1 Introduction

In today’s era, there is requirement of the image acquisition devices to capture the images for various practical applications ranging from medical to surveillance [11, 14]. But, sometimes captured images are not suitable for processing due to improper imaging conditions like non-uniform illumination, luminescence level etc. Since, illumination is crucial to improve the optimum image quality, so, there is need of enhancement for non-uniformly illuminated images.

Several image enhancement techniques exist to improve the quality of images in terms of both visual appearance and quantitative measures [1, 4, 13, 20, 35, 36, 50, 51]. These are further divided into the spatial and frequency domain techniques. In spatial domain techniques, the operation is performed directly on the pixels while in frequency domain techniques, the image is transformed into the frequency domain before applying any operation on the image [11, 14]. The transformation has been done using the Discrete Fourier Transform (DFT), Wavelet Transform (WT), Discrete Cosine Transform (DCT), Fractional Fourier Transform (FrFT), Discrete Fractional Cosine Transform (DFrCT), Linear Canonical Transform (LCT) etc. [2, 10, 11, 14, 15, 28, 33, 37]. There are various image enhancement techniques for low contrast and non-uniformly illuminated images such as Histogram Equalization (HE), Single Scale Retinex (SSR), Multi Scale Retinex (MSR), Homomorphic Filtering (HF), Linear Contrast Adjustment (LCA), Contrast Limited Adaptive Histogram Equalization (CLAHE) etc. [4, 11, 14, 50, 51]. Moreover, the issue of non-uniform illumination can be resolved by using the frequency domain technique such as Homomorphic Filtering (HF). In addition to image enhancement, it also sharpens the edges of the image. It is applied in various applications such as medical images, underwater images, face recognition etc. [1, 7, 13, 16, 19, 27, 29, 31, 35, 36, 45, 47].

Sheet et al. [35] modified the Brightness Preserving Dynamic Histogram Equalization (BPDHE) technique by computing the fuzzy histogram to perform smoothing before dividing the image into sub-histograms. It increases the ability of technique to preserve brightness and contrast enhancement with additional advantage of less computation time in comparison to BPDHE [13]. Median-Mean Based Sub-Image-Clipped Histogram Equalization (MMSICHE) preserves images. But, these methods can be used only for images with substantial peaks in the histogram. So, HF has been introduced for the enhancement of non-uniformly illuminated images. In [45], Tseng and Lee used image fusion in addition to the DCT based HF to combine the various enhanced images having different exposures to get the final output image. In [19], Lee and Tseng presented a DCT based matrix homomorphic filtering technique on the grayscale and color images.

Nowadays, image enhancement is mostly done with the fractional derivatives because they are able to enhance the low frequency details in smooth areas and sharpen the high frequency details. Thus, fractional derivatives have been used in various signal and image processing applications [3, 5, 8, 9, 12, 17, 18, 21, 30, 38, 39]. In [30], YiFeiPU−2 is considered to be better among the six fractional derivative masks and algorithms analyzed by Pu et al. for the texture enhancement on the basis of error analysis. Garg and Singh [9] improved GL based fractional differential operator for enhancing the textural information of an image that depends on the intensity factor and order of fractional operator. Besides, some recent works such as [40, 41] on image enhancement are based on Deep Neural Networks (DNNs). Actually, the image enhancement techniques based on fractional derivatives by using Deep Neural Network (DNN) is still not common in the existing literature. Moreover, the existing literature does not provide a comparison between DNN based image enhancement methods and fractional derivative based image enhancement methods. This may be due to the fact that the DNN approaches are designed for different scenarios. For instance, DNN methods are useful, when we need to deal with very large datasets with a large number of features and complex classification, thereby increasing the computation cost or execution time. Moreover, most of the existing recent image enhancement techniques such as [5, 12] are based on this concept. Therefore, in this paper, the proposed scheme is evaluated by considering most of the recent image enhancement methods based on fractional derivatives in order to make the comparison feasible.

In [19, 45] DCT has been used for the transformation of image into the frequency domain in HF. Although, DCT provides more accumulation of energy as compared to other transforms. But, it is unable to extract the local spectral features. In this paper, two techniques are presented for enhancement of the low contrast and non-uniformly illuminated images as well as sharpening of the edges. The first technique employs the fractional derivative (GL) instead of high pass filter while the second technique involves a combination of fractional derivative with FrFT to take the advantage of two extra degrees of freedom. FrFT is used instead of DCT in the proposed technique as its energy is also concentrated in the central region [24]. The performance of the proposed technique is evaluated and compared with other image enhancement techniques on the basis of various image assessment parameters [1, 11, 35, 36, 45]. The proposed scheme provides better image visual quality and image information in terms of average PSNR and entropy values.

The paper is organized as follows: Section 2 discussed about the preliminaries used in the paper. Section 3 depicts the proposed HF technique based on the fractional derivative as well as the combination of fractional derivative with FrFT. Section 4 discussed the simulated results of the proposed work and comparison with the existing techniques. The conclusion and future scope are presented in Section 5.

2 Preliminaries

2.1 Fractional derivative

Fractional Order Calculus (FOC) is a generalization of the integer order calculus. FOC has the capability to model systems more accurately in comparison to the integer orders. The commonly used fractional order derivatives are Riemann–Liouville (RL), Grünwald–Letnikov (GL), Caputo etc. Due to the discrete nature, GL derivative is used in most of the applications. The GL based derivative of a function z(t) is given as [22, 23]:

$$ {}_c{}D{}_t{}^{\vartheta }\ z(t)=\underset{h\to 0}{\lim}\frac{1}{\Gamma \left(\vartheta \right){h}^{\vartheta }}\sum \limits_{k=0}^{\left(\frac{t-c}{h}\right)}\frac{\Gamma \left(\vartheta +k\right)}{\Gamma \left(k+1\right)}z\left(t- kh\right) $$
(1)

where, c and t are lower and upper limits of the integration. Here, ϑ ε R+ (real numbers) such that m − 1 < ϑ < m, where, m is the operation order (integer). Here, Γ(.) is the Euler’s gamma function and h is the sampling period, where, \( \left(\frac{t-c}{h}\right) \) is an integer and k ranges from 0 to \( \left(\frac{t-c}{h}\right) \).

2.2 Fractional Fourier Transform (FrFT)

Fractional Fourier Transform (FrFT) is an important signal processing tool that rotates the signal in the time-frequency plane by an angle ‘α’ [33]. The FrFT of the signal z(t) is given by [25, 26]:

$$ {Z}_{\upalpha}\left({u}_{\upalpha}\right)={\int}_{-\infty}^{\infty }z(t){K}_{\upalpha}\left(t,{u}_{\upalpha}\right) dt $$
(2)

where, 0 < |a| < 2, α=aπ/2 and Kα(t, uα) represents the Kernel function defined as:

$$ {K}_{\upalpha}\left(t,{u}_{\upalpha}\right)=\left\{\begin{array}{c}\sqrt{\frac{1-i\ \cot \alpha }{2\ \pi }}\exp \left[i\left(\frac{t^2+{u_{\alpha}}^2}{2}\right)\mathit{\cot}\alpha -i\ {u}_{\alpha }t\ csc\alpha \right]\kern1em if\ \alpha \ne n\pi \\ {}\kern5em \delta \left(t-u\right)\kern13em if\ \alpha =2 n\pi \\ {}\kern7em \delta \left(t+u\right)\kern13em if\kern0.5em \alpha +\pi =2 n\pi \end{array}\right. $$
(3)

and δ(t) represents the Dirac’s delta function. The signal is restored by taking the FrFT with the rotation angle of ‘-α’, i.e., by replacing ‘α’ with ‘-α’ in eq. (2) and (3). The two-dimensional FrFT is required to process the images in the frequency domain. The two-dimensional FrFT is taken separately in x and y directions. The separable two-dimensional FrFT has two orders α and β for x and y directions, i.e., 0 < α < π/2 and 0 < β < π/2 is given below:

$$ {Z}_{\alpha, \beta}\left({u}_{\upalpha},{v}_{\upbeta}\ \right)={\int}_{-\infty}^{\infty }{\int}_{-\infty}^{\infty }z\left(x,y\right){K}_{\upalpha, \upbeta}\left(x,y,{u}_{\upalpha},{v}_{\upbeta}\right) dx\ dy\kern0.75em $$
(4)

where, α and β are the rotation angles.

