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Relationship between entropy and SNR changes in image enhancement

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Abstract

There are many techniques of image enhancement. Their parameters are traditionally tuned by maximization of SNR criterion, which is unfortunately based on the knowledge of an ideal image. Our approach is based on Hartley entropy, its estimation, and differentiation. Resulting gradient of entropy is estimated without knowledge of ideal images, and it is a subject of minimization. Both SNR maximization and gradient magnitude minimization cause various settings of the given filter. The optