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superpixel based filtering for image noise reduction anna egorova samara national research university samara russia 2358anna gmail com abstract the paper presents a superpixel based image is further designated as ...

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                                            Superpixel-Based Filtering for Image Noise 
                                                                                                                             Reduction 
                                                                                                                                         Anna Egorova 
                                                                                                                   Samara National Research University 
                                                                                                                                        Samara, Russia 
                                                                                                                                 2358anna@gmail.com 
                              Abstract—The  paper  presents  a  superpixel-based  image                                                                   is      further  designated  as  “superpixel  threshold”.  This 
                       filtering algorithm for additive white Gaussian noise (AWGN)                                                                       algorithm  is  chosen  due  to  low  computational  complexity 
                       reduction.  The  algorithm  processes  an  image  by  connected                                                                    and  ease  of  setup  (one  input  parameter)  compared  to  the 
                       homogeneous  regions  of  small  size  (superpixels).  Each                                                                        popular graph superpixel segmentation algorithms [6-8] and 
                       superpixel  is  restored  using  the  least  squares  method.  The                                                                 the clustering algorithms [4, 9, 10]. 
                       mean square error (MSE) between a reconstructed image and 
                       an  ideal  image  provided  by  the  proposed  algorithm  is                                                                               III.        THE PROPOSED SUPERPIXEL-WISE IMAGE NOISE 
                       compared with the MSE provided by the Wiener filter. The                                                                                                                 FILTERING ALGORITHM 
                       experimental part shows that the proposed superpixel filtering                                                                            Let                           be  an  original  image  and                                               be  a 
                       algorithm outperforms the Wiener filter, providing lower MSE                                                                                        x (n ,n )                                                                    v(n ,n )
                                                                                                                                                                             0     1     2                                                                    12
                       values.                                                                                                                            random noise (AWGN). Then an observed image x(n ,n )  is 
                                                                                                                                                                                                                                                                     12
                                                                                                                                                          modeled                   as            x(n ,n ) x (n ,n )                        v(n ,n ),                 where 
                              Keywords—additive  white  Gaussian  noise,  filtering,  least                                                                                                             1     2           0     1     2              1     2
                       squares method, mean square error, noise reduction, superpixel,                                                                                                                      , and                        is size of the original 
                                                                                                                                                           nN 1,..,             ,  nN 1,..,                           NN
                                                                                                                                                                                                                            12
                       Wiener filter                                                                                                                         1122
                                                                                                                                                          image. Let a partition of the observed image  x(n ,n )  into 
                                                                                                                                                                                                                                                                12
                                                                  I.        INTRODUCTION                                                                  superpixels  is  given.  Denote  DD                                                             a  set  of  all 
                                                                                                                                                                                                                                       m mM1,..,
                              Various random noises are introduced in images at the                                                                       superpixels, where M  is the total number of superpixels of 
                       forming  and  transmitting  stages  [1].  Noises  decrease  the                                                                    the image  x(n ,n ) . 
                       visual quality of images and negatively affect the result of                                                                                                   12
                       image processing and analysis. Thus, the problem of image                                                                                 The task of image reconstruction is to design a filter that 
                       noise reduction is important today.                                                                                                takes as input the observed image  x(n ,n )  and outputs an 
                                                                                                                                                                                                                                         12
                              In practice, the most widespread is additive white noise                                                                    estimate  x n ,n                         that  is  close  to  the  original  image 
                                                                                                                                                                                             
                       [2]. Most of existing image filtering algorithms are aimed at                                                                                                  12
                       reducing noise having a Gaussian distribution since such a                                                                          x0 (n1, n2 )  [1].  The  proposed  algorithm  filters  the  image 
                       model  well  approximates  many  noises.  The  most  popular                                                                       superpixel-wise  and  finds  for  each  superpixel  a  linear 
                       algorithm  for  reducing  white  Gaussian  noise  (AWGN)  in                                                                       combination of some functions  fi, i  1,.., I,  where  I  is the 
                       images  is  the  Wiener  filtering.  It’s  the  optimal  linear                                                                    number of functions: 
                       processing technique for minimizing, in the statistical sense, 
                       the mean square error (MSE) between a restored image and                                                                                                                        I 1
                                                                                                                                                                               x n ,n                      a f       n , n       ,     n , n        D                       
                                                                                                                                                                                                                                               
