattern is the one with which the winning output node is associated. 3. MOMENT-BASED FEATURE EXTRACTOR Moments are extracted features that are derived from raw measurements. In practical imagery, images are subject to various geometric distortions or pattern perturbations. It is therefore necessary that features that are invariant to orientations be used for purposes of recognition or classification. For 2D images, moments have been used to achieve Rotation (R), Scaling (S), and Translation (T ) invariants. The moment transformation of an image function f (x, y) is given by ∞ ∞ m pq = xp y q f (x, y)dxdy p, q = 0, 1, 2, . . . ∞ . (9) −∞
−∞
However in the case of a spatially discretized MXN image denoted by f (i, j), Eq. (9) is formulated using an approximation of double summations, mpq =
M N i=0 j=0
ip j q f (i, j) .
(10)
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The so-called central moments are given by µpq =
N M (i − ˆi)p (j − ˆj)q f (i, j)
(11)
i=0 j=0
where ˆi = m10 , ˆj = m01 . (12) m00 m00 The central moments are still sensitive to R and S transformations. The scaling invariant may be obtained by further normalizing µpq or by forming µ pq , p + q = 2, 3, . . . . ηpq = (13) p+q +1 µ00 2 From Eq. (13), constraining p, q ≤ 3, and using the tools of invariant algebra, a set of seven RST invariant features as shown in Table 1 may be derived. However, though the set of invariant moments shown in Table 1 are invariant to translation, inspite of being computed discretely, the moments cannot be expected to be strictly invariant under rotation and scaling changes. Table 1.
Moment-based RST invariant features.
φ1 = η20 + η02 φ2 = (η20 − η02 )2 + 4η11 2 φ3 = (η30 − 3η12 )2 + (3η21 − η03 )2 φ4 = (η30 + η12 )2 + (η21 + η03 )2 φ5 = (η30 − 3η12 )(η30 + η12 )[(η30 + η12 )2 − 3(η21 + η03 )2 ] + (3η21 − η03 )(η21 + η03 )[3(η30 + η12 )2 − (η21 + η03 )2 ] φ6 = (η20 − η02 )[(η30 + η12 )2 − (η21 + η03 )2 ] + 4η11 (η30 + η12 )(η21 + η03 ) φ7 = (3η21 − η03 )(η30 + η12 )[(η30 + η12 )2 − 3(η21 + η03 )2 ] − (η30 − 3η12 ) (η21 + η03 )[3(η30 + η12 )2 − (η21 + η03 )2
An investigation11 by the authors revealed that in the definition of µpq , the contribution made by a pixel had been overlooked. The modified µpq definitions have been presented in Table 2. The moment-based invariant functions are computed using the new definitions as before, from the raw measurements of each image. Those images, which are similar, are classified as belonging to the same class. In other words, images which are perturbed (rotated, scaled or translated) versions of the given nominal pattern are all classified as belonging to a class. 4. RECOGNITION OF MONOCHROME IMAGES The problem of identifying air planes, tanks and helicopters from a satellite discussed by Wang et al.,12 is the test suite problem. The experiments conducted could
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S. RAJASEKARAN & G. A. V. PAI Table 2.
Revised µpq definitions.
µ00 = M (M ass) µ10 = 0 µ01 = 0 1 f (xi , yj ) xi 2 + 12 j=1 i=1 n n 1 = f (xi , yj ) yj 2 + 12 j=1 i=1
µ20 = µ02
µ21 = µ12 = µ11 = µ30 = µ03 =
n n
n n
j=1 i=1 n n j=1 i=1 n n j=1 i=1 n n j=1 i=1 n n
f (xi , yj )(xi 2 yj ) f (xi , yj )(xi yj 2 ) f (xi , yj )(xi yj ) f (xi , yj )(xi 3 ) f (xi , yj )(yj 3 )
j=1 i=1
be categorized as under: IMAGE
TRAINING SET
TESTING SET Nominal Patterns
(Noisy)
Monochrome
Nominal (ideal)
Rotated/Scaled/Translated/
(Noise free)
Patterns
Combinations Rotated/Scaled/Translated/
(Noisy)
Combinations
Figure 2 illustrates a set of sample nominal patterns that are trained with SFAM. The results were observed for varying vigilance parameter values, 0.5 ≤ ρ < 1. The number of training epochs was kept fixed to a paltry 3. The model was tested for a set of 50 patterns absolutely noise free, but rotated or scaled or translated, or for combinations of one or more or all of these. Figure 3 illustrates a sample set of inference patterns. Table 3 shows the results of the experiments. Table 3.