The Kernel function Kα, β(x, y, uα, vβ) is defined as:

$$ {K}_{\upalpha, \upbeta}\left(x,y,{u}_{\upalpha},{v}_{\upbeta}\right)={K}_{\upalpha}\left(x,{u}_{\upalpha}\right)\ {K}_{\upbeta}\left(y,{v}_{\upbeta}\right)\kern0.75em $$
(5)
$$ =\frac{1}{2\ \pi}\sqrt{1-i\ \cot \kern0.3em \alpha}\sqrt{1-i\ \cot \kern0.3em \beta}\kern0.3em \exp \left[i\left(\frac{x^2+{u_{\alpha}}^2}{2}\right)\mathit{\cot}\kern0.3em \alpha -i\ {u}_{\alpha }x\ \csc \kern0.3em \alpha \right]\exp \left[i\left(\frac{y^2+{v_{\upbeta}}^2}{2}\right)\mathit{\cot}\beta -i\ {v}_{\upbeta}\ y\ \csc \kern0.3em \beta \right]\kern0.5em $$
(6)

The signal is restored by taking FrFT with the rotation angle of ‘-α’ and ‘-β’, that is, by replacing ‘α’ with ‘-α’ and ‘β’ with ‘-β’ in eq. (4)–(7). The proposed work uses two-dimensional Discrete Fractional Fourier Transform (2D-DFrFT) [26]. The (K, L)-point 2D-DFrFT is given as:

$$ {Z}_{\alpha, \beta}\left({u}_{\upalpha},{v}_{\upbeta}\ \right)=\sum \limits_{y=0}^{L-1}\left[\sum \limits_{x=0}^{K-1}z\left(x,y\right)\exp \left[i\left(\frac{x^2+{u_{\alpha}}^2}{2}\right)\mathit{\cot}\kern0.3em \alpha -i\ {u}_{\alpha }x\ \csc \kern0.3em \alpha \right]\right]\exp \left[i\left(\frac{y^2+{v_{\upbeta}}^2}{2}\right)\mathit{\cot}\beta -i\ {v}_{\upbeta}\ y\ \csc \kern0.3em \beta \right] $$
(7)

The 2D-DFrFT for matrix K × L is determined by applying the one-dimensional DFrFT to each row of the matrix and afterwards to the column of the resultant.

2.3 Homomorphic Filtering

The HF technique is based on the illumination-reflectance model. Illumination refers to the amount of source illumination which is incident on the scene to be viewed denoted by Ɨ(x, y). Reflectance refers to the amount of illumination that is reflected by the entities present in scene denoted by Ʀ(x, y). Intensity of image Ƒ(x, y) at spatial coordinates (x, y) is given by:

(8)

where, 0 < Ɨ(x, y) < ∞ and 0 < Ʀ(x, y) < 1. The nature of illumination depends on the source of illumination while the reflectance depends on the attributes of the image entities. Reflectance is bounded by zero and one which means, total absorption and total reflectance, respectively. In this technique, the logarithm of the original image is taken, which maps the image from multiplicative domain into the additive domain. Image obtained after the logarithm operation is transformed into the frequency domain, after which the linear filtering is done that amplified the high frequencies while attenuating the low frequencies. Then, the enhanced image is obtained by taking the exponential of inverse transform of the image which is filtered also as shown in Fig. 1.

Fig. 1
figure 1

Generalized homomorphic filtering technique [14]

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The HF method used the High Pass Filter (HPF) for the enhancement of image which is the procedure to capture the important properties such as geometry, reflectivity, and illumination. The basic ideal high pass filters are used in the modified form in this technique. This modification is done by including two parameters γL and γH in the equation of ideal high pass filter such that γL < 1 while, γH > 1 as shown in eq. (9):

$$ \mathrm{H} \left(u,v\right)=\left({\gamma}_H-{\gamma}_L\right)\ast (HPF)+{\gamma}_L $$
(9)

Here, Н(u, v) is modified equation for HPF in the frequency domain. The parameters γH and γL decreases the contribution made by low frequencies, whereas, increases the contribution made by the high frequencies. This technique increases the contrast of images as well as sharpens the edges of the images [11].

3 Proposed scheme

On the basis of HF technique [45], an improved HF technique is presented to achieve better visual quality and more information detail from the enhanced images. DCT transform [19, 45] provides accumulation of energy but it doesn’t provide the local spectral features. The improved HF technique used fractional derivative and FrFT to enhance the high frequency features. In this paper, two techniques are proposed for enhancement of the low contrast and non-uniformly illuminated images. The first technique employs the fractional derivative (GL) instead of high pass filter while the second technique involves a combination of fractional derivative with FrFT to take the advantage of two extra degrees of freedom. The significance of the proposed algorithms is to achieve the enhancement of the low contrast and non-uniformly illuminated images as well as sharpening of the edges with the utilization of the fractional derivative and fractional derivative in combination with FrFT.