                       an ideal image. It efficiently removes AWGN, but the degree                                                                                                     1     2          ii 1 2                             12 m
                       of blurring of restored images can exceed the values allowed                                                                                                                    i  0
                       by the task [2].                                                                                                                    ai  are the expansion coefficients. 
                              In this paper, an algorithm for image AWGN filtering by                                                                     Then it uses the least squares method [11] to reconstruct each 
                       superpixels  –  perceptually  meaningful  connected  disjoint                                                                      superpixel: 
                       regions [3] is proposed. It has several advantages over pixel-                                                                                          S                     [ x    n , n         x n ,n             ]2  min
                                                                                                                                                                                                                                                                           
                       based noise reduction algorithms. First, it processes images                                                                                                                           1     2               1     2                 a
                                                                                                                                                                                         n ,n   D                                                           i
                                                                                                                                                                                              
                       by objects or their parts, since no superpixel should include                                                                                                      12 m
                       pixels  of  more  than  one  object  [4],  whereas  pixel-based                                                                           To  find  the  expansion  coefficients  ai  at  which 
                       algorithms often process images by “sliding window”, which                                                                         minimum of (2) is achieved, equate the partial derivatives 
                       may  consist  of  pixels  belonging  to  various  objects  with                                                                    taken of (1) to zero, differentiate and obtain the following 
                       different characteristics. Secondly, the number of superpixels                                                                     system of linear equations: 
                       of  the  image  is  much  less  than  the  number  of  pixels. 
                       Consequently,  the  computational  complexity  of  the  noise                                                                                           I 1
                                                                                                                                                                                     a                   f     n ,,n       f      n    n       
                                                                                                                                                                                                                                         
                                                                                                                                                                               
                       filtering task is reduced.                                                                                                                                      i                  i      1     2     j      1     2
                                                                                                                                                                               i 0,n n           D
                                                                                                                                                                                               
                                                                                                                                                                                           12 m
                                                                                                                                                                                                                                                                               
                                                      II.  SUPERPIXEL ALGORITHM                                                                                                                  x n ,n            f     n , n        ,   0  j  I  1
                                                                                                                                                                                                                                  
                                                                                                                                                                                                        1     2      j     1     2
                                                                                                                                                                                    n ,n D
                              For obtaining a superpixel representation of an image, the                                                                                             12 m
                       threshold  region  detection  algorithm  [5]  is  used.  The                                                                              In matrix form, the system (3) can be written as follows: 
                       algorithm in the order of progressive scanning divides the                                                                                                                                BA C
                       image  into  spatially  connected  disjoint  homogeneous  in                                                                                                                                                                                           
                       intensity areas (superpixels) in such a way that the spread of 
                       pixel intensity values inside each of them is within the range 
                       of 2  , where   is the input parameter of the algorithm that 
                       Copyright © 2020 for this paper by its authors. 
                       Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0) 
                     Image Processing and Earth Remote Sensing 
                                                                                                              I 1                        of  the  original  image  and  D  is  the  noise  variance.  The 
                                                                                                                                                                                                 v
                                                    I 1   
                                                           
                     where  B  b                                          f    n ,,n      f     n    n                 is      a 
                                                                                                     
                                                           
                                                ij  ij,0                  i     1    2     j     1    2                                 following  d  values were considered: 10 dB, 15 dB, 20 dB, 
                                                              n ,n D
                                                           
                                                               12 m
                                                           
                                                                                                              ij,0                       30 dB, 50 dB, 100 dB, 200 dB, 500 dB, and 1000 dB. For 
                     symmetric                            matrix,                        A  a I1                          and           each pair of values ( ,d )  10 images were generated.  
                                                                                                ii0
                                                                                         I 1
                                    I 1 
                                         
                                                                                              are  column-vectors 
                      C  c                             x n ,,n        f    n    n
                                                                                
                                         
                                               
                                 ii12 1 2
                                    i  0
                                            n ,n D
                                         
