Introduction

The images are generally designed for the purpose of exhibiting fruitful data. It is often found that the original images are habitually spoilt on account of certain disarray at the time of image acquirement. Hence, the vital motive of the image restoration is concerned with the function of either “compensating for” or “undoing” the inherent deficiencies which ultimately results in the degradation of the image concerned [1]. The degradation generally appears in many versions like the motion blur, noise, and camera mis-focus. In the case of the motion blur, it is easy to arrive at an excellent evaluation of the actual blurring function and carry out the process of “undoing” the blur to effectively regain the original image. However, where the image is tainted by noise, the only possible option to perform the process which duly compensates for the degradation already happened. In the current investigation, it is planned to launch novel methods and execute various approaches extensively used in the domain of the image processing with the intention of effectively regaining the images [2].

In the ever-zooming realm of the image processing, the image restoration enacts a very vital role for the purpose of attaining excellent quality image from the noise-polluted or tainted image. In this regard, the denoising appears on the arena as the most appealing image restoration approach applied to get rid of the noise persisting in the input image [3]. The superfluous noise in the images has to be drastically reduced so as to improve the efficiency of the image restoration performance well-ahead of the restoration procedure in the image pre-processing technique. In fact, the image de-noising contributes its mite by performing very vital functions in various facets [4].

It is with the ultimate objective of eliminating the noise from a degraded image in the course of attainment and communication that the image de-noising is habitually utilized. While taking due care in preserving the vital signal features intact, the image de-noising is elegantly executed to eliminate the additive noise [5]. The most critical feature for any noise-polluted data is concerned with the deployment of an ideal de-noising technique for the purpose of compensation. An added issue cropping up in the process of the de-noising is to appropriately address the ‘staircase effect’ simultaneously conserving the sharpness and convexity of the image [6].

In this connection, the Wavelet transform (WT) projects itself as an ideal candidate dedicated for the purpose of the disintegration and rebuilding of the multi-dimensional signals for their investigation, resolution enrichment and additional processing [7]. The noise elimination by means of diverse versions of the low pass filtering approaches has established itself as a very significant topic of exploration in the realms of the digital image and signal processing. Incidentally, the term ‘image restoration’ refers to the process of denoising a tainted image. The fundamental objective of the restoration technique is concerned with the regaining of the original image from the perceived tainted image. Nowadays, the image restoration techniques are extensively employed to design the degradation procedure and perform a roughly inverse procedure to the tainted image for the purpose of regaining the original image [8].

The efficacy of the corresponding restoration methods is squarely dependent on the accessibility and totality of the knowledge regarding the impulse deprivation procedure and on the configuration of the filtering technique. The linear filter is sufficient enough to get rid of the noise in respect of a bandwidth constrained additive noise such as the Gaussian noise from the tainted images. For the purpose of effectively eradicating the related deficiencies, the non-linear filters such as the median filter are habitually utilized to denoise the Salt & Pepper noise from the tainted image. The median filter, in essence, represents a computationally effectual nonlinear filter intended for the denoising of the impulse noise in addition to conserving the edges. A feast of diverse image restoration approaches have been fascinatingly flagged off in the literary arena which is intended for the purpose of the digital image processing [9].

Related works

Lei Yang et al. [10] elegantly envisaged an innovative feature-preserving non-local means approach for denoising the tainted images so as to perk up the feature revival and particle recognition. The most widely employed non-local means filter was found to be unsuitable for the noise-polluted bio-logical images encompassing trivial facets of significance as the image noise invariably thwarted the appropriate estimation of the accurate coefficients for the purpose of averaging, resulting in the over-smoothing and parallel artifacts. They successfully tackled the issue by generating a particle feature probability image, and illustrated the fact the novel filter was competent to realize superior levels of the peak signal-to-noise ratio in the denoised images and was well-equipped with the requisite skills in effectually detecting the weak particles when applied to artificial data. Further, they unequivocally established the fact that their feature-preserving non-local means filter was able to considerably cutback the threshold of imaging conditions essential for the achievement of significant data.

Tzu-Chao Lin et al [11] charismatically launched a novel decision-based fuzzy averaging (DFA) filter noise detector. The novel filter was competent to efficiently address the impulsive noise, and a mixture of Gaussian and impulsive noise. The combination confidence value represented the decision rule in respect of the D–S noise detector. Further a fuzzy averaging technique was also brilliantly brought in, where the weights were built by means of a pre-determined fuzzy set, for the purpose of attaining the noise abolition. Further, a straightforward second-pass filter was effectively utilized to augment the efficiency of the ultimate final filtering accomplishment. The enthusing test upshots effectively exhibited the exemplary efficacy of the novel DFA filter in the compression of the suppressing impulsive noise and also the mixture of the Gaussian and impulsive noise, together with a significant enhancement in the apparent image excellence.

Chul Lee et al. [12] were instrumental in efficiently launching an innovative nonlocal minimum mean square error (MMSE) image denoising technique. In their novel approach, they investigated the nonlocal neighbors from an external database and also the complete input image to boost the efficiency in execution even in cases where a noise-polluted block is not likely to contain identical blocks within the image. In view of the fact that the extensive search range necessitated a greater computational load, they deftly devised a probabilistic tree-based search technique to considerably cutback the computational complication. The remarkable replication results have revealed the fact that the novel technique was well-geared to exhibit further exemplary denoising performance in relation to that of the traditional nonlocal means filter.

An innovative and powerful pattern for the purpose of the image and video de-noising, de-blurring, and super resolution reconstruction was effectively green-signaled by Haichao Zhang et al. [13]. The overall test outcomes on the mutual single images and pragmatic video progressions illustrated with perfect precision that the novel structure gained a convincing edge over the earlier investigations both in qualitative and quantitative aspects.

A Fenchel duality root from dictionary learning (FD-DL) technique for the restitution of the noise-tainted images was brilliantly brought to limelight by Shanshan Wang et al. [14]. By way of the inhibited concern for the additive white Gaussian noise, the sparse image illustration was planed as an l2-l1 reducing challenge, deftly deploying an oversimplification of Fenchel’s duality thesis and effectively resolved depending on the improved Lagrangian frame-work whose dual configuration was established. Further, the fantastic technique was assessed and compared with four diverse sophisticated approaches such as the integration of the local pixel grouping-principal element examines, K-singular value disintegration, the method of optimal directions, on grayscale natural images and the beta process factor examine. It was crystal from the outstanding outputs released by various algorithms that the FD-DL technique was able to reach the top of the list by effectively enriching the excellence of the image and also because of its inherent skills in regard to the restoration of the tainted image which when during the comparison with the other four popular techniques, scaled a significant edge over them.

Wishing Dong et al. [15] deeply discussed the issue of the sparse coding noise and efficiently elucidated how the ultimate motive of the image restoration was able to put a hold on the sparse coding noise. Further they effectively discouraged nonlocal self-similarity of the image to successfully achieve the superior appraisal of the sparse coding coefficients of the genuine image for integrating the sparse coding coefficients of the experimental image to the related evaluation, for achieving the purpose. While their extensive experiments on diverse classes of image restoration challenges such as the de-noising, de-blurring and super-resolution, authenticated the generalization and hi-tech performance of the projected NCSR technique, the self-styled non-locally centralized sparse representation (NCSR) pattern emerged as simple as the typical sparse representation model.

The Wavelet-based sparse reduced-rank regression (WSRRR) technique for the hyper-spectral image restoration was brilliantly brought to limelight by Behnood Rastiet al. [16]. The innovative approach was dependent on alleviation of a sparse stipulation hassle linked to an orthogonality constraint. With the intention of scaling down the dilemma a cyclic descent-type technique was elegantly employed. Taking inspiration from Stein’s unprejudiced risk inference, they pressed the green light for a new approach for selecting the tuning constraints. The suggested technique was efficiently evaluated by means of the signal-to-noise ratio and spectral angle distance for a simulated corrupted data set which were classified as efficiencies for a bona fide data set. By means of employing a minimal number of sparse components it was established that the hyper-spectral image could be regained. In their projected hypothesis, two distinct classifiers such as the support vector machines and random forest were significantly employed. Their anticipated method was compared and contrasted with the parallel restoration approaches which upheld the superiority and significance of the WSRRR for the replicated tainted data set.

Proposed image restoration technique

The innovative technique flows through the following five phases.

  1. 1.

    The preprocessing

  2. 2.

    The denoising employing the OGHP

  • The optimization employing the optimized Cuckoo Search technique

  1. 3.

    The denoising employing the shrinkage

  2. 4.

    The evaluation employing the DWT.

  3. 5.

    The image restoration employing the AGA

Preprocessing

The noise represents the adverse impact generated in the image which ultimately leads to the diminution in the quality of the image. Hence, it is all the more essential to get rid of the noise from the image. In the novel approach, for the purpose of steering clear of the noise, two distinct and effective filters are employed which are shown below.

  • The Adaptive Median Filter

  • The Adaptive Fuzzy Switching

Prior to the preprocessing of the input image, two diverse categories of noise are added to the image and they are known as the salt & pepper and the Gaussian noise.

Adaptive median filter

The Adaptive Median Filtering characterizes a nonlinear low pass filter which is sufficient for the purpose of eliminating the outliers which constitute the intense pixel values in an image. The input image \( {R}_d=\left\{{f}_{i^1},{f}_{i^2},{f}_{i^3}\cdots {f}_{i^j}\right\};j=1,2,3,\cdots N \) is fed to the Median filter.

The adaptive median filter is performed on the images rd(f, s) which are tainted by the (Gaussian noise) and subsequently a noiseless image is achieved as an output. The gradual procedure of the adaptive median filtering in the noise elimination is effectively pictured as follows.

  1. Step 1:

    Initialize the window w size wz.

  2. Step 2:

    Examine whether the center pixel pcen(r, s) within w is noise-contaminated. If the pixel pcen(f, s) is noise-polluted proceed to step 3. Else, slide the window to the successive pixel and replicate step 1.

  3. Step 3:

    Organize the whole pixels within the window win an increasing order and locate the minimum (pmin(f, s)), median (pmed(f, s)), and maximum (pmax(f, s)) values.

  4. Step 4:

    Evaluate whether pmed(r, s) is noise-polluted. (i.e.) pmin(f, s) < pmed(f, s) < pmax(f, s).

If the median value range falls in between the minimum and maximum means the pixel is free from noise and then proceed to step 5. Else, pmed(f, s) is a noise-polluted pixel and hence proceed to step 6.

  1. Step 5:

    Substitute the related centre pixel in output image with pmed(f, s) and proceed to step 8.

  2. Step 6:

    Examine whether all the other pixels are noise-polluted. If yes then enlarge the window size by 2 and return to step 3. Else, proceed to step 7.

  3. Step 7:

    Substitute the center pixel of the image with the noiseless pixel which is the nearest one of the median pixel pmed(f, s).

  4. Step 8:

    Reorganize window size wz and center of window to the succeeding pixel.

  5. Step 9:

    Replicate the above steps till all the pixels are processed.

Fuzzy switching method

Of late, in the literature regarding the denoising, the fuzzy switching median filter has emerged as one of the most widely used techniques. Habitually, it contains two specific phases. The initial phase is the noise recognition phase which is essential before proceeding to the fuzzy switching technique. In accordance with the noise identification, it is possible to differentiate between the “suspicious noise pixels” and the “noise free pixels”. Then, the fuzzy switching technique is performed for the cancellation module as detailed below.

$$ d\left(i+k+1\right)=\Big[X\left(i+k,j+1\right)-X\left(i,j\right) $$
(1)

Then the constraint D(i, j) characterizing the local data is defined as the maximum absolute luminance difference in the filtering window as shown in Eq. 4 given below.

$$ D\left(i,j\right)=\max \left\{d\left(i+k,j+1\right)\right\} $$
(2)

In accordance with D(i, j) pixels in the image X is segmented into three diverse categories such as the “noise free pixels”, “suspicious noise free pixels” and the “suspicious noise pixels”. The fuzzy reasoning used is illustrated in Fig. 1 and the function f(i, j) is illustrated by means of Eq. 5 shown as follows.

