Pergamon Pattern Recognition, Vol. 31, No. 2, pp. 159-167, 1998

© 1997 Pattern Recognition Society. Published by Elsevier Science Ltd Printed in Great Britain. All rights reserved

0031-3203/98 $17.00 + .00

PII: S0031-3203 (97)00040-X

RECOGNITION OF FACIAL IMAGES WITH LOW RESOLUTION USING A HOPFIELD MEMORY MODEL

YING DAI and YASUAKI NAKANO*

Department of Information Engineering, Shinshu University 500 Wakasato, Nagano, 380 Japan

(Received 3 September 1996; accepted 25 March 1997)

Abstract--In this paper, a new method to recognize facial images with low resolution by utilizing the Hopfield model is presented.

In this method, a Hopfield memory model for the facial images is organized and the optimal procedure of the unlearning is determined. Based on the composed Hopfield memory model, the relation between the reliability of the recalls and the number of faces memorized in the Hopfield memory is analyzed. Finally, the method for the face recognition using the Hopfield memory model combined with the pattern matching is proposed. The experimental results for face recognition are also represented and analyzed. © 1997 Pattern Recognition Society. Published by Elsevier Science Ltd.

Face recognition Hopfield model Associative memory

1. I N T R O D U C T I O N

Recently, with the potential applications in many fields, the study of the face recognition techniques has given rise to the more and more interests for re- searchers. The techniques have many applications such as security systems, criminal identifications, tele- conferences and so on.

Many researchers have reported face recognition techniques. As a pioneer of the research, the method for face recognition by using the facial profile was presented in (1). In some other methods, (2-4), the facial parts were extracted and the relation among them was used. In reference (5), by rotating the facial 3D data in the 3D space, the facial profile was made use of in face recognition.

Recently, the methods using the statistical features of the facial images with gray levels were presented. In reference (7), the higher order local autocorrelation features, which are shift-invariant and additive, were extracted for in the face recognition. In reference (8), a sub-space classification method and the robust feature extraction based on K - L expansion of an invariant feature space were combined to compose a face identification system.

As the features for recognition, the global or local shapes of the facial images were used in the methods mentioned above. Consequently, it is necessary to use the facial images with high resolution. From the view- point of the practicability, it is necessary to recognize the facial images with low resolution, but it is not clear for these methods to be able to recognize the facial images with low resolution efficiently.

* Author to whom correspondence should be addressed.

On the other hand, a hierarchical neural network gave a good result of the result of face recognition, in spite of the low resolution of the faces such as 12 x 12 pixels. (9) Accordingly, the recognition capability of the neural network is remarkable and to be studied further.

In this paper, a new method for recognizing the facial images with low resolution by utilizing the Hop- field model is presented.

In this method, a Hopfield memory model for the facial images is organized and the the optimal proced- ure of the unlearning is determined. Based on the composed Hopfield memory model, the relation between the reliability of the recalls and the number of faces memorized in the Hopfield memory is ana- lyzed. Finally, a method for the face recognition using the Hopfield memory model combined with the pattern matching is proposed. The experimental results for face recognition are also represented and analyzed.

2. HOPFIELD NET FOR FACE IMAGES

2.1. Hopfield Net

The Hopfield net (10) has only one layer of units. These units play a triple role as input, output, and processing units. The units are globally inter- connected and every unit is thus connected to every other units. The total input Ui of the ith neuron is given by the linear combination:

N

Ui = ~ WqVj + Ii, (1) j = l

where N is the number of units, V) the output of thejth unit, W~j a component of the connection matrix W,

159

160 Y. DAI and Y. NAKANO

and I~ the bias of input to the ith neuron. Equation (3) can be expressed by matrix notation as

U = WV + I. (2)

The output V~ of the ith unit is given as:

1 V~ = f ( U 0 = 1 + exp(-22Ui)" (3)

Here, ,I is the parameter which determines the gradi- ent of the sigmoid function defined by equation (3).

