2011 20th European Conference on Circuit Theory and Design (ECCTD)

Investigation of Recall Image by Partitioned
Hopfield Neural Network
Tomoya Shima, Chihiro Ikuta, Yoko Uwate and Yoshifumi Nishio
Dept. Electrical and Electronic Eng, Tokushima University
Tokushima 770-8506, JAPAN
Email: {s-tomoya, ikuta, uwate, nishio}@ee.tokushima-u.ac.jp
the energy of the network becomes minimum. Thereby, the
HNN recalls a similar image to the whole input image fromsome stored images. However, the HNN operating for the
associative memory alone can not realize the specific exampleof the memory mechanism.
In order to realize the memory mechanism, we propose
Partitioned HNN (PHNN). In the field of image processing,the method of the divided HNN has been already proposed [8].
However, in our study, the PHNN is used as the associative
memory modeled the high-functional information processingof the human brain. In the PHNN, several small HNNs arearranged in the whole image regularly without overlaps. The
PHNN recalls similar patterns to partial input image only onsmall separate regions. By computer simulations, we confirm
that the PHNN can recall the whole image from the partial
characteristic of the images.

Abstract—In this study, we propose Partitioned Hopfield Neural Network (PHNN) to realize the memory mechanism of thehuman brain. The PHNN is realized by arranging cell which
has a small HNN in the whole image regularly without overlap.
By computer simulations, we confirm that the PHNN recalls thewhole image from the partial characteristic of the stored images.

I. Introduction
Computer can execute processing which requires a hugeamount of computational cost. Human brain can not executesuch processing. However, in other cases, the human brain
can be superior to the computer, for example, the patternrecognition, the data mining and so on. Furthermore, thehuman brain have broad utility, which can apply in various
fields. Hence, it is imperative that the human brain functions
are clarified. In this study, we focus on “memory” which isone of the human brain functions. The memory mechanism of
the human brain consists of three processes; “memorization”
is to commit new experience to memory, “store” is to retain
the memorized information, and “recall” is to remember thestored information. The memory mechanism is interesting
information-processing capability [1]. In order to show a
specific example of the memory mechanism, we use image
A and B (see Fig. 1). The image A includes landscape anda triangle pattern. Meanwhile, the image B includes somepatterns. However a triangle pattern in the image B is exactly
same as one of the image A.
We assume that a human memorizes the image A, after thatthe human watches the image B. Then, the human remembers
the whole image A by watching the triangle pattern in theimage B. The example relates to three processes of the memory
mechanism. First, when the human watches the image A,
information of the image A is transmitted to the humanbrain. Then, the human brain “memorizes” and “stores” theinformation by watching the image A. Second, the humanremembers the image A from the triangle pattern in the image
B. Namely, the human can recall the whole image from thepartial image. If such memory mechanism will be applied inengineering, we expect that much information will be able tobe obtained from little information [2]-[5].
We consider that the memory mechanism can be represented
by Hopfield Neural Network (HNN) [6]. Because the HNN
operating for an associative memory is similar to the memory
mechanism [7]. The HNN memorizes and stores some patterns
by determining the weight parameter, and recalls a similar
pattern to an input pattern from some stored patterns byrepeating update of the internal state of all neurons such that

978-1-4577-0616-5/11/$26.00 ©2011 IEEE

(a)

(b)

Fig. 1. Example images for the memory mechanism. (a) Image A. (b) Image
B.

II. HNN Working as Associative Memory
The HNN based on the model of biological neurons was
proposed by J.J. Hopfield in 1982 [6]. Every neurons in the
HNN is connected to each other neuron in the network withno self-connection.
The HNN is used as the associative memory by exploiting the property that the network has multiple stable states.
Namely, if the parameters of the network can be decided insuch a way that the patterns to be stored become stable statesof the network, the network produces a stored pattern thatis similar to an input pattern. The process of the associative
memory by the conventional HNN is described as follows.
[Step1 (HNN)]: If we apply the conventional HNN for the
associative memory, one pixel in an image corresponds with
one neuron.
[Step2 (HNN)]: The HNN stores images by determining theweight parameter. The weight parameter is given by 1 P (p) (p)x x(i  j)(1)
wi j = P p=1 i j
0(i = j),41

where P is the number of patterns, w is the weight parameterand x is the pattern of memory. It turns out that the weightparameter is symmetrical by Eq. (1).
[Step3 (HNN)]: The neurons is initialized by an unknown
image. Namely, the network is given the input. The state ofeach neuron is determined by Eq. (2).
ui (t) =n

wi j (t)x j (t),

(2)

j=1

where u is the internal state of the neuron and x is the input
or output. The value of u is binary.
[Step4 (HNN)]: Neuron’s output is determined by a linear
function. The equation of the determining Neuron’s output isgiven as follows.
1,(ui (t) ≥ 0)
(3)
xi (t + 1) =−1,(ui (t) < 0).

