Sensitivity Analysis of Hopfield Neural Network in Classifying Natural RGB Color Space

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1 Sensitivity Anaysis of Hopfied Neura Network in Cassifying Natura RGB Coor Space Department of Computer Science University of Sharjah UAE Abstract: - This paper presents a study of the sensitivity anaysis of the artificia Hopfied Neura Network (HNN) when segmenting natura coor images. The coor distinction or vision system reies on two step process which, first cassifies the different regions in the scene into a given number of custers, and then assigns to each custer a coor that is ikey to one of its corresponding region in the raw image. The cassification process is performed using the minimization of an energy function typicay the Sum of Squared Errors (SSE). The optimization process is found sensitive to the step taken by the network in its way to the goba minimum. The coor assignment to the custers is performed based on combination of information from the coor paette used in the raw image and the ast distribution of the pixes among custers. Appying the system to a god standard coor image, the resuts show that HNN natura coor segmentation accuracy can be significanty improved if we contro its step size when modifying its weights between its neurons each iteration. The coor matching process shows a ot of consistency when tested with natura coor images as shown in the resuts presented here.. Key-Words: - Hopfied Neura Network, Sensitivity anaysis, Segmentation, Natura Coor Image matching, RGB Coor Space 1 Introduction The division of natura images ike rock, stone, couds, ice, or vegetation into casses based on their visua simiarity is a common task in many machine vision and image anaysis soutions. Cassification of natura images is demanding, because in the nature the objects are sedom homogenous. For exampe, when the images of rock surface are inspected, there are often strong differences in directionaity, granuarity, or coor of the rock; even the images represented the same rock type. These kinds of variations make it difficut to cassify these images accuratey [1]. There are many papers deaing with segmentation of images using coor, see [2] survey. Severa authors are appying different techniques for coor in order to improve the fina resut of the segmentation and some of them are mentioned in [3]. In [4], we have used HNN for segmentation of pathoogica iver coor images obtained using neede biopsy. The segmentation resuts have been appreciated by pathoogists as it heped to provide quantitative diagnosis of iver cancer. The agorithm is as foows: The HNN cassifier structure consists of a grid of N M neurons with each row representing a pixe and each coumn representing a custer. The network cassifies the image composed of N pixes with P features among M custers, in a way that the assignment of the pixes minimizes the foowing criterion function: N M E = R k V k (1) 2 k = 1 = 1 Where R k is the Mahaanobis distance measure between the k th pixe and the centroid of cass. R k is aso equivaent to the error committed when a pixe k is assigned to a cass. Note that we have removed the term of white noise from the equation (2) in [4] in order to remove random effects in this study. The minimization is achieved using HNN by soving the motion equations satisfying: U k E = μ ( t) (2) t V where U k is the input of the k th neuron, and μ(t) is a scaar positive function of time, used as heuristicay motivated stopping criterion of HNN, and is defined as in [5] by: μ ( t) = t( T t) (3) s k ISSN: Issue 9, Voume 8, September 2009

