A Novel Multi-diagonal Matrix Filter for Binary Image Denoising
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1 Columbia International Publishing Journal of Advanced Electrical and Computer Engineering (2014) Vol. 1 No. 1 pp Research Article A Novel Multi-diagonal Matrix Filter for Binary Image Denoising Gholamreza Anbarjafari 1*, Hasan Demirel 2, and Ahmet E. Gokus 2 Received 10 January 2014; Published online 22 February 2014 The author(s) Published with open access at Abstract Binary Image denoising is one of the most frequently used image pre-processing. In this paper, a new filtering technique for eliminating the salt and pepper noise in binary images is introduced. The proposed technique is based on the multiplication of multi-diagonal binary matrix with the noisy binary image followed by thresholding. The technique is superior in quality and computational cost to the state of art filtering techniques such as median and morphological filtering in binary images. Keywords: Image denoising; Filtering; Binary image processing; Image processing; Multi diagonal matrix operations 1. Introduction Binary image processing techniques are widely used in different phases of many image processing applications such as masking for face detection (Demirel and Anbarjafari, 2008), tracking (Mei et al. 2004), and biomedical image processing (Rodrigues et al. 2006). Image denoising has been always important in image processing (Tasmaz et al. 2012). Binary image filtering is one of the most important pre-processing step required in binary image processing (Junyi et al. 2006, Zhang et al. 2009, Shao and Barner 2006). Binary image processing performance can be affected by the presence of salt and/or pepper noise where the salt noise corresponds to on noise and pepper noise corresponds to off noise (Ibrahim et al. 2008, Graf et al. 2003). Min filter and max filter can be used to remove only the presence of the salt and pepper noise respectively. However, when both types of noise are included in the binary image min or max filters fails, hence filters such as median filter or morphological operations based filters are required. *Corresponding sjafari@ut.ee 1* IMS Lab, Institute of Technology, University of Tartu, Tartu 50411, Estonia 14 2 Department of Electrical and Electronic Engineering, Eastern Mediterranean University, Famagusta, TRNC, via Mersin 10, Turkey
2 Morphological operations erosion and dilation can be used to remove either salt or pepper noise according to the structuring element. However, when both types of noise are included then combination of erosion and dilation operations can be used to remove noise. Another effective method which is used for filtering in binary images is median filter. Median filter requires sorting within the pixel neighbourhood specified by the filter size. Sorting is not a computationally cheap process, which increases the computational burden as the size of the filter increases. In this paper, a new binary image filtering technique has been introduced which is based on multi-diagonal matrix multiplication followed by thresholding to perform removal of salt and/or pepper noise. Standard spatial domain filtering techniques employ a filter mask within the neighbourhood of the processed pixel to determine the new value of the pixel (Gonzalez and Woods 2010). Therefore, the computational cost of these filtering processes will depend on the size of the image as well as the size of the filter. However, in the proposed technique the computation cost of the filtering process only depends on the size of the image. 2. Multi-diagonal Matrix Multiplication based Filtering A ξ-diagonal matrix is defined to be a symmetric square matrix, ψ kxk(ξ), which has value 1 on its ξ diagonals and 0 on the remaining members, expressed below: 1 i j kk i, j 1,, k, k 0 otherwise (1) In this paper, a ξ-diagonal matrix is denoted by ψ kxk(ξ), where p shows the size of square matrix ψ 1 and ξ is indicating the total number of diagonals including the main diagonal, above and 2 1 below of the main diagonal, e.g. ψ7x7(5) is shown below: (2) Consider an mxn binary image matrix, A mxn, and ψ nxn(ξ) as follows: 15
3 Amn a1 1 a1 j1 a1 j a1 j1 a1 n ai 1 1 ai 1 j 1 ai 1 j ai 1 j 1 a i1 n ai 1 ai j1 ai j ai j1 ai n ai1 1 ai1 j1 ai1 j ai1 j1 ai1 n am 1 am j1 am j am j1 am n, nn (3) The result of multiplication of A by ψ nxn(3) is: a1 1 a1 2 a1 j1 a1 j a1 j1 a1 n1 a1 n 1 Amn nn 3 ai 1 ai 2 ai j1 ai j ai j1 ai n1 a i n am 1 am 2 am j1 am j am j1 am n1 am n (4) In X 1 each element of A has been replaced by addition of the pixel with its left and right neighbours in a row. Consider X 2 calculated by multiplication of A T by ψ mxm(3) : a1 1 a2 1 a j1 1 a j 1 a j1 1 an1 1 an 1 T 2 Amn mm 3 a1 i a2 i a j1 i a j i a j1 i an1 i a n i (5) a1 m a2 m a j1 m a j m a j1 m an1 m an m Here each element of X 2 is addition of a pixel with its neighbours in a row of A T. The addition of X 1 and X 2 T will result in a new matrix X given below: 2a1 1 a1 2 a2 1 a1 j1 2a1 j a2 j a1 j1 a1 n1 a2 n 2a1 n T 1 2 2ai 1 ai1 1 ai 2 ai1 1 ai j1 ai1 j 2ai j ai1 j ai j1 ai n1 2ai n ai1 n a i1 n (6) 2am 1 am1 1 am 2 am j1 2am j am1 j am j1 am n1 2am n am1 n According to equation (6) a ij, which represents the pixels in a binary image, is replaced with a new value which is the sum of its horizontal and vertical neighbourhood. The number of pixels in each 1 neighbourhood direction (above, below, right, and left) is a function of ξ which is denoted by. 2 16