Fig. 3.

Step2: Memorizing and storing images.

[Step5 (HNN)]: The HNN recalls a similar image to the input
image from the stored images by repeating Step3 to Step4.
Fig. 4.

III. PHNN Working as Associative Memory
In this section, we explain the process of the PHNN for
associative memory. One pixel in an image corresponds with
one neuron as same as the conventional HNN. In the PHNN,
we separate an image into small square regions. We call one
small square region, “cell”. Every cell is composed of the
pixels and has one small HNN. The cells are arranged in
the whole image regularly without overlap. The every cell in
the PHNN is not connected each other. Figure 2 shows thecomposition of the PHNN. In Fig. 2, every cell has 2×2 pixels.
This means every cell has the small HNN of 2×2 neurons.

Fig. 2.

Step3: Inputting an initial image into the PHNN.

images. Therefore, while the recalled image by the HNN
includes only information of one stored image, the recalled
image by the PHNN includes information of a number ofstored images.

The composition of the PHNN.
Fig. 5.

The process of the PHNN for the associative memory is
explained as follows.
[Step1 (PHNN)]: One pixel in an image corresponds with one
neuron in the PHNN.
[Step2 (PHNN)]: The PHNN memorizes and stores the partial
images on every cell by determining the weight parameter asshown in Fig. 3.
[Step3 (PHNN)]: An initial image is inputted to the PHNN
as shown in Fig. 4.
[Step4 (PHNN)]: The PHNN recalls similar patterns to thepartial input image on every cell from the partial stored images
by repeating updating rule as shown in Fig. 5. Thereby, the
PHNN can work associative memory locally. The recalled
image by the PHNN is composed patterns of the local stored

Step4: Recalling patterns on every cell.

IV. Simulation Results
We apply the PHNN for two types of simulations, “recalling image” and “detecting sign”. For comparison, we also
investigate the performance of the conventional HNN. The
conventional HNN has no cells, namely every neuron of the
HNN are connected each other. We focus on the Hamming
distance (HD) for the two simulations. The HD between two
images is the number of corresponding pixels. The HD is
obtained between the stored image and the input image insimulations. Thereby, if the HD is small, the stored image
is similar to the input image. Hence, the conventional HNN
recalls the stored image which has the smaller HD than otherstored images. In this study, we use the HD as the methodof confirmation for recalling ability. In the two simulations,42

images size is set to 64×64 pixels and every cell has 4×4 or
8×8 neuron units.

the conventional HNN depends on the HD. Figure 8 (b) isrecalled image by the PHNN. The PHNN clearly recalls thesign in Fig. 7 (a).
By applying [StepI (PHNN)] and [StepII (PHNN)], we
obtain the recalled whole image as shown in Fig. 7 (c)Therefore, Fig. 8 (b) has more cells of the same patterns as
Fig. 7 (a) than Fig. 7 (b) and the PHNN recalls Fig. 7 (a).
Hence, the PHNN can recall Fig. 7 (a) by the sign in Fig. 7 (c)without dependence of the HD.

A. Recalling Image
In this section, we investigate whether the PHNN can recall
the whole image from the partial image. In this simulation, we
add two more steps for the PHNN after [Step4 (PHNN)] in
order to make a judgment about what stored image is recalledby the PHNN as the whole image. The additional steps for the
PHNN is described as follows.
[StepI (PHNN)]: We compare the recalled image at [Step4(PHNN)] with the stored images for every cell and count the
number of cells which have the exact same pattern as eachpartial stored image in the recalled image as shown in Fig. 6.

(a)

(b)

(c)

Fig. 8. Simulation results for “recalling image”. (The number of stored
images is two.) (a) Recalled image by the HNN. (b) Recalled partial image
by the PHNN. (c) Recalled whole image by the PHNN.

Next, we investigate the recalling ability when the numberof stored image is increased. We add the two stored images asshown in Figs. 9 (c) and (d). Figure 10 shows the simulation
result for four stored images. Even if stored image increases,
the simulation result is same as previous case (Fig. 8). Fromthese results, we can confirm that the PHNN can recall thewhole image from the partial image.