3.1 Proposed Algorithm 1: Fractional derivative based HF

In this algorithm, the fractional derivative is used to enhance the low contrast and non-uniformly illuminated images and sharpening the edges of image. The block diagram of the proposed algorithm based on fractional derivative is shown in Fig. 2.

Fig. 2
figure 2

Proposed HF technique based on fractional derivative

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In this technique, the logarithm of intensity as given in eq. (8) is taken before applying the transform because the transform of product of two functions is not separable.

(10)

where, z(x, y) is the logarithm of Ƒ(x, y).

(11)

Here, and Ζ(u, v) refers to the DFT of z(x, y). ƑƗ(u, v) and ƑƦ(u, v) is the DFT of ln(Ɨ(x, y)) and ln(Ʀ(x, y)) respectively. The GL fractional derivative is used for the analysis in DFT.

The precise form of the GL based fractional operator [30] is given by the following equation:

$$ \frac{d^{\vartheta }}{dt^{\vartheta }}\ z(t)=\underset{h\to 0}{\lim}\frac{1}{\Gamma \left(-\vartheta \right){h}^{\vartheta }}\sum \limits_{k=0}^{\left(n-1\right)}\frac{\Gamma \left(k-\vartheta \right)}{\Gamma \left(k+1\right)}z\left(t+\frac{\vartheta h}{2}- kh\right) $$
(12)

The GL based fractional derivative is derived by inserting the values of signals on the non-nodes assuming ϑ = 0, ± 2, ± 4, …, considering the nodes at z(t + h − kh), z(t − kh) and z(t − h − kh).

The interpolation is done using the Lagrange’s 3-point interpolation method [9, 30]:

$$ z\left(\tau \right)\cong \frac{\left(\tau -t+ kh\right)\left(\tau -t+h+ kh\right)}{\ 2{h}^2}\ z\left(t+h- kh\right)-\frac{\left(\tau -t-h+ kh\right)\left(\tau -t+h+ kh\right)}{h^2}\ z\left(t- kh\right)+\frac{\left(\tau -t+ kh\right)\left(\tau -t-h+ kh\right)}{\ 2{h}^2}\ z\left(t-h- kh\right)\kern0.75em $$
(13)

Let\( \tau =\left(t+\frac{\vartheta h}{2}- kh\right) \) and interpolating it, the equation comes out to be:

$$ z\left(t+\frac{\vartheta h}{2}- kh\right)\cong \left(\frac{\vartheta }{4}+\frac{\vartheta^2}{8}\right)\ z\left(t+h- kh\right)+\left(1-\frac{\vartheta^2}{4}\right)\ z\left(t- kh\right)+\left(-\frac{\vartheta }{4}+\frac{\vartheta^2}{8}\right)\ z\left(t-h- kh\right) $$
(14)
$$ \frac{d^{\vartheta }}{d{t}^{\vartheta }}\ z(t)\cong \frac{1}{\Gamma \left(-\vartheta \right){h}^{\vartheta }}\sum \limits_{k=0}^{\left(n-1\right)}\frac{\Gamma \left(k-\vartheta \right)}{\Gamma \left(k+1\right)}\left[{z}_k+\frac{\vartheta }{4}\left({z}_{k-1}-{z}_{k+1}\right)+\frac{\vartheta^2}{8}\left({z}_{k-1}-2{z}_k+{z}_{k+1}\right)\right] $$
(15)

The coefficients of the filter obtained from eq. (15) are in the spatial domain. So, the DFT of the fractional derivative will be obtained in order to compute the frequency domain coefficients required in the HF technique. Then, the fractional derivative Н(u, v) is applied on the obtained Fourier Transform coefficients.

(16)

where, \( \mathcal{S}\left(u,v\right) \) is the DFT of the result obtained after filtering operation. The obtained coefficients after IDFT is given by:

(17)

where, and . The enhanced image obtained after the exponential operation is given by:

(18)

where, and .