Fig. 1
figure 1

Architecture of the Proposed Image Restoration Technique

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$$ {\displaystyle \begin{array}{l}f\left(i,j\right)=\Big\{0,D\left(i,j\right),{T}_1\\ {}f\left(i,j\right)=\Big\{\frac{D\left(i,j\right)-{T}_1}{T_2-{T}_1};{T}_1\le D\left(i,j\right)\le {T}_1\\ {}f\left(i,j\right)=\Big\{1,D\left(i,j\right)\ge {T}_2\end{array}} $$
(3)

The consequential noiseless image is thereafter furnished to the image denoising procedure.

Denoising optimized gradient histogram preservation (GHP)

Here, the image denoising is modeled by the gradient histogram preservation together with the sparse non-local regularization, and an effective histogram specification technique to resolve the anticipated model for the purpose of the texture enhanced image denoising.

In respect of a specified noiseless image n(x), the noisy observation n(y) of n(x) is habitually represented by means of the following Eq. 6.

$$ n(y)=n(x)+g(n) $$
(4)

Here g(n) represents the noise with zero mean and standard deviation σ. The vital motive of the image denoising is to assess the desired image n(x) from n(y).With the result, there emerges a well-ordered iterative histogram specification technique to resolve the model in Eq. (2). Further, from the shrinkage dependent noise riddance process, the denoised image I2 is effectively attained. As a result the optimized Gradient Histogram Preservation (OGHP) dependent denoised images I1 and shrinkage dependent denoised image I2 are incorporated by the Discrete wavelet transform. Obtaining the denoised image using GHP process is detailed in the previous paper except the optimization process.

Gradient histogram preservation optimized using MCSA

Principle behind modified cuckoo search algorithm

Each cuckoo bird lays a solitary egg at a time when it is abandoned into an arbitrarily selected nest. The optimum nest with the excellent quality eggs is carried over to the successive generations. The number of host nests is observed to be stagnant and a host is capable of locating an alien egg with a probability (Pa) [0, 1], whose existence ultimately leads to either discarding the egg or nest by the host bird. It is pertinent to note that each egg in a nest characterizes a solution while a cuckoo egg symbolizes a novel solution in which the ultimate motive is to substitute the feeble fitness solution by a novel one.

The flowchart for the CSA is effectively exhibited below together with the vital steps:

  1. Step 1:

    Bring in a random population of n host nests, Id

  2. Step 2:

    Achieve a cuckoo arbitrarily by Levy flight conduct, d

  3. Step 3:

    Evaluate its fitness function, Fd

  4. Step 4:

    Choose a nest arbitrarily among the host nests termed as j and evaluate its fitness, Fj

  5. Step 5:

    If Fd < Fj then substitute j by the new solution otherwise let j be the solution.

  6. Step 6:

    Abandon a tiny proportion of Ax of the worst nest by constructing new ones at the new locations by employing the Levy flights.

  7. Step 7:

    Maintain the current optimum nest, Return to Step (2) if T (Current Iteration) < MI (Maximum Iteration).

  8. Step 8:

    Locate the optimum solution.

In this regard, the most vital phases involved in the CSA include the Initialization, where an arbitrary population of n host nest (Id = 1, 2, 3...n). ii) is brought in, and the Levy Flight Behaviour in a cuckoo is achieved by means of the Levy flight behaviour equation as illustrated in the following Eqs. 5 and 6 respectively.

$$ {I}_d\left(\mathrm{t}+1\right)=\kern0.5em \left(\mathrm{t}\right)+\alpha \oplus levy\left(\lambda \right),\alpha >0 $$
(5)
$$ levy\left(\lambda \right)=t\left(-\lambda \right),0 to\alpha $$
(6)

In the innovative customized cuckoo search technique, arbitrary values are chosen for λ.