Recognition of noise free monochrome images.
No. of Training Epochs
Vigilance Parameter
Training Set
Testing Set
Nature of the Testing Set
Recognition Rate
3
0.5 ≤ ρ < 1
3 Exemplars(one in each category
50 patterns
Rotated/Scaled/ Translated/ Combinations
100%
IMAGE RECOGNITION USING SIMPLIFIED FUZZY ARTMAP
Fig. 2.
Fig. 3.
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Training patterns of SFAM (monochrome, nominal).
Sample inference patterns (monochrome, noise free) correctly identified by SFAM.
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Fig. 4.
Sample noisy patterns correctly identified by SFAM.
In the next stage, for the same training set, a set of noisy patterns — nominal, rotated, scaled, displaced or a combination — was presented, to test for the recognition capability. Figure 4 illustrates a sample set of noisy patterns. The activation values of the top–down weight nodes and the correctness of the classification were observed for varying noise levels. Since the experiment pertains to monochrome (binary) images, the noise level was determined in terms of the Hamming distance. The recognition flag is set to 1 or 0 depending on whether the recognition is correct or incorrect, respectively. Figure 5 illustrates the behavior of the model for varying noise levels, when a nominal pattern was subjected to random noise. Figure 6 does the same for a scaled and translated image and Fig. 7 for a rotated, scaled and translated noisy image.
IMAGE RECOGNITION USING SIMPLIFIED FUZZY ARTMAP
Fig. 5.
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Performance of SFAM during the recognition of noisy patterns (monochrome–noisy).
Fig. 6. Performance of SFAM during the recognition of noisy patterns (monochrome, scaled and translated).
5. RECOGNITION OF COLOR IMAGES In this phase, a similar set of experiments was repeated with respect to colored images. The problem pertained to the recognition of sample colored test patterns. The experiments performed could be categorized as under:
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Fig. 7. Performance of SFAM during the recognition of noisy patterns (monochrome, rotated, scaled and translated).
IMAGE
TRAINING SET
TESTING SET Nominal Patterns
(Noisy)
Color
Nominal (ideal)
Rotated/Scaled/Translated/
( Noise free)
Patterns
Combinations Rotated/Scaled/Translated/
(Noisy)
Combinations
Figure 8 illustrates a sample set of nominal patterns and Fig. 9 a set of noise free patterns but subject to perturbations — rotation, scaling, translation and combinations of the three. Table 4 shows the results of the experiment. Table 4.
Recognition of noise free color images.
No. of Training Epochs
Vigilance Parameter
Training Set
Testing Set
Nature of the Testing Set
Recognition Rate
3
0.5 ≤ ρ < 1
4 Exemplars (one in each category)
93 patterns
Rotated/Scaled Translated Combinations
100%
In the case of noisy patterns, a sample of which is illustrated in Fig. 10, the performance of the model for varying noise levels was observed. As before, the activation value of the top–down weight vectors and the recognition capability of the model for varying noise levels was under observation. Figure 11 illustrates the behavior when nominal patterns were subjected to random noise. Figures 12 and 13 illustrate the behavior for scaled and translated noisy image, and rotated, scaled and translated noisy image, respectively.
IMAGE RECOGNITION USING SIMPLIFIED FUZZY ARTMAP
Fig. 8.
Fig. 9.
Sample training patterns for SFAM (color, nominal).
Sample inference patterns (color, noise free and perturbed).
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Fig. 10.
Fig. 11.
Sample noisy patterns (color) correctly identified by SFAM.
Performance of SFAM during the recognition of noisy patterns (color, nominal).
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Fig. 12. Performance of SFAM during the recognition of noisy patterns (color, translated and scaled).
Fig. 13. Performance of SFAM during the recognition of noisy patterns (color, rotated, translated and scaled).
6. CONCLUSION In this paper, the pattern recognition capability of SFAM has been discussed. The architecture augmented with a moment-based feature extractor exhibits an excellent capability to recognize patterns by working on the RST invariant feature vectors of the patterns rather than the patterns themselves.
