Heron and Alpha-Centroidal Mean Filters with Exponential Factors For

International Journal of Recent Technology and Engineering (IJRTE) ISSN: 2277-3878, Volume-9 Issue-1, May 2020

Heron and Alpha-Centroidal Mean Filters with Exponential factors for Deep Space Images Vivek B A, Keshav V Bharadwaj, Shankar Anbalagan, Abhinav Narayan, Sampath Kumar R

 white respectively, this is an 8-bit representation. There are floats and double representations for more accurate image Abstract: In this article, we discuss empirical methods that can description. Digital Image can be represented as a matrix be add-on to enhance spatial image filters. Discussing on the function mathematically. Consider a function f with two implementation of such methods for Deep-Sky images from space th th telescopes. Spatial image filtering houses mean filters that use parameters m row and n column. Fig. 1 shows are kernel method to filter a digital image in the spatial domain. To presentation of the function f in image pixel space. The improve the performance of the existing filters and aid the new Mathematical function f can be a grey image or a color image. upcoming filter, a technique is developed that can be added to the Where in a grey image contains pixel values from 0 to 1. filter to improve their performance. This study gives a detailed Color images have three channels Red (R), Blue (B), Green analysis of an empirical weight factor and exponential factoring methods for the mean filters that are proposed and implemented. (G) or simply known as RGB. Each channel can have values The exponential factoring is applied to the recent mean filters from 0 to 255 regions for an 8-bit image. Typically, the values Heron, Centroidal and Inverse-Contraharmonic. Performance of p and q are taken to be 3×3, 5×5, 7×7, and 9×9. The p × q analysis of both methods and comparison to existing mean matrix is called a kernel. Larger the kernel more filtering of filtering performance are represented. The images are classified noise, but the more blurring of the image takes place. The into white or black based background and analyzed for performance parameters, comparing the results from the filters surrounding pixels affect the center pixel. For a noised image and filters with the exponential factoring. The empirical methods the spatial filter is applied to the kernels, traversing through were proven to improve the efficiency in denoising of images of the image. Fig.1 the existing mean filters and the new Heron and Centroidal mean filters, hence can be used as add-ons to existing spatial mean filters. One of the empirical techniques, exponential factoring is promising in denoising high density noised images. This novel method is implemented to filter noise from Deep Space images taken by space telescopes. Ultra-high-resolution images are filtered and implemented on Single Board Computers for remote handling, with testing done on MATLAB. Keywords: Centroidal filter, Deep- Sky images, Empirical weighted factoring, Exponential factoring, Heron filter, Spatial Mean filters, Structural Similarity Index.

I. INTRODUCTION Fig. 1. Image Kernel and masking matrix. Digital Image Processing technique evolved from the 1960s in various institutes across the world. It gained thrust Transmission of image data through a channel induces from the late 1970s during the birth of the digital computer various noises. Spatial image filtering can be done on images era. The expense of circuitry for denoising is handled by which are corrupted with additive and multiplicative noises. digital signal processors, and currently PC- CPU’s are One of the highlighting properties of spatial image filtering is powerful enough to process them. Digital Image Processing “Complete removal of noises and preservation of edges”. involves the manipulation of image pixels using computer algorithms. The values of these pixels can be in different data A. Noise formats like integer format with 0 to 255 values for black to 1) Gaussian Noise: Usually termed as the noise in an amplifier. The signal is additive, the intensity is independent in each pixel. Statistically, the noise is Revised Manuscript Received on May 21, 2020. Gaussian distribution curve Fig. 2(a). Vivek B A, Electronics and Communication, RNS Institute of 2) Poisson Noise: Generated in electronic devices Technology, Bengaluru, India. Email: [email protected] referred as shot noise. Discrete nature of electric Keshav V Bharadwaj, Electronics and Communication, RNS Institute of Technology, Bengaluru, India. Email: [email protected] charge flow is the reason for this noise Fig. 2(b). Shankar Anbalagan, Computer Science, RNS Institute of Technology, Bengaluru, India. Email: [email protected] Abhinav Narayan, Electronics and Communication, RNS Institute of Technology, Bengaluru, India. Email: [email protected] Dr. Sampath Kumar R, Mathematics, RNS Institute of Technology, Bengaluru, India. Email: [email protected]

Published By: Retrieval Number: A2866059120/2020©BEIESP 2333 Blue Eyes Intelligence Engineering DOI:10.35940/ijrte.A2866.059120 & Sciences Publication

Heron and Alpha-Centroidal Mean Filters with Exponential factors for Deep Space Images

3) Salt and Pepper Noise: A Mathematical model is as in image after filtering. Mathematically it is given by (1). (3). If the probability matches, it replaces that pixel by salt or pepper. Otherwise, the original pixel value 1 mn11 MSE[ s ( ,  ) t (  ,  )]2 (1) is retained. Original image and Salt & Pepper noised pq*  image can be compared in Fig. 2(c). uv00 4) Speckle Noise: Speckle is multiplicative type of noise. 2) Peak Signal to Noise Ratio (PSNR): Ratio for extreme It is one of the tough noises to be removed. Speckle values of signal and noise. A higher value means the noise is a major issue in medical imaging and SAR (Synthetic Aperture 2 Radar). Original image and reconstructed image is said to be good. Mathematically it Speckle image can be compared in Fig. 2(d). is given by (2).

r 2 10log (dB ) (2) 10 MSE

3) Mean Absolute Error (MAE): Shows how much the predicted value differs from observed values. Mathematically given by (3). (a) Hilbert curve Gaussian Noise 1 N (3)  prdict   observed N i1

4) Coefficient of Correlation (COC): Estimation of one variable when other variable changes. (4).

abii (b) Cameraman Poisson Noise r  (4) 22 abii

II. RELATED WORK H Hu and G. De Haan in [1] gave insight into how hybrid filters are performing better compared to linear filters. The Hybrid filter is designed with combinational vectors and adaptive filtering. Sampath Kumar and K M Nagaraja have (c) Hilbert curve Speckle Noise elaborated on rth oscillatory means in [2]. They have also contributed in bringing up of convexities of generalized Heron means and convexity of Invariant Contra-harmonic means, discussed in [5],[8]. They laid the foundation for the upbringing of Inverse Contra-harmonic and Heron means. Along with Geetha and Satish Kumar, they have analyzed the performances of these means in [9]. The performance values of Normal means in [9] are taken for comparative analysis in this work. Improvements to the (d) Hilbert curve Salt & Pepper Noise methods and equations used in [9],[10] have been proposed and implemented in this work. K M Nagaraja with others in Fig. 2. Original and noised images with Hilbert Curve [4] has discussed the mean inequalities of harmonic and and Cameraman test images. Contra-harmonic mean. In [7] the authors have shown paths for new means by proposing Alpha Centroidal mean. Azadeh Hilbert curve image is chosen to serve as the standard Noori Hoshyar in [3] discusses the implementation of spatial image for comparison. mean filters in the identification of skin cancers. This work has proved the uses of a mean filter in denoising of images. The mean filters used in [9] are discussed in below subsection. B. Parameters Statistical methods used to analyze the performance of A. Mean Filters mean filters and the proposed empirical methods. 1) Harmonic Mean: For p × q dimension harmonic mean is represented as in Fig. 1. 1) Mean Square Error (MSE): The computation of average for squares of errors. In image processing, it is used to compare the pixels of noiseless image with the pixels of

Published By: Retrieval Number: A2866059120/2020©BEIESP Blue Eyes Intelligence Engineering DOI:10.35940/ijrte.A2866.059120 2334 & Sciences Publication International Journal of Recent Technology and Engineering (IJRTE) ISSN: 2277-3878, Volume-9 Issue-1, May 2020

Function k is the kernel function accessing the s row and t Contra-harmonic means. Weighted factors are introduced in column. The product of total rows and columns p × q is in the form of fractions and exponentials. product with the general harmonic mean. Heron mean formula is introduced with 2/3 and 1/3 weight values. pq* 1 ˆ ˆ 2 1 1 pq* f (,)  (5) f(,)  k (,) s  ( k (,)) s  (10) 1 3pq * 3   (,),s    (,),s    (,),s    ks(,) Equation (10) can be used for multidimensional arrays. Such 2) Contra-harmonic Mean: Mathematically represented as as in filtering of RGB images, which contain in (6). multidimensional arrays in three dimensions. The Centroidal mean is also introduced with the weighted fractional values. Q1  ks(,) Mathematically, it can be represented as in (11). fˆ(,)  (,),s    (6) ks(,) Q  Q1  (,),s     ks(,)  1(,),s    2pq * (11) fˆ(,)  1 3) Heron Mean: Heron mean is a multiplicative denoising 33Q   ks(,)  ks(,) technique. As a result, consumes more memory and power. (,),s    (,),s    Resulting in good denoising. Mathematically it can be represented as in (7). Discussed in [7] Inverse Contra-harmonic mean has been upgraded with the weighted factors in its exponents. It is represented as in (12). 1 1 Q ˆ pq* (7) ks(,) 2 f(,)  k (,)( s  k (,)) s   pq*   (,),s     pq* (,),s    (,),s    ˆ  (12) f(,)*(,)   k s   ks(,) Q1 (,),s    4) Centroidal Mean: Similar to calculating the centroid of a  (,),s    regular shape. It is a combination of Contra-harmonic and harmonic means. Q values give second-order effects. (8). Discussed in [7]. IV. EMPIRICAL METHOD OF EXPONENTIAL FACTORING

Q1  ks(,) Using a Kernel of different dimensions can result in  different levels of denoising. In the novel new method, the ˆ (,),s    pq* (8) f (,) emphasis is given to effects of the surrounding pixels to the 1 Q center pixel. While considering higher dimensions for the  ks(,)  ks(,) (,),s    (,),s    kernel an exponentially decaying factor is induced. Pixels further than the center doesn’t resemble the center pixel as 5) Inverse Contra-harmonic Mean: Can be obtained with compared to nearer pixels. In the standard approach, it gives a properties of means with Contra-harmonic and Heron means. linear priority for all kernel dimensions. This method decays ( 1) (  2) ( n ) The Q values have second-order effects. Mathematically this the inﬂuence of the outer pixels. e, e , ...., e as the mean can be represented as in (9). Discussed in [7]. dimensions go to n. Fig. 3. The Q factor is used as a dependent parameter for the exponential decaying effect. The means  which allow second orders have this additional feature. ks(,) Q 1  pq* ˆ (,),s     (9) f(,)*(,)   k s  Q1 (,),s     ks(,) (,),s   

Inverse Contra-harmonic terms with powers of Q and Q+1 are taken as reciprocal. They further aide in developing the exponential factoring.

