IDENTIFYING GEOMETRICAL OBJECTS USING IMAGE ANALYSIS

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1 IDENTIFYING GEOMETRICAL OBJECTS USING IMAGE ANALYSIS Fathi M. O. Hamed and Salma F. Elkofhaifee Department of Statistics Faculty of Science University of Benghazi Benghazi Libya and ABSTRACT One of the major challenges in computer vision is to extract the objects automatically from an image. This work seeks to detect objects in image. Herein, the objects are supposed to be circles. This paper presents an algorithm to determine the main characteristics to define the circles in image. The main ingredients of the algorithm are thresholding, scale-space and median filter methods. The identification is made by estimate the three characteristics which are number, sizes and locations of the object. A proposed algorithm has been applied to synthetic and real datasets and yields good results. KEYWORDS Segmentation, Threshold, scale-space, median filter, Bandwidth. 1 INTRODUCTION Image analysis combines techniques to extract meaningful information from image, in particular from digital image. Early work in statistical imaging was carried out by [1] and [2]. Image analysis tasks can be as simple as reading barcoded tags or as sophisticated as identifying a person from his face. One of the major challenges in computer vision is to extract the objects automatically from an image. This work seeks to detect object(s) in image. This paper suggests some approaches which help in identify geometrical objects in image. The first suggested method is thresholding method, which enables us to select ranges of pixel values in Grayscale and colour image that separate the objects from the background. In this study the threshold method is used to extract the foreground from the background in the image. Secondly, scale-space method is applied to smooth the density curve of thresholded image. This method is needed to find optimum bandwidth (smoothing parameter) which is used to find optimum smoothing density curve. Then the optimum thresholding is calculated from the optimum density curve. The optimum thresholded image may include some noise such as isolated pixels, to remove these unwanted pixels median filter approach has been proposed. 2 METHODS OF ANALYSIS IN THIS PROJECT 2.1 Thresholding Method Thresholding often provides an easy and convenient way to perform segmentation on the basis of the different intensities or colours of the foreground and background region of an image. [3] had used threshold to segment region to regions. [4] used thresholding to segmentation the soil image into different regions. 2.2 Scale-space Method The method introduced by [5], who proposed the term "scale-space" which is simply a family of Gaussian curve smoothers indexed by bandwidth (smoothing-parameter). See [6] and [7] for more details. [8] used scale-space method to find optimum smoothing parameter. [7] used the scale space texture classification approach to ultrasound liver tissue characterization for discrimination of ISBN: SDIWC 174

2 normal liver from cirrhosis yields promising results. The scale-space method will be used in this paper to smooth the curve and then to determine the optimum bandwidth. It is then used to identify optimum number and value of the thresholds. The diagram in Figure 1, presented by Hamed in 2005, shows the relation between the location of thresholds, on the y-axis and various bandwidths. The number and values of the thresholds are changed with the change in the bandwidth. As the bandwidth increasing the number of threshold decreasing, till becomes no threshold. The optimum number of the threshold is considered based on number of the threshold remains longer. Thus the optimum bandwidth is selected to be in the middle of the threshold, that staying after the rest of the thresholds dies out. Number of Threshold Bandwidths Figure 1: Diagram illustrates the scale-space 2.3 Median Filter Method Median filtering in signal processing has been established by Tukey (1974), and it has now become a standard technique in image processing. The median filter used extensively for image noise reduction and smoothing, it is especially good for removing impulsive noise from images. In the binary image, some isolated pixels (unwanted pixels) may appear. For instance, white pixels in black region and black pixels in white region. This is called salt and pepper noise. The median filter method is proposed to deal with this kind of noise. [9] used it to remove isolated pixels. [10] introduced a new solution for impulsive noise detection in colour images. [11] described the definition and testing of a new type of median filter for images. The new type is topological median filter. [12] presented a basic tool for analyzing structures at different scales. 3 THE PROPOSED ALGORITHM The proposal algorithm is combined of methods, it consists of thresholding approach, which is used to extract the foreground from background, and scale-space method is adopted to determine the optimum bandwidth that used to obtain optimum number and value of the thresholding. Then the median filter method may be needed to remove salt and pepper noise. These methods are applied sequentially in the proposed algorithm; some computation steps have been added to achieve our goals, which are determined numbers, sizes and locations of the objects in image. This new algorithm will be evaluated by applied to simulated data, if the good performance is achieved then same technique will be applied to real data. This processing is diagrammatically shown in Figure 2. Input image Threshold Output Thresholded image with unimportant detail Median filter Scale-space Optimum Threshold Re-threshold data with optimum threshold Thresholded image (With salt and pepper noise) Figure2: Diagram of image processing ISBN: SDIWC 175