Fitness calculation

At the outset, the fitness is evaluated by means of the fitness function with the intention of achieving an optimum solution. Now an arbitrary nest is selected, which is termed as j.

fitness = max(PSNR)

Subsequently, the fitness of the cuckoo egg (new solution) is appraised and contrasted with those of the host eggs (solutions) existing in the nest. If the value of the fitness function of the cuckoo egg falls within or is equivalent to that of the arbitrarily selected nest, then the arbitrarily selected nest (j) is substituted by the new solution.

Termination

If the algorithm is terminated subsequent to attainment of the maximum number of generations, it is possible that a suitable solution is achieved or not.

Combining image using DWT

The optimized Gradient Histogram Preservation (OGHP) and shrinkage are fused by the discrete wavelet transform. The denoised images are represented as I1andI2.The discrete wavelet transform (DWT) represents a linear transformation which functions on a data vector whose length constitutes an integer power of two, converting it into a numerically different vector of the identical length. It represents a device that divides the data into various frequency segments, and thereafter investigates each component with resolution harmonized to its scale. The DWT [3] is evaluated with a cascade of filtering accompanied by a factor 2 sub sampling.

In wavelet image fusing technique, the source images I1(x, y) and I2(x, y), are decayed into the approximation and comprehensive coefficients at the preferred level with the help of the DWT.The integrated image If(x, y) is attained by taking the inverse discrete wavelet transform (IDWT). The current and the upcoming sections are also detailed in the previous paper.

Image restoration using AGA

The noiseless image If' attained from the hybrid filter appears to be distorted, hence it is highly essential to fine-tune the quality of the image. With an eye on boosting the noiseless image quality, we have resorted to the deployment of AGA method along with Richardson-Lucy (R-L) algorithm. In the image restitution procedure, the all-powerful Point Spread Function (PSF) is entrusted with the task of carrying out successful restitution. The optimized PSF by means of our new-fangled AGA is advantageous to the RL algorithm for restoring the distorted image. The modus operandi of AGA regarding PSF calculation is detailed as follows:

Initialization

At the outset, a chromosome hi; 0 ≤ i ≤ nc is engendered, where nc represents the total number of produced chromosomes. The chromosomes gene values arbitrarily are created in the range of [0, 1]. In this case, 10 matrices are randomly generated to give shape to the preliminary population. The engendered chromosomes are assessed in accordance with their fitness function. The estimations of fitness function for the engendered chromosomes are carried out as follows:

Fitness function

The chromosomes fitness function is evaluated by means of the following Relation:

\( SDME=-\frac{1}{b_1.{b}_2}\sum \limits_{i=1}^{b_1}\sum \limits_{j=1}^{b_2}20\ln \mid \frac{p_{\max, j,i}-2{p}_{cen,j,i}+{p}_{\min, j,i}}{p_{\max, j,i}+2{p}_{cen,j,i}+{p}_{\min, j,i}}\mid \)

It represents an enhancement measure integrating the theory of the second-derivative. Let us presume that the image If' is divided into b1 × b2 blocks, and pmax, j, i, pmin, j, i represent the maximum and minimum values of the pixels in each block independently, and pcen, j, i represents the intensity of the center pixel in each block.

Probability of crossover and mutation

When the fitness function evaluation is completed, the chromosomes are crossed and mutated in accordance with their crossover and mutation rates. In the novel AGA, both these rates are evaluated by employing their fitness values from the population. The newly created probability based crossover and mutation rates are effectively utilized in the crossover and mutation rates evaluation process.

Termination

If the technique is terminated after attainment of the maximum number of generations, it is possible that a fitting solution is achieved or not. As a result, the optimized PSF from the AGA is effectively deployed in the RL technique for regaining the tainted image further proficiently.

Richardson-Lucy algorithm

The Richardson-Lucy (R-L) technique effectively utilizes an innovative probabilistic technique for regaining the tainted image. At this juncture, the contaminated image df' is fed as input, and an image k which maximizes the chances of supervising the image df' is found out. Considering the image as an analysis of a Poisson process, the probability function is defined as:

From the above equation, a functional to be deducted, \( L\left(\hat{k}\right)=-\log\;p\left({I}^{\hbox{'}}|k\right) \) is obtained, representing the maximum probability estimate as:

An iterative technique is expected to be achieved from the above functional. It is known as the Richardson-Lucy technique and is expressed by:

The relative technique comes to an end after a preset number of iterations. When the de-convolution is ill-posed, which habitually occurs in the genuine applications, the signal-to-noise ratio exhibits a tendency to emerge as extremely inferior to the number of iterations n → ∞. In RL technique, a set of functions is carried out for a fixed number of iterations and at last, we come face to face with the anticipated, enhanced and reclaimed image \( {d}_f^{\hbox{'}} \).