The N units are updated asynchronously at ran- dom times:

V~ +1 =f(U~), (4)

where t denotes the time. It is known that when the matrix W~i is symmetric,

the dynamics of equation (3) minimizes a bounded energy function, defined as

1 N N N

: - z l z E -i i- j - i=1

u 1 ('v, + , ~ l ~ J o f - l ( V ) d V (5)

and the network converges to one of the minimal energy states.

It is also known that the Hopfield net can be used for pattern recalling. The process of recalling informa- tion from a Hopfield net begins by putting the net- work into an initial state which represents an input key pattern. The network then iterates from this initial state until it converges to a final stable state, which corresponds to the recalled pattern from the key.

Namely, when the output of a unit iterates based on equation (3) in a Hopfield net:

(1) the energy must decrease with time's passing; (2) the state of all the units becomes stable when the

energy ceases to decrease.

2.2. Associative memory model ~11)

In this section, the connection matrix W is deter- mined, in order to store facial images in a Hopfield net and enable the net a perfect recall.

According to the Markov random field (MRF) model, the posterior distribution of the given image data is analyzed to find the likeliest generators of the image data. In the MRF model approach, the condi- tional distribution on all variable values reduces to the conditional distribution on neighbors, so that iterative computations are bounded into the local neighborhoods.

The MRF model of images can be expressed as a kind of globally interconnected nets. In the simplest MRF model of images, one pixel corresponds to one processing module. The state of the module is deter- mined by the interaction between the output and the processing modules si- 1. i, & + 1, j , Si, j - 1, Si, j + 1 , which correspond to pixels of its nearest neighborhood. This

W(i, j - l),(i,j)

input

Fig. 1. Simplest MRF model.

simplest interconnecting network of the MRF model is shown in Fig. 1. Values of weights connecting a certain unit and its four neighborhood ones are W(i, j ) ( i - 1 , j ) , W(i,j)(i+ l , j ) , W(i , j ) ( i , j - 1), W(i,j)(i , j+ 1) respe- ctively, and all other weights are 0.

It is known that the MRF model can store patterns, each of which can be recalled. In other words, it can work as an associative memory. In order to give the network the capability of the associative memory, the weights are determined as below:

1 M

W"'i~"" J'~ = -N ZlS~"J'i~m= s~'7,,i,~, (6)

where, (i,j) and (i',j') are the units which are linked to each other, N the total number of units, and s m the (i,j) value of the (i,j)th pixel of the mth pattern. M indi- cates the number of patterns to be memorized.

It is clear from equation (6) that the weights satisfy the conditions of a Hopfield net, because the weights are symmetrical and the connection to oneself is 0. This kind of the Hopfield net is called the associative memory model with recalling capability. For simpli- city, this model is called Hopfield memory model in our paper.

2.3. The memory and recall of facial images by Hopfield memory model

M faces can be stored in the Hopfield memory by giving the weight W~,i)li, j, ~ according to equation (6). There, one pixel (i, j) corresponds to one unit (i, j). This process corresponds to learning of M faces, each of which will be recalled by the process explained below.

In our previous work, a2~ the bound of the lower facial image size for recognition is 16 x 20 pixels, so this image size is used. Actually, they are obtained by blurring and re-sampling the facial images if they are with comparatively high resolution. As a result, our networks is encoded with 320 units.

When a facial image is given as an input to the Hopfield memory, the units of the system converge to a stable state finally, if they are updated asyn- chronously at random times. The value of all the units at the stable states give an associated recall pattern from the face given as input.

When M is equal to l, the pattern which is memor- ized in the Hopfield memory is only the standard face of a person. In this case, the system recalls the original

Recognition of facial images 161

Table 1. Relation between d and M

M 2 4 6 d 0.10 0.08 0.05

Fig. 2. Learned pattern and recall from it (M = 1).

,ii

Fig. 4. Facial image and its recall (M = 4, k = 0.01).

Fig. 3. Test pattern of the same person and recall from it (M = l).

image completely. In other words, the similarity be- tween the input pattern and the recalled one is equal to unity.

