Recibido el 09 de Diciembre 2003

En forma revisada el 28 de Febrero de 2005

1. Introduction

Humans use color, shape and texture to understand and recollect the contents of a pattern. Therefore, it is natural to use features based on these attributes for pattern recognition [8, 9, 16, 17]. The effectiveness of using simple color features for pattern recognition is demonstrated in [15]. Colombo et al. described a system for pictorial content representation and recognition based on color distribution features [8] in which the distribution of chromatic content in a pattern is described through a set of color histograms and a pattern matching strategy using this set. Mojsilovic et al. recently determined the basic categories (vocabulary) used by humans in judging similarity of color patterns, their relative importance and relationsh ips, as well as the hierarchy of rules (grammar) [17].

In this paper a color pattern recognition approach composed by a learning algorithm and a retrieval procedure for the multiple classes RNN is proposed. We shall use each class to model a color. We present a backpropagation type learning algorithm for the recurrent multiple classes RNN model using gradient descent minimization of a quadratic error function when a set of input-output pairs is presented to the network. Our model is defined for nC parameters for the whole network, where C is the number of primary colors, n is the number of pixels of the image, and each neuron is used to obtain the color value of each pixel in the bit map plane. The primary colors create different colors according to the RGB model. Thus, our learning algorithm requires the solution of a system of nC non-linear equations each time the n-neurons network learns a new input-output pair (n-pixels image with C primary colors). In addition, we propose a progressive retrieval process with adaptive threshold value.

The Random Neural Network (RNN) was proposed by Gelenbe in 1989 [11, 12, 13]. This model does not use a dynamic equation, but uses a scheme of interaction among neurons. It calculates the probability of activation of the neurons in the network. Signals in this model take the form of impulses that mimic what is presently known as inter-neural signals in biophysical neural networks. The RNN has been used to solve optimization [1, 2, 4] and pattern recognition problems [3, 5, 7]. Gelenbe considered a learning algorithm for recurrent random neural network model [14], and we have proposed modifications of this algorithm for combinatorial optimization problems [4] and an evolutionary learning for combinatorial optimization and recognition problems [1, 5]. Fourneau et al. have proposed an extension of the RNN, called multiple classes random neural network model [10].

This work is organized as follows, in section 2 we present the multiple classes RNN. Section 3 presents our recognition algorithm (learning and retrieval processes) for multiple classes RNN. In section 4, we present some applications. Remarks concerning future work and conclusions are provided in section 5.

2. The Multiple Classes Random Network Model

The neural network is composed of n neurons and receives exogenous positive (excitatory) and negative (inhibitory) signals as well as endogenous signals exchanged by the neurons. As in the classical model [11, 12, 13], neurons send excitatory and inhibitory signals when they fire, to other neurons in the network or to outside world. In this model, positive signals may belong to several classes and the potential at a neuron is represented by the vector K_i=(K_i1,..., K_iC), where K_ic is the value of the “class c potential” of neuron i, or its “excitation level in terms of class c signals”; negative signals only belong to a single class. The total potential of neuron i is K_i= ∑_c=1^C K_ic. The arrival of an excitatory signal of some class increases the corresponding potential of a neuron by 1, while an inhibitory signal’s arrival decreases it by 1. That is, when a positive signal of class c arrives at a neuron, it merely increases K_ic by 1, while when a negative signal arrives at it and if K_i>0, the potential is reduced by 1, with the class of the potential to be reduced chosen randomly with probability K_ic/K_i for any c=1,..., C. A negative signal arriving at a neuron whose potential is zero has no effect on its potential.

∑ _(j,φ)p⁺(i, c; j, φ) + ∑ _jp^-(i, c; j) + d(i, c) = 1     for "i =1,n and c= 1,C 

Let K(t) be the vector representing the state of the neural network at time t and K =(K₁,..., K_n) be a particular value of the vector. We shall denote by p(K, t) = Pr[K(t)=K] the probability distribution of the state. The main property of this model is the excitation probability of the “class φ potential” of neuron j, q(j,φ), which 0 < q(j,φ) < 1 satisfies the non-linear equation [10]:

q(j,φ) = λ⁺ (j,φ) / (r(j,φ)+λ^-(j))      (1)

where,  

λ⁺ (j,φ) = ∑ _(i,c) q(i,c)r(i, c)p⁺(i, c; j, φ) + Λ(j,φ)

λ^-(j) = ∑ _(i,c) q(i,c)r(i, c)p^-(i, c; j) + λ(j)