2 where t is the iteration step and T s is the prespecified convergence time of the network which has been found to be 120 iterations [5]. The network cassifies the feature space, without teacher, based on the compactness of each custer cacuated using Mahaanobis distance measured between the k th pixe and the centroid of cass as given by: 1 Rk = X k X = ( X k X ) ( X k X ) 1 (4) 1 k N and1 M Where X k is the P-dimensiona feature vector of the k th pixe (here P = 3 with respect to the RGB coor space components), is the P-dimensiona centroid vector of cass, and is the covariance matrix of cass. The segmentation agorithm is described as foows in our previous work [4]: Step1: Initiaize the input of the neurons to random vaues. Step2: Appy the foowing input-output reation, estabishing the assignment of each pixe to ony one cass: if U km = Max[ U k ( t), ] then Vkm ( t + 1) = 1, ese Vkm ( t + 1) = 0, (5) 1 k N, and1 M Step3: Compute the centroid X and the covariance matrix Σ of each cass, respectivey, as foows: N X = X kvk n (6) k = 1 1 M T ( X X ) n 1 = V (7) k k where n is the number of pixes in cass, and the covariance matrix is then normaized by dividing 1 each of its eements by [ ] p. Step 4: Update the inputs of each neuron by soving the set of differentia equations in (2) using Euer s approximation: du k U k ( t + 1) = U k ( t) + dt (8) 1 k N, and 1 M Step5: If t < T s repeat from Step2, ese terminated. 2 The Sources Ony natura coor images have been considered in this work with a god standard coor image to check the accuracy of the method. Each image can be thought of as a set of points in a three dimensiona Eucidean space. Each pixe is represented as a point in this Eucidean space, where the three coordinates are the RGB components of the pixe coor in the RGB coor space. HNN cassifies the pixes among a given number of custers based on the mean and covariance matrix of each custer without training data set. Figure1 shows the god standard coor image formed with five homogenous rectanguar regions. 3 Segmentation Resut We have appied the above agorithm to segment the coor image in Figure1 with a fixed number of custers to five and the resut is shown in Figure2. As it is seen in this resut, HNN segments the image into three cear and homogenous custers, and two other different regions non homogenous with the same coor and other pixes dispersed with a specia coor among the two custers. Figure3 shows the curve of HNN energy function in its way to the convergence state after 120 iterations. For this reason we decided to anayse the above agorithm s degrees of freedom in order to find the range of these parameters where we can ensure homogenous and accurate segmentation of the different objects of a natura scene in an RGB coor space. 3.1 Sensitivity Anaysis of HNN to its Degrees of Freedom The above segmentation method of HNN has the desirabe feature of rapid convergence to oca optimum cose to the goba one without being trapped in eary oca optima. The convergence speed and ocation are controed by two parameters [or degrees of freedom (DOF)] which are the initiaization of neurons inputs (Step1), and the update of neurons weights (Step4). Here, we chose a fixed random initiaization of the neurons input and we focus ony on the anaysis of the gradientbased update of HNN weights, given in equation (9). This updating of the neurons input is a process for a contro agorithm to find the optima soution of the segmentation probem in a reiabe manner. In our tria to improve the segmentation resuts of HNN to the god standard image in Figure1, we have introduced a new contro goba term β ( E) to equation (3) in HNN agorithm as foows: U t k E = μ ( t) β ( E) (9) V 1 β ( E) = (10) m 1+ ( Log( E) / t) k ISSN: Issue 9, Voume 8, September 2009

3 Where E is the tota Error at iteration t, and m contros the sope of the function β ( E). Figure5 shows the curves of HNN energy function during the segmentation process of the image in Figure1 with the same initiaization matrix of neurons inputs and with different vaues of the sope contro parameter. The interva [2.1, 2.3] incudes convenient vaues of the sope contro m that may insure better quaity of the segmentation process. Figure6 shows the segmentation resuts corresponding to some of the curves in Figure5, it is cear from these resuts that the vaue 2.3 of the sope contro parameter m is appropriate to give a better segmentation resut with HNN. Energy Function of HNN Cassifier Tota Error 2.50E E E E E E Iteration No. New Weight: Based Euer's Approximation Fig.3 Energy function of HNN during the segmentation of the RGB coor image in Fig.1. Fig.1 Ground truth coor image with five homogenous regions. Zooming in the HNN Energy Function Curve Tota Error 6.00E E E E E E E+06 ' Iteration No. Series1 Fig.4 Zooming in the HNN Energy Function Curve for the ast twenty iteration during the segmentation of the RGB coor image in Fig.1. Fig.2 Resut of HNN Segmentation to the ground truth coor image in Figure1 with five custers. ISSN: Issue 9, Voume 8, September 2009

4 Curves of HNN Energy Function with different Step Contro vaues 1.40E E+07 Tota Error 1.00E E E E+06 m=3 m=2.9 m=2.8 m=2.7 m=2.6 m=2.5 m=2.4 m=2.3 m=2.2 m=2.1 m=2 2.00E E+00 ' ' Iteration No. Fig.5 Curves of HNN energy function during the segmentation process of the image in Figure1 with the same initiaization matrix of neurons inputs and with different vaues of the sope contro parameter. Fig.6(a) Fig.6(b) ISSN: Issue 9, Voume 8, September 2009

