Learning algorithm effect on multilayer feed forward artificial neural

Learning algorithm effect on multilayer feed forward artificial neural
Journal of Engineering Science and Technology
Vol. 2, No. 2 (2007) 188 - 199
© School of Engineering, Taylor’s University College
LEARNING ALGORITHM EFFECT ON MULTILAYER FEED
FORWARD ARTIFICIAL NEURAL NETWORK PERFORMANCE
IN IMAGE CODING
OMER MAHMOUD1, FARHAT ANWAR1, MOMOH JIMOH E. SALAMI2
1
Department of Electrical and Computer Engineering, Faculty of Engineering,
International Islamic University Malaysia, MALASIA.
2
Department of Mechatronics Engineering, Faculty of Engineering,
International Islamic University Malaysia, MALAYSIA.
*Corresponding Author: [email protected]
Abstract
One of the essential factors that affect the performance of Artificial Neural
Networks is the learning algorithm. The performance of Multilayer Feed
Forward Artificial Neural Network performance in image compression using
different learning algorithms is examined in this paper. Based on Gradient
Descent, Conjugate Gradient, Quasi-Newton techniques three different error
back propagation algorithms have been developed for use in training two types
of neural networks, a single hidden layer network and three hidden layers
network. The essence of this study is to investigate the most efficient and
effective training methods for use in image compression and its subsequent
applications. The obtained results show that the Quasi-Newton based algorithm
has better performance as compared to the other two algorithms.
Keywords: Image Compression /Decompression, Neural Network, Optimisation
1. Introduction
The need for effective data compression is evident in almost all applications
where storage and transmission of digital images are involved. For examples an
1024 X1024 color image with 8 bit/pixel generates 25Mbits data, which without
compression requires about 7 minutes of transmission time over 64 kbps line. A
Compact Disk with storage capacity of 5 Gbits can only hold about 200
uncompressed images. Reducing the image size by applying compression
techniques is usually possible because images contain a high degree of spatial
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Multilayer Feed Forward Artificial Neural Network Performance
189
Nomenclatures
B
E
e
g
H
l
n
p
w
Xi
X'i
Approximate inverse Hessian matrix
Error function over all neurons in output layer
Error signal at the output layer of the ith neuron at iteration n
Gradin vector
Hessian matrix
Index of the image blocks
Number of iterations
Direction vector
Vector of all weights
Desired output
Actual output
Greek Symbols
ή
Learning rate
α
Constant
redundancy due to correlation between neighbouring pixels. There are two basic
types of image compression: lossless compression and lossy compression. A
lossless scheme encodes and decodes the data perfectly, and the resulting image
matches the original image exactly. Lossy compression schemes allow redundant
and nonessential information to be lost and it has higher compression ratio.
Compression ratio is simply the size of original image divide be the size of the
compressed one. Typically with lossy compression methods there is a tradeoff
between compression ratio and obtained image quality. Apart from the existing
technology on image compression represented by series of JPEG, MPEG
standards, new technology such as neural networks and genetic algorithms are
being developed to explore the future of image coding [1, 2]. Oja [3] proposed
the use of a simple neural network that can perform nonlinear principal
component analysis as a transform-based method in image compression. Since
this pioneering work, several new learning algorithms have been proposed for
extending this approach [4, 5].
Learning algorithms has significant impact on the performance of neural
networks, and the effects of this depend on the targeted application. The choice of
suitable learning algorithms is therefore application dependent. This paper
presents a comparative study that shows the effect of using different learning
algorithms on the performance of Multilayer Feed Forward Artificial Neural
Network (MFFANN) with Error Back- propagation Algorithm in images coding.
The paper starts with an Introduction followed by an overview of Multilayer Feed
Forward Artificial Neural Network with Error Back propagation Algorithm
approach in image compression. It then explains the functionalities of three
different learning algorithms. Finally the results obtained from the simulation
study are presented.
