3. A Recurrent Fuzzy Coupled Cellular Neural Network System with Automatic
3.3. Learning Algorithms for the RFCCNN
(t y
Ai i is ai0,0yi(t). The details of the function of each node of the RFCNN are described in Section 2 of Chapter 2.
3.3. Learning Algorithms for the RFCCNN
Similarly, two types of learning, structure and parameter learning, are used concurrently for the RFCCNN. The structure-learning algorithm of the RFCCNN is the same as those of the RFCNN. For the details of structure-learning algorithm of RFCCNN based on on-line ICA mixture model are described in Section 3.1 of Chapter 2. For parameter-learning algorithm of RFCCNN, they are described in the
Layer 1 Layer 5
Layer 4
Layer 3
Layer 2
x1 x9
yout
R1 R2
. . .
∑
1 1u I B +
yi
A1
−1
Z
∑
2 2u I B +
yi
A2
−1
Z
∑
Ii
u Bi +
i iy A
−1
Z
Feedback Layer
Ri
. . .
Context Node
Feedback Term Node
y1
y2
yi
Figure 3.1 Structure of the proposed RFCCNN.
rest of this section.
After the network structure is adjusted according to the current training pattern, the network then enters the parameter identification phase to adjust the parameters of the network optimally based on the same training pattern. Notice that the following parameter learning is performed on the whole network after structure learning; no matter whether the nodes (links) are newly added or are existent originally. Since the RFCCNN is a dynamic system with feedback connections, the backpropagation learning algorithm cannot be applied to it directly. Also, due to the on-line learning property of the RFCCNN, the off-line learning algorithms for the recurrent neural networks, like backpropagation through time and time-dependent recurrent backpropagation [12], cannot be applied here. Instead, the ordered derivative [14],
which is a partial derivative whose constant and varying terms are defined using an ordered set of equations, is used to derive our learning algorithm. The ordered set of equations, described in Section 2 in each layer, is summarized in (3.5)-(3.10). Our goal is to minimize the error function
2 forward pass is used to compute the activity levels of all the nodes in the network to obtain the current output yout(t+1). In the followings, dependency on time will be omitted unless emphasis on temporal relationships is required.
Summarizing the node functions defined in Section 2, the function performed by the network is
∑
and (3.1) is redefined as the following equation for clarity:]
With the above formula and the error function defined in (3.4), we can derive the update rules for the free parameters in the RFCCNN as follows.
Update rule of aij (the parameter of feedback template of the ith CNN) is
Similarly, the parameter bij (the parameters of control template of the ith CNN) is updated by
)]
1 )(
1 ( )[
1 ( ) ( ) 1
( i i i(6) i(6)
i t z t t h o o
z + = −ηε + + − . (3.19)
As shown in (3.14) to (3.16), the update rules are in recursive form. The value
i
aj
y ∂
∂+ / is equal to zero initially. For the rest free parameters in the RFCCNN, they are obtained during the structure-learning phase by the on-line ICA mixture model algorithm proposed in Section 3.1 of Chapter 2.
3.4. Experimental Results and Discussions
For comparison to the RFCNN described in Chapter 2, the capability of the proposed RFCCNN is demonstrated on the same defect inspection problems for color filter. As mentioned earlier, the difficulties in the defect inspection of color filter are its complex texture and demand for high-speed processing. For the reasons of high-speed processing, and that different kinds of defects in color filter need different CNN templates and some complex defects cannot be detected by a single CNN, the proposed RFCCNN is a good alternative to detect defect of color filter images. The way to train the RFCCNN is the same as the RFCNN. The training image and corresponding desired output are shown in Figs. 2.8(a) and 2.8(b). As mentioned in Section 3, there are no rules (and no CNNs) in the RFCCNN initially. They are created dynamically as learning proceeds upon receiving on-line incoming training data by performing the learning processes. When the learning processes are done, six clusters (six fuzzy rules and CNN templates) were obtained. For the example of color filter, it takes about 90 seconds to learn the structure (interconnection set) and the parameters with Pentium IV 2.0G PC. However, the training can be done off-line, so it is not a problem for the on-line processing of CNN. For simulation in this chapter, it causes about 9 seconds. After the proposed RFCCNN is implemented by analog circuit in the future, we believe that the processing times will be much faster than
those of simulation.
Figure 3.2 shows the output of RFCCNN and the outputs of Layer 3, 4, and Feedback Layer for the training image. Figures 3.2(a) shows the output of the RFCCNN. Figures 3.2(b) to 3.2(g) show the outputs of the six Layer-4 nodes, respectively, i.e., the outputs of the six CNNs in the Feedback Layer multiplied by the outputs of the six Layer-3 nodes (i.e., firing strength of each rule), respectively.
Figures 3.2(h) to 3.2(m) show the outputs of the six CNNs in the Feedback Layer, respectively. Figures 3.2(o) to 3.2(t) show the outputs of the six Layer-3 nodes, respectively (firing strength of each rule). The sum of the outputs of the six Layer-4 nodes (i.e., Figs. 3.2(b) to 3.2(g)) forms the RFCCNN final output (Fig. 3.2(a)). From Figs. 3.2(b) to 3.2(g), we can see that CNNs 4 and 5 take care of the defect texture on the right side of the training image, CNNs 1 and 6 mainly take care of the defect textures on the left side of the training image, and the other CNNs balance the output of the RFCCNN. The template of each learned CNN is given in (3.20).