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The augmented architecture can handle both symmetric and asymmetric patterns. In the case of asymmetric patterns the RST invariant functions φ1 − φ7 turn out to be the same for all perturbations of a given pattern. Hence SFAM has no difficulty in recognizing perturbed patterns since it only calls for associating the same feature vectors with the top–down weight vectors which results in the invocation of the same category node. This results in the correct identification of the perturbed pattern. However, in the case of symmetric patterns it is essential that only distinct portions of the images be trained. This is so since in the case of doubly symmetric or in general, multisymmetric images, their RST invariant feature vectors φ2 − φ7 acquire values which are very close to 0, and φ1 tends to 1. This consequently results in feature vectors which are almost similar leading to a misclassification of patterns. Hence in the case of multisymmetric patterns, it is sufficient to consider 1 2n th portion of the image. REFERENCES 1. J. C. Bezdek and S. K. Pal (eds.), Fuzzy Models for Pattern Recognition, IEEE Press, Piscataway, NJ, 1992. 2. G. A. Carpenter and S. Grossberg, Pattern Recognition by Self Organizing Neural Networks, MIT Press, Cambridge, MA, 1991. 3. G. A. Carpenter, S. Grossberg, N. Markuzon, J. H. Reynolds and D. B. Rosen, “Fuzzy ARTMAP: a neural network architecture for incremental supervised learning of analog multidimensional maps,” IEEE Trans. Neural Networks 3, 5 (1992) 698–713. 4. G. A. Carpenter, S. Grossberg and J. H. Reynolds, “ARTMAP: supervised real time learning and classification of non stationary data by a self organizing neural network,” Neural Networks 4 (1991) 565–588. 5. A. Kandel, Fuzzy Techniques in Pattern Recognition, Wiley, NY, 1982. 6. T. Kasuba, “Simplified fuzzy ARTMAP,” AI Expert, November (1993) 18–25. 7. B. Kosko, Neural Networks and Fuzzy Systems: A Dynamical Systems Approach to Machine Intelligence, Prentice Hall, Englewood Cliffs, NJ, 1992. 8. S. K. Pal and D. K. D. Majumdar, Fuzzy Mathematical Approach in Pattern Recognition, Wiley Eastern Ltd., New Delhi, India, 1986. 9. S. Rajasekaran and G. A. V. Pai, “Application of simplified fuzzy ARTMAP to structural engineering problems,” All India Seminar on Application of NN in Science, Engineering and Management, Bhubaneswar, June 1997. 10. S. Rajasekaran, G. A. V. Pai and J. P. George, “Simplified fuzzy ARTMAP for determination of deflection in slabs of different geometry,” Nat. Conf. NN and Fuzzy Systems, Chennai, 1997, pp. 107–116. 11. S. Rajasekaran and V. Pai, “Simplified fuzzy ARTMAP as a pattern recognizer,” J. Comput. Civil Engin. ASCE 14, 2 (2000) 92–99. 12. Y. F. Wang, J. B. Cruz Jr. and J. H. Mulligan Jr., “Two coding strategies for bidirectional associative memory,” IEEE Trans. Neural Networks 1, 1 (1990) 81–92.
IMAGE RECOGNITION USING SIMPLIFIED FUZZY ARTMAP S. Rajasekaran is Professor Emeritus of AICTE at the Department of Civil Engineering, PSG College of Technology, Coimbatore, India. He obtained his Ph.D. in civil engineering from the University of Alberta, Edmonton, Canada in 1971 and D.Sc. (civil engineering) from the Bharathiar University, India in October 1999. He was a visiting Professor at the University of Alberta, Canada, University of Sydney, Australia, and the Alexander von Humboldt Guest Professor at the University of Stuttgart, Germany. He is a recipient of the ISTE National Award for his outstanding research work in engineering and technology in the year 1991, Tamilnadu Scientist Award by TNSCST, Government of Tamilnadu in 1966 and NAGADI award for his book “Finite Element Analysis in Engineering Design” by the Association of Consulting Civil Engineers(ACCE) in 1996 and the Vocational Excellence award by the Coimbatore West Rotary in December 1999 and ISTE Anna University National Award for the outstanding Academic for the year 1999–2000. Rajasekaran is the principal investigator of many projects sponsored by AICTE, ARDB, BARC, DST, ISRO and MHRD. He is a Fellow of the Institution of Engineers, The Institute of Valuers and a Member of American Society of Civil Engineers and Computer Society of India and Indian Society of Technical Education. He has published more than 230 research papers in national and international journals and conferences, besides fifteen books. He is listed in the American Biographical Research Institute and the eminent personalities of India. His specific interests include finite element analysis, boundary integral element method, nonlinear analysis, neural networks, genetic algorithms and fuzzy systems.