Here, the similarity is defined as

E x E r ( f (x, y) - f ) ( t ( x , y) - f ) b l - -

X/ZxYy(f(x, y) -Y)22xY,(t(x, y) - F)'

where f ( x , y) and t (x , y) indicate two patterns and f and f do the average values of the patterns, respec- tively. Figure 2 shows the facial image used in the learning and its associated recall when M is 1 in the Hopfield memory.

As is shown in Fig. 2, the memorized face is recalled completely except the difference of the image's gray level. As stated above, the similarity between the facial image and its recall is equal to the unity, showing that two patterns are same except the averages and the proportional constants.

Figure 3 shows the test pattern of the same person learned in the model in Fig. 2 and its recall by the Hopfield memory when M is l.

The similarity between the standard pattern and the test one of the person is equal to 0.73, while the one between these recalls is equal to 0.84. Thus, the similarity between the recalls is larger than the one between the original images, and this observation is supported in most cases.

For M > 1, the smaller the value of M is, the more similar the recalled pattern of a test pattern becomes to its standard pattern. In order to inspect the conjec- ture above, we define

Us ~ U d = - - (7)

Us

as the distance measure of how the recall of the test pattern is close to its standard pattern and separated from the other patterns. Here, us indicates the sim- ilarity between the recalls of the test pattern and the standard pattern of the same class (person), and u does the one between the recalls of the test pattern and another classe's standard pattern. If the value of d is large, we will have higher reliability on the decision that the test pattern belongs to the same class, not to the other classes.

Table 1 shows the relation between d and M for the test pattern shown in Fig. 3. In this case, the test pattern is limited to the normal front face. The result shown in Table 1 verified the expectation that there is a tendency for the recall of the test pattern to be close to its standard pattern, when M is small. It can be expected d will become very small for a much larger M. From the practical viewpoint, however, we do not need its verification.

When the value of M is larger than 1, more than one facial images are stored in the Hopfield memory. Thus, the recall has false attractors caused by the influence of the mixed memories. This phenomenon is called "induced false memory". In order to remove the induced false memory, the so-called "unlearning" is utilized. This is done by the modification of W,,j),,.j,~, after its initial value is selected.

The processes of unlearning of the Hopfield mem- ory model goes as follows. From the initial state which represents the input key pattern, the network iterates until it converges to a final stable state. Then, by using the stable values of states s t = (s], . . . , s~), the weight modification is calculated according to the formula

A W,, j) , , , j , j = - ks{i.j) s[i,,j,), (8)

where, k is called as an unlearning parameter. It is expected that the induced false memory is obliterated, if AW,,j),, ~,) is added to the weight W,,~),,,~,).

Figure 4 shows a certain facial image and its asso- ciated recall by the Hopfield memory which was

162 Y. DAI and Y. NAKANO

obtained after applying the unlearning, when M is equal to 4 and the unlearning parameter k is 0.01.

3. PERFORMANCE OF THE HOPFIELD MODEL

3.1. Optimizing the unlearning parameter k

As stated in the previous section, after a certain standard pattern is memorized in the Hopfield mem- ory, the pattern can be recalled by a test pattern given to it. This characteristics of associative memory of the Hopfield memory model will serve for face recogni- tion.

It is necessary to adopt the unlearning in order to remove the induced false memory. Along with the removal of the induced false memory, however, the useful information for the associate recall may be ruined also. This damage is disadvantageous to face recognition. In order to remove the induced false memory and keep the valid information of the face recall as much as possible, it is necessary to select the value of k optimally. In this section, the relation between the value of k and the recognition perfor- mance of the associate recall of the facial image is described.

The similarity between the memorized facial pat- tern and its associate recall is denoted as u. A system- atic set of experiments was done when the number of facial images memorized in the Hopfield memory is 4 to give the relation between the values u and k shown in Fig. 5. Two faces are used as the test inputs. In the figure, lines ~ and x - - x corre- spond to the facial images Nos 1 and 2, respectively.