Thus, p(K, t), the stationary probability distribution of network state, satisfies

p(K, t) = ∏_i=iⁿ ∏ _c=i^C [1-q(i,c)] q(i,c)^kic     (2) 

The synaptic weights for positive (w⁺(i, c; j, φ)) and negative (w^-(i, c; j)) signals are defined as:

w^-(i, c; j) = r(i, c)p^-(i, c; j)

and, if d(i, c)=0, the fire rate r(i, c) will be

r(i, c) = [ ∑_(j,φ) w⁺(i, c; j, φ) + ∑_j w^-(i, c; j)]       (3)

3. Color pattern recognition algorithm on the multiple classes random neural network model

We now show how the multiple classes RNN can be used to solve the Color Pattern Recognition problem, based on an associative memory technique [3, 5]. In our approach, a “signal class” represents obviously each color. To design such a memory, we have used a single-layer RNN of n fully interconnected neurons. For every neuron i the probability that emitting signals depart from the network is d(i, c)=0. We suppose a pattern composed by n pixels (m, k) in the plane (for m= 1,..., J and k=1,..., K), each of which has associated a neuron N(i) (for i=1,..., n). The state of neuron N(i) can be interpreted as the color intensity value of the corresponding pixel. On the other hand, we suppose three classes to represent the primary colors (red, green, and blue) according to the RGB model. This model allows to create different colors with the combination of different intensities of the primary colors. For example, to represent a pixel with red color the neuron value is (1, 0, 0), the black color is (1, 1, 1), the pink color is (0.5, 0, 0), etc. We suppose possible values of 0, 0.5 and 1 for each class on every neuron. In this way, we can represent geometric Figures with different combinations of colors. We have used this model because it agrees better with human chromatic perception [8], but the proposed approach can use another model like this one to represent the colors of a given pattern. The parameters of the neural network will be chosen as follows:

a) p⁺(j, φ ; i, c) = p⁺( i, c; j, φ) p^-( i, c; j) = p^-( j, c; i) for any i,j=1,..., n and c,φ=1,..., C.

b) Λ(i, c)=Lic and λ(i)=0, where Lic is a constant for the class c of the neuron i.

The values in (a) were chosen because the neural network has a symmetric relationship between the neurons to guarantee the associative memory behavior of our approach. On the other hand, the values in (b) were chosen since an exogenous signal is sufficient to guarantee the network stability, and the value of Lic must be chosen accordingly (Equation 2).

3.1. Learning Algorithm

X = {X₁,..., X_m}

where, X_k= {X_k(1,1),..., X_k (n, C)}, and X_k(i, c) is the c^th class on the neuron i for the k^th pair

X_k(i, c) = {Λ_k(i, c), λ_k(i)}

and the successive desired outputs are

Y = {Y₁,..., Y_m}

where, Y_k = {Y_k(1,1),..., Y_k(n, C)}, and Y_k(1,1) = {0, 0.5, 1}

The values Λ_k(i, c) and λ_k(i) provide the network stability. Particularly, in our model Λ_k(i, c) and λ_k(i) are initialized as have been defined previously. Typically, arrival rates of exogenous signals are chosen as follows:

Y_k(i, c)>0 => X_k(i, c) = (Λ_k(i, c), λ_k(i)) = (Lic, 0)

Y_ik(i, c)=0 => (Λ_k(i, c), λ_k(i)) = (0, 0)

E_k = 1/2 ∑ⁿ_i=1 ∑^C_c=1 [q_k(i,c)- Y_k(i,c)]²    (4)

is minimized. The rule to update the weights may be written as:

w_k⁺(u,p; v,z) = w_k-1⁺(u,p; v,c)- µ∑ⁿ_i=1 ∑^C_c=1 (q_k(i,c)- y_k(i,c))[ δq(i,c) / δw⁺(u,p; v,z)]_k

w_k^-(u,p; v) = w_k-1^-(u,p; v)- µ∑ⁿ_i=1 ∑^C_c=1 (q_k(i,c)- y_k(i,c))[ δq(i,c) / δw^-(u,p; v)]_k    (5)

where, µ > 0 is the learning rate (some constant).

q_k(i) is calculated using

X_k, w⁺_k(u, p; v, z) = w⁺_k-1 (u, p; v, z)

and w^-_k(u, p; v) = w^-_k-1(u, p; v) in (1)

[ δq(i,c) / δw⁺(u,p; v,z)]_k and [ δq(i,c) / δw^-(u,p; v)]_k are evaluated using the values:

The complete learning algorithm for the network is:

- Initiate the matrices W₀⁺ and W₀^- in some appropriate manner. Choose a value of µ in (2).

