The work proposes a novel edge detection algorithm for noisy images which constitutes two phases, one for making noise free image noise free and the other for finding edges. The identification of edges is based on the occurrence of changes in the grey level in the regions. In particularly, the salt and pepper noise has been considered in this research for proposed edge detection method is represented in Fig. 1.
Fig. 1. Flow diagram of proposed edge detection
III.1. Adaptive Bilateral Filtering (ABF)
The main advantage of the classical bilateral filtering method is that it allows for consideration of both the spatial locality and neighboring points with similar amplitudes and time thus helps for preserving the image edges and textures better than the conventional linear filtering algorithms [15].
The Eq. (1) implies the dependence of behavior of the classical bilateral filtering method on the selection of the two parameters { , }.
However, these two parameters used for controlling the spatial and intensity weight, are constant for the entire image signal.
In other words, the local filters used in classical bilateral filtering method used to change depending on the underlying image signal. The changing degree is usually constrained by a fixed set of parameters{ , }.
There is not much theoretical research reports available on the method of selection of the optimal values for { , } in the filtering process.
Therefore, the two empirical parameters are usually selected. The analysis of the two parameters as a function of noise variance for image denoising applications has been studied already [16] and two conclusions about selection of parameters were given: one is the value should be approximately between [1.5 ~ 2.1] and the other is that the optimal value of is linearly proportional to the standard deviation of the noise under the sense of mean square error (MSE). The two parameters determined will remain unchanged in the filtering process for the entire image signal.
A typical non-stationary signal which consisting of smooth, edge and texture regions represents the natural image. As each region will have different statistics characteristics, the best image filtering performance can be achieved by either the optimal values of { , } or the optimal weights for those that changes based on the local statistics characteristics of the natural image change.
Therefore, spatially adaptive bilateral filtering method is proposed in this study to change the values of { , }as per changes in the spatial location during the filtering process. The spatially adaptive bilateral filtering method is represented as follows:
( ) =1 ( , ) ( , , ) ( )
∈
(1)
where the new intensity weight is given by:
( , , ) = {− | ( ) − ( )| / } (2)
The new normalization constant:
= ( , )
∈
( , ) (3)
The parameter , shown above represents a small constant, the noise variance and the local standard deviations of the image signal respectively. The intensity parameter in classical bilateral filtering plays a crucial role in the filteringprocess. For the given spatial parameter , the largest parameter will lead to more flatness of Gaussian shape of intensity weight whereas smaller the parameter will lead to sharpness of the shape of intensity weight. For the natural image signal in the smooth regions, the shape of intensity weight will be expected to be more flat as it could effectively smooth the noise. For the edge and texture regions, the shape of intensity weight is expected to be sharper for the well preservation of the edges and textures.
Image
Converted into pixel and form matrix
Edge
Fuzzy neural network based Fuzzy cognitive map for edge
detection
Calculate feature vectors values for each pixel
Entropy
Gradient Variance
Non-edge Removing the noise using adaptive bilateral filtering
T. Karthikeyan,N. P. Revathy
Beyond the adaptation of the local characteristics of the image signal, the intensity weight also depends on the noise level. When the signal to noise ratio is low, the shape of intensity weight should be more flatness to better suppress the noise.
Therefore, the optimal parameter should adaptive with the local characteristics of the image signal and the noise level.Considering the computational complexity, we use the local deviation to reflect the local statistical characteristics of the image. Comparing the new intensity weight with the classical intensity weight, it can be seen that the classical intensity parameter is replaced by:
= / (4)
III.2. Features Selected From Image
In this section, we will describe the four different features for image pixels that we use.
Pixel Features
The information of an image pixel can be best obtained by comparing the pixel's features with those of its neighboring pixels. This can be done by extracting 3 × 3 matrix of the neighboring pixels surrounding thepixel in question. (Other odd-number-sized matrices like 5 × 5 7 × 7 are also possible.) For any [ , ], its neighborhood matrix contains 9 : [ − 1, − 1], … . , [ + 1, + 1].
We use a 3 × 3 neighborhoodmatrix for extracting of the features of variance, entropy, and gradient, for each pixel in the image. These attributes hold special properties to determine edge and non-edge pixels of the image. For a grayscale image, the color intensity or grayness of a pixel [ , ] will be denoted as ( , ). the value of ( , ) rangesfrom 0 to 255.
Variance
Statistical properties like mean and variance contain important information about pixels. Variance is a common measure of how far the numbers lie from the mean. Low variance indicates small variation in grayness and high variance means large variation in grayness. So pixel with high variance is candidate to be an edge pixel. The mean ( , ) of grayness for 3 × 3 neighborhood matrix centered at the pixel [ , ] is computed as: ( , ) = (1 9⁄ ) ( + , + ) , (5) ( , ) = (1 9⁄ ) ( + , + ) − ( , ) , (6) Entropy
From information theory, we know that that the smaller the local entropy is, the bigger the information
gain and the rate of change in the intensity is. Thus, we can conjecture that the smaller the local entropy is, the bigger the dispersion is.