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O. Mahmoud et al
2. Image Compression Using MFFANN
This section presents the architecture of a Feed Forward Neural network that is
used to compress images in the research work. The MFFANN consists of one
Input Layer (IL) with N neurons, one Output Layer (OL) with N neurons and one
(or more) Hidden Layer (HL) with Y neurons where, the network is trained using
Error Back propagation Algorithm. The network is designed in a way such that
N>Y, where N is input layer/output neurons and Y is hidden layer neurons as
shown in Fig. 1
X1
X'1
X2
X'2
.
XN
WNY
Y
WYN
X'N
Fig. 1. Three layers Neural Network.
Compression Yi = ∑ Wji * Xi, decompression X'j= ∑W’ij * Yj
When MFFANN is used for image compression the process require the following
three steps
2.1 Training the MFFANN
The input image is split up into blocks or vectors of 4×4, 8×8 or 16×16 pixels.
These vectors are used as inputs to the network. The network is provide by the
expected (or the desired) out put, and it is trained so that the coupling weights,
{wji}, scale the input vector of N-dimension into a narrow channel of Y-dimension
(Y < N) at the hidden layer and produce the optimum output value which makes
the quadratic error between output and the desired one minimum. In fact this part
represents the learning phase, where the network will learn how to perform the
task. In this process of leering a training algorithm is used to update network
weights by comparing the result that was obtained and the results that was
expected. It then uses this information to systematically modify the weight
throughout the network till it finds the optimum weights matrix. As explained in
section 3
Divide the
training image
into blocks
Scale each block
Apply it to IL and
get the output of
OL
Adjust the weights
to minimize the
difference between
the output and the
desired output
Repeat until
the error is
small
enough
Fig. 2. Block Diagram of MFFANN Training
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2.2 Encoding
The trained network is now ready to be used for image compression which, is
achieved by dividing or splitting the input images into blocks after that scaling
and applying each block to the input of Input Layer (IL) then the out put of
Hidden layer HL is quantized and entropy coded to represent the compressed
image. Entropy coding is lossless compression that will further squeeze the
image; for instance, Huffman coding code be used here. Fig. 3 shows the
encoding steps
Input image
Divide the image
into blocks
Scale each block
And use it as
input to IL and
take HL
Quantizing &
entropy coding
Compressed
image
Fig. 3. Block Diagram of the Encoding Steps.
2.3 Decoding
To decompress the image; first decode the entropy coding then apply it to the out
put of the hidden layer and get the out put of the OL scale the it and reconstruct
the image. Fig. 4 show the decoder block diagram.
Decompressed
image
Scale it and
Reconstruct the
image
Set it to the
output of the HL
and get out put
of OL
Entropy decode
Compressed
image
Fig. 4. Block Diagram of the Decoding steps
3. Weight Adjustment
The networks weights need to be adjusted in order to minimise the difference or
the error between the output and the expected output. This is explained in the
equations below.
The error signal at the output layer of the ith neuron at iteration n is given by
ei (n) = X i (n) − X 'i (n)
(1)
where Xi represent the desired out put and X'i represent the actual out put. The
error function over all neurons in output layer is given by Eq. (2).
El (n) = ∑ ei (n)
2
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The error function, over all input vectors in the training image, is
E = ∑ El , El = ( X ' , w)
(3)
where l indexes the image blocks (inputs vector), X' is the vector of outputs, and
w is the vector of all weights. In order to minimise the error function with respect
to weight vector (w) it is necessary to find an optimal solution (w*) that satisfy
the condition
E ( w*) ≤ E ( w)
(4)
The necessary condition for the optimality is
∆E ( w) = 0
(5)
where ∆ is gradient operator
∆ = [∂ / ∂w]
(6)
and ∆E (w) is gradient vector (g) of error function is defined as follows
∆E ( w) = ∂E / ∂w
(7)
The solution can be obtained using a class of unconstrained optimization
methods based on the idea of local iterative descent. Starting with initial guess
denoted w(0), generate a sequence of weight vectors w(1),w(2)… such that the
error function is reduced for each iteration
E ( wn +1 ) ≤ E ( wn )
(8)
In this study three optimisation methods will be considered, Gradient Descent,
Conjugate Gradient method and Quasi Newton method. These methods are the
most popular techniques used for iteratively solving unconstrained minimization
problems.