3.2 Proposed Algorithm 2: Fractional derivative FrFT based HF

In this technique, the fractional derivative is used in combination with DFrFT to enhance the edges of the low contrast and non-uniformly illuminated images. The DFrFT is applied in eq. (11) for transforming the image into frequency domain. This technique also uses GL fractional derivative as discussed in Algorithm 1. Instead of computing the DFT, DFrFT of eq. (15) will be obtained in order to compute the frequency domain coefficients required in the HF technique. Similarly, IDFrFT is used to obtain the spatial domain coefficients instead of IDFT. The block diagram of the proposed algorithm based on fractional derivative in combination with FrFT is shown in Fig. 3.

Fig. 3
figure 3

Proposed HF technique based on the combination of fractional derivative and FrFT

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figure d

4 Experimental results

The adequacy of proposed techniques is confirmed using the MATLAB R2016a on a system with an Intel® CPU 2.7 GHz processor with 16 GB RAM. The effectiveness of the proposed scheme is also evaluated with different images. The comparison of the proposed algorithms is done with the existing techniques that is, HE [11], BDPFHE [35], MMSICHE [36], HF [1] and DCT based HF [45].

4.1 Performance analysis for images from standard datasets

In this section, the performance of proposed techniques is evaluated on various original images of different sizes from different datasets namely The USC-SIPI Image Database, TraitImage, Segmentation by Regions, Image Databases and in-built MATLAB images as shown in Fig. 4 [11, 34, 42, 43].

Fig. 4
figure 4

Test images used for simulation of different sizes

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The enhanced images for the test images obtained using both Proposed Algorithm 1 and Algorithm 2 and existing techniques [1, 11, 35, 36, 45] are shown in Fig. 5. The performance of the enhancement of images is evaluated with the different performance metric parameters such as Peak Signal to Noise Ratio (PSNR), Mean Square Error (MSE), Information Entropy, Structural Similarity Index Measure (SSIM) and Universal Image Quality Index (UIQI).

Fig. 5
figure 5figure 5

Enhanced test images obtained with proposed algorithms in comparison to the existing techniques

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The MSE formula is given by [32]:

$$ MSE=\frac{1}{kl}\ \sum \limits_{i=0}^{k-1}\sum \limits_{j=0}^{l-1}{\left[A\left(i,j\right)-O\left(i,j\right)\right]}^2 $$
(19)

where, A and O are the images for comparison with size k × l.

The PSNR is given by [32]:

$$ PSNR=10\ {\mathit{\log}}_{10}\left(\frac{{{\mathit{\operatorname{Max}}}_I}^2}{MSE}\right) $$
(20)

where, MaxI is the maximum value of a pixel.

Table 1 shows the Average PSNR for the different test images for various techniques. The optimal order of fractional derivative and transform order ‘a’ is mentioned in Table 1 that provides the best results. Maximum average PSNR is obtained for the proposed scheme for fractional derivative order of range 0.11–0.124 for transform order 0.99. It is interpreted that the enhanced images obtained with the fractional derivatives and DFrFT are more similar to the original images. Average PSNR is high for proposed scheme for all the considered images because the fractional derivatives enhance the high-frequency information present in the images. The amplitude and phase information of the image in DFrFT depends on its transform order [24]. The results of PSNR in Algorithm 2 are better as DFrFT preserves the details of the texture of the image with an increase in transform order. Moreover, it is clear from the visual perception that the proposed scheme is better than the existing techniques [1, 11, 35, 36, 45] as shown in Fig. 5. The average PSNR obtained is maximum for Algorithm 2 as it exploited the advantage of both fractional derivative and DFrFT. From the eq. (19) and (20), it is clear that lesser the MSE, more the PSNR, better the quality of enhanced image. Table 2 shows the MSE of different test images for a proposed scheme in comparison to the existing techniques.

Table 1 Average PSNR (dB) for the test images of different sizes
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Table 2 MSE for the test images of different sizes
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On the basis of Shannon’s information, the entropy of an image is given by [44]:

$$ E=-\sum \limits_{i=0}^{255}{P}_i{\mathit{\log}}_2{P}_i\kern1em $$
(21)

where, E is entropy, Pi is i’s probability in the image. Table 3 shows the information entropy for the different test images for different techniques.

Table 3 Information entropy for the test images of different sizes
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Information entropy of the proposed scheme is more as compared to the existing techniques [1, 11, 35, 36, 45] for all the considered images because it enhances the high frequency information while preserving low and medium frequency details. As observed from Table 3, Algorithm 2 provides more detailed information about the image. The information entropy for the images of Aquitaine and Line3 is almost comparable for both proposed algorithms.