So, the pixel with big local entropy is more likely to be an edge pixel [17].For a given pixel [ , ], we take a 3 × 3 neighborhood matrix with that pixel at the center.
The entropy for the pixel [ , ] is calculated as:
( , ) = − (7)
= ( , )/ ( , ) (8)
Gradient
The gradient is the directional change in grayness of an image. The magnitude of the gradient tells us how quickly the image is changing, while the direction of the gradient tells us the direction in which the image is changing most rapidly. We use the similar gradient measure as the one used in the Sobel method [18].
The gradient ( , ) for a 3 neighborhood matrix centered at the pixel [ , ] is computed as:
( , ) = ( , ) + ( , ) (9)
( , ) is the mask in direction and ( , ) is the mask in direction respectively.
III.3. Edge Detection using Hybrid Fuzzy Cognitive Map based on Fuzzy Neural Network
Classification
In this section we present a new way to detect edges by using the hybrid fuzzy cognitive map based on fuzzy neural network [19]. The decision needed in this case is between "the pixel is part of an edge” or "the pixel is not part of an edge". In order to obtain this decision we must extract the information from the images since the entire image is not useful as the input to the hybrid fuzzy cognitive map based on fuzzy neural network (HFCM).
In order to obtain this decision we must extract the information needed from the images. In this work a vector is formed for each pixel given the difference between this one and the pixels in a 3 × 3 neighborhood around it. This way a components vector such as entropy, gradient, and variance are calculated at each pixel except for the border of the image, because in this case the differences cannot be calculated. This vector is used as input to the HFCM both in the training process and when we use the trained HFCM over real images.
The existing WSVM differs from the new approach in detecting the edge without considering the denoising method or by removing the noise from the image, this work remove the noise from image after feature extraction and edge detection for image processing.
T. Karthikeyan,N. P. Revathy
Copyright © 2014 Praise Worthy Prize S.r.l. - All rights reserved International Review on Computers and Software, Vol. 9, N. 9 An FCM based FNN method is used which helps in
improved performance of edge detection in image processing and further reduce the execution time of the existing methods.
Before performing the edge detection task first important features in the images are extracted as mentioned above, then perform edge detection task with improved FCM with FNN methods. So first study the basic FCM method how it is applicable to the edge detection, the major issue of the FCM method how it is solved by using the FNN for edge detection. FCM methods are a signed directed graph which consists of a set of features which corresponds to same feature for the image represented by nodes and a set of weights represented by directed arcs for features.
The active degree of features is described by state value ∈ [0, 1], either edge or non edge features which changes over time in inference process.
The value of the weight specifies how strongly the causal feature affects the effect feature . Whilea positive value of represents a proportional edge detection effect, a negative value represents an inversely proportional to edge detection results effect and zero represents the absence of an edge detection results effect.
A graphical representation of FCM for edge detection is depicted in Fig. 2.
Fig. 2. Flow diagram of proposed edge detection
The inference mechanism of FCM is described by the following formula: ⎩ ⎪ ⎨ ⎪ ⎧ ( ) = ( ( )) × , = 1,2, … . , ( + 1) = ( ) × = 0,1,2, … . , (10)
where is the iteration step, and ( ) indicates the state value of feature at iteration . ( )indicates the system is edge or non edge detection state at iteration, and is a threshold function. From the Eq. (10), it is inferred that the process of an FCM is an iterative process and will be applied to the scalar product and threshold function.
The discrete-time series of system state could be generated only until the FCM converges yields based on the following:
A fixed point: In this case ( + 1) = ( )( ∈ ) where ( ) is the state of detection of final edge or non edge. A limit cycle of this case is ( + ∆ ) =
( )( ∈ )in which ( ) is the final state. It is meant that the system falls in a loop of a specific period and after a certain number of inference steps △ , the same final state ( ) will be reached. A chaotic attractor in this case is the change in FCM state vector based on iteration. The states are repeated until there is no detection of edge or non-edge detection states.
FCM Based on OWA Operators
FCM uses simple weighted sum which is very less to represent the edge or non edge detection results for image. Hence, this paper introduces OWA operators instead of FCM, to replace weighted sum and threshold functions . An n-dimensional OWA operator is a mapping : → that has an associated weighting vector = ( , , … , ) of having the properties [20]-[21]:
+ + ⋯ + =
= 1 ≤ ≤ 1, = 1,2, … , (11)
and such that:
( , , . . , ) = (12)
where is the jth
largest element of the collection of the aggregated features { , , . . , }.The measure of orness of the aggregation is defined as:
( ) = 1
− 1 ( − ) (13a)
By introducing OWA operators to FCM the following reasoning formula were represented as:
( + 1) = ( ) ( ), , … , ( ) ( ) (13b) From this formula, it is found that OWA-FCM contains two types of weights information: weights for measuring causal link strength and weight vector for measuring And/Or relationships among data source.