3.1 Gradient descent (GD)
In this method of Gradient descent (also known as steepest descent) [7] the
successive adjustments applied to the weight vector are in the direction of the
steepest descent that is the direction opposite to the gradient vector ∆ E (w)
wn +1 = wn + ∆wn
(9)
∆wn = −η n g n
(10)
where g is gradient vector and η is a small positive number called the learning
rate, which is the step size needed to take for the next step. Gradient descent only
indicates the direction to move, however the step size or learning rate needs to be
decided as well. Too low a learning rate makes the network learn very slowly,
while too high a learning rate will lead to oscillation. One way to avoid oscillation
for large η is to make the weight change dependent on the past weight change by
adding a momentum term,
∆wn +1 = −η n g n + α∆wn
(11)
That is, the weight change is a combination of a step down the negative gradient,
plus a fraction α of the previous weight change, where 0 ≤ α < 0.9
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3.2 Conjugate gradient methods (CG)
The conjugate-gradient method [7, 8] uses a direction vector which is a linear
combination of past direction vectors and the current negative gradient vector. In
so doing the conjugate-gradient method reduces oscillatory behavior in the
minimum search and reinforces weight adjustment in accordance with previously
successful path directions.
Let p(n) denotes the direction vector at the nth iteration of the conjugate-gradient
method algorithm. Then the weight vector of the network is updated as a linear
combination of the previous weight vector and the current direction vector
according to.
(12)
wn +1 = wn + η n pn
The initial direction vector, p(0), is set equal to the negative gradient vector,
g(0) at the initial weight w(0).
Successive direction vectors are computed as a linear combination of the current
negative gradient vector and the previous direction vector as shown below
(Fletcher -Reeves version).
pn +1 = − g n +1 + bn pn
(13)
where
[
][
bn = g nT+1 g n +1 / g nT g n
]
(14)
3.3 Quasi Newton methods (QN)
Quasi-Newton methods [7] are based on Newton’s method The basic idea behind
Quasi-Newton methods is to use an approximation of an inverse Hessian in place
of true inverse as required in Newton’s method. A typical iteration for this
method is
wn +1 = wn + η n d n ; Where d n = − Bn g n
(15)
Where Bn is a positive definite matrix (approximate inverse Hessian matrix which
is adjusted from iteration to iteration) chosen so that the directions dn tend to
approximate Newton’s direction. The step size ηn is usually chosen by a line
search. The important idea behind the methods is that two successive iterates xn
and xn+1 together with the gradients ∆f(xn) and ∆f(xn+1) contain curvature (i.e.,
Hessian) information, in particular,
∆f ( xn +1 ) − ∆f ( xn ) ≈ H ( xn )( xn +1 − xn )
(16)
Therefore, at every iteration it would be necessary to choose Bn+1 to satisfy
Bn +1qn = zn
(17)
Where
z = xn +1 − xn ; q = ∆f ( xn +1 ) − ∆f ( xn )
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In the most popular class of Quasi- Newton methods the matrix Bn+1 is obtained
from the previous Bn , vector q and z by suing the following equation. (BFGS
version).
Bn+1 = Bn + zz T / qT q − Bn qqT Bn / qT Bn q
+ qBn qT + z / z T z − Bn q / qT Bn q
(19)
4. Results and Discussion
The MFFANN with three layer and five layers is implemented. The network is
trained using Error Back propagation Algorithm that utilizes optimization
methods described in section 3. The obtained results are presented in the
subsequent sections.
4.1 Learning algorithms performance during training
In order to find the best method for training a neural network that performs image
compression and decompression. The above mentioned methods used to update
the weights in Feed forward neural network with error back propagation
algorithms. The Initial layers weight matrix was the same for all methods. Table 1
shows the details obtained for each method and Fig. 5 show performances (RMSE
vs. Number of Epochs) of the three methods during training. The data shown in
table 1 and Fig. 5 belong to a network trained using the image in Fig. 6a.
Furthermore, similar results were obtained using different sample images, for
instance MIR images, and different number of epochs (e.g. 700, 1000).
Table. 1. Performance Parameters During Training.