Fig. 6. StepI: Comparing the recalled image with each partial stored images.

[StepII (PHNN)]: The PHNN recalls the whole stored image(a) in Fig. 6 which has more same cells as the recalled image
from some stored images.
For the simulation of “recalling image”, three binary images
are prepared as shown in Fig. 7. Figures 7 (a) and (b) arestored images and Fig. 7 (c) is an input image. Each stored
image includes a pattern of sign. The sign in Fig. 7 (c) isthe exact same pattern as partial Fig. 7 (a). The HD between
Fig. 7 (a) and (c) is 2161, and the HD between Fig. 7 (b)and (c) is 1297. In this simulation, every cell has 4×4 neuron
units.

(a) HD = 2161.

(b) HD = 1297.

(d) HD = 1836.

(c) HD = 1989.

(e)Fig. 9. Simulation images for “recalling image”. (The number of stored
images is four.) (a) Stored image (COOL!). (b) Stored image (GO!). (c) Stored
image (NOISE). (d) Stored image (PATTERN). (e) Input image.
(a) HD = 2161.

(b) HD = 1297.

(c)

B. Detecting Sign
Next, we investigate whether the PHNN can recall the signin the stored image even if position of the sign in the input
image is changed. In this simulation, we do not add two more
steps for the PHNN after [Step4 (PHNN)]. We prepare four
stored images and two input images to confirm the behaviorof the PHNN as shown in Fig. 11. Figure 11 (a) shows thestored image which is disposed same sign on the image space.

Fig. 7. Simulation images for “recalling images”. (The number of stored
images is two.) (a) Stored image (COOL!). (b) Stored image (GO!). (c) Input
image.

The simulation results of “recalling image” is shown in
Fig. 8. Figure 8 (a) is recalled image by the conventional HNN.
We can see that the conventional HNN recalls stored image
which has smaller HD than the others. Hence, the ability of

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(a)

(b)

(c)

(a)

(b)

(c)

(d)

Fig. 10. Simulation results for “recalling image”. (The number of stored
images is four.) (a) Recalled image by the HNN. (b) Recalled partial image
by the PHNN. (c) Recalled whole image by the PHNN.

Figure 11 (b) is the stored image which is a similar image tothe input image 1. Figure 11 (c) is the stored image which
is a similar image to the input image 2. Figure 11 (d) is thestored image with regular pattern.
In this simulation, two types of input images are prepared asshown in Figs. 11 (e) and (f). Figure 11 (e) is the input image 1
which has the smallest HD to Fig. 11 (b) and the exactly same
sign of Fig. 11 (a) is included on the outside of the object.
Figure 11 (f) is the input image 2 which has the smallest HD
to Fig. 11 (c) and the exactly same sign of Fig. 11 (a) is
included on the inside of the object. We compare the recalled
images obtained by the PHNN and the conventional HNN for
the two input images. In the case of the PHNN, every cell has
8×8 neuron units.

(a) HD1 = 2294.
HD2 = 2105.

(d) HD1 = 1606.
HD2 = 1937.

(b) HD1 = 839.
HD2 = 1640.

(e)Fig. 12. Simulation results for “detecting sign”. (a) Recalled image by the
HNN for input image 1. (b) Recalled image by the HNN for input image 2. (c)Recalled partial image by the PHNN for input image 1. (d) Recalled partial
image by the PHNN for input image 2.

V. Conclusions
In this study, we have proposed the PHNN and applied the
PHNN to associative memory. In our method, the PHNN has
recalled a similar pattern to the partial input image only on thecell. First simulation results showed that the PHNN recalls thewhole image from the partial image. Second simulation results
showed that the PHNN recalls the sign in the stored image
even if position of the sign in the input image was changed.
From these results, we can say that associative memory of the
PHNN is closer the memory mechanism of the human brain
than the HNN.
Acknowledgment
This work was partly supported by The Nakajima Foundation.

(c) HD1 = 1756.
HD2 = 355.

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(f)Fig. 11. Simulation images for “detecting sign”. (a) Stored image 1. (b)Stored image 2. (c) Stored image 3. (d) Stored image 4. (e) Input image 1.
(f) Input image 2.

Figures 12 (a) and (b) are recalled by the conventional
HNN from Figs. 11 (e) and (f). The conventional HNN
recalls stored images which have the smallest HD to the input
images. Figures 12 (c) and (d) are recalled by the PHNN from
Figs. 11 (e) and (f). From this simulation results, we can seethat the PHNN can recall the stored image with the sign even
if the position of the sign in the input image is changed.

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