F rom Fig. 5, it can be seen that the value of u varies as a parabola with the increase of k. For the face No. 1, the similarity u becomes maximal at the value k of 0.01. The curve can be recognized as the compro- mise of the removal of the induced false memory, which requires larger k, and the reservation of the face information, which requires smaller k.

Figure 5 also shows that the relation between u and k for the face No. 2 is similar to that of No. 1 except the optimal value of k; in this case, u is maximal at k = 0.02. Namely, the optimal unlearning parameter k is different for the different face to make u maximal.

Figure 6 shows the facial images No. 2 and its asso- ciated recall for various k values. It is felt that the associate recall of face No. 2 corresponding to k = 0.02 is most similar to the original images. Al- though the evaluation is subjective, it is in accordance with the situation observed in Fig. 5.

Figure 7 shows the relation between u and k for the faces No. 1 and No. 2 when M is increased to 11. In this case, the relation between the u and k is similar to the one for M = 4, but, the optimal value of k is different from the one when M is 4, and the maximum value itself is decreased.

For the other faces, there is also a similar relation between the u and k such as for faces Nos 1 and 2, w h e n M = 4 o r M = 11.

0.95

0.9

U

0.85

0.8 I I I I I I

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

k

*--o: corresponding to face #1 x - -x : corresponding to face #2

Fig. 5. Similarity between the u and k(M = 4).

or ig ina l k = 0.01

k = 0.02 k = 0.04

Fig. 6. Facial image and its recall (M = 4 with k's changing).

F rom the observation above, the connection matrix W for the Hopfield memory is modified by the un- learning as follows, in order to remove the induced false memory and keep the valid information of the face as much as possible.

First, the initial state of the units of the network is set to the input key pattern. Then, the network iterates from the initial state until it converges to the final stable state. By using the stable values s" = (s~ . . . . . s~v),

Recognition of facial images 163

0.85

0.80

0.75

0.70

- 1 1

I I I I I I

0.00 0.01 0.02 0.03 0.04 0.05 0.06

k

0.07

*--*: corresponding to face #1 x- -x : corresponding to face #:

Fig. 7. Similarity between the u and k(M = 11).

1.0

0.9

0.8

0,7 I I I I I I I I I I I

1 2 3 4 5 6 7 8 9 t O l l

M

12

• - - • : corresponding to fact, # 1

x - - x : corresponding to fitce # 2

Fig. 9. Relation between the similarity u and the number of memorized faces M.

1.00

0.95

0.90

u . . . . . ,,11.. " ' - u . " . - - " . . . . . . • in ....

Q s tep=5

o s tcp~3

u s t ep=7

I I I I I I No. 1 No.2 No.3 No.4 No. 5 No. 6

pattern

Fig. 8. Relation between the similarity and the unlearning's steps.

the modification values of weights are calculated ac- cording to the formula:

t ,r ~r AW~i,~, , j ,~ - k s~i,j) • , , j ,) . (9)

Next, the stable state obtained above is set as the initial state in turn, and the procedure stated above is iterated, until the variance of the stable state

~ 7 " 2 0 ~ ' 1 6 e - 2 D~ = Li= 1Lj= l t s i j - s i j ) becomes minimal. Here, ggj is a mean of the state s'. The Wi,/s at final modification are adopted as the connection matrix of the Hopfield memory model.

In equation (9) k' is set according to the number of the memories. When M = 6, k' is set as 0.3.

Figure 8 shows the similarities between some re- calls and their original patterns with the different iteration steps. From Fig. 8, we can see that the per- formance of recalling is optimal as a whole for the iteration step 5, where the variance at the stable state of the network is minimal. Therefore, the weight

matrix W of the Hopfield memory model modified by the unlearning procedure is considered to work satis- factorily.

3.2. The number o f the memories, M

As stated in Section 3.1, with the number of the facial memories M becoming larger, the similarity between the recall and its original pattern declines. In this section, the relation between the number M and the similarity u will be discussed further.

Figure 9 shows the change of u when M is varied. The curves represented as e - - o and x x corre- spond to the faces of No. 1 and No. 2, respectively.