Fig. 3. Kernel Denoising III. EMPIRICAL WEIGHTED FACTORING Methods of factoring for mean ﬁlters to improve their performance. The weighted factors improve the denoising property of the Heron, Centroidal, and Inverse

Published By: Retrieval Number: A2866059120/2020©BEIESP 2335 Blue Eyes Intelligence Engineering DOI:10.35940/ijrte.A2866.059120 & Sciences Publication

Heron and Alpha-Centroidal Mean Filters with Exponential factors for Deep Space Images

3   ks(,)  1(,),s    2 3*3   1 332   ks(,)  (,),s    ks(,) (,)   (,),s    3*3*3  3   ks(,)  0.5 1(,),s    2 5*5 e *  331 ks(,) 2   ks(,) (,),s    (,),s     3*3*3   3  Fig. 4. Exponential factoring for 5×5 and 7×7  ks(,)  0.5 1(,),s    2 3*3 e *  331 ks(,) 2  The exponential factoring involves higher dimension  ks(,) (,),s    (,),s    image pixels. For instance, using a 5x5 matrix of image pixels  3*3*3 is taken and the pixels external to the 3x3 matrix is factored  3  with an exponential weight. This allows for a much smoother  ks(,)  0.75 1 (,),s    2 7*7 transition of the pixels, represented in Fig.8. This improves e *  3 3 1 the performance of the mean filter in terms of PSNR and ks(,) 2     ks(,) (,),s    (,),s    MAE. Represented Mathematically in (13), (14).  3*3*3 ˆ  f (,) ks(,) 3  pq,3   (,)    0.75 1(,),s    2 5*5 2*3*3 e *  (13) 331 ks(,) 2  ˆˆ  ks(,) (16) ff(,)(,)    (,),s    (,),s    0.5  p, q 5 p , q 3   e 3*3*3 2*3*3  The empirical exponential factoring uses weighted methods as the base equation. The exponential factors applied 3   ks(,) can be a three layer kernel, five layer and seven layers. 1(,),s    2 3*3   1 Equation (15) depicts the 7 layer exponential factor that is 332   ks(,)  applied as an add on to weighted factoring which can be in (,),s    ks(,) (,)   (,),s    2*3*3 turn added to any spatial mean filter to improve performance.  Equation 16 is the implementation of the 7 layer empirical 3  (14)  ks(,) exponential factoring to Centroidal mean. Equation 18 is the 0.5 1(,),s    2 5*5 e * use of Heron mean in exponential factoring and (17) depicts 331 ks(,) 2   ks(,) for Inverse Contra-harmonic mean. Gamma enhanced (,),s    (,),s     2*3*3 exponential filtering. Factoring exponentially and usage of gamma functions as a multiplier. Introducing gamma 3   ks(,) multiplier to the exponential factoring has been proven to 1(,),s    2 3*3 e0.5 *   1 improve the PSNR of the denoised image at very high 332   ks(,)  percentages of noise (above 50 percent). This method can be (,),s    ks(,) (,),s    applied with all the mean filters as in the case of exponential 2*3*3 factoring, represented in (19).

V. KERNEL METHODS ˆ f (,) Spatial image filtering starts with capturing a digital image. (,)   pq,3 3*3*3 (15) This image is then run through the various algorithms to ffˆˆ(,)(,)    enable noise filtering. In the testing procedure, we induce  p, q 5 p , q 3  0.5 e  noise to a regular image and apply the filtering techniques. 3*3*3 The data sets are then analyzed for performance parameters and limits. ffˆˆ(,)(,)     p, q 7 p , q 5  e0.75  3*3*3 

Published By: Retrieval Number: A2866059120/2020©BEIESP Blue Eyes Intelligence Engineering DOI:10.35940/ijrte.A2866.059120 2336 & Sciences Publication International Journal of Recent Technology and Engineering (IJRTE) ISSN: 2277-3878, Volume-9 Issue-1, May 2020

2 1 1 1 k( s , )  ( k ( s , ))3*3 3 3*3 3  (,)   (,),s    (,),s    3*3*3 1 0.5 2 1 1  e* k ( s , )  ( k ( s , ))5*5 3 5*5(,),s    3 (,),s     3*3*3  1 0.5 2 1 1  e* k ( s , )  ( k ( s , ))3*3 3 3*3(,),s    3 (,),s    3*3*3 1 0.75 2 1 1  e* k ( s , )  ( k ( s , )) 7*7 3 7*7(,),s    3 (,),s     3*3*3  (18) 1 0.75 2 1 1  e* k ( s , )  ( k ( s , ))5*5  3 5*5(,),s    3  (,),s     3*3*3

(  ,  )  (  ,  )*    where 'w' is initially '1' (19)