3 4 THE ALGORITHM Suppose that I is a matrix of noisy image, with gray levels, it contains n circles as objects. 1. Segment the image by using threshold method according to the following steps, Find the histogram for the image intensities, and let f (x) denote to the frequency, where x is a sequence of gray levels from 0 to 1. Calculate g (x), where g( x) f ( x 1) f ( x) Observe the signs of each value in the vector g (x), if g (x) <0 and g ( x 1) >0 then the value of x 1will be stored in vector U. 2. Use Scale-Space approach to fined optimum smoothing parameter (Bandwidth) for Histogram curve as explain in the following steps: Create a sequence of values with initial bandwidth in the middle of this sequence. The difference between the values should be as small as possible. Each value in the sequence is used, as a bandwidth, to plot histogram of the data. Then in every plotted histogram count number of minima (thresholds), as illustrated in the first step and store the results in vector. Some of the minima are staying longer Y T than other, and this leads to some values in vectory are repeated more the other. T The median of these values, which are staying longest in the vectory, is T considered as the optimum bandwidth, (say). hopt 3. Apply the Median Filter Method, to remove isolated pixels, which may be appeared in the thresholded image. Suppose that the matrix of the obtained image is I( i, j ). Then the median filter method is as follows: The median filter considers each pixel in the image in turn and looks at its nearby neighbors to decide whether or not it is representative of its surroundings. It replaces the value of pixel by the median of the gray level in the neighborhood of that pixel: f ( i, =median { I ( i, }, where f ( i, is filtered image with size NM. 4. Set a new matrix, y( i, with zeros values, where i 1,2,..., N. j 1,2,..., M. 5. Compute mean square differences (MSD), MSD= N M i1 j1 6. Set i 1, j 1 in y ( i, 7. Set y ( i, =1 f ( i, j ) y( i, NM 8. Recalculate MSD with new value: MSD= N M i1 j1 f ( i, j ) y( i, NM 9. Define another zeros matrix, x, of size N M and set x ( i, j ) MSD 10. If MSD<MSD, then keep y( i, 1 and MSD=MSD, otherwise y( i, If j M then put j j 1 and go to step (7) 12. If i N then put i i 1 go back to step (7). 13. Calculate mean of each row, in matrix x (i, and store the results in vector R 14. Set i If R( i 1) R( i), store i in vector A 16. If ( i 1) N set i i 1 and go back to step 15. l A k1 Ak 1, k 2,3..., Length of (A), 17. Set vectors a, a, Let a1 A1, a Ak, If 2 2. ISBN: SDIWC 176

4 l 1,2,..., length of ( a ). Store A k in A k in a. 1 a and 5 APPLICATIONS 5.1 Application to a synthetic data 18. Calculate mean of each column in the matrix x(i,) and store the results in vector C. 19. Set j If C( j 1) C(, store j in vector B 21. If ( j 1) M set j j 1 and go back to step Set vectors b,b, Let b1 B1, b l B ' k then, If B B 1, k' 2,3,..., Length of ( B ), k' 1 ' k l 1,2,..., length of (b), Store in b B k ' Suppose that one circle is simulated in 2-D, n=1. The center of this circle ( x, y) is generated from the uniform distribution on the region (0,1), while the radiusr is simulated from lognormal 2 2 distribution with parameters, =-3, = That is x~u ( 0,1 ), y~u ( 0,1 ) and r~logn ( 3,0.1 ). The simulated circle has been plotted in Figure 3. and in b B k ' Number of elements in vector a represents number of objects circles in image. where length of a a b b 24. compute radii of circles, r l, as following a l a r l l, l 1,2,... length (a), or 2N b l b r l l, l 1,2,... 2M length (b) Figure 3: Simulated data The simulated circle is converted to a binary image as shown in Figure 4. The resolution of the image is pixels. 25. Calculate coordinates locations "centers" of the circles, ( c l, c l ), where c l is the first coordinate, c l cl al al, 2N is the second coordinate, 26. Use these results r l, c l, c "reconstruct" the object. c l bl bl 2M l to plot circles, Figure 4: Binary Image for the circle ISBN: SDIWC 177