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G. A. Vijayalakshmi Pai is a Lecturer (Senior Grade) in computer applications in the Department of Computer Applications, PSG College of Technology, Coimbatore. She completed her Masters Degree and Master of Philosophy in applied mathematics, specializing in computer science, in 1984 and 1989, respectively, both degrees awarded by the Faculty of Engineering, Bharathiar University. She obtained her Ph.D. in computer science from the Department of Civil Engineering, PSG College of Technology in 1999. Her research interests include neural networks, fuzzy logic, genetic algorithms and logic for artificial intelligence.
† Department
The capability of Kasuba’s Simplified Fuzzy ARTMAP (SFAM) to behave as a Pattern Recognizer/Classifier of images both noisy and noise free has been investigated in this paper. This calls for augmenting the original Neuro–Fuzzy model with a modified moment-based RST invariant feature extractor. The potential of the SFAM based Pattern Recognizer to recognize patterns — monochrome and color, noisy and noise free — has been studied on two experimental problems. The first experiment which concerns monochrome images, pertains to recognition of satellite images, a problem discussed by Wang et al. The second experiment, which concerns color images, deals with the recognition of some sample test colored patterns. The results of the computer simulation have also been presented. Keywords: Simplified fuzzy ARTMAP; moments; feature extraction; pattern recognition.
1. INTRODUCTION Pattern Recognition (PR) which is a science that deals with the description or classification (recognition) of measurements has turned out to be an important component of a dominant technology such as Machine Intelligence. Various approaches to pattern recognition include statistical (or decision theoretic), syntactic (or structural) and neural approaches. 1 Of the three major approaches, neural technology is emerging quickly as a powerful means to solve PR problems. Fuzzy Logic (FL), which has turned out to be an excellent computational methodology has significantly contributed to the solution of PR problems.2 – 6 The fusion of Neural Networks and Fuzzy Systems,7 termed Neuro–Fuzzy Systems have also been applied for the solution of various PR applications. ARTMAP is a class of Neural Network architectures that perform incremental supervised learning of recognition categories and multidimensional maps in response to input vectors presented in arbitrary order.8 The first ARTMAP9 system classified input patterns represented as binary values. Carpenter et al.4 refined the system to a general one by incorporating Fuzzy ART dynamics and termed it Fuzzy ARTMAP. Fuzzy ARTMAP responds to both analog and fuzzy patterns. Kasuba6 propounded the Simplified Fuzzy ARTMAP (SFAM) system which is a vast simplification of Fuzzy ARTMAP. The network is a step ahead of Fuzzy ARTMAP in reducing the computational overhead and architectural redundancy of Fuzzy ARTMAP. 1081 International Journal of Pattern Recognition and Artificial Intelligence, Vol. 14, No. 8 (2000) 1081–1095 c World Scientific Publishing Company
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However, Kasuba’s SFAM despite its simplicity, does not display the characteristic of tolerance to pattern perturbations or noise while processing images. 9,10 The aim of the investigation is therefore to enhance the pattern recognition capability of SFAM by retaining its simple architecture but by augmenting it with a moment-based feature extractor to enable it display tolerance to pattern perturbations and noise. The feature extractor extracts features from patterns, which are “preprocessed” inputs to SFAM. Digital approximations of moment-based invariants1 have been employed. However, these approximations which are invariant to translation of patterns are not strictly invariant to rotation and scaling changes. The authors therefore, in a parallel investigation11 have mathematically modified the properties before putting it to use for SFAM. The SFAM augmented with the above-said modified feature extractor, has been investigated for its recognition of monochrome and color images. In this paper, the architecture of SFAM has been presented in Sec. 2. The moment-based feature extractor has been reviewed in Sec. 3. The recognition of monochrome and color images by SFAM has been presented in Secs. 4 and 5, respectively. Results of the computer simulation have also been presented in the sections. 2. SFAM — A REVIEW SFAM comprises of two layers viz. an input and an output layer [see Fig. 1]. The input to the network flows through the complement coder where the input string is stretched to double the size by adding its complement as well. The complement coded input then flows into the input layer. Weights (W ) from each of the output category nodes reach down to the input layer. The category layer is just an area to hold the names of the M number of categories that the network has to learn. The other mechanisms of the network architecture are primarily for network training. ρ is the vigilance parameter, which can range from 0 to 1. It controls the granularity
Fig. 1.