In the image processing community, few standard images are used for analysis. Digital image of JPEG format is taken as input. To test for noises we induce the four different noises one at a time. Create storage for the noised images. Apply the Mathematically formulated mean filters to the image data. For statistical analysis, parameters are used. Comparing the input Fig. 5. Algorithm implementation image, noised image, and reconstructed image Fig. 5. As the kernel slides through the image, it leaves out the edges of the  ks(,) 2 2 image. To overcome this, a copy of the same layer is made to  3*3 (,),s     *(,)ks aid the kernel filtering by this method the edges of the image  3 (,),s    are preserved. Along with this, there are other techniques  ks(,) (,)   (,),s    which optimize edge denoising Fig. 4. Image set also contains 3*3*3 variants based on the overall background of an image. If the  background color features range is towards black it is ks(,) 2 2  5*5 classified as a black background image, otherwise as a white 0.5 (,),s     e**(,) k s  background image. The denoising analysis is also done on  ks(,) 3 (,),s     these variants and compared for results. The exponential (,),s     factoring, weighted factoring, and plain mean filters are also 3*3*3  considered as three base parameters to further classify.  ks(,) 2 2  3*3 0.5 (s,),    VI. ANALYSIS e **(,)ks  ks(,) 3 (,),s    Referring to Table-I, The results are compared between  (,),s    images denoised with Heron, Centroidal and Inverse  3*3*3 Contraharmonic filters. With normal and weighted factors   implemented. Four types of standard noises are applied; they  ks(,) 2 2   7*7 are Gaussian, Poisson, Speckle and Salt & Pepper. Statistical 0.75 (,),s     e**(,)  k s   parameters are MSE, PSNR, COC, and MAE, referring to  ks(,) 3 (,),s     noise and parameter sections respectively.Overall results (,),s     show that weighted factoring has dramatically improved the 3*3*3 performance of the filter at noise rates of 10 percent. PSNR  2 values changed straight from 12 to 23 for Heron filter after ks(,) 2  5*5 0.75 (,),s     weighted factoring. MSE values are ideal if they are closer to e**(,) k s   zero. For Speckle noise, Heron filter MSE value dropped ks(,) 3 (,),s     (17) from 0.0178 (normal filter) to 0.0041 (weighted factoring (,),s    filter).There has been a highly notable improvement for 3*3*3 Inverse Contra-harmonic mean filter in terms of MSE and PSNR values after weighted factoring.

Published By: Retrieval Number: A2866059120/2020©BEIESP 2337 Blue Eyes Intelligence Engineering DOI:10.35940/ijrte.A2866.059120 & Sciences Publication

Heron and Alpha-Centroidal Mean Filters with Exponential factors for Deep Space Images

This improvement is seen for denoising of Salt & Pepper the high resolution images of nebulas and star clusters. The noise. Three types of prominent noises for the test are further sections are discussed with the implementation of the considered. The analysis is done in noise range of 60 percent methods to RGB images and their outcomes. How they have for all the three noises. The overview states that exponential improved denoising in high density noised Deep-Sky images. factoring improves performance significantly at higher noise rates. Standard test image of Cameraman are utilized to compare the performances of the spatial filters with empirical weighted factors and exponential factors. Fig. 6(a), 6(b) shows the cameraman denoising of Gaussian noise with Heron filter, applied with the two empirical formulas. Fig. 7(a), 7(b) shows the application of the empirical formula to denoise with Centroidal mean. Black background images are filtered better than white counterparts. For white background images, standard Camera Man images are used. As for the black background images the Deep-Sky images of Jupiter is used. These two serve as standard test images for the empirical factors performance analysis in different image backgrounds. Referring to Table-II, Standard image used for the comparisons is a Deep-Sky photograph of planet Jupiter. The performance of these images is discussed and depicted with Table-I: Performance improvement due to empirical weighted mean factoring. Comparison of weighted method to results in [7]

Gaussia Filter Parameter n Poisson Speckle Salt & Pepper

Weighted Norma Weighte Norma Weighte Norma Weighted Norm Empirical Method: Factors l d l d l Factors al

MSE: 0.0042 0.0165 0.0030 0.0108 0.0041 0.0178 0.0127 0.0215 PSNR: 23.723 12.394 25.157 14.102 23.849 12.072 18.9736 11.261

Heron

COC: 0.9649 0.9199 0.9745 0.9613 0.9655 0.9151 0.8981 0.8876

MAE: 0.0413 0.1005 0.0307 0.0805 0.0418 0.0962 0.0659 0.1017

MSE: 0.0074 0.0092 0.0059 0.0020 0.0076 0.0154 0.0287 0.0111 PSNR: 21.3345 14.9115 22.3161 21.5356 21.2117 12.7121 15.4209 14.1306

Centroidal

COC: 0.9475 0.9276 0.9525 0.9840 0.9434 0.8830 0.7467 0.9188

MAE: 0.0564 0.0759 0.0385 0.0327 0.0516 0.0252 0.0930 0.0805

MSE: 0.0074 0.0110 0.0058 0.1942 0.0076 0.0091 0.0296 0.2049

Inverse PSNR: 21.2937 14.1537 22.3649 1.5593 21.2184 15.0040 15.2915 1.4829

Contra-harmonic

COC: 0.9470 0.9193 0.9529 0.7392 0.9441 0.9285 0.7374 0.6709

MAE: 0.0567 0.0801 0.0383 0.3942 0.0516 0.0755 0.0953 0.3972

Published By: Retrieval Number: A2866059120/2020©BEIESP Blue Eyes Intelligence Engineering DOI:10.35940/ijrte.A2866.059120 2338 & Sciences Publication International Journal of Recent Technology and Engineering (IJRTE) ISSN: 2277-3878, Volume-9 Issue-1, May 2020