From Fig. 9, we know that u declines when M be- comes larger. That is to say that the more the number of the facial memories is, the more seriously the facial recall is obstructed by the other memories. The fewer the number of the memories is, the better the facial recall keeps the states of the original pattern. This phenomenon is considered as a matter of course, but the numerical values obtained will be valuable.

In order to evaluate the association ability of the Hopfield memory, the measure d defined in (7) is examined further.

Figure 10 shows the change of the measure d for the 40 facial test patterns. Here, the values of d are the average values of d obtained for all test patterns. In these experiments, the standard patterns are used as the test set.

It is natural that with increasing M, the value of d declines and the recall ability of the Hopfield mem- ory becomes low. Accordingly, the fewer the learned patterns in the Hopfield memory is, the higher the classification performance for the unknown patterns becomes. What is more interesting is that the value of d saturates when M > 8. In other words, the classi- fication performance can't be much improved by

164 Y. DAI and Y. NAKANO

0.4

d

0.2

I I I I I I I I i I 2 3 4 5 6 7 8 9 10 11 12

M

Fig. 10. Relation between the M and d.

0.10

0.05

" ,%

" . , , '...,

d 0 .00 g ................................... ,,,,

. ~ . , ........ ?:... . . . " / ~ . . . " " . , :< . . . . . . . .

: : "-... ... • . . . . . . . -0.05 ." .., .... -... - 1 .. ".... ....

d , . " "--,. %-.. # ' " - . . . , ' ......

4) .10 I I t t I ..... ?

l 2 3 4 5 6

M . : size d m n g t a : expression change

• : rolati0a

Fig. 11 Relation between the d and the facial test patterns.

decreasing the number of the stored patterns in the Hopfield memory, when the decreased number is still large. The value of M at which the measure d saturates is called the critical point Mc of the Hopfield memory. The Me will be used in the next section.

The kinds of the test patterns discussed so far are limited to the normal front faces. We will discuss the robustness of the recalls for various deformation of the test patterns furthermore.

Figure 11 shows the relation between the measure d and the number of learned facial patterns using various distorted test patterns, such as the expression change, wearing the glasses or not, the inclination and the rotat ion of the face and so on. The number of learned patterns M used was 1, 2, and 6. The case M = 1 is equivalent to the one in which the test pattern is matched to the standard patterns directly.

F rom Fig. 11, it can be seen that for dl > 0, which means the first candidate of the test pattern is correct, the values of d increases when M decreases. For d~ < 0, which means the correct class of the test face is

not the first candidate, but the second, the values of d also increase when the M decreases except the case o f M = 1. The value ofd for M = 1 is less than the one for M = 2. It is indicated that the recall of the test pattern by using the Hopfield model is closer to its standard pattern than the test pattern itself, though sometimes the first candidate of the test pattern is not its standard pattern. This observation that there is the tendency for the recall of the test pattern to approach to its standard pattern was confirmed by many sam- pies, in condit ion that their deformation is limited in some range; the angle of the facial rotation is less than +30 °, the size of the face is within the range of

[0.9, 1.2] as the ratio to the standard pattern, and the variation between the center of the test and the stan- dard faces is within one pixel.

4. FACE RECOGNITION BY THE HOPFIELD MODEL

In this section, a new method for the face recogni- tion using the Hopfield memory model combined with the pattern matching method is proposed.

simple pattern matching

constructing I Hopfield net

1 pattern matching using recalls by Hopfield net

1 !

adopting C1 or C2 I as output consulting

d2 andeo

Fig. 12. Flow chart of the face recognition.