Obtainment of OWA-FCM Weights
The weights of OWA operator are determined by the maximum entropy of the edge detection result which formulates the problem of weights of OWA operators based on the solution of the following mathematical programming problem [22]:
( ) = − (14)
T. Karthikeyan,N. P. Revathy
From the above method, the weight value is assigned without consideration of the edge result values. In order to overcome this problem and to consider the result based on weight assignment, the weight coefficient is obtained by fuzzy neural network training as described in the following sections.
Fuzzy Neural Network for Improved FCM Learning The Structure of the Proposed Fuzzy Neural Network
To solve the problem of the FCM random assignment of the weight value, this work proposes a fourlayer fuzzy neural network for realizing the automatic identification of weight value for each feature in the images for edge detection as shown in Fig. 3.
Fig. 3. The representation of the structure of the HFCM –FNN
Input Layer
In this layer, each input node represents a feature for each pixel in the image. Input values = [ , ,· · · , ,· · · , ] are directly transmitted to the next layer. So, the net input ( ) and net output ( ) of the ithnode (weight value of the feature for edge detection):
( )
= , ( )= ( ) (16)
Fuzzification Layer
Nodes in this layer represent the linguistic-term set of feature inputs. The th linguistic term of the input
feature variable for pixel it is denoted as ( = (1, 2,· · · , )), which is expressed as “small (S),” “medium (M),” or “large (L),” etc. In the proposed FNN, the fuzzy set is modeled by a symmetric Gaussian- membership with center and spread , which is represented by = , . Therefore, the net output ( ) of the th linguistic node are given by:
( ) = ( ) = ( ) (17)
Quantification and Defuzzification Layer
In this layer, the causalities among features and defuzzification are realized. The linguistic term of output variable is represented in the layer which indicates the same semantic symbol expressed by .
, is mutual between , to
measure the similarity between them. Fuzzy weights 1 − , represent the causal effect relationship from input linguistic term to output linguistic term
Therefore, the net input
, ( )
and net output
( )
of the output linguistic term are expressed:
, ( )
=
, ( )1 −
, (18) ( )=
∑
, ( )∑
, ( ), ≠ ,
= 1,2, … . , (19) Output LayerIn this layer, the net input ( )and net output ( )of the jth output are calculated by:
( )
= , ( ) (20)
( )
= = ( ) (21)
where , is the crisp weight from the output linguistic
term OL to the output variable y .In order to improve the speed of the FNN in the FCM ,in this work presents a Levenberg- Marquart (L-M) Training Algorithm , which combines the gradient falling algorithm with Newton method.
The iterative formula of L-M algorithm is as follows [23]:
Input layer Fuzzification layer Quantization and defuzzification layer Output layer X (concept 1) (concept i) X (concept N)
T. Karthikeyan,N. P. Revathy
Copyright © 2014 Praise Worthy Prize S.r.l. - All rights reserved International Review on Computers and Software, Vol. 9, N. 9
= − ( + ) (22)
in which is Jacobian matrix of derivatives of network function error, is the error that is calculated as the difference between desired edge detection and estimated edge detection outputs, is the identity matrix with proper dimensions. The variation of network weights and deviation:
∆ = −( + ) (23)
IV.
Experimental Results
The proposed edge detection method is simulated using MATLAB for different images. It is observed that this proposed method provides much more distinct marked edges as compared to other edge detection algorithms such as SVM and weighted SVM.
The performance metrics used for analyzing the proposed method are Mean square error (MSE), Peak signal to noise ratio (PSNR).
IV.1. Peak Signal to Noise Ratio
The peak signal to noise ratio is represented by the ratio between the maximum possible powers to the power of corrupting noise.
It is also referred as the logarithmic function of peak value of image and mean square error and hence represented as:
= 10 ( / ) (24)
IV.2. Mean Square Error
Mean square error (MSE) of an estimator is to quantify the difference between an estimator and the true value of the quantity being estimated:
= 1 [ ( , ) − ( , )] (25)
TABLEI PERFORMANCE COMPARISON
Methodology MSE PSNR Number of edges detected
SVM 0.3259 34.258 989
WSVM 0.2369 37.259 2045
HFCM-FNN 0.2145 40.256 3125
The graphical representation of comparison of MSE and PSNR of the proposed method with the existing methodologies such as cluster algorithms, SVM and WSVM has been represented graphically in Fig. 4 and Fig. 5. The results inferred suggest that the proposed method removes noise from image samples using ABF.
Hence, the performance of the proposed methodology using HFCM-FNN is found to be far better than the existing edge detection algorithms such as clustering, WSVM and SVM.
Fig. 4. Comparison of MSE of various edge detection methods with HFCM-FNN
Fig. 5. Comparison of PSNR of various edge detection methods with HFCM-FNN
Fig. 6. Comparison of number of edges between various edge detection methods
0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 SVM WSVM HFCM-FNN M SE(%) Methodologies