Architecture of SFAM network.
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of the output node encoding. Thus, high vigilance values make the output node much fussier when deciding how to encode input patterns whereas low vigilance allows much more relaxed matching criteria for the output node. The “match tracking” portion of the network lets itself adjust its vigilance during learning from some base level, in response to errors in classification during the training phase. It is through match tracking that the network adjusts its own learning parameter to decide when to sprout new output nodes or reshape its decision regions. During training, match tracking is evoked when the selected output node does not represent the same output category corresponding to the input vectors given. 2.1. Input Normalization Complement coding is used for input normalization and it represents the presence of a particular feature in the input pattern and its absence. For example, if a is the given input pattern vector of d features, the complement coded vector a represents the absence of each feature, where a is defined as a = 1 − a .
(1)
The previous equation is valid since, just as in fuzzy logic, all SFAM input values must be within the range 0 to 1. Therefore, the complement coded input vector I internal to SFAM is given by the two-dimensional vector: I = (a, a ) = (a1 , a2 , . . . ad , a1 , a2 . . . ad ) .
(2)
An interesting side-effect of the complement coding is the automatic normalization of input vectors such that |I| = |(a, a )| =
d
ai + (d −
i=1
d
ai )
(3)
i=1
where the norm || is defined as |p| =
d
pi .
(4)
i=1
2.2. Output Node Activation When SFAM is presented an input pattern whose complement coded representation is I, all output nodes become active to some degree. This output activation is denoted by Tj for the jth output node, where Wj is the corresponding weight vector given by: Tj (I) =
|I ∧ Wj | . α + |Wj |
(5)
Here, α is kept as a small value close to 0 usually about 0.0000001. The winning output node is the node with the highest activation Winner = max(Tj ) .
(6)
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If more than one Tj is maximal, the output node j with the smallest index is arbitrarily chosen to break the tie. The category associated with the winning output node is thus, the networks classification of the current input pattern. The match function given below helps determine if learning should occur. |I ∧ Wj | . (7) |I| When used in conjunction with the vigilance parameter, the match function value states whether the current input is a good enough match to a particular output node to be encoded by that output node, or instead, whether a new output node should be formed to encode the input pattern. If the match function value is greater than the vigilance parameter the network is said to be in a state of resonance. Resonance means that output node j is good enough to encode the input I, provided that output node j represents the same category as input I. A network state called “mismatch reset” occurs if the match function is less than vigilance. This state indicates that the current output node does not meet the encoding granularity represented by the vigilance parameter and therefore cannot update its weights even if the input patterns’ category is equal to the category of the winning output node. Once a winning output node j has been selected to learn a particular input pattern I, the top–down weight vector Wj from the output node is updated according to the equation. Wjnew = β(I ∧ Wjold ) + (1 − β)Wjold
(8)
where 0 < β ≤ 1. Once SFAM has been trained, the equivalent of a “feed forward” pass for an unknown pattern classification consists of passing the input pattern through the complement coder and into the input layer. The output node activation function is evaluated and the winner is the one with the highest value. The category of the input pattern is the one with which the winning output node is associated. 3. MOMENT-BASED FEATURE EXTRACTOR Moments are extracted features that are derived from raw measurements. In practical imagery, images are subject to various geometric distortions or pattern perturbations. It is therefore necessary that features that are invariant to orientations be used for purposes of recognition or classification. For 2D images, moments have been used to achieve Rotation (R), Scaling (S), and Translation (T ) invariants. The moment transformation of an image function f (x, y) is given by ∞ ∞ m pq = xp y q f (x, y)dxdy p, q = 0, 1, 2, . . . ∞ . (9) −∞
−∞
However in the case of a spatially discretized MXN image denoted by f (i, j), Eq. (9) is formulated using an approximation of double summations, mpq =
M N i=0 j=0
ip j q f (i, j) .