5 The image is contaminated by Gaussian noise with mean equal to zero and standard deviation equal to 0.1, the noisy image is plotted in Figure 5. Then the algorithm will be used to identify the circle in the noisy image. Number of Threshold Bandwidth Figure 7: Scale-space method Figure 5: Noisy image for the circle The density curve of the noisy image is obtained and plotted in Figure 6. Although some negligible minima, which are resulted from the noise, the density curve can be considered as a bimodal curve, and one threshold should be enough to segment the image. The optimum bandwidth is used to find the optimum smoothing density as plotted in Figure 8, which shows the density curve with only one threshold att Pixels Intensities Figure 8: Optimum smooth Histogram Pixels Intensities Figure 6: Histogram of noisy image The optimum smoothing parameter is needed to smooth the unimportant minima in the density curve, then the scale-space method will be used. In Figure 7 the resulted graph of scale-space method is presented. It yields optimum bandwidth equal to The obtained threshold (T=0.36) is used on the noisy image and the resulted thresholded image is plotted in Figure 9. The thresholded image shows to contain some isolated pixels (salt and pepper noise). This noise could be seen as white pixels in the black object and black pixels in the white background. ISBN: SDIWC 178

6 The values in vector A are successive values, that is, there is no gap between the values. This means the image has one circle. Figure 9: Thresholded image with salt and pepper noise The isolated pixels can be removed via median filter method (step 3 in the algorithm). Figure 10 displays the filtered image Mean of rows of MSD Figure 11: Mean of rows of MSD matrix Likewise, the same procedure has been conducted on the columns in the matrix x ( i,, the results of mean of each column are stored in vector C, Figure 12 shown these results. Figure 10: Filtered image. According to the steps 4-12 in the algorithm, the matrix x ( i, has been obtained. Mean of each row in the matrix x( i, will be computed and stored in vector R. Figure 11 shows the plot of vector R. The values in the vector R are constants at the rows, which have no objects and they will be decreased at the rows have pixels of foreground. The ranks of the rows of the decreasing values are stored in vector A as, A= (36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56,57, 58,59, 60, 61,62, 63, 64). Mean of columns of MSD Figure 12: Mean of columns of MSD matrix The ranks of the decreasing values, in the vector C, are stored in vector B, ISBN: SDIWC 179

7 B =(47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69,70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88). Once more, the vector B confirmed that the image has one circle. Radius of the circle, will be calculated as explained in step 24 in the general algorithm, the radius, r, of the circle can be obtained by using the first and the last values in the vector A or B, r A 0.14 or r B , Since the object in this example is a circle then r A and r B must be equal. By applying step 21 in the algorithm, the center of the circle is equal to ( 0.5,0.45). The obtained results can be summarized as: number of objects is one, the radius of the object is 0.14 and the location is at ( 0.5,0.45). According to these results the object has been reconstructed in Figure Application to a Real data In this example a slide of blood cells is taken from the internet through the cite: And it has been displayed in Figure 14. MATLAB software (version.5.0) is used to read the image as numerical data. Since the interesting information can be obtained entirely from a grayscale representation, the data is converted to gray scale image. Figure 14: Slide of blood cell This image has been converted to grayscale image and it is shown in Figure 15 with pixels, where N 87 and M 86. Figure 13: Reconstruction dataset The result indicates that the algorithm yielded good achievement to identify one circle. This can be seen in this example, where the algorithm detects number, size and location of the object in the image. This result encourages us to apply the algorithm to real data, as in the following example. Figure 15: Grayscale image for blood cell image ISBN: SDIWC 180

8 To segment the grayscale image, the histogram curve is plotted in Figure 16. This histogram needs to smooth by optimum bandwidth. Density Density T= Pixels Intensities Figure 18: optimum Smoothing Histogram Pixels Intensities Figure 16: Histogram curve for grayscale The optimum smoothing parameter has been obtained by using scale-space method as illustrate in Figure 17, with the arrow points to the optimum smoothing value, and it is equal to 9.6. The grayscale image is segmented using the obtained threshold. The thresholded image is presented in Figure 19. Figure 19: Thresholded image Bandwidth Figure 17: Scale-space diagram Figure 18 shows smoothing histogram, the value of threshold can be seen visually in this histogram as a single threshold and it is determined at (T=163.72). The thresholded image shows some isolated pixels, which can be removed via median filter method. The filtered image is displayed in Figure20. ISBN: SDIWC 181