Epochs
GD
400
Time
(sec)
17.22
RMSE
149.83 *10-5
Image size
(pixels)
291x240
Compression
ratio (IL/HL)
4:1
CG
400
39.70
18.925 *10-5
291x240
4:1
QN
400
41.09
11.110 *10-5
291x240
4:1
In Fig. 5, the x-axis represent the training time in seconds and the y-axis
represents the performance of the network in term error between in input and
output. Image size in pixels refers to the size of image used in the training. Fig. 5
and Table 1 show that GD method takes less time as compared to CG and QN
methods. However in term of error between the input and the output QN has
performed better than other two methods. Despite the fact that QN methods take
longer time during training it the suitable among the three methods since it has
least RMSE error. As results the decompressed image has higher quality
compared to the other two methods. CG training time is slightly less than QN
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time. The CG decompressed images quality could be enhanced by increasing the
number of training epochs.
Fig. 6 show the resulted output from the previous training for each methods of
weight updating
RMSE
Number of Epochs
Fig. 5. Training Performance.
An important feature of neural networks is the generalization ability which, refers
to the performance ability of the network with new data that were not used during
training [7]. To test the network generalization ability the pervious Network has
been used to compress the image shown in Fig. 7a corresponding decompressed
images are shown in Fig. 7b and Fig. 7c. The image in Fig. 7.c has higher quality
compared to image in Fig. 7b which indicates that QN has better performance
with respect to processing new images.
4.2 Still image compression /decompression
The following sections present the results obtained for MFFANN with one hidden
layer and three hidden layers. Networks trained using error back propagation with
QN optimization methods to update the weights. The compression ratio shown
here represents ratio between input layer and hidden layer. Without entropy
coding, that can compress the image further, up to three or four times, with out
affecting its quality.
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a. Original image
b. Decompressed image 4:1using GD
c. Decompressed Image 4:1e using CG
d. Decompressed Image 4:1 using QN
Fig. 6. Original Image and Decompressed Image.
The results in Fig. 8 show that it possible to achieve good decompressed image
with ratio 8:1 using neural networks with different number of neurons and layers.
For instance, the image in Fig. 8b is obtained using network with 16 neurons in
the input/output layer and single hidden layers with 2 neurons while the image in
Fig. 8c obtained using network with 64 neurons in the input/output layer and
single hidden layers with 8 neurons. Fig. 8d is obtained using network with five
layers, two input and output layers with 64 neurons and three hidden layers with 16, 8 and
16 neurons.
Generally the training time will increase for increased number of neurons and layers. It
is also observed, in general, that single hidden layer network compressed image has a
better quality compared to the three hidden layers network compressed one.
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a. Original Image.
197
b. CG Decompressed Image.
d. QN Decompressed Image.
Fig. 7. Generalization Test
4.3 Video compression
This section discusses an extension of the MFFANN compression approach to
compress video frames. In order to handle video compression, the MFFANN
compression applied to each frame in a sequence which resulted in a compressed
video sequence similar to the technique used in Motion-JPEG [10]. Fig. 9 shows
the results obtained by using this approach on video frames Fig. 9a show original
frame. Fig. 9b is the corresponding decompressed frame
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a. Original Image.
b. Decompressed Image 8:1
RSME=0.0518
Original Image Sampled 4×4 Pixels
(One Hidden Layer)
c. Decompressed image 8:1
RMSE = 0.0431
Original Image Sampled 8×8 Pixels
(One Hidden Layer)
d. Decompressed Image 8:1
RMSE = 0.0630
Original Image Sampled 8×8 Pixels
3 Hidden Layers (16, 8 and 16 neurons)
Fig. 8. Decompressed Images of Single and Three Hidden Layers Networks
a. Original Frame.
b. Decompressed Frame.
Fig. 9. Video Frames
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5. Conclusion s
In this paper neural network has been used for image compression. Based on the
obtained results the gradient descent takes less time during training as compared
to Conjugate Gradient methods and Quasi Newton methods. However Quasi
Newton performs better in term of minimizing the error as such the image
compressed by Quasi Newton has a higher quality and better generalization
ability. It is also observed that, in general, one hidden layer network compressed
image has better quality opposed to the three hidden layers network compressed
one.
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