Fig. 8. The significant procedures for discrimination.
transformation of a linear combination of input variables with their weights and biases by a transfer function f produces the neuron output y:
y = f
n i = 1
w
i
·x
i
− u 4
where x
i
, input variables; w
i
, weights of input variables; u, bias; and y: neuron output.
Several types of transfer functions have been developed. Choice of the transfer function depends upon applications. A typical sigmoid non-linear transfer function
is shown in Fig. 7b. The ANN consists of a large number of highly inter-connected processing
elements neurons. It is therefore more robust than the other methods in dealing with complex problems. The most frequently used ANN is a back propagation
neural network BPNN. The BPNN is a typical supervised neural network whose learning rules are to keep adjusting the weights and biases of the network to
minimize the sum of squared errors of the network. The minimization is done by continually changing the values of the network weights and biases in the direction
of the steepest decent with respect to the sum of errors. This is called a gradient descent procedure. Changes in each weight and bias are proportional to element’s
effect on the sum-squared error of the network. Eventually, we use these weights and biases when the training finished to build the classification model for rotifers.
3. Results and discussion
A flow chart showing significant procedures for a complete discrimination model is shown in Fig. 8. It still followed the strategy of the two-stage model as mentioned
in Section 2.5, but a redundant information reduction procedure was inserted for better efficiency. Several experimental data were used to verify the model pro-
grammed with MATLAB version 5.1, The Mathworks Inc..
3
.
1
. Separation of debris from rotifers Debris were separated from rotifers in this stage. Rotifers and debris were not
easily distinguished using moment invariants. Fig. 9a shows a scatter plot of moment invariants, f
8
and f
9
. Data for rotifers crowded around point 0, 0 and data for debris spread more widely than the rotifers. A close up view of data near
point 0, 0 in Fig. 9a is presented in Fig. 9b. We could easily conclude that the entire range in f
8
and f
9
of debris has covered the rotifers. This situation was
Fig. 9. Plot of moment invariants f
8
and f
9
: on the legend, the hollow square
represents only the debris and solid symbols , and
represent rotifers without eggs, rotifers with 1 egg, and rotifers
with 2 eggs, respectively. a Full scope of f
8
and f
9
, including three types of rotifers as well as debris distributed in the plane. From this plot, we can find that the data regarding rotifers are crowded around
point 0, 0. b To make the neighborhood near point 0, 0 clear, we magnified the part of full scope near point 0, 0.
Table 3 Experimental results for separating debris from rotifers in SP-1, SP-2, and SP-3 using similarity
measurement approach Rotifers
ActualPredicted as Debris
SP-
1:
7 162
Rotifers 27
7 Debris
SP-
2:
7 325
Rotifers Debris
15 28
SP-
3:
13 331
Rotifers 22
13 Debris
found in other moment invariants, which resulted in poor separating condition of the debris from the rotifers.
Tables 3 and 4 show the results obtained using the method introduced in Section 2.5.1. Here SP-1, SP-2 and SP-3 are specimens from Table 1. The tables reported
both correct and incorrect discriminations with the two approaches. For SP-2 in Table 3, 325 rotifers were correctly classified as rotifers, 7 rotifers as debris, 15
debris as rotifers, and 28 debris as debris. The misclassifications resulted from an overlap in the moment invariants as mentioned previously. A comparison between
these two methods is shown in Table 5, where the error count is the sum of misclassifications. In Table 5, the misclassification rate is the percentage of error
counts over the total number in each sample batch. As pointed out in this table, misclassification rate with the degree of membership approach 2.61 is lower than
the one with the similarity measurement 6.48. However, the degree of member- ship approach reflects the real distribution characteristics of each group. The
similarity measurement approach represents the relationship merely by the distance to the class center that is linearly proportional. Therefore, the former approach
should be better than the latter.
Table 4 Experimental results for separating debris from rotifers in SP-1, SP-2, and SP-3 using degree of
membership approach Rotifers
Debris ActualPredicted as
SP-
1:
167 2
Rotifers 4
Debris 30
SP-2: 3
329 Rotifers
6 Debris
37 SP-3:
339 5
Rotifers 5
Debris 30
Table 5 Comparison of misclassifications between similarity measurement and degree of membership approach
Similarity measurement approach Degree of membership approach
Error count Error count
Sample Misclassification
Misclassification rate
rate 203
6 14
2.96 6.90
375 9
22 2.40
5.87 10
2.64 6.86
379 26
Total: 29
2.61 6.48
957 62
3
.
2
. Selection of moment in6ariants There are two problems when moment invariants were used to classify rotifers
and debris: redundant information and magnitude differences for moment invari- ants. The problem of the magnitude differences among moment invariants is
discussed in the next section. We propose a method to select efficient invariant features for redundant information reduction in this section.
The bases in the kernel of Fourier – Mellin transform are not orthogonal Sheng and Arsenault, 1986. This causes the moment invariant f
i
, derived by Fourier – Mellin transform, correlated or redundancy existed among moment invariants. A
subset of these invariant features which is sufficient to accomplish the pattern recognition is necessary to reduce the complex correlation. It is difficult to follow
a particular method to determine the appropriate set of invariants and to analyze the redundancy of information. In general, this selection depends on processed
images.
