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III. Neuro-fuzzy system
Neuro-fuzzy system is a combination of neural network and fuzzy system in such a way that neural network learning algorithms, is used to determine parameters of the fuzzy system [20]. ANFIS is a neuro-fuzzy model proposed by Jang [11]. The structure of ANFIS with five layers is shown in Fig. 3. x and y are the inputs for ANFIS. Note that the input layer is not calculated as an ANFIS layer.


Figure 3: ANFIS Architecture
For learning rule of ANFIS, hybrid learning algorithm [4,5] which combines the gradient descent and least-squares method is used to find a feasible set of parameters.

Table 1 shows the hybrid learning procedure for ANFIS. Further information can be obtained from [10, 11, 12, and 20]

Table 1: Two passes in the hybrid learning procedure for ANFIS

	Forward Pass	Backward pass
Premise Parameters	Fixed	Gradient Descent
Consequest Parameters	Least squares estimate	Fixed

However, ANFIS itself only suitable for single output system. For a system with multiple outputs, ANFIS will be placed side by side to produce a Multiple ANFIS (MANFIS) [12]. The number of ANFIS required depends on the number of required output. Fig. 4 shows a MANFIS with five outputs. Since the input data remains the same for each ANFIS, they also have the same initial parameter such as initial step size? , membership function (MF) type and number of MF.


Figure 4: MANFIS with five output
IV. Experimental results
In order to examine the performance of this system, we have selected five 3D objects for this recognition. Some examples are shown in Fig. 5. As we mentioned earlier, each object will has 72 conditions. We choose odd condition (1, 3, 5, … 71) as a training data and the even condition (2, 4, 6,… 72) for the testing, so that the views of the testing images have never appeared in the training process at all. Hence, for five objects, we will have 180 data for training set and 180 testing data set. MANFIS with five outputs was used to perform this task.


Figure 5: Example of objects used in the experiment
We have analyzed the MANFIS performance using different initial parameter set. To find the best, first, we run our system using MF=2 with initial step size, ? =0.01, 0.05, 0.10, 0.25, and 0.35. Increasing the initial step size value will increase the learning rate for the ANFIS.

However, if the step size is set too large (i.e. 0.35), the system will fail to learn properly. Table 2 summarized the system performance.

Table 2: System performance using MF=2 with different step size

Step size	Maximum accuracy(%)
0.01	82.78
0.10	82.22
0.25	67.78
0.35	70.00

We also analyzed the system performance with the number of MF=3 and 4. MF=5 and above are not suitable for the analysis since the number of data is smaller than the number of adjustable parameters in the network. Table 3 and 4 summarized the results for each number of MF.

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