(10)
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The so-called central moments are given by µpq =
N M (i − ˆi)p (j − ˆj)q f (i, j)
(11)
i=0 j=0
where ˆi = m10 , ˆj = m01 . (12) m00 m00 The central moments are still sensitive to R and S transformations. The scaling invariant may be obtained by further normalizing µpq or by forming µ pq , p + q = 2, 3, . . . . ηpq = (13) p+q +1 µ00 2 From Eq. (13), constraining p, q ≤ 3, and using the tools of invariant algebra, a set of seven RST invariant features as shown in Table 1 may be derived. However, though the set of invariant moments shown in Table 1 are invariant to translation, inspite of being computed discretely, the moments cannot be expected to be strictly invariant under rotation and scaling changes. Table 1.
Moment-based RST invariant features.
φ1 = η20 + η02 φ2 = (η20 − η02 )2 + 4η11 2 φ3 = (η30 − 3η12 )2 + (3η21 − η03 )2 φ4 = (η30 + η12 )2 + (η21 + η03 )2 φ5 = (η30 − 3η12 )(η30 + η12 )[(η30 + η12 )2 − 3(η21 + η03 )2 ] + (3η21 − η03 )(η21 + η03 )[3(η30 + η12 )2 − (η21 + η03 )2 ] φ6 = (η20 − η02 )[(η30 + η12 )2 − (η21 + η03 )2 ] + 4η11 (η30 + η12 )(η21 + η03 ) φ7 = (3η21 − η03 )(η30 + η12 )[(η30 + η12 )2 − 3(η21 + η03 )2 ] − (η30 − 3η12 ) (η21 + η03 )[3(η30 + η12 )2 − (η21 + η03 )2
An investigation11 by the authors revealed that in the definition of µpq , the contribution made by a pixel had been overlooked. The modified µpq definitions have been presented in Table 2. The moment-based invariant functions are computed using the new definitions as before, from the raw measurements of each image. Those images, which are similar, are classified as belonging to the same class. In other words, images which are perturbed (rotated, scaled or translated) versions of the given nominal pattern are all classified as belonging to a class. 4. RECOGNITION OF MONOCHROME IMAGES The problem of identifying air planes, tanks and helicopters from a satellite discussed by Wang et al.,12 is the test suite problem. The experiments conducted could
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Revised µpq definitions.
µ00 = M (M ass) µ10 = 0 µ01 = 0 1 f (xi , yj ) xi 2 + 12 j=1 i=1 n n 1 = f (xi , yj ) yj 2 + 12 j=1 i=1
µ20 = µ02
µ21 = µ12 = µ11 = µ30 = µ03 =
n n
n n
j=1 i=1 n n j=1 i=1 n n j=1 i=1 n n j=1 i=1 n n
f (xi , yj )(xi 2 yj ) f (xi , yj )(xi yj 2 ) f (xi , yj )(xi yj ) f (xi , yj )(xi 3 ) f (xi , yj )(yj 3 )
j=1 i=1
be categorized as under: IMAGE
TRAINING SET
TESTING SET Nominal Patterns
(Noisy)
Monochrome
Nominal (ideal)
Rotated/Scaled/Translated/
(Noise free)
Patterns
Combinations Rotated/Scaled/Translated/
(Noisy)
Combinations
Figure 2 illustrates a set of sample nominal patterns that are trained with SFAM. The results were observed for varying vigilance parameter values, 0.5 ≤ ρ < 1. The number of training epochs was kept fixed to a paltry 3. The model was tested for a set of 50 patterns absolutely noise free, but rotated or scaled or translated, or for combinations of one or more or all of these. Figure 3 illustrates a sample set of inference patterns. Table 3 shows the results of the experiments. Table 3.
Recognition of noise free monochrome images.
No. of Training Epochs
Vigilance Parameter
Training Set
Testing Set
Nature of the Testing Set
Recognition Rate
3
0.5 ≤ ρ < 1
3 Exemplars(one in each category
50 patterns
Rotated/Scaled/ Translated/ Combinations
100%
IMAGE RECOGNITION USING SIMPLIFIED FUZZY ARTMAP
Fig. 2.
Fig. 3.
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Training patterns of SFAM (monochrome, nominal).
Sample inference patterns (monochrome, noise free) correctly identified by SFAM.
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Fig. 4.
Sample noisy patterns correctly identified by SFAM.