                                             12 m
                                         
                                                                                        i  0
                     of the sought coefficients and absolute terms of the system, 
                     respectively. 
                           Let consider polynomials as expansion functions. 
                             If  the  degree  of  polynomials  I  1 ,  the  proposed 
                                  superpixel-based image filtering represents  intensity 
                                  averaging operation inside each superpixel:                                                             a)                                                    b)                                                     
                                                                  f     n ,1n      
                                                                  0    1     2                                                   
                                                               b                     1                                            
                                                                  00          
                                                                           n ,n D
                                                                            12 m
                                                         c                    x n ,n                                              
                                                                                             
                                                            0                         1    2
                                                                    n ,n D
                                                                     12 m
                                                                                 x n ,n
                                                                                             
                                                                                      12
                                                                    n ,n D
                                                                     12 m
                                                        a0                                                                        
                                                                                       1
                                                                              
                                                                           n ,n D
                                                                            12 m                                                                                         c)                                                     
                             If  the  degree  of  polynomials  I  3 ,  the  proposed 
                                  algorithm solves the system of linear equations to find                                                 Fig. 1.  Example of generated piecewise-constant images: a)    0.90 ,  
                                  the coefficients  a                  :                                                                  b)    0.95 , c)    0.99 . 
                                                                i
                                                                  f    n ,1n        
                                                                  0    1     2                                                   
                                                                f    n , n         n                                              
                                                                  1    1     2        1
                                                                f     n , n       n                                               
                                                                  2    1     2        2
                                   
                                                  1                      nn
                                   
                                                                                      
                                                                          12
                                      n ,n   D             n ,n   D                n ,n   D                a
                                                                                                      
                                   
                                       1  2     m            1  2     m               1   2    m                0
                                                                           2                                
                                   
                                                   n                     n                        n n         a      
                                                    1                   1                        1  2        1
                                                                                                            
                                   
                                      n1 , n2  Dm        n1 , n2  Dm           n1 , n2  Dm
                                                                                                            
                                   
                                                                                                              a2
                                                   n                     n n                      n2        
                                   
                                                    2                   1   2                    2
                                   
                                      n1 , n2  Dm        n1 , n2  Dm           n1 , n2  Dm
                                   
                                                                                                                                                                                                                                             
                                      
                                                      x n ,n                                                                              Fig. 2.  The  dependence  of  superpixel  threshold  values    minimizing 
                                                                        
                                                            12                                                                           MSE  between  the  reconstructed  image  and  the  ideal  image  on  noise 
                                          n ,n D
                                          12 m                                                                                          standard deviation               . 
                                                                                                                                                                       v
                                                    x n ,n          n .
                                                                   
                                                            1    2     1
                                       n ,n D                            
                                              
                                           12 m
                                                                                                                                              First  of  all,  the  effectiveness  of  the  proposed  filtering 
                                                     x n ,n          n 
                                                                   
                                                            1    2      2                                                                algorithm was tested. To automate the stage of searching for 
                                       n ,n D                            
                                              
                                           12 m
                                       the superpixel threshold values  minimizing  MSE  the 
                                               IV.        EXPERIMENTAL RESEARCH                                                           dependence of superpixel threshold values on noise standard 
                                                                                                                                          deviation    was  investigated.  Note  the  MSE    
                           For experimental research, piecewise-constant images of                                                                                   v 
                     size 512×512 were generated. Such images represent a set of                                                          between  a  reconstructed  image  and  an  ideal  image  was 
                     regions with random intensity values formed by dividing the                                                          calculated as follows: 
                     plane by random lines [12]. The experiments were carried                                                                                                                                                         1/ 2
                                                                                                                                                                                NN
                                                                                                                                                                                  12                                              2
                                                                                                                                                                  
                     out on three sets of synthesized data, each of which included                                                                                      1
                                                                                                                                                           x n,,n x n n
                                                                                                                                                                                                                                                    
                                                                                                                                                                  
                                                                                                                                                                                                                                
                     images  with  a  fixed  value  of  the  correlation  coefficient                                                                               NN 0 1 2                                          1    2
                                                                                                                                                                               nn11
                                                                                                                                                                       12
                                                                                                                                                                  12
                     between  neighboring  pixels   :  0.90,  0.95,  and  0.99.  An                                                            Superpixel  segmentation  was  performed  at  various 
                     example of generated piecewise-constant images is shown in                                                           threshold values   from 2 to 25 in increments of 1. Fig. 2 
                     Fig. 1.                                                                                                              illustrates that the dependence is linear. To figure out the 
                           The  source  images  were  noised  by  putting  into  them                                                     dependence experimental data were approximated using the 
                     AWGN with zero mean. Further,  the  signal-to-noise  ratio                                                           least  squares  method.  The  obtained  dependence  has  the 
                     (SNR) is denoted as d  D / D , where D  is the variance                                                                                                        1.9            2.
                                                                                                                                          following  form:                                                    Thus,  the  higher  the 
                                                                       xv x                                                                                                         
                                                                                                                                                                                   vv
                     VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020)                                                                                                                                          2 
                      