Based on the learned structure and parameters of the RFCCNN, we test several images and some of those images as shown in Fig. 3.3. Figs. 3.3(a), 3.3(d), and 3.3(g) are input testing images. Figs. 3.3(b), 3.3(e), and 3.3(h) are corresponding detection results of the RFCCNN. Figs. 3.3(c), 3.3(f), and 3.3(i) are corresponding detection results of the uncoupled RFCNN. From Figs. 3.3(b), 3.3(e), and 3.3(h), we can see that the learned structure and CNN templates of the RFCCNN are well suited to detect the defects of color filer images. It has also been confirmed that detection results are still good if the images are shifted or rotated. One of simulation results is shown in Fig. 3.4.
Figure 3.2 The outputs of Layer 3, 4, and Feedback Layer for the training image. (a) The output of the RFCCNN. (b)~(g) The outputs of the six Layer-4 nodes, respectively. (h)~(m) The outputs of the six CNNs in the Feedback Layer, respectively.
(n)~(s) The outputs of the six Layer-3 nodes, respectively (firing strength of each rule).
(b) (c) (d) (e) (f) (g)
(h) (i) (j) (k) (l) (m)
(n) (o) (p) (q) (r) (s)
(a)
.
From Fig. 3.3, we can see some differences between RFCCNN (coupled RFCNN) and uncoupled RFCNN for defect inspection of color filter. First, as shown in Figs.
3.3(g) to 3.3(i), the results of the coupled RFCNN is better than those of uncoupled RFCNN for detecting the large defects on the top-left of test image shown in Fig.
3.3(g). Second, as shown in all test images and corresponding detection results, the results of the coupled RFCNN are better than those of uncoupled RFCNN for detecting the black defects. The results of the coupled RFCNN are better for detecting the large defects in that the coupled RFCNN, like coupled CNN, has fully outputs feedback and will take care of further neighboring pixels. Some quantity comparisons are shown in Table 3.1. Here detection rate is defined as the ratio of detected defect pixel number to real defect pixel number. As we can see from Table 3.1, the detection rates of the RFCCNN are better than those of the RFCNN, no matter for the all defects, large defects, or black defects. More testing images and corresponding
detection results of the RFCCNN are shown in Fig. 3.5 to 3.8.
Figure 3.3 Simulation (Testing) results of the learned RFCCNN and RFCNN. (a), (d), and (g) are input testing images. (b), (e), and (h) are corresponding detection results of RFCCNN. (c), (f), and (i) are corresponding detection results of uncoupled RFCNN.
Table 3.1 Comparison of detection rate.
Detection rate of the RFCCNN
Detection rate of the RFCNN
All defects 56% 43%
Large defects 76% 52%
Black defects 72% 51%
Figure 3.4 Simulation results of shifted and rotated images. (a), (c), and (e) are input testing images of original, shifted, and rotated ones, respectively. (b), (d), and (e) are corresponding detection results of RFCCNN.
Figure 3.5 Other testing results of the learned RFCCNN and RFCNN, part 1. (a), (d), and (g) are input testing images. (b), (e), and (h) are corresponding detection results of RFCCNN. (c), (f), and (i) are corresponding detection results of uncoupled RFCNN.
Figure 3.6 Other testing results of the learned RFCCNN and RFCNN, part 2. (a), (d), and (g) are input testing images. (b), (e), and (h) are corresponding detection results of RFCCNN. (c), (f), and (i) are corresponding detection results of uncoupled RFCNN.
Figure 3.7 Other testing results of the learned RFCCNN and RFCNN, part 3. (a), (d), and (g) are input testing images. (b), (e), and (h) are corresponding detection results of RFCCNN. (c), (f), and (i) are corresponding detection results of uncoupled RFCNN.
Figure 3.8 Other testing results of the learned RFCCNN and RFCNN, part 4. (a), (d), and (g) are input testing images. (b), (e), and (h) are corresponding detection results of RFCCNN. (c), (f), and (i) are corresponding detection results of uncoupled RFCNN.
The main idea of the proposed RFCCNN is an integrated system of FIS and CNNs, which can construct fuzzy rules and CNN templates automatically. There are two general approaches to realizing the RFCNN model by real-time hardware. First, it can be “coded” and run in the CNN universal machine (CNNUM). The CNN-UM can handle analog and digital signals with built-in converters and memories by instructions. It is generally considered as a general-purpose image-processing computer. In fact, a fuzzy-rule-based image-processing algorithm [7] has been successfully implemented on the CNN-UM. Another approach is to design application-specific RFCNN circuits for particular applications with prelearned fuzzy rules and CNN templates. To achieve this, the circuit design technique of multilayer CNN [43, 44] called MLCNN can be applied to implement the multilayer structure of the proposed RFCNN model in Figure 3. A CNN-based Gaussian function circuit as designed in [45] can realize the Gaussian membership function required in Layer 2.
The fuzzy logic operations in Layers 3 and 4 can be realized by analog CNN circuits as studied in [7, 11]. Therefore, it is very promising and feasible to implement the RFCCNN using high-speed analogic circuits.
3.5. Concluding Remarks
In this chapter, we extended our previous work, called RFCNN, from considering uncoupled CNNs to coupled CNNs, called RFCCNN, for automatically constructing a multiple-CNN integrated neural system. This CNN-based fuzzy neural network can automatically learn its proper network structure and parameters simultaneously. In order to verify the capability of the RFCCNN, a defect inspection problem has been demonstrated and compared to the RFCNN. The simulation results show that the performance of the proposed RFCCNN is better than that of RFCNN. In addition, a scheme for hardware realization of the RFCCNN has been proposed. We believe such an integrated CNN system, the RFCCNN, has potential to solve more complex intelligent problems.