- For each successive value of m:

• Select the input-output pair (X_k, Y_k)

• Repeat

   - Solve equation (1) with these values

   - Using (5) and the previous results update the matrices W_k⁺ and W_k^-

Until the change in the new values of the weights is smaller than some predetermined valued.

For more details about this learning algorithm, see [6].

Once the learning phase is completed, the network must perform as well as possible the completion of noisy versions of the training vectors. In this case, we propose a progressive retrieval process with adaptive threshold value. Let X’= {X’(1, 1),..., X’(n, C)} be any input vector in which each X’(i, c) (i=1,..., n and c=1,..., C), may take values 0, 0.5 or 1. In order to determine the corresponding output vector Y= {Y(1,1),..., Y(n, C)} we first compute the vector of probabilities Q=(q(1, 1),..., q(n, C)). We consider the q(i, c) values such that 1-T<q(i, c)<T/2 or 1-T/2<q(i, c)<T, with for instance T=0.8, belong to the uncertainty interval Z. When the network stabilizes to an attractor state, the number NB_Z of neurons whose q(i, c)ÎZ is equal to 0. Hence, we first treat the neurons whose state is considered certain to obtain the output vector Y⁽¹⁾= (Y⁽¹⁾(1,1),..., Y⁽¹⁾(n,C)), with:

where Fz is the thresholding function by intervals. When q(i, c) > T we can guarantee than q(i, c) =1, when T/2 ≤ q(i, c) ≤1-T/2 we can guarantee than q(i, c)=0.5, and when q(i, c) < 1-T we can guarantee than q(i, c)=0. If NB_Z = 0 this phase is terminated and the output vector is Y=Y⁽¹⁾. Otherwise, Y is obtained after applying the thresholding function f_β as follows:

where β is the selected threshold. Each value q(i, c)ÎZ is considered as potential thresholds. That is, for each q(i, c)ÎZ:

Eventually, Z can be reduced by decreasing T (for T>0.666). For each potential value of β, we present the vector X’⁽¹⁾(β)= f_β(Q) to the network. Then, we compute the new vector of probabilities Q⁽¹⁾(β) and the output vector Y⁽²⁾(β) = Fz(Q⁽¹⁾(β)). We keep the cases where NB_Z=0 and X’⁽¹⁾(β)= Y⁽²⁾(β). If these two conditions are never satisfied, the initial X’ is considered too different of any training vector. If several thresholds are candidate, we choose the one which provides the minimal error (difference between q(i, c) and Y(i, c), for i=1, n and c=1,..., C):

E(β) = 1/2 ∑ⁿ_i=1 [q(i, c)⁽¹⁾(β)- Y(i, c)⁽¹⁾(α)]²     (6)

4. Experimental Results

In this section we present several examples to evaluate the quality of our recognition algorithm for different pattern types. We will give various geometric Figures as inputs to a Multiple Classes Random Neural Network and train the network to recognize them. To evaluate our approach, we use three Figure groups: the first group (group A) includes the set of Figures shown in Figure 1, where blackened boxes represent blue colors, gray boxes represent green colors and white boxes represent red colors. The next group (Group B), which is composed of the black and white Figures shown in Figure 2, is used to compare our approach with the recognition algorithm based on RNN proposed in [3], and the evolutionary learning approach proposed in [5]. The last group (group C) is composed by the set of patterns used in [17] (Figure 3). For group C extended experiments are presented to evaluate the performance of our method according to the relationship between the problem features (number of patterns, pixels and colors), recognition rates and processing time.

For the first and second cases, each Figure is represented by a 6*6 grid of pixels. For example, the seventh pattern in Figure 2 can be represented as shown in Figure 4. According to the RGB model, the black boxes are represented as (1, 1, 1), while white boxes are represented as (0, 0, 0). Hence, we can represent geometric Figures with different combinations of colors (for example, in Figure 2, if we suppose black boxes correspond to red colors, and white boxes to blue colors, neurons for black boxes are equal to (1, 0, 0) and for white boxes are equal to (0, 0, 1)). Thus, for these cases we use a single-layer multiple classes RNN composed by 36 neurons (n=36) and 3 classes (C=3).