A Euclidean distance approach was proposed to reduce the dimension of the feature space Johnson and Wichern, 1998. The coefficient l stands for the grade
of separation which is the ratio of inter-class distance to within-class distance: l =
inter-class distance within-class distance
. 5
Generally, the inter-class distance is greater than the within-class distance to reduce the overlapping condition introduced in Section 2.5.1.1. Table 6 shows a
preliminary test for the whole set of features f
1
– f
12
. Ten patterns in each rotifer class were chosen randomly to calculate l for different features and to examine
their effectiveness of discrimination. The above discussed method can be formu- lated as
l = B
i, j
W
k
\ 1
6 B
i, j
: inter-class distance between groups i and j, W
k
: within-class distance of group k, for i, j = 0, 1, 2; k = i or j.
In Table 6, each l was examined for every i, j pair in columns. ls of f
1
, f
2
,
47 C
.Y .
Yang ,
J .J
. Chou
Aquacultural
Engineering
24 2000
33 –
57
Table 6 Effective feature selection to reduce the feature space dimension
a
f
5
f
6
f
7
f
8
f
9
f
10
f
11
f
12
f
1
f
2
f
3
f
4
2.99E+00 2.90E+00
3.25E+00 5.92E−01
1.98E+00 2.79E+00
1.38E+00 2.72E+00
2.68E+00 7.42E+00
1.76E+00 2.41E−01
B
0, 1
2.99E+00 2.91E+00
2.96E+00 2.50E−01
1.16E+00 2.66E+00
2.47E+00 B
1, 2
8.20E+00 9.22E−02
7.98E−01 1.03E+00
2.77E+00 1.51E−03
1.17E−02 2.89E−01
3.42E−01 8.17E−01
1.33E−01 8.57E−02
2.56E−01 3.53E−01
7.77E−01 B
2, 0
1.49E−01 9.67E−01
9.10E−03 3.56E−02
6.23E−01 2.64E−02
5.32E−02 2.01E−02
8.29E−03 W
6.22E+00 1.11E−02
6.76E−02 3.16E−01
1.08E−01 2.79E+00
1.96E+00 5.23E+01
1.01E−01 5.37E−01
1.27E+00 1.63E+00
1.45E+00 6.00E+01
9.41E−01 3.58E−01
3.48E−02 W
1
6.56E−03 W
2
4.44E−02 2.15E−01
4.71E−02 1.76E−01
1.44E−01 1.90E−01
4.03E+01 1.76E−02
1.58E−01 3.25E−01
4.04E−02 328.21
81.47 5.21
22.45 37.16
139.25 4.38
328.58 24.82
1.19 26.09
21.66 B
0,1
W l
1.07 1.48
0.06 5.89
3.68 2.20
1.88 B
0,1
W
1
0.12 6.92
4.93 1.47
1.65 1.07
1.49 0.06
2.49 2.16
2.10 1.70
1.70 0.14
1.10 2.23
2.65 B
1,2
W
1
455.69 B
1,2
W
2
65.56 13.78
5.32 6.58
18.53 12.96
0.20 5.23
5.05 3.18
68.55 0.23
0.26 1.35
7.27 4.64
0.92 2.12
1.35 1.09
0.02 B
2,0
W
2
8.46 6.12
0.17 B
2,0
W 0.33
0.46 12.96
15.37 6.62
30.89 0.12
13.38 14.29
1.12 0.79
a
An effective feature should satisfy conditions in Eq. Eq. 24. A set of features {f
1
, f
2
, f
3
, f
8
, f
9
, f
11
} was finally selected
Table 7 Comparison between inter-class distances and within-class distances in full 12-D feature space
Rotifers with 0 egg Rotifers with 1 egg
Rotifers with 2 eggs and 1 egg
and 0 egg and 2 eggs
S B
0,1
Inter-class S
B
1,2
S B
2,0
10.83 1.63
10.71 distance
Within-class S
W
1
S W
1
S W
2
S W
2
S W
S W
distance 6.26
79.77 79.77
40.34 40.34
6.26 S
B
1,2
S W
1
S B
1,2
S W
2
S B
0,1
S W
S B
2,0
S W
2
l S
B
2,0
S W
S B
0,1
S W
1
0.14 0.27
0.04 0.26
1.71 0.13
f
3
, f
8
, f
9
and f
11
are greater than 1. As a result, a set of features {f
1
, f
2
, f
3
, f
8
, f
9
, f
11
} was selected to reduce feature space dimension from 12-D to 6-D. For rotifer discrimination, it was preferable to use a subset of lower-order moments
than using the entire set of moments. The corresponding results are listed in Table 7 and Table 8 for comparison after
redundant information reduction. Here, B
i, j
and W
k
are defined as B
i, j
=
n
B
i, j,f
n
2 12
7 W
k
=
n
W
k,f
n
2 12
8 for i, j, k = 0, 1, 2, and f
n
moment invariant set.