4 Each pixel of a connected component with 4-connectivity in X will accumulate a value less than ξ, if and only if the width and height of the component is less than ξ. By choosing a threshold value equal to ξ the connected components with sizes less than ξ will be removed. Equation (7) shows how each pixel will be thresholded to generate the final image, A. 1 ij, A i 1,, m, j 1,, n 0 otherwise If binary image A is corrupted with binary noise and ξ is chosen such that it is greater or equal than the size of the noise, then A represents the filtered image. Fig. 1 illustrates the block diagram of the proposed system. (7) Salt and Pepper noise Ψ n(ξ): ξ-diagonal matrix Binary Image with noise A mxn + X Ξ 1 + ij 1, A 0 otherwise Filtered Binary Image Am n Transpose X Ξ 2 Transpose Ψ m(ξ): ξ-diagonal matrix Fig 1. The block diagram of the proposed multi-diagonal matrix filter. Fig. 2 shows step by step process of the proposed binary filtering of an image with 12x12 pixels processed by 3-diagonal matrix, by ψ nxn(3). (a) (b) (c) (d) Fig 2. (a) Original binary image, (b) corrupted image, (c) the generated intermediate image after multiplications and addition, and (d) filter image after thresholding. Fig. 3 show the visual performance of the proposed filtering method on 214x214 check-board image corrupted with salt and pepper noise with density of 0.15 compared with the median and morphological filters. The results show that the proposed filtering technique with 9-diagonal matrix clears the salt and pepper noise while preserving the shape information. On the other hand, the median filter with 9x9 pixel neighborhood clears the noise; however the shape information of the image is distorted. Finally, the morphological filter using opening followed by closing with 9x9 17
5 square structuring element, not only loses the shape information but also fails to remove the noise completely. (a) (b) (c) (d) (e) Fig 3. Original binary image (a), salt and pepper corrupted image with density 0.15 (b), output of median filter with filter size of 9x9 (c), output of morphological filter with filter size of 9x9 (d), and filtered image using proposed 9-diagonal matrix(e). As Fig.3 only contains vertical and horizontal geometrical structures, Fig. 4 has been introduced to include objects with round and sharp corners and curvatures. Fig. 4(a) is a binary image with 440x428 pixels and Fig 4(b) is the corrupted image by salt and pepper noise with a density of 0.1. The proposed method using 3-diagonal matrix filter performs marginally better than the median filter with 3x3 filter size as median filter leaves few salt and pepper noise pixels in the filtered image. The morphological filter using opening followed by closing with 3x3 square structuring elements distorts the output image. (a) (b) (c) 18
6 (d) (e) Fig 4. Original binary image (a), salt and pepper corrupted image with density 0.1 (b), output of median filter with filter size of 3x3 (c), output of morphological filtering with filter size of 3x3 (d), and filtered image using proposed 3-diagonal matrix (e). (a) (b) (c) (d) (e) Fig 5. Original binary image (a), salt and pepper corrupted image with density 0.02 (b), output of median filter with filter size of 3x3 (c), output of morphological filtering with filter size of 3x3 (d), and filtered image using proposed 3-diagonal matrix (e). 3. Computational Complexity Analysis of the Proposed Filtering Technique The computational complexity based on the required number of additions, multiplications and comparisons of the proposed filter compared with the other conventional filtering techniques are given in Table 1. Given a noisy binary image, A mxn, the proposed technique requires cross product of two matrices (2mn multiplications and 2mn additions) followed by thresholding (mn comparisons). Among the well-known filtering techniques; the median filter requires mn(p 2 )log(p 2 ) iterations (comparisons and swapping) for sorting where the filter mask is chosen to be in the pxp 19