4.2. Analysis of Results 

In order to test associative memories, we have evaluated the recognition rates of distorted versions of the training patterns (Tables 1 and 2). We generated 20 noisy versions of each training image, for a given distortion rate. The result of the learning stage is used as the initial neural network for this second stage (retrieval stage). We have corrupted them with noise rates of 10%, 20% and 30% distortion, by modifying bit values at random. A pattern is recognized if the residual error rate is less than 3%. The results are presented in tables 1 and 2. These values represent the average of 8 processes for each set S_i of images. The performance degrades when the noise rate is important (memories are then more discriminating). The results for the first group are presented on Table 1. Our algorithm provides a good recognition rate because we recognize a large number of patterns despite corrupted parts of the Figures (even when the noise rate is large (30%)). Particularly, the recognition rate of the sets S₄ and S₆ remain good for our approach. Concerning S₁₀ and 30% of noise rate, recognition rate decreases.

Table 1

Table 2

Recognition rate of noisy versions of Group B 

Acknowledgment

This work was partially supported by CONICIT grant AP-97003817, CDCHT-ULA grant I-503-95-A05 and CeCalCULA (High Performance Computing Center of Venezuela).

References

1. Aguilar J. Evolutionary Learning on Recurrent Random Neural Network, In Proceeding of the World Congress on Neural Networks, 1995, pp. 232-236.        [ Links ]

2. Aguilar J. An Energy Function for the Random Neural Networks, Neural Processing Letters, 1996, 4: 17-27.        [ Links ]

3. Aguilar J. A Recognition Algorithm using the Random Neural Network, In Proceeding. of the 3rd. International Congress on Computer Science Research, 1996, pp. 15-22.        [ Links ]

4. Aguilar J. Definition of an Energy Function for the Random Neural to solve Optimization Problems, Neural Networks, 1998, 11: 731-738.        [ Links ]

5. Aguilar J., Colmenares A. Resolution of Pattern Recognition Problems using a Hybrid Genetic/Random Neural Network Learning Algorithm, Pattern Analysis and Applications, 1998, 1: 52-61.        [ Links ]

6. Aguilar J. Learning Algorithms for the Multiple Classes Random Neural Network, Lecture Notes in Artificial Intelligence, 2000, 1821: 561-566.        [ Links ]

7. Atalay V., Gelenbe E., Yalabik N. The random neural network model for texture generation, Intl. Journal of Pattern Recognition and Artificial Intelligence, 1992, 6: 131-141.        [ Links ]

8. Colombo C., A. Del Bimbo (1999), Color-induced image representation and retrieval, Pattern Recognition, 1999, 32: 1685-1695.        [ Links ]

9. Del Bimbo A., Pala P. Visual image retrieval by elastic matching of user sketches, IEEE Trans. Pattern Anal. Machine Intelligence, 1997, 19: 223-234.        [ Links ]

10. Fourneau M., Gelenbe E., Suros R. G-networks with Multiple classes of negative and positive customers, Theoretical Computer Science, 1996, 155: 141-156.        [ Links ]

11. Gelenbe E. Random neural networks with positive and negative signals and product form solution, Neural Computation, 1989, 1: 502-511.        [ Links ]

12. Gelenbe E. Stability of the random neural networks, Neural Computation, 1990, 2: 239-247.        [ Links ]

13. Gelenbe E. Theory of the random neural network model, In E. Gelenbe ed., Neural Networks: Advances and Applications, North-Holland, Pays-Bas, 1991.        [ Links ]

14. Gelenbe E. Learning in the recurrent random neural network, Neural Computation, 1993, 5: 376-389.        [ Links ]

15. Jain A., Vailaya A. Image Retrieval using Color and Shape, Pattern Recognition, 1996, 29: 1233-1244.        [ Links ]

16. Minka T., Picard R. Interactive Learning with a Society of Models, Pattern Recognition, 1997, 30: 1370-1381.        [ Links ]

17. Mojsilovic A., Kovacevic J., Kall D., Safranek R., Ganapathy K. The Vocabulary and Grammar of Color Patterns, IEEE Trans. on Image Processing, 2000, 9: 417-431.        [ Links ] ]]> 1 1995 232-236 2 1996 4 17-27 3 1996 15-22 4 1998 11 731-738 5 1998 1 52-61 6 2000 1821 561-566 7 1992 6 131-141 8 1999 19 99 32 1685-1695 9 1997 19 223-234 10 1996 155 141-156 11 1989 1 502-511 12 1990 2 239-247 13 1991 14 1993 5 376-389 15 1996 29 1233-1244 16 1997 30 1370-1381 17 2000 9 417-431