Table- II: Performance improvement of empirical exponential factoring to weighted factoring

Noise: Gaussian Poisson Speckle

Empirical Method: Exponential Weight Factors Exponential Weight Factors Exponential Weight Factors

White & White & White & White & White & White & Image Background: Black-BG Black-BG Black-BG Black-BG Black-BG Black-BG

Filter Parameter

MSE: 0.096 0.020 0.161 0.040 0.098 0.001 0.180 0.001 0.069 0.026 0.161 0.040

Heron

PSNR: 10.169 16.807 7.908 13.973 10.068 27.25 7.436 29.558 11.592 15.812 6.727 12.960

MSE: 0.034 0.087 0.062 0.028 0.033 0.0012 0.066 0.0049 0.034 0.0012 0.068 0.419 Centroid al

PSNR: 14.100 10.579 12.076 5.5068 14.7212 29.268 11.7607 23.1165 14,655 29.176 11.645 3.773

MSE: 0.449 0.0054 0.576 0.0055 0.644 0.0051 0.575 0.040 0.743 0.0058 0.743 0.0109

ICHM PSNR: 3.471 22.654 2.392 22.587 1.986 22.910 2.397 13.939 1.285 22.399 1.288 19.621

(a) Empirical weighted factors

(b) Empirical exponential factors

(b) Empirical exponential factors Fig. 7. Denoising Gaussian noise with a) weighted and b) exponential factors on Centroidal mean respectively. Fig. 6. Denoising Gaussian noise with a) weighted and b) exponential factors on Heron mean respectively

A. Graphical Analysis Comparison of results for normal filter, weighted filter, and exponential factoring. Analysis on how the factoring has improved the performance for higher noise induced. Fig. 8 shows the graph of Centroidal mean filter with normal factors and weighted factoring. The PSNR values on the y-axis show Fig. 8. Centroidal mean PSNR values. improvement for Gaussian and Speckle noise after additional weighted factoring. (a) Empirical weighted factors

Published By: Retrieval Number: A2866059120/2020©BEIESP 2339 Blue Eyes Intelligence Engineering DOI:10.35940/ijrte.A2866.059120 & Sciences Publication

Heron and Alpha-Centroidal Mean Filters with Exponential factors for Deep Space Images

Fig. 9 represents the graph of MSE values after denoising the four standard noises for Centroidal mean ﬁlter. The scaling is in the logarithm domain. MSE values for Speckle noise is signiﬁcantly brought down.

Fig. 12. PSNR analyses of Heron mean ﬁlter for weighted and normal ﬁltering.

Fig. 15 shows the logarithm scale of MSE values in the two Fig. 9. Logarithmic scaling of MSE values for Centroidal factoring methods. MSE values have been reduced for noises mean. compared to previous methods.

Fig. 13. Log scaling of MSE values of Heron mean filter. Fig. 10 is the analysis of MSE values for normal and weighted Exponential factoring further improves performance factoring for Inverse Contra-harmonic mean filter. Compared compared to weighted factors. A significant change is only at to the other filters, weighted factoring has had the most impact higher noise rate of 60 percent and above. Figure 15 shows on Inverse Contra-harmonic mean. Improved by a factor of denoising PNSR values for exponential factoring at 60 10. percent Gaussian noise. We can observe that it has improved Fig. 10. Inverse Contra-harmonic MSE log scaling for Heron and Centroidal means. Performance is in the same analysis. Values for weighted and normal filtering. level for Inverse-Contraharmonic mean. The analysis is done on images with black-based backgrounds.

Fig. 11. PSNR analysis for weighted and normal filtering for Inverse Contraharmonic filter. Fig. 14. Exponential and weighted filtering analysis for Heron mean filter results are shown in Fig. 12. Weighted Black background images. factoring had dramatically improved performance for Gaussian, Speckle, Salt & Pepper noises. Poisson denoise White backgrounds based filtering has similar results. doesn’t show very high improvements. Exponential factoring has improved filtering and results in higher PSNR, as shown in Fig. 15.

Published By: Retrieval Number: A2866059120/2020©BEIESP Blue Eyes Intelligence Engineering DOI:10.35940/ijrte.A2866.059120 2340 & Sciences Publication International Journal of Recent Technology and Engineering (IJRTE) ISSN: 2277-3878, Volume-9 Issue-1, May 2020

Fig. 15. Exponential and weighted ﬁltering analysis for White background images.

Fig. 17. Planetary image of Jupiter and its moon’s. Gaussian noise is induced and denoised with the different ﬁlters for comparison. Heron, Centroidal, and Inverse Contra-harmonic means are added with exponential factoring.