In the next stage, for the same training set, a set of noisy patterns — nominal, rotated, scaled, displaced or a combination — was presented, to test for the recognition capability. Figure 4 illustrates a sample set of noisy patterns. The activation values of the top–down weight nodes and the correctness of the classification were observed for varying noise levels. Since the experiment pertains to monochrome (binary) images, the noise level was determined in terms of the Hamming distance. The recognition flag is set to 1 or 0 depending on whether the recognition is correct or incorrect, respectively. Figure 5 illustrates the behavior of the model for varying noise levels, when a nominal pattern was subjected to random noise. Figure 6 does the same for a scaled and translated image and Fig. 7 for a rotated, scaled and translated noisy image.
IMAGE RECOGNITION USING SIMPLIFIED FUZZY ARTMAP
Fig. 5.
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Performance of SFAM during the recognition of noisy patterns (monochrome–noisy).
Fig. 6. Performance of SFAM during the recognition of noisy patterns (monochrome, scaled and translated).
5. RECOGNITION OF COLOR IMAGES In this phase, a similar set of experiments was repeated with respect to colored images. The problem pertained to the recognition of sample colored test patterns. The experiments performed could be categorized as under:
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Fig. 7. Performance of SFAM during the recognition of noisy patterns (monochrome, rotated, scaled and translated).
IMAGE
TRAINING SET
TESTING SET Nominal Patterns
(Noisy)
Color
Nominal (ideal)
Rotated/Scaled/Translated/
( Noise free)
Patterns
Combinations Rotated/Scaled/Translated/
(Noisy)
Combinations
Figure 8 illustrates a sample set of nominal patterns and Fig. 9 a set of noise free patterns but subject to perturbations — rotation, scaling, translation and combinations of the three. Table 4 shows the results of the experiment. Table 4.
Recognition of noise free color images.
No. of Training Epochs
Vigilance Parameter
Training Set
Testing Set
Nature of the Testing Set
Recognition Rate
3
0.5 ≤ ρ < 1
4 Exemplars (one in each category)
93 patterns
Rotated/Scaled Translated Combinations
100%
In the case of noisy patterns, a sample of which is illustrated in Fig. 10, the performance of the model for varying noise levels was observed. As before, the activation value of the top–down weight vectors and the recognition capability of the model for varying noise levels was under observation. Figure 11 illustrates the behavior when nominal patterns were subjected to random noise. Figures 12 and 13 illustrate the behavior for scaled and translated noisy image, and rotated, scaled and translated noisy image, respectively.
IMAGE RECOGNITION USING SIMPLIFIED FUZZY ARTMAP
Fig. 8.
Fig. 9.
Sample training patterns for SFAM (color, nominal).
Sample inference patterns (color, noise free and perturbed).
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Fig. 10.
Fig. 11.
Sample noisy patterns (color) correctly identified by SFAM.
Performance of SFAM during the recognition of noisy patterns (color, nominal).
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Fig. 12. Performance of SFAM during the recognition of noisy patterns (color, translated and scaled).
Fig. 13. Performance of SFAM during the recognition of noisy patterns (color, rotated, translated and scaled).
6. CONCLUSION In this paper, the pattern recognition capability of SFAM has been discussed. The architecture augmented with a moment-based feature extractor exhibits an excellent capability to recognize patterns by working on the RST invariant feature vectors of the patterns rather than the patterns themselves.