The SSIM is represented by the given formula [49]:

$$ SSIM\left(x,y\right)=\frac{\left(2{\mu}_x{\mu}_{\mathrm{y}}+{K}_1\right)\left(2{\sigma}_{xy}+{K}_2\right)}{\left({\mu}_x^2+{\mu}_y^2+{K}_1\right)\left({\sigma}_x^2+{\sigma}_y^2+{K}_2\right)}\kern0.75em $$
(22)

where,μx and μy represent the mean intensities, σx and σy represent the contrast while K1 and K2 represent the constants. Table 4 shows the SSIM for the different test images for various techniques.

Table 4 SSIM for the test images of different sizes
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SSIM value corresponds to the structural similarity between the original and reconstructed image. Enhancement of the low contrast and non-uniformly illuminated images results in the change in the structure of the image, thus reducing its SSIM. Although, SSIM of the proposed algorithms is less, but, it is still comparable to the existing techniques [1, 11, 35, 36, 45]. SSIM is high for Algorithm 1 in case of Aquitaine, Line3 and Motion as the change in structure is less after enhancement as compared to other techniques. It has kept the structure almost similar in addition to the enhancement of images. The Mean Structural Similarity Index Measure (SSIM) has also been calculated, but, it is observed that MSSIM is maximum for HF than the proposed techniques.

UIQI possess the ability to measure the structural distortion occurred during the process of degradation of an image. It indicates similarity and dissimilarity. It considers the three components for computing distortion while SSIM considers only one component, i.e., structure. In this, the comparison between two images is done by dividing it further into the three comparisons, that is, luminance L(x, y), contrast C(x, y) and structural comparison S (x, y) given by [48]:

$$ UIQI=L\left(x,y\right)\ C\left(x,y\right)\ S\ \left(x,y\right)=\frac{4{\mu}_x{\mu}_y{\mu}_{xy}}{\left({\mu}_x^2+{\mu}_y^2\right)\left({\sigma}_x^2+{\sigma}_y^2\right)} $$
(23)

The L(x, y), C(x, y) and S (x, y) is given by:

$$ {\displaystyle \begin{array}{l}L\left(x,y\right)=\frac{2{\mu}_x{\mu}_y}{\mu_x^2+{\mu}_y^2}\\ {}C\left(x,y\right)=\frac{2{\sigma}_x{\sigma}_y}{\sigma_x^2+{\sigma}_y^2}\\ {}S\left(x,y\right)=\frac{2{\sigma}_{xy}}{\sigma_x+{\sigma}_y}\end{array}} $$

where,μx and μy represent the mean intensities of original and distorted images, σx and σy represent the standard deviation of original and distorted images while σxy represent the covariance of both images. Table 5 shows the UIQI for different test images for various techniques.

Table 5 UIQI for the test images of different sizes
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It is almost comparable to other techniques, but as illustrated in Table 5, the Algorithm 2 has the highest UIQI for all considered test images of different sizes as compared to the existing techniques as well as the Algorithm 1. This may happen due to the use of DFrFT transform in combination with fractional derivatives because it enhances high frequency information as well as contrast while preserving the low and medium frequency details.

The effectiveness of the proposed techniques for the enhancement of the low contrast and non-uniformly illuminated images as well as strengthening of edges is demonstrated with the various image assessment parameters in Fig. 6. PSNR of the proposed techniques (Algorithm 2) is improved by 2.86–50.37 dB, thus, indicating that the images enhanced by proposed algorithms are of higher quality in comparison to existing algorithms [1, 11, 35, 36, 45] as shown in Fig. 6. The PSNR shows the improvement of 9.59 dB, 11.71 dB, 6.3 dB, and 3.4 dB in case of proposed techniques when compared to BPDFHE, MMSICHE, HF and DCT based HF for test image of Lena. MSE for HE and MMSICHE is not shown in Fig. 6 as it is large as compared to other techniques. Information Entropy shows the improvement of 0.3–20% when the comparison of proposed algorithm is done with the existing techniques for test image Lena [1, 11, 35, 36, 45]. Information entropy is improved by 3–42% with the Algorithm 2 for different images. The UIQI of all the images is close to one, indicating that even after enhancement the images are almost similar to original ones. Therefore, in Fig. 6, it is perceived that the proposed algorithms provide more enhancement and information details in comparison to the existing techniques [1, 11, 35, 36, 45]. The average time elapsed for execution of code ranges from 9.01 to 39.476 s for Algorithm 1 and 9.17 to 52.09 s for the Algorithm 2 for different images. The execution time varies in accordance with the number of pixels in the image.