Fig. 16. Performance of methods with respect to noise In Figure 18, Filtering of Gaussian noise of 30 percent. The percentage. results are compared and it is evident that Heron and Inverse Contra-harmonic means are predominant. Fig. 16 shows the PSNR variation with the increase in noise Whereas the other filters are inducing more noise into the percentage. The variants of filtering with and without system by denoising. Salt and pepper filtering are done better factoring are compared. Exponential factoring decays slowly by Inverse Contra harmonic and Heron means. As seen in Fig. as compared to the other two methods. Hence the exponential 21, the above two filters are giving results at par with method has more stability and noise rejection. harmonic filter. In Fig. 20, the Hilbert Curve is chosen as a standard image. VII. DENOISING DEEP-SKY IMAGES This image represents the major shades of from white to black. This serves as a good template for comparing results. Standard test images and other Deep-Sky images are taken for analysis. As the filters work best on black background, denoising stellar photography would be well suited. High resolution images have been rendered and tested for performance of the filters. They have been proved to be an optimal and faster technique of denoising. Furthermore, RGB images of 3 layers have also been adopted. Image filtering is done on them with the filters working in all the three layers. The results have exceptional improvement from the base form of the filter. A. Grayscale Images of Deep-Sky Grayscale images are simpler to analyze and denoise. Fig. 17 shows the results of denoising Gaussian noise. Jupiter with its 3 moons is the original image of 1000x1000 pixels resolution Fig. 18. Gaussian noise of 30%, and exponentially are the original images. This image shows all the existing filters and the exponential factoring capability filters, at once. factored denoising for a star clusters image. Concluding Heron and Centroidal filters produce image closer to the original. Inverse Contra harmonic induces additional contrast to the rendered image.

Published By: Retrieval Number: A2866059120/2020©BEIESP 2341 Blue Eyes Intelligence Engineering DOI:10.35940/ijrte.A2866.059120 & Sciences Publication

Heron and Alpha-Centroidal Mean Filters with Exponential factors for Deep Space Images

higher noise percentage. Inverse Contra-harmonic has better boundary restoring capability.

Fig. 23. Heron mean denoising with exponential factoring.

Fig. 19. Salt & Pepper noise of 20%, denoised with different mean ﬁlters.

Gaussian noise of 70 percent is applied to the standard Hilbert Curve image as depicted in Fig. 21. The denoising is done with two ﬁlters Inverse Contra-harmonic and Heron.

Fig. 24. Speckle noise induced at 20%.

Fig. 24 is the standard Hilbert Curve image with induced Speckle noise of 40 percent. Inverse Contra-harmonic ﬁlter is

used to denoise. The resultant image is as in Fig.25. In the Fig. 20. Hilbert Curve, used as standard image for rendered image one can observe the much better distinction of grayscale boundaries where the shades of gray are varying.

Fig. 21. Gaussian Noise induced at 70% on Hilbert Curve. Fig. 25. Denoising Speckle noise by Inverse-Contraharmonic mean with exponential factoring.

B. RGB Images of Deep-Sky RGB images have three layers; this makes the processing three times more complex. The mean ﬁlters are applied to the layers individually and their results are optimally taken into Fig. 22. Inverse Contra-harmonic denoising with consideration.

exponential factoring.

The results are in Fig. 22 and Fig. 23 respectively. Inverse Contra-harmonic ﬁlter is removing the noise better than Heron at this percentage of noise. Both the means are equipped with exponential factoring to improve PSNR at

Published By: Retrieval Number: A2866059120/2020©BEIESP Blue Eyes Intelligence Engineering DOI:10.35940/ijrte.A2866.059120 2342 & Sciences Publication International Journal of Recent Technology and Engineering (IJRTE) ISSN: 2277-3878, Volume-9 Issue-1, May 2020

Fig. 29. Exponential Heron mean ﬁltering.

Fig. 26. Hubble Deep-Sky image introduced with 60% Gaussian Noise.

Fig. 26, Hubble Deep-Sky image of a galaxy cluster, induced with 60% Gaussian noise. The boundaries of the galaxies and stars have faded away. Fig. 27 is the result of denoising the Fig. 30. Centroidal denoising with exponential factors Fig. 26 with exponentially factored Heron mean filter. The and Speckle noise at 50% applied onto Hubble edges are well represented, this aide in further object Deep-Sky image. recognition steps. There is a very low percentage of false-color of green and retaining background features of the Fig. 30 is the comparison of noised images and Centroidal image. Exception image filtering can be analyzed in the above denoising with exponential factor aid. In this RGB image, two images for highly intensive noise in the background. after the three-layer analysis, the red layer is retaining some amount of noise. Centroidal filter otherwise is providing with one of the best boundary and feature restoration.Image of Jupiter and its moons is induced with 30 percent Gaussian noise as in Fig.31. Exponential Heron denoising is done; Fig. 32 is the resulting image. There is a small shift towards the blue layer for the overall image.

Fig. 27. Exponential factoring in multidimensional Heron mean denoising, for all channels in RGB

Speckle denoising is a tough process. Heron filtering is applied to Fig. 28, with exponential factoring. Resulting filtered image is as shown in Fig. 29. The boundaries are restored. Two major spots on the image can be taken for observation, the central galaxy and a bright yellow mid-right Fig. 31. Gaussian noise at 30% induced in planetary star. In the denoised image, the arms of the galaxy and image of Jupiter. boundaries of the star are hazy. Filtering has resulted in optimal feature restoration. This is one of the Hubble deep The clarity is restored, the moons are visible distinctly. Even field images which have very high resolution. after removing such background noises, tiny specs of the moons are retained.Applying exponentially factored Centroidal mean filter, to a 70 percent Gaussian induced noise image. At such high noise background, the filtering is very efficient. After even being through 70 percent of noise, the speck-like moons are restored (can be seen in a line below Jupiter). As shown in Fig. 33.

Fig. 28. Speckle noise at 20% causing hazy boundaries of stars.