Recognition of facial images

Table 2. Decision strategy based on the order of the candidates

165

M Combination of 1st and 2nd candidates at each step

1 C1, C2 C1, C2 Cl, C2 Mc Including other(s) Cx, C2 C1, Cz 2 CI, C2 C2, C1 Decision Reject [d2[ < [de[ ~ C2 C2

Id21 > Idol ~ Ca

C~, C2 C1, C2 C2, C1 C2, Cl C2, CI Cl, C2

C2 R~ect

The flow chart of the proposed face recognition scheme is shown in Fig. 12. First, the pattern match- ing is used directly in order to extract candidate classes. The similarity between the test pattern and each facial template is calculated and the maximum is determined. Only one template for each class is regis- tered. A class is discarded, if the difference between the maximal similarity and the similarity for the class is larger than a threshold Oa. If the number of the re- maining classes is one, and the similarity between the class and the unknown pattern is larger than another threshold 0a, this class is outputted as the recognition result for the unknown input. The values Oa and 0a are determined experimentally.

If the number of the remaining classes is more than one, the Mc templates for whose similarities are ran- ked in front are stored in the Hopfield memory based on the equations (6) and (9). The class corresponding to the maximal similarity is defined as the first candi- date Ca, and the one to the next similarity as the second candidate C2.

Next, calculating similarities between the recalled patterns associated from the input and each template stored in the Hopfield memory, the first and second candidate are selected newly from those Mc classes. If Ca and C2 do not coincide with the first or second candidates, the input pattern is rejected. Otherwise, the values of d in equation (7) are calculated, and a new Hopfield net is organized to store templates for C1 and Cz. Then the similarities between the recalls of the input pattern and each of the two patterns stored in the new Hopfield memory are calculated to deter- mine the new first and second candidates and give the value of d.

Table 2 shows the algorithm for deciding the class of the unknown face. To simplify the description, we denote the value of d for M = 2 as d2, and that for M = Mc as dc. In the case that C2 is still the second candidate for M = 2 and M = Me, C2 is outputted as the recognition result, if [d2[ < [dd, while the C1 is outputted, if [d2[ > [de[.

On the other hand, in the case that C2 becomes the first candidate for M = Me, the C2 is outputted as the recognition result, if the C2 is still the first candidate for M = 2; otherwise, the unknown pattern is rejected. In the case that C2 becomes the first candidate for M = 2, C2 is outputted as the recognition result.

It may look strange to adopt C2 as the result in the second case; i.e. C2 is the second candidate through-

out three steps, in condition that [dz[ < Idol. This strat- egy was adopted by observing the characteristics of similarities illustrated in Fig, 11.

5. EXPERIMENTAL RESULTS AND ANALYSIS

5.1. The robustness to the facial change

In this section, results of the experiment of face recognition based on the proposed method are shown. The size of the facial images for test is set as 16 × 20 pixels. The face set for the experiments in- eludes 10 persons from our laboratory and 30 persons selected randomly from the face database supported by ORL (Olivetti Research Laboratory). The total number of the persons is 40.

Next, one facial image selected respectively from each of the 40 persons constitutes the learning set. The remaining 300 facial images from the 40 persons com- pose the test set.

The test set includes the samples under various distortions, such as the expression change, wearing the glasses or not, the facial rotation, and so on. But, the angle of the facial rotation is limited within _+ 30 °, the size of the face in [0.9, 1.2] times of the templates, and the variation between the center of the test and the standard faces within one pixels. The examples of expression change are shown in Fig. 13, which include the case of laugh and anger. The examples of glasses are shown in Fig. 14, including glasses with metal and black frames. Although the reduced facial images with 16 x 20 pixels are used in the experiment, the original images with high resolution are shown in Figs 13 and 14 for the readers' convenience.

Based on the method presented in our paper, the recognition results for the test set are shown in Table 3.

From the results, it can be seen that for a little distortion of the size change, which indicates the size change of face, age change and glasses' wearing, the face recognition rate is high. The number of the sub- stitution or rejection errors for these cases is small. It is noted that the proposed method is robust to many variation such as the face with glasses, etc. As for the facial expression change and the rotation less than _+ 30 °, the recognition rate is comparatively high. As

for the facial position variation, however, there were three substitution or rejection errors among the 10 test samples. On the whole, the correct recognition

166 Y. DAI and Y. NAKANO

~ 1 Table 3. Experimental results by the proposed method

Condition Number of Number of Number of samples recognized rejection

Expression 80 72 5 Age 10 9 Glasses 80 76 3 Size 80 77 2 Rotation 40 34 4 Position 10 7 2 Sum 300 275 16

Fig. 13. Example of the facial expression change (by the face database from ORL).