Experimental results and discussion

The novel heuristic method is performed in the working platform of the MATLAB version 7.14 together with the classification GUI (Graphical User Interface) which is furnished for both the Gaussian and salt and pepper noise categories. The input image is denoised with the help of the innovative hybrid Adaptive median filter and the adaptive fuzzy switching with the AGA technique. Further, the denoised image is regained by the AGA (Richard-Lucy technique). The sample input images employed in the new-fangled approach are effectively exhibited in Fig. 2.

Fig. 2
figure 2

Sample input images

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Well-before the start of the noise elimination procedure, the impulse noise and Gaussian noises are included in the image. The figure appearing below illustrates the samples of noise-added images.

Figures 3 and 4 Noise Added Images (3) Image with Gaussian Noise (4) Image with impulse noise. These noise affected images are de-noised using optimized texture and Shrinkage. Thus the noise removed images are shown in Fig. 5.

Fig. 3
figure 3

Gaussian noise added images

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Fig. 4
figure 4

Salt and pepper noise added images

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Fig. 5
figure 5

De-noised Images (i) Gaussian Noise Removed Image (ii) Impulse Noise Removed Image

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Now the noise-free images are furnished to the DWT for merging and later on subjected to the AGA image restoration procedure. The image obtained from the AGA restoration procedure is effectively exhibited in Fig. 6.

Fig. 6
figure 6

Restored Images (i) Gaussian Noise Removed Image (ii) Impulse Noise Removed Image

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Performance analysis

The functioning of the novel method is effectively appraised with the help of two efficiency metrics such as PSNR, and SSIM by duly modulating the Gaussian and impulse noises as 0.02, 0.05 and 0.07. In addition, the feat of the novel method is assessed and contrasted with those of the modern filter and optimization approaches. In this connection, the noise elimination feat of the novel technique together with those of the modern filter based approaches with respect to their PSNR values are effectively furnished in Table 1.

Table 1 Proposed and different filtering methods PSNR value of three Different Noise Variance levels 0.02, 0.05 and 0.07 (i) Gaussian Noisy images (ii) Impulse Noisy images
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The average PSNR value computed from the above tables are represented as graphs in Fig. 7.

Fig. 7
figure 7

Comparison of noise removal performance of the proposed technique with that of the existing techniques (i) Gaussian Noise (ii) Impulse Noise

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Figure 7 illustrates the performance of the noise removal process of both the existing and the proposed technique at the regular intervals of noise levels 0.02, 0.05and 0.07. While looking the Fig. 7(i), the PSNR value of the proposed and that of the existing technique looks like similar. But the PSNR value of the proposed technique is slightly higher than the existing technique except noise level at 0.07. Figure 7(ii) shows the PSNR value obtained while varying the impulse noise levels. Here the PSNR value of the proposed technique beats the existing technique’s PSNR value. The PSNR value of the proposed technique is nearer to 90 which is greater than the existing technique.

The noise removal performance of both the proposed method and the existing methods by varying the noise levels at noise removal stage, merging stage and restoration stage in terms of their PSNR value is given in Table 2.

Table 2 PSNR value at Different Noise Variance levels 0.2, 0.3, 0.5, 0.7, and 0.9 (i) Gaussian Noisy images (ii) Impulse Noisy images
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In Fig. 8(i) and (ii), shows the comparison graph of proposed method, and the existing methods in the Gaussian and impulse noise removal process. The graphs are drawn by taking the average values of the PSNR values mentioned in Table 2. From the Fig. 8, we can see that the PSNR average PSNR value of the proposed technique is higher than that of the existing technique. Even though the PSNR value decreases as the noise level increases, proposed technique’s PSNR value is still higher. Thus we proved that the proposed technique performance is better than the existing technique in the noise removal process.

