Superpixel-Based Filtering for Image Noise Reduction

Abstract

-The paper presents a superpixel-based image filtering algorithm for additive white Gaussian noise (AWGN) reduction. The algorithm processes an image by connected homogeneous regions of small size (superpixels). Each superpixel is restored using the least squares method. The mean square error (MSE) between a reconstructed image and an ideal image provided by the proposed algorithm is compared with the MSE provided by the Wiener filter. The experimental part shows that the proposed superpixel filtering algorithm outperforms the Wiener filter, providing lower MSE values.

Keywords

• additive white Gaussian noise
• filtering
• least squares method
• mean square error
• noise reduction
• superpixel
• Wiener filter

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INTRODUCTION

Various random noises are introduced in images at the forming and transmitting stages [ 1 ]. Noises decrease the visual quality of images and negatively affect the result of image processing and analysis. Thus, the problem of image noise reduction is important today.

In practice, the most widespread is additive white noise [ 2 ]. Most of existing image filtering algorithms are aimed at reducing noise having a Gaussian distribution since such a model well approximates many noises. The most popular algorithm for reducing white Gaussian noise (AWGN) in images is the Wiener filtering. It’s the optimal linear processing technique for minimizing, in the statistical sense, the mean square error (MSE) between a restored image and an ideal image. It efficiently removes AWGN, but the degree of blurring of restored images can exceed the values allowed by the task [ 2 ].

In this paper, an algorithm for image AWGN filtering by superpixels – perceptually meaningful connected disjoint regions [ 3 ] is proposed. It has several advantages over pixelbased noise reduction algorithms. First, it processes images by objects or their parts, since no superpixel should include pixels of more than one object [ 4 ], whereas pixel-based algorithms often process images by “sliding window”, which may consist of pixels belonging to various objects with different characteristics. Secondly, the number of superpixels of the image is much less than the number of pixels. Consequently, the computational complexity of the noise filtering task is reduced.

II. SUPERPIXEL ALGORITHM

For obtaining a superpixel representation of an image, the threshold region detection algorithm [ 5 ] is used. The algorithm in the order of progressive scanning divides the 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   is further designated as “superpixel threshold”. This algorithm is chosen due to low computational complexity and ease of setup (one input parameter) compared to the popular graph superpixel segmentation algorithms [ 6-8 ] and the clustering algorithms [ 4, 9, 10 ].

III.

THE PROPOSED SUPERPIXEL-WISE IMAGE NOISE

FILTERING ALGORITHM

Let x0 ( n1 , n 2 ) be an original image and v ( n1 , n 2 ) be a random noise (AWGN). Then an observed image x ( n1 , n 2 ) is modeled as x ( n1 , n 2 )  x0 ( n1 , n 2 )  v ( n1 , n 2 ), where n1  1, .., N 1 , n 2  1, .., N 2 , and N 1  N 2 is size of the original image. Let a partition of the observed image x ( n1 , n 2 ) into superpixels is given. Denote D   D m  m 1,..,M a set of all superpixels, where M is the total number of superpixels of the image x ( n1 , n 2 ) .

The task of image reconstruction is to design a filter that takes as input the observed image x ( n1 , n 2 ) and outputs an estimate x  n1 , n 2  that is close to the original image x0 ( n1 , n 2 ) [ 1 ]. The proposed algorithm filters the image superpixel-wise and finds for each superpixel a linear combination of some functions f i , i  1, .., I , where I is the number of functions:

I 1  x  n1 , n 2    a i f i  n1 , n 2  ,   n1 , n 2   D m  i  0  a i  are the expansion coefficients.

Then it uses the least squares method [ 11 ] to reconstruct each superpixel:

S 

  n1 ,n2  Dm [ x  n1 , n 2   x  n1 , n 2 ]2  m in  ai 

To find the expansion coefficients  a i  at which minimum of (2) is achieved, equate the partial derivatives taken of (1) to zero, differentiate and obtain the following system of linear equations:

I 1  a i i  0

  n1 ,n2  D m f i  n1 , n 2  f j  n1 , n 2      x  n1 , n 2  f j  n1 , n 2  , 0  j  I  1

 n1 ,n2  D m In matrix form, the system (3) can be written as follows:

B A  C  and of the original image and D v is the noise variance. The following d values were considered: 10 dB, 15 dB, 20 dB, 30 dB, 50 dB, 100 dB, 200 dB, 500 dB, and 1000 dB. For each pair of values ( , d ) 10 images were generated. where symmetric matrix,

A   a i  iI01  I 1 C   ci  iI01   x  n1 , n 2  f i  n1 , n 2   are column-vectors   n1 ,n2  Dm  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: f 0  n1 , n 2   1  b00 

For experimental research, piecewise-constant images of size 512×512 were generated. Such images represent a set of regions with random intensity values formed by dividing the plane by random lines [ 12 ]. The experiments were carried out on three sets of synthesized data, each of which included images with a fixed value of the correlation coefficient between neighboring pixels  : 0.90, 0.95, and 0.99. An example of generated piecewise-constant images is shown in Fig. 1.

The source images were noised by putting into them AWGN with zero mean. Further, the signal-to-noise ratio (SNR) is denoted as d  D x / D v , where D x is the variance         a)  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. using the proposed superpixel-based filtering algorithm ( I  1) , c) the image fragment reconstructed using the Wiener filtering.

By comparing the proposed filtering algorithm with the Wiener filter, the following conclusions can be drawn.    

At signal-to-noise ratio d  5 0 dB, the Wiener filter provides lower MSE values (however, they are high), whereas at d  5 0 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 Wiener filter at   0 .9 5 .

The higher the correlation between the original image pixels, the lower MSE, regardless of the filtering method used.

An example of a noisy image fragment reconstructed by each of the compared algorithms is shown in Fig. 4. The reconstruction errors of the proposed algorithm are local and are observed at the boundaries of similar in intensity regions. In turn, the Wiener filtering is characterized by a blurring of reconstructed images.

CONCLUSION

The paper presents a superpixel-based filtering algorithm and compares it with the Wiener filtering. The experimental part of the research shows that at signal-to-noise ratios higher than 50 dB, the proposed superpixel-based filtering algorithm provides lower reconstruction errors than the Wiener filter. Moreover, unlike the Wiener filter, the proposed method proved to be good at various values of the correlation coefficient between the pixels of the original image. The superpixel-based filtering algorithm is more efficient than the Wiener filter at the correlation coefficient between neighboring pixels less than 0.95.

It’s also shown that it’s sufficient to approximate superpixels with polynomials of the first degree, since at higher degrees the reduction in MSE between the reconstructed image and the ideal image isn’t significant.

The disadvantage of the proposed algorithm is the effect of the obtaining superpixel representation stage on the final result. In other words, an incorrectly selected superpixel threshold results in pixels of different objects are merged into a single superpixel. Conversely, when the noise level in the observed image is high, the oversegmentation may occur, and, as a result, the noise after filtering remains partially.

ACKNOWLEDGMENT

The research was supported by RFBR projects № 19-3790116, № 19-07-00474, and № 20-37-70053.

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