                Image Processing and Earth Remote Sensing 
                value of noise standard deviation, and, therefore, the lower 
                the SNR, the higher the superpixel threshold   value that 
                minimizes reconstruction error. It was also found that the 
                threshold values of the superpixel segmentation algorithm 
                [5], which provides the minimum MSE, don’t depend on the 
                correlation between the pixels of the original image. 
                                                                                                            a)                                        b)                                          
                         a)                                                                                                        c)                                            
                                                                                                            Fig. 4.      Piecewise-constant  image  reconstruction:  a)  the  noisy  image 
                                                                                                            fragment  ( 0.95, d       200 dB) ,  b)  the  image  fragment  reconstructed 
                                                                                                            using  the  proposed  superpixel-based  filtering  algorithm  (I  1) ,  c)  the 
                                                                                                            image fragment reconstructed using the Wiener filtering. 
                                                                                                                 Fig.  3  also  illustrates  the  dependence                           for  the 
                                                                                                                                                                               ()d
                                                                                                            Wiener  filter.  It’s  worth  noting  that  the  Wiener  filter 
                                                                                                            reconstruction  error  can  be  calculated  using  the  power 
                                                                                                            spectral  density  of  the  image  and  noise.  It’s  known  that 
                                                                                                            piecewise-constant  images  have  an  isotropic  exponential 
                                                                                                            autocorrelation function [13]. The calculation of the energy 
                                                                                                            spectrum of such signals is presented in [14]. 
                         b)                                                                                      By comparing the proposed filtering algorithm with the 
                                                                                                            Wiener filter, the following conclusions can be drawn. 
                                                                                                                   At signal-to-noise ratio d  50  dB, the Wiener filter 
                                                                                                                      provides lower MSE values (however, they are high), 
                                                                                                                      whereas  at  d  50  dB  the  proposed  superpixel 
                                                                                                                      filtering performs better regardless of the value of  I . 
                                                                                                                   The  higher  the  value  of  the  correlation  coefficient 
                                                                                                                      between  the  pixels  of  the  original  image   ,  the 
                                                                                                                      smaller MSE obtained for the proposed algorithm and 
                                                                                                                      the Wiener filter. 
                                                                                                                   The  proposed  algorithm  is  more  efficient  than  the 
                         c)                                                                                           Wiener filter at    0.95 . 
                                                                                                                   The higher the correlation between the original image 
                Fig. 3.  The dependence of MSE   between the reconstructed image and                                 pixels,  the  lower  MSE,  regardless  of  the  filtering 
                the ideal image on the signal-to-noise ratio  d : a)    0.90 , b)    0.95 ,                       method used. 
                c)    0.99 .                                                                                   An example of a noisy image fragment reconstructed by 
                                                                                                            each of the compared algorithms is shown in Fig. 4. The 
                     Fig. 3 shows the dependence of MSE on the signal-to-                                   reconstruction errors of the proposed algorithm are local and 
                noise  ratio  for  the  proposed  superpixel  filtering  algorithm                          are  observed  at  the  boundaries  of  similar  in  intensity 
                with threshold values   defined in the previous step. It can                               regions. In turn, the Wiener filtering is characterized by a 
                be seen that the proposed algorithm can be applied to filter                                blurring of reconstructed images. 
                piecewise-constant images at d  50  dB. Approximation by 
                polynomials of degree  I  3  isn’t much more efficient than                                                               V.      CONCLUSION 
                approximation  by  polynomials  of  degree  I  1 .  Thus,  to                                   The paper presents a superpixel-based filtering algorithm 
                reconstruct  piecewise-constant  images  by  the  proposed                                  and compares it with the Wiener filtering. The experimental 
                filtering  algorithm,  it’s  sufficient  to  use  a  polynomial  of                         part of the research shows that at signal-to-noise ratios higher 
                degree  I  1 .                                                                             than  50  dB,  the  proposed  superpixel-based  filtering 
                                                                                                            algorithm  provides  lower  reconstruction  errors  than  the 
                VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020)                                                                                          3 
                 