Fig. 8
figure 8

Comparison of proposed and existing methods in terms of PSNR value by varying noise levels (i) Gaussian Noise (ii) Impulse Noise

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After that, the image restoration performance is evaluated by computing SSIM measure on the restored images. The restored images results obtained from the different noise variance levels are given in Table 3.

Table 3 SSIM value of our proposed method and the different existing methods results for different noise level (i) Gaussian noise denoised image (ii) Impulse noise denoised image
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Figure 9 illustrates the performance of image restoration by taking the average of SSIM given in Table 3 for the existing and the proposed technique at the regular intervals of high level noise 0.2, 0.3, 0.5, 0.7 and 0.09. On looking at the graph, we can say that the SSIM value of the proposed technique is higher than that of the existing technique.

Fig. 9
figure 9

Comparison of SSIM value of the proposed technique with that of the existing techniques by varying the high level noise range (i) Gaussian noise denoised image (ii) Impulse noise denoised image

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The average PSNR value of our proposed method and those of the different existing methods for the different noise variance level comparison graph are illustrated in Fig. 10.

Fig. 10
figure 10

Comparison of SSIM value of the proposed technique with that of the existing techniques by varying the low level noise range (i) Gaussian noise denoised image (ii) Impulse noise denoised image

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In Fig. 10(i) and (ii), shows the comparison graph of proposed method, and the existing methods in the image restoration process. The graphs are drawn by taking the average values of the SSIM values mentioned in Table 4. From the Fig. 10, eventhough the SSIM values of both the existing and the proposed technique look similar, there is a slight variation between them.

Table 4 Proposed and different filtering methods SSIM value of three Different Noise Variance levels 0.02, 0.05 and 0.07 (i) Gaussian Noisy images (ii) Impulse Noisy images
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Table 5 shows the proposed technique performance in terms of PSNR, SDME and SSIM. Moreover, our proposed technique performance is compared with the Existing GA and optimization methods (Cuckoo search Algorithm).

Table 5 PSNR, SSDM, SSIM value of proposed and existing techniques with noise variance 0.02 and 0.2(i) Gaussian Noisy images (ii) Impulse Noisy images
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Figure 11 shows the comparison of image restoration performance of our proposed modified cuckoo search algorithm and other optimization methods GA and Cuckoo search Algorithm. The picture is depicted by taking the average values taken from Table 5. When compared to the conventional CS and GA algorithms, our proposed modified CS algorithm attained higher value in all the measures such as PSNR, SSIM and SDME. Higher value indicates higher performance. Thus high value of PSNR, SSIM SDME shows the better image restoration performance.

Fig. 11
figure 11

Comparison of image restoration performance of the proposed and the existing techniques at noise level 0.02 (i) Impulse noise added image (ii) Gaussian noise added image

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Figure 12 shows the image restoration performance of our proposed AGA and other optimization methods GA and Cuckoo search Algorithm. The high value of PSNR SSIM SDME shows the better image restoration performance. With our proposed denoising technique with AGA, and Cuckoo search Algorithm method has given high image restoration performance than the other filtering methods. In our proposed restoration technique, AGA has given high PSNR SSIM SDME value than the other existing techniques.

Fig. 12
figure 12

Comparison of PSNR, SSDM, SSIM values of the proposed and conventional algorithms CS and GA at noise level 0.2 (i) Gaussian Noisy images (ii) Impulse Noisy images

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Conclusion

In the proposed image restoration technique, initially the preprocessing is carried out by two filters adaptive median filter, adaptive fuzzy switching, and that preprocessed image is given to next step of noise removal process such as (OGHP) and SURE shrinkage. During OGHP noise removal process, GHP parameters are optimized by using modified Cuckoo Search Algorithm. All this process has improved the performance of the noise removal and restoration techniques. The results have shown that the proposed technique has achieved higher PSNR values than the existing and optimization methods. Thus, the proposed techniques has offered better performance in de-noising all type of noisy images with higher de-noising PSNR ratio and restore all images with high quality.