Table 8 Comparison between inter-class distances and within-class distances after redundant information
reduction of the feature space, when f
1
, f
2
, f
3
, f
8
, f
9
, f
11
are used Rotifers with 1 egg
Rotifers with 0 egg Rotifers with 2 eggs
and 0 egg and 2 eggs
and 1 egg S
B
2,0
S B
1,2
S B
0,1
Inter-class distance
1.39 3.03
4.09 S
W
1
S W
2
S W
2
S W
Within-class S
W S
W
1
distance 1.85
0.45 0.45
0.33 0.33
1.85 S
B
1,2
S W
2
S B
0,1
S W
S B
2,0
S W
2
S B
2,0
S W
S B
0,1
S i
1
S B
1,2
S W
1
l 4.23
3.11 6.78
1.64 12.45
2.21
Fig. 10. The BPNN model for rotifer classification.
In most cases, ls in Table 8 were greater than those in Table 7. In other words, increase in the distance between groups made the discrimination easier.
3
.
3
. Classification of rotifers A BPNN described in Section 2.5.2 was proposed as a classifier for various
rotifer types. To establish the model, a batch of training samples with reduced features f
1
, f
2
, f
3
, f
8
, f
9
, f
11
for arbitrary types of rotifers was necessary. With the six moment invariants and an object area A as input variables, the BPNN
model produced appropriate weights and biases for each node upon finishing the training. Since the magnitude of the invariant features in lower-order was often
larger than the features in higher-order, the input variables were unbalanced. Using a weighting operation normalization on the features could counterbalance this
unbalancing situation. A convenient way to normalize the features is to divide each coefficient by their maximum value. A schematic diagram of the three-layer BPNN
model with five neurons in the hidden layer is represented in Fig. 10.
One hundred and eighty-five rotifer samples mentioned in Section 2.5.1.1 have been trained. The experimental results for three validation batches are shown in
Table 9. Transfer functions for the neurons in the hidden layer and output layer were sigmoid functions:
fx = 1
1 + e
− x
. 9
In Table 9, the rows with condition ‘‘debris not discarded in stage 1’’ represent the objects misclassified in the first stage, as described in Table 4. The total
classification rate in stage 2 is estimated to be about 93.15 on an average Table 10. If the condition ‘‘debris not discarded in stage 1’’ was omitted, it reached up
to 94.92. The high accuracy indicates that the ANN approach is effective in the second stage of discrimination. Table 9 shows that the classification rate decreases
50
C .Y
. Yang
, J
.J .
Chou Aquacultural
Engineering
24 2000
33 –
57
Table 9 Three experimental results for classification stage 2, reduced features: f
1
, f
2
, f
3
, f
8
, f
9
, f
11
and area were used as inputs
a
Total Classification rate
Rotifer without egg ActualPredicted as
Rotifer with 1 egg Rotifer with 2 eggs
SP-
1:
Rotifer without egg 98.96
96 95
1 85.71
30 5
35 Rotifer with 1 egg
36 83.33
Rotifer with 2 eggs 1
5 30
4 Debris not discarded in stage 1
3 1
90.64 95
30 30
171 Correct count
SP-
2:
242 97.93
Rotifer without egg 5
237 73
95.89 Rotifer with 1 egg
70 2
1 78.57
Rotifer with 2 eggs 3
14 11
4 2
6 Debris not discarded in stage 1
335 94.93
237 Correct count
70 11
SP-
3:
211 97.16
2 Rotifer without egg
4 205
63 96.83
61 Rotifer with 1 egg
1 1
65 87.69
Rotifer with 2 eggs 57
1 7
1 1
3 5
Debris not discarded in stage 1 344
93.90 61
Correct count 57
205
a
Samples were from SP-1, SP-2, and SP-3.
Table 10 Total classification rate in stage 2
a1
Batch Comparison
SP-3 Average
SP-1 SP-2
94.93 93.90
90.64 93.15
Condition: ‘‘debris not discarded in stage 1’’ was included
92.81 Condition: ‘‘debris not discarded in stage 1’’
96.66 95.28
94.92 was excluded
a
System performance was examined and compared when the condition ‘‘debris not discarded in stage 1’’ was included or excluded.
as the number of rotifer eggs increases. It indicates that the rotifer shape becomes complex when it carries multiple eggs. The complex shapes resulted in more
overlapping in the discrimination space of moment invariants. Table 11 shows the classification results using a full set of features. The
classification rate was less accurate compared to the classification in Table 9. The classification rate was degraded due to serious misclassification of rotifers with two
eggs. Although the classification rate increased slightly for the first two classes in Table 11, the total classification rate did not increase accordingly due to the serious
decrease for the last class. This consequence followed the assumption in Section 3.2, and validated the importance of the removal of non-effective features. The accuracy
for each stage was summarized in Table 12. The average accuracy in the three batches was about 97.34 in the first stage, 93.15 in the second stage, and 92.49
overall. The processing time was less than 0.1 s for a frame of image if parameters in the system had already tuned.
4. Conclusions