Fig. 6
figure 6

Comparison of various image enhancement techniques on the basis of image assessment parameters for Kids and Lena Image

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4.2 Performance analysis for images with different contrast and illumination conditions

In this section, the performance of proposed techniques is analyzed on datasets [6, 46] containing test images with different contrast and illumination conditions as shown in Fig. 7.

Fig. 7
figure 7

Test images with different contrast and illumination conditions

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It is difficult to add all the images due to the space constraint. So, the enhanced images for some test images with different contrast and illumination conditions are shown in Fig. 8. It is clearly perceived from Fig. 8 that proposed algorithm results in more clarity in the enhanced images in comparison to the existing techniques. Tables 6, 7, 8, 9 and 10 depicts various performance parameters for different contrast and illumination conditions of various test images of different sizes.

Fig. 8
figure 8

Enhanced test images for different contrast and illumination conditions obtained with proposed algorithms in comparison to the existing techniques

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Table 6 PSNR (dB) of test images of different sizes for different contrast and illumination conditions
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Table 7 MSE of test images of different sizes for different contrast and illumination conditions
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Table 8 Information Entropy of test images of different sizes for different contrast and illumination conditions
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Table 9 SSIM of test images of different sizes for different contrast and illumination conditions
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Table 10 UIQI of test images of different sizes for different contrast and illumination conditions
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It is worth noting that in the case of lossy compression such as JPEG, the PSNR value is constrained to 50 dB [20]. However, the PSNR value of greater than 50 dB is achieved in the case of proposed scheme because the test images (from datasets [11, 34, 42, 43]) used to evaluate the proposed scheme are in lossless file format. Secondly, this may also happen because the proposed scheme is based on fractional derivatives that results in the enhancement of high and mid frequency components non-linearly while preserving the low frequency components.

Nevertheless, in order to confirm the above mentioned fact, the proposed scheme is further evaluated by considering the dataset images [46] in the JPEG file format. It is observed from Table 6 that maximum PSNR value achieved for the proposed scheme is of value 49.5650 dB (i.e. statue5 image) because the considered images are in JPEG file format. The PSNR value greater than 50 dB is achieved for lossless images such as TIFF, PNG file formats as shown in Table 1. It is also noted from Tables 6 to 10 that Algorithm 1 and Algorithm 2 provides high PSNR, information entropy, and UIQI in comparison to existing techniques even in the case of different contrast and illumination conditions for the same scene. The improvement in PSNR is 0.2635–42.2162 dB for Algorithm 1 and 0.2777–42.2273 dB for Algorithm 2 in case of images with different contrast and illumination conditions for the same scene. Furthermore, the improvement in information entropy is 0.02–32.63% for Algorithm 1 and 0.04–32.65% for Algorithm 2. It provides less SSIM but still it is comparable to existing techniques. Thus, the analysis done on the basis of different contrast and illumination conditions for the same scene confirms the efficacy of the proposed algorithms.

5 Conclusion

In this paper, two techniques based on fractional derivative and FrFT have been implemented. These techniques sharpened the edges of the image as well as enhanced the low contrast and non-uniformly illuminated images. The improvement in average PSNR of 2.86–20.49 dB has been obtained for the test images on comparison with the HF and DCT based HF. While, for HE, BPDFHE and MMSICHE improvement in PSNR is 2.44–50.37 dB for the different test images. The improvement of about 3–42% has been achieved in the information entropy for proposed techniques when compared with the HE, HF, and DCT based HF techniques. In the case of images with different contrast and illumination conditions for the same scene, the improvement in PSNR is 0.2635–42.2273 dB while for information entropy is 0.02–32.65% for the proposed algorithms. The analysis of proposed techniques on basis of various image assessment parameters shows more enhancement in comparison to the existing techniques. Thus, it has been observed that techniques based on fractional derivative and FrFT outperform the existing techniques. The future work involves the use of fractional derivative operators and FrFT for more image processing applications. Moreover, the future work will be devoted to perform the comparative analysis of fractional derivative based enhancement methods with DNN based image enhancement methods, which would further confirm the capability of the proposed technique.