9 The different points in Figure 21 and Figure 22 are stored in vectors A and B respectively. The values in A represent the rows which contain object, while the values in B represent the columns have object, A=(38,39,40,41,42,43,44,45,46,47,48,49,50,51,52, 53,54,55), Figure 20: Filtered image Figure 21 displays the mean of rows of the MSD matrix (matrix x ( i, ). This figure shows that number of circles in the image is one. Mean of columns of MSD Mean of rows of MSD Mean of Row of MSD Figure 21: Mean of rows of MSD matrix While Figure 22 presents the mean of the columns of the MSD matrix (matrix x ( i, ). Both Figures detect a single object in the image. Figure 22: Mean of columns of MSD matrix B=(24,25,26,27,28,29,30,31,32,33,34,35,36,37,38, 39). The values in vector A and vector B are in sequence, without any gap, this means number of objects in the image is one. The radius of the circle has determined as following: r A or r B 286 The center coordinate of the circle is found equal to ( 0.53,0.37). By employing these results n 1, r 0.09 and center = ( 0.53,0.37) the object can be reconstructed as in Figure 23. Figure 23: Reconstructed image The well identification of the object, in the real image, has been achieved by using the algorithm. So, the algorithm works well in case of one circle in the image. ISBN: SDIWC 182

10 6 SUMMARY This paper tried to identify geometrical objects in an image. This identification made by determining three characteristics, number, location and size of the objects. Herein, the object is supposed to be a circle. A number of approaches were suggested to extract the object from the image. The concept of thresholding has been used to separate the objects from background. To obtain optimum threshold, the scale-space approach is adopted to smooth the density of image. Some isolated pixels were arisen in thresholded image, median filter method was suggested to remove the isolated pixels. These methods are combined and presented in algorithm. The algorithm consisted of the three approaches in addition to some steps to determine the three characteristics and steps to reconstruct the objects. The algorithm evaluated by simulated dataset, it yielded well identification for the simulated circle, then same technique was applied to real data. The algorithm provided good results and achieved the aims of this paper. 7. Gangeh, M.J., Duin, R.P.W., Eswaran, C., Rommeny, B.M.: Scale Space Texture Classification Using Combined Classifiers with Application to Ultrasound Tissue Characterization. In: Proc International Conference on Biomedical Engineering, BioMed (2006). 8. Hamed, F.M.: Geometrical Modeling and Identification of Structure in Image Data. Ph.D. thesis. University of Leeds (2005). 9. Tukey, J.K.: Nonlinear (nonsuperposable) methods for smoothing data. In: Proc Congr Rec. EASCOM, pp. 673 (1974). 10. Lianghai, J., Dehua, L.: A switching vector median filter based on the CIELAB color space for color image restoration. In: Signal Processing. 87, pp (2007). 11. Senel, H.G., Peters, R.A., Dawant, B.: Topological Median Filters, in IEEE Transactions on Image Processing, 11, pp (2002). 12. Lindeberg, T.: Scale-space theory: A basic tool for analyzing structures at different scales, Statistics and Images, 21, pp (1994). 7 REFERENCES 1. Geman, S., Geman, D.: Statistical analysis of dirty pictures. In: Mardia, K.V., Kanji, G.K. (eds.) Advances in Applied Statistics, 20, pp (1993). 2. Besag, J.: On the statistical analysis of dirty pictures (with discussion). In: Journal of the Royal statistical society B 48, pp (1986). 3. Jahne, B.: Digital Image Processing, Springer, New York (2003). 4. Glasbey, C. A., Horgan, G.W.: Image Analysis for the Biological Sciences, John Wiley and sons, Hall, Chichester (1995). 5. Witkin, A.P.: Scale-space filtering. In: Proc Eight International Joint Conference on Artificial Intelligence, Karlsruhe, Germany, pp (1983). 6. Marron, J.S., Chaudhur, P.: Scale-Space View of Curve Estimation. In: The Annals of statistics, 28, pp. 4o (2000). ISBN: SDIWC 183