7 Required number of iterations pixel neighbourhood. Morphological filter (opening followed by closing) using a pxp square structuring element requires 2mnp 2 AND and 2mnp 2 OR operations to filter the image. The proposed filtering technique is independent of the value of ξ which is chosen based on the size of the noise components. Hence, changing the value of ξ does not change the computational complexity of the proposed filter. However, both in median and morphological filters changing the value of filter mask, which is based on the value of p, changes the computational cost that is proportional to the square of p. Table 1 shows, the computational cost of the proposed filter versus median and morphological filters. Furthermore, the table includes figures in seconds required to run proposed, median and morphological filters to de-noise a given binary image corrupted with salt and pepper noise with changing densities. A binary image of circles as shown in Fig. 5 corrupted with random salt and pepper noise with a size of 242x308 pixels is used to generate the time in seconds required to rum the proposed, median and morphological filters on a HP Pavilion dv6500 platform with Intel Core 2 Duo CPU T7700 at 2.40 GHz with 4.00 GB RAM. The results are average of 500 runs on randomly corrupted noisy images. The results given in Table 1 show that the total number of iterations required by the proposed method is significantly less than median and morphological filtering techniques. However, the types of the operations used in different methods are not common. For example the proposed method requires addition, multiplication and comparison operations where in median filtering requires sorting (comparison and swapping). Due to these differences the time analysis has been studied. Required time in seconds to perform filtering on the given corrupted image in Fig 5 has been shown in Table 1. Table 1 The computational cost and the required time to filter the corrupted image in Fig. 6 with 242x308 pixels Density of the salt and pepper noise Minimum value of ξ in order to remove all noise Minimum value of p of median filter in order to remove all noise Minimum value of p of morphological filter in order to remove all noise Addition 2mn 149, , , , , , ,072 Proposed Multiplication Filter 2mn 149, , , , , , ,072 Comparison mn 74,536 74,536 74,536 74,536 74,536 74,536 74,536 Median Filter Comp. + swapping mn(p 2 )log(p 2 ) 8.653x x x x x x x10 7 OR-ing Morphologica 2mnp 2 3,726,800 3,726,800 3,726,800 3,726,800 3,726,800 3,726,800 3,726,800 l Filter AND-ing 2mnp 2 3,726,800 3,726,800 3,726,800 3,726,800 3,726,800 3,726,800 3,726,800 Proposed Filter Required time (s) Median Filter Morphological Filter
8 According to the time analysis, the proposed method is approximately 3 times faster than the median filter and 8 times faster than the morphological filter on the tested image. 4. Conclusion In this work we have proposed a new filtering technique, which is used to eliminate the salt and pepper noise in binary images. This technique is based on the multiplication of multi-diagonal binary matrix with the noisy binary image followed by thresholding. The results on the test images show that, the technique is superior in quality and computational cost to the well-known filtering techniques such as median and morphological filtering in binary images. Acknowledgements This work is partially supported by ERDF program Estonian higher education information and communications technology and research and development activities state program (ICT program). References Demirel, H., Anbarjafari, G., Pose Invariant Face Recognition Using Probability Distribution Functions in Different Color Channels, IEEE Signal Processing Letters, 2008, Vol. 15, pp Gonzalez, R. C., Woods, R., Digital Image Processing, 4th edition, 2010, Prentice Hall. Graf, A. B. A., Smola, A. J., Borer, S., Classification in a normalized feature space using support vector machines, IEEE Transactions on Neural Networks, 2003, Vol. 14, Issue 3, pp Ibrahim, Z., Khalid, N. K., Ibrahim, I., Abidin, M. S. Z., Mokji, M. M., Syed Abu Bakar, S. A. R., A Noise Elimination Procedure for Printed Circuit Board Inspection System, Second Asia International Conference on Modeling & Simulation, 2008, pp Junyi, Z., Chunhui, Z., Quan, P., Wei, L., A Novel Binary Image Filtering Algorithm Based on Information Entropy, 6th world congress on intelligent control and automation, 2006, Vol. 2, pp Mei, H., Sethi, A., Wei, H., Yihong, G., A detection-based multiple object tracking method, International Conference on. Image Processing, 2004, Vol. 5, pp Rodrigues, R., Castillo, P. J., Guerra, V., Suarez, A. G., Izquierdo, E., Two robust Techniques for segmentation of biomedical Images, Comutation y Sistemas, 2006, Vol. 9, No. 4, pp Shao, M., Barner, K. E., Optimization of partition-based weighted sum filters and their application to image denoising, IEEE Transactions on Image Processing, 2006, Vol. 15, Issue 7, pp Tasmaz, H., Demirel, H., and Anbarjafari, G., Satellite image enhancement by using dual tree complex wavelet transform: Denoising and illumination enhancement, 20th IEEE Signal Processing and Communications Applications Conference, 2012, pp Zhang, W., Yu, F., Guo, H. M., Improved adaptive wavelet threshold for image denoising, IEEE Conference on Control and Decision, 2009, pp
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