5 making a crisp cassification for homogenous coor regions. To dispay the segmentation resut with its five custers with coors simiar or identica to those in the raw image, we compute the sum of each coor component of the pixes assigned to each custer, the average of each component is considered as fina or matched coor of that custer. Fig.6(c) Fig.6(d) Fig.6 (a), (b), (c), and (d) are the segmentation resuts of HNN with vaues of the parameter m in the contro goba term, m=2.1, m=2.2, m=2.3, m=2.6, respectivey 4 Discussion of HNN Sensitivity & Natura Coor Matching The resuts obtained using the contro goba term β ( E) with HNN show that the contro parameter can drive the network through a better path to reach a position coser to the goba optima. The use of a ground truth coor image can be considered as a piece of evidence for the capabiity of HNN in Fig.7 (a) and (b) show a raw coor image (the same as in Figure1) and its corresponding segmented image, respectivey, after matching their regions' coors based on the average of the coor space components of a pixes in the custer. As it is cear from the resuts, the matching technique is perfect 100% for a regions in the raw image, in that region(a) has the same coor as its corresponding custer (F) in the segmented image, aso the same for the rest of regions. The noise that appears in the ast rows of the segmented image is an indication that HNN did improve the resut of the segmentation process by converging to optima coser but not equa to the goba optima. Fig.8 shows anther coor image (a) and its corresponding resut (b) of our segmentation and coor matching processes obtained with seven as custers number given to HNN cassifier. The discontinuity seen in the segmentation resut is due to two main facts: the first fact is that the network did not reach a goba minimum during the segmentation task, and the second is the intensity variation among the pixes of the same region in the raw images. The intensity variation did not exist in the ground truth coor image with five homogenous regions shown in Figure1 and that is the reason why its corresponding segmentation and matching resut, shown in Fig.7 (b), is much smoother than the resut shown in Fig.8 (b) when compared to its corresponding raw image Fig.8 (a) Fig.9 shows another coor image and its corresponding segmentation resuts with different custers number. This image contains big intensity variation among its regions, as it is cear in Fig.9 (a), the raising of the custers number to ten did not hep HNN to put the beak of the bird in a specific custer, however, If we focus in the beak region in the raw image, we find that even this sma region contains a ot of intensity variation among its representative pixes. For this reason, to improve the image segmentation resuts it is necessary to use a technique to maximize pixe consistency within true regions before we segment the whoe image. ISSN: Issue 9, Voume 8, September 2009

6 A B C D E F G H K M (a) (b) Fig.7 (a) and (b) show the raw coor image (the same as in Fig.1) and its corresponding segmentation resut after matching the custers' coor with their regions in the raw image. (a) (b) Fig. 8 (a) and (b) show, respectivey, a raw coor image and its corresponding segmentation resut with 7 custers after automaticay matching the custers' coors with their regions in the raw image. ISSN: Issue 9, Voume 8, September 2009

7 (a-raw image) (b-5custers) (c-8 custers) (d-10 custers) Figure 9 shows a coor image (a) and its corresponding segmentation resuts obtained using our method (b) with 5 custers, (c) with 8 custers, and (d) 10 custers. ISSN: Issue 9, Voume 8, September 2009

8 4 Concusion Herein we have proposed a new method to contro the direction of HNN in its seeking of the goba minimum when segmenting coor images. As it is proven by the use of a ground truth coor image, HNN did not reach the goba optima, but produced a cear and better segmentation resut when it is used with a step contro parameter. After trying a arge range of the contro parameter m, we came to a concusion that the atter is not the ony parameter responsibe or which decides the convergence point of HNN, but aso the random initiaization matrix used to initiaize the inputs of HNN. From the case studies presented above it is cear that the matching approach added natura touches to the segmentation resuts by giving the custers coors cose to the raw image, this information can hep in the pattern recognition fied. In our future work, we wi focus on the effects of neura network weights initiaization and study their effect in seeking or reaching the goba optimum. References: [1] L. Leena, K. Livari, and V. Ari, Rock image cassification using coor features in Gabor space, Journa of Eectrica Imaging, Vo.14, No. 4, 2005, pp [2] H. Jiang, Y. Sun, J. Wang, Coor image segmentation: advances and prospects, Pattern Recognition, Vo. 34, 2001, pp [3] J. Mena, B. Mapica, Coor image segmentation using the dempster-shafer theory of evidence for the fusion of texture, ISPRS Archives, Vo. XXXIV, Part3/W8, 2003, pp [4] M. Sammouda, R. Sammouda R, N. Niki, Liver cancer detection system based on the anaysis of digitized coor images of tissue sampes obtained using neede biopsy. In Internationa Journa of Information Visuaization. Vo. 1, 2002, pp [5] R. Sammouda R, N. Niki, H. Nishitani, A Comparison of Hopfied Neura Network and Botzmann Machine in Segmenting MR Images of the Brain, IEEE Transactions on Nucear Science, Vo.43, No. 6, 1996, pp ISSN: Issue 9, Voume 8, September 2009