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The augmented architecture can handle both symmetric and asymmetric patterns. In the case of asymmetric patterns the RST invariant functions φ1 − φ7 turn out to be the same for all perturbations of a given pattern. Hence SFAM has no difficulty in recognizing perturbed patterns since it only calls for associating the same feature vectors with the top–down weight vectors which results in the invocation of the same category node. This results in the correct identification of the perturbed pattern. However, in the case of symmetric patterns it is essential that only distinct portions of the images be trained. This is so since in the case of doubly symmetric or in general, multisymmetric images, their RST invariant feature vectors φ2 − φ7 acquire values which are very close to 0, and φ1 tends to 1. This consequently results in feature vectors which are almost similar leading to a misclassification of patterns. Hence in the case of multisymmetric patterns, it is sufficient to consider 1 2n th portion of the image. REFERENCES 1. J. C. Bezdek and S. K. Pal (eds.), Fuzzy Models for Pattern Recognition, IEEE Press, Piscataway, NJ, 1992. 2. G. A. Carpenter and S. Grossberg, Pattern Recognition by Self Organizing Neural Networks, MIT Press, Cambridge, MA, 1991. 3. G. A. Carpenter, S. Grossberg, N. Markuzon, J. H. Reynolds and D. B. Rosen, “Fuzzy ARTMAP: a neural network architecture for incremental supervised learning of analog multidimensional maps,” IEEE Trans. Neural Networks 3, 5 (1992) 698–713. 4. G. A. Carpenter, S. Grossberg and J. H. Reynolds, “ARTMAP: supervised real time learning and classification of non stationary data by a self organizing neural network,” Neural Networks 4 (1991) 565–588. 5. A. Kandel, Fuzzy Techniques in Pattern Recognition, Wiley, NY, 1982. 6. T. Kasuba, “Simplified fuzzy ARTMAP,” AI Expert, November (1993) 18–25. 7. B. Kosko, Neural Networks and Fuzzy Systems: A Dynamical Systems Approach to Machine Intelligence, Prentice Hall, Englewood Cliffs, NJ, 1992. 8. S. K. Pal and D. K. D. Majumdar, Fuzzy Mathematical Approach in Pattern Recognition, Wiley Eastern Ltd., New Delhi, India, 1986. 9. S. Rajasekaran and G. A. V. Pai, “Application of simplified fuzzy ARTMAP to structural engineering problems,” All India Seminar on Application of NN in Science, Engineering and Management, Bhubaneswar, June 1997. 10. S. Rajasekaran, G. A. V. Pai and J. P. George, “Simplified fuzzy ARTMAP for determination of deflection in slabs of different geometry,” Nat. Conf. NN and Fuzzy Systems, Chennai, 1997, pp. 107–116. 11. S. Rajasekaran and V. Pai, “Simplified fuzzy ARTMAP as a pattern recognizer,” J. Comput. Civil Engin. ASCE 14, 2 (2000) 92–99. 12. Y. F. Wang, J. B. Cruz Jr. and J. H. Mulligan Jr., “Two coding strategies for bidirectional associative memory,” IEEE Trans. Neural Networks 1, 1 (1990) 81–92.
IMAGE RECOGNITION USING SIMPLIFIED FUZZY ARTMAP S. Rajasekaran is Professor Emeritus of AICTE at the Department of Civil Engineering, PSG College of Technology, Coimbatore, India. He obtained his Ph.D. in civil engineering from the University of Alberta, Edmonton, Canada in 1971 and D.Sc. (civil engineering) from the Bharathiar University, India in October 1999. He was a visiting Professor at the University of Alberta, Canada, University of Sydney, Australia, and the Alexander von Humboldt Guest Professor at the University of Stuttgart, Germany. He is a recipient of the ISTE National Award for his outstanding research work in engineering and technology in the year 1991, Tamilnadu Scientist Award by TNSCST, Government of Tamilnadu in 1966 and NAGADI award for his book “Finite Element Analysis in Engineering Design” by the Association of Consulting Civil Engineers(ACCE) in 1996 and the Vocational Excellence award by the Coimbatore West Rotary in December 1999 and ISTE Anna University National Award for the outstanding Academic for the year 1999–2000. Rajasekaran is the principal investigator of many projects sponsored by AICTE, ARDB, BARC, DST, ISRO and MHRD. He is a Fellow of the Institution of Engineers, The Institute of Valuers and a Member of American Society of Civil Engineers and Computer Society of India and Indian Society of Technical Education. He has published more than 230 research papers in national and international journals and conferences, besides fifteen books. He is listed in the American Biographical Research Institute and the eminent personalities of India. His specific interests include finite element analysis, boundary integral element method, nonlinear analysis, neural networks, genetic algorithms and fuzzy systems.
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G. A. Vijayalakshmi Pai is a Lecturer (Senior Grade) in computer applications in the Department of Computer Applications, PSG College of Technology, Coimbatore. She completed her Masters Degree and Master of Philosophy in applied mathematics, specializing in computer science, in 1984 and 1989, respectively, both degrees awarded by the Faculty of Engineering, Bharathiar University. She obtained her Ph.D. in computer science from the Department of Civil Engineering, PSG College of Technology in 1999. Her research interests include neural networks, fuzzy logic, genetic algorithms and logic for artificial intelligence.