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Heron and Alpha-Centroidal Mean Filters with Exponential factors for Deep Space Images

Fig. 32. Exponential Heron mean filtering. Blue shift in Fig. 34. Denoising by Inverse Contra-harmonic with the image. Jupiter’s moons are retained after added exponential factors. Trade off to red shift can denoising. be observed. Moons of Jupiter are retained with High-level denoising comes with a trade-off, retaining of red slight boundary haziness. layer noise. Even though exponential Centroidal mean has a Fig. 35 and Fig. 36 are the pairs of Speckle noised Deep-Sky better edge in denoising Black background high-resolution image and filtering. The denoising is done with RGB images, due to the red noise layer retention it is not Inverse-Contraharmonic mean. As mentioned and observed a advisable to use it into highly sensitive imaging. Such slight blue shift is present, for RGB images. situations can be handled by Inverse Contra-harmonic mean. The filtered image is as shown in Fig. 34. Inverse-Contraharmonic induces slight blue shift, as the images are Black background based, for human visual acuity dark blue to black won’t have large deviation. Unlike in the situation of retaining red noise layer. As it comes with a price, Inverse Contra-harmonic is not up to the level of Centroidal in terms of PSNR and MSE. The designer must evaluate for the best to be used for image filtering.

Fig. 35. Speckle noise at 20% on Hubble Deep Field image.

Fig. 33. Gaussian noise at 70% applied to the Jupiter Deep-Sky image. Fig. 36. Denoising via Inverse-Contraharmonic mean along with exponential factoring. Restoring the boundaries of galaxies and stars in the Deep Field Hubble image.

Published By: Retrieval Number: A2866059120/2020©BEIESP Blue Eyes Intelligence Engineering DOI:10.35940/ijrte.A2866.059120 2344 & Sciences Publication International Journal of Recent Technology and Engineering (IJRTE) ISSN: 2277-3878, Volume-9 Issue-1, May 2020

Deep field images are of very high resolutions. The image in multiple noises, also how different means can be combined Fig. 37 is originally 3920 × 4120 pixels. This image is for a given noise to achieve better reconstruction. Splitting the induced with 30 percent Speckle noise. By applying image into segments, filtering the respective segments and exponentially factored Heron mean filter, it has been later stitching them back. These are a few developments that denoised. Part of the noised image in the red rectangle of Fig. can be worked on. This paper represents an insight into 37 is shown in Fig. 38. The same area after denoising is empirical weighted and exponential factoring methods which displayed in Fig. 39. are proven to be fruitful techniques to the existing spatial filters and the freshly introduced Heron, Centroidal and VIII. CONCLUSION Inverse-Contraharmonic filters. With the better filtering techniques developed, they can assist object recognition and The spatial mean filters are proven to be good choices for tracking in real-time. Taking the image processing and moderate filtering. To achieve higher orders filtering in high computer vision impact in automation and technological density noise images adaptive filters or neural networks are developments to new heights. being used. The empirical method of weighted factors here shows the improvement to the existing spatial filters and the other new upcoming spatial filters. The weighted factoring has adapted to the Heron, Centroidal and Inverse-Contraharmonic filters, the results show their performance improvement. Second method of exponential factors implemented on top of the weighted factors take the performance charts another level higher. The filtering of Deep-Sky images form exponential factoring method is profound for high density noise and give a chance for the spatial mean filter to work at par with adaptive mean filters and neural networks. Image filtering is a challenging domain. Fig. 37. Heron filter with exponential factoring for To enhance the impact of these mathematical means in image denoising the Deep-Sky image of Hubble with 30% processing, new techniques of measuring parameters have to be designed. The current statistical methods are not Gaussian Noise completely dependable. There are cases where PSNR value is high and MSE is low but for human perception, the image looks noisy and not denoised. Using techniques like structural similarity index measurement (SSIM), image quality index (IQI) and other complex machine learning techniques depending on computer vision techniques will aid in analysis. Multiple noises and multiple means can be applied. The results can be analyzed for the behavior of the filter for

Fig. 38. High resolution Hubble Deep Field image of 3920x4120. Area under the red-box is selected for better comparison.

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Heron and Alpha-Centroidal Mean Filters with Exponential factors for Deep Space Images

Fig. 40. Exponentially factored Heron mean denoising. Fig. 39. Gaussian noise of 30% induced on the high Faint stars and galaxies are restored. resolution image.