                                                Image Processing and Earth Remote Sensing 
                                                Wiener  filter.  Moreover,  unlike  the  Wiener  filter,  the                                                                                                                                                                                                                   [4]                R. Achanta, A. Shaji, K. Smith, A. Lucchi, P. Fua and S. Susstrunk, 
                                                proposed method proved to be good at various values of the                                                                                                                                                                                                                                         “SLIC Superpixels compared to state-of-the-art superpixel methods,” 
                                                correlation  coefficient  between  the  pixels  of  the  original                                                                                                                                                                                                                                  IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 
                                                image.  The  superpixel-based  filtering  algorithm  is  more                                                                                                                                                                                                                                      34, no. 11, pp. 2274-2282, 2012. 
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                                                between neighboring pixels less than 0.95.                                                                                                                                                                                                                                                         compression  method,”  Automatic  Control  and  Computer  Sciences, 
                                                                                                                                                                                                                                                                                                                                                   vol. 12, no. 3, pp. 75-77, 1978. 
                                                               It’s  also  shown  that  it’s  sufficient  to  approximate                                                                                                                                                                                                       [6]                P.F.  Felzenszwalb  and  D.P.  Huttenlocher,  “Efficient  graph-based 
                                                superpixels  with  polynomials  of  the  first  degree,  since  at                                                                                                                                                                                                                                 image segmentation,” International Journal of Computer Vision, vol. 
                                                higher  degrees  the  reduction  in  MSE  between  the                                                                                                                                                                                                                                             59, no. 2, pp. 167-181, 2004. 
                                                reconstructed image and the ideal image isn’t significant.                                                                                                                                                                                                                      [7]                J. Shi and J. Malik, “Normalized cuts and image segmentation,” IEEE 
                                                                                                                                                                                                                                                                                                                                                   Transactions on Pattern Analysis and Machine Intelligence, vol. 22, 
                                                               The disadvantage of the proposed algorithm is the effect                                                                                                                                                                                                                            no. 8, pp. 888-905, 2000. 
                                                of the obtaining superpixel representation stage on the final                                                                                                                                                                                                                   [8]                M.Y. Liu, O. Tuzel, S. Ramalingam and R. Chellappa, “Entropy rate 
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                                                                                                                                                                                                                                                                                                                                                   Pattern Recognition, pp. 2097-2104, 2011. 
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                                                a single superpixel. Conversely, when the noise level in the                                                                                                                                                                                                                                       clustering,”  IEEE  Conference  on  Computer  Vision  and  Pattern 
                                                observed  image  is  high,  the  oversegmentation  may  occur,                                                                                                                                                                                                                                     Recognition, pp. 1356-1363, 2015. 
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                                                                                                                                                                                                                                                                                                                                                   using  edge-weighted  centroidal  Voronoi  tessellations,”  IEEE 
                                                                                                                                    ACKNOWLEDGMENT                                                                                                                                                                                                 Transactions on Pattern Analysis and Machine Intelligence, vol. 34, 
                                                               The research was supported by RFBR projects № 19-37-                                                                                                                                                                                                                                no. 6, pp. 1241-1247, 2012. 
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                                                VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020)                                                                                                                                                                                                                                                                                                                                                                                                                                                  4 
                                                 
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...Superpixel based filtering for image noise reduction anna egorova samara national research university russia gmail com abstract the paper presents a is further designated as threshold this algorithm additive white gaussian awgn chosen due to low computational complexity processes an by connected and ease of setup one input parameter compared homogeneous regions small size superpixels each popular graph segmentation algorithms restored using least squares method clustering mean square error mse between reconstructed ideal provided proposed iii wise with wiener filter experimental part shows that let be original outperforms providing lower x n v values random then observed modeled where keywords nn partition into i introduction given denote dd set all m mm various noises are introduced in images at total number forming transmitting stages decrease visual quality negatively affect result processing analysis thus problem task reconstruction design important today takes outputs practice mos...

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