. . . . ~

!

i:

Table 4. Experimental results by pattern matching

Condition Number of Ranked 1st Ranked samples 1st or 2nd

Expression 80 60 74 Age 10 7 9 Glasses 80 65 76 Size 80 65 77 Rotation 40 26 35 Position 10 6 8 Sum 300 229 279

Fig. 14. Example of the glasses wearing.

rate to the test pattern set is 92%, and the rejection rate is 5%.

5.2. Comparison with the pattern matching method

In the so-called pattern matching, the similarity between the test pattern and the every one of the standard template set is calculated for each class, and the class which corresponds to the maximal similarity is selected as the recognition result.

For the test set used in Section 5.1, the recognition results by the pattern matching are shown in Table 4. The figures in the third column denote the number of

faces which were identified by adopting the first candi- date as the recognition result. The figures in the fourth column show numbers of patterns whose classes were ranked at the first or the second. From the values at the third column, the number of substitution errors is about 10 times higher than that of the method pro- posed in this paper. The recognition rate is only 76%. On the other hand, if the first and second candidates are adopted as the identification results, the identifica- tion rate becomes 93%. It is near to the one obtained by using the proposed method, i.e. 92%. In other words, almost all patterns are recognized by our method, if the correct class is picked up among the best two candidates at the first stage. Accordingly, the proposed method is considered to utilize the tendency that the recalls of the unknown pattern approach to its standard pattern with decreasing M'.

Comparing with the results in Tables 3 and 4, it is obvious that the face recognition performance is im- proved remarkably by the proposed method.

6. C O N C L U S I O N

In this paper, the Hopfield memory model for the facial images was constructed and the incorporation of unlearning was described. By analyzing the perfor- mance of the unlearning parameter k, the optimal procedure of the unlearning was determined. Based on the composed Hopfield memory model, the rela- tion between the reliability of the recalls and the number of faces memorized in the Hopfield memory was analyzed. Next, the method for the face recogni- tion by using the performance of the Hopfield

Recognition of facial images 167

memory model combined with the pa t te rn ma tch ing was proposed. The exper imental results by the 300 test pa t te rns for 40 persons showed tha t the proposed me thod was robus t to the facial change, especially, to the glasses wearing, the age change and expression change. The face recogni t ion rate for the test pa t te rns was 92% with the 5% rejection rate.

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About the Author--YING DAI was born in Xian, China. She received the B. S. and M. S. degrees in information and control from Xian Jiaotong University, China in 1985 and 1988, respectively. She received the Ph.D. degree in Information Engineering from Shinshu University, Japan, in 1996. From 1988 to 1991, she was a research associate and then a lecturer in Xian Jiatong University. From 1991 to 1992, she was a lecturer in Huanan Technology University, China. In 1993, she entered into the Doctoral Program at Shinshu University. Currently, she is a research associate fed by Nippon Gakujutsu Shinkokai. Her research interests include computer vision, pattern recognition and image processing.

About the Author--YASUAKI NAKANO received the B.S., M.S. and Ph.D. degrees in mathematical engineering from the University of Tokyo in 1961, 1963 and 1975, respectively. From 1963 to 1989, he was with the Central Research Laboratory, Hitachi, Ltd., where he did research on speech synthesis, speech recognition, character recognition and document understanding. In 1989, he was appointed as a professor in the Department of Information Engineering, Faculty of Engineering, at Shinshu University. Currently, he is responsible for research and education in pattern recognition and artificial intelligence. He wrote many papers in these studies and his current concern is pattern recognition, including handwritten character and cursive wod, neural networks and natural language processing. Professor Nakano is a senior member of IEEE and belongs to many academic societies in Japan. He will serve as the Program Chair at ICDAR'97. He is also an associate editor of the Pattern Recognition Journal.