Images. IEEE T. Instrumentation and Measurement. 60. 584-597. REFERENCES 10.1109/TIM.2010.2052478. 17. https://docs.opencv.org 1. H Hu and G. de Haan, ’Classification-based hybrid filters for image 18. https://homepages.inf.ed.ac.uk/rbf/HIPR2/filtops.htm processing. Proc.SPIE, Visual Communications and Image Processing’, 19. https://docs.opencv.org/3.0-beta/modules/imgproc/doc/filtering.html 6077, 607711.1-607711.10,2006. 20. https://in.mathworks.com/help/images/linear-filtering.html 2. K. M. Nagaraja and R. Sampathkumar, ’Schur convexities of r th 21. https://hubblesite.org/images/gallery oscillatory mean and its dual’, Proc. Jangjeon Math. 17(3), (2014), 22. http://hubblesource.stsci.edu/sources/illustrations/gallery pp.383-392. 3. Azadeh Noori Hoshyar a, Adel Al-Jumailya, Afsaneh Noori Hoshyar b, AUTHORS PROFILE ’Comparing the Performance of Various Filters on Skin Cancer Images’, International Conference on Robot PRIDE 2013-2014 - Medical Vivek B A is currently pursuing Bachelor of andRehabilitation Robotics and Instrumentation, Engineering in Department of Electronics and ConfPRIDE2013-2014. Communication from RNS Institute of Technology, 4. K. M. Nagaraja, Murali K, and Lakshmi Janardhana R C, ’Improvement Bengaluru, India since 2016. Actively coordinating of harmonic and Contra-harmonic Mean Inequality Chain’, Ijpam., Vol. workshops in fields of electronics. His research interests 114(4),(2017). include Computer vision, Robotics, Quantum 5. Sampathkumar.R, R.C.LakshmiJanardhana, K.Murali and K. M. Na- Computing, and Optoelectronics. garaja, Schur Convexities and Concavities of Generalized Heron Means, International Journal of Pure and Applied Mathematics, 117 (2017)13, pp.69-79. Keshav V Bharadwaj is currently pursuing Bachelor of 6. G. Toaderand S. Toader, ’Greek means and Arithmetic mean and Engineering in Department of Electronics and Geometricmean’,RGMIAMonograph,(2005),Australia. Communication from RNS Institute of Technology, 7. K.M.Nagaraja and P.S.K.Reddy, ’Alpha-Centroidal mean and its dual’,ProceedingsoftheJangjeonMath.Soc.15(2012),No.2,pp.163-170. Bengaluru, India since 2016. Actively coordinating 8. R. Sampath Kumar and K.M. Nagaraja, The Convexities of Invariant workshops in fields of electronics. His research interests Contra-harmonic Mean with Respect to Geometric Mean, International include Robotics, Quantum computing and Computer JournalofPureandAppliedMathematics,116(2017)22,pp.407-412. vision. 9. Sampath Kumar R, Satish Kumar T, K M Nagaraja and Geetha G, ’Application of Different Filters for Noise Removal in Digital Images’ Shankar Anbalagan is currently pursuing Bachelor of Int. Conf. on Signal, Image Processing Communication and Engineering in Department of Computer Science from Automation, ICSIP. 10. Sampath Kumar R: ’Mathematical Means and their Convexities’. PhD RNS Institute of Technology, Bengaluru, India since thesis, Bharathiar University,2019. 2016. Actively coordinating workshops in fields of 11. J. Arnal and L. Scar, Hybrid Filter Based on Fuzzy Techniques for computer algorithms and programming. His interests Mixed Noise Reduction in Color Images, Applied Sciences, vol. 10, no. include full stack web development, mobile application 1, p. 243, Dec.2019. development and cloud computing. 12. Anchal, Anchal&Budhiraja, Sumit & Goyal, Bhawna & Dogra, Ayush& Agrawal, Sunil. (2018). An Efficient Image Denoising Scheme for Abhinav Narayan is currently pursuing Bachelor of Higher Noise Levels Using Spatial Domain Filters. Biomedical and Pharmacology Journal. 11. 625-634.10.13005/bpj/1415. Engineering in Department of Electronics and 13. Tallapragada, V.V.S., Manga, N.A., Kumar, G.V.P. et al. Mixed image Communication from RNS Institute of Technology, denoising using weighted coding and non-local similarity. SN Appl. Sci. Bengaluru, India since 2016. His research interests are

2, 997(2020). in signal processing and communications, image 14. Fan, L., Zhang, F., Fan, H. et al. Brief review of image denoising processing, computer vision and machine learning. techniques.Vis.Comput.Ind.Biomed.Art2,7(2019). 15. B. Gopalan, A. Chilambuchelvan, S. Vijayan and G. Gowrison, ”Perfor-

mance Improvement of Average Based Spatial Filters through Multilevel Preprocessing using Wavelets,” in IEEE Signal Processing Letters, vol. 22,no.10,pp.1698-1702,Oct.2015. 16. Jarabo-Amores, Maria-Pilar & Rosa-Zurera, Manuel &Mata-Moya, David &Vicen, Ral& Maldonado-Bascn, Saturnino. (2011). Spatial- Range Mean-Shift Filtering and Segmentation Applied to SAR

Published By: Retrieval Number: A2866059120/2020©BEIESP Blue Eyes Intelligence Engineering DOI:10.35940/ijrte.A2866.059120 2346 & Sciences Publication International Journal of Recent Technology and Engineering (IJRTE) ISSN: 2277-3878, Volume-9 Issue-1, May 2020

Dr. Sampath Kumar R is currently working as Assistant Professor in the Department of Mathematics in RNS Institute of Technology, Bengaluru, India. He specializes in Engineering Mathematics with vast experience of 27 years. He has completed post graduation from Bangalore University and obtained his M.Phil from Alagappa University, Puducherry. He has been awarded doctorate from Bharathiar University in 2019 for mathematical means and convexities. He has publish more than 15 research papers in recognized international journals. His research interests are real analysis, mathematical means, graph theory, and linear algebra.

Published By: Retrieval Number: A2866059120/2020©BEIESP 2347 Blue Eyes Intelligence Engineering DOI:10.35940/ijrte.A2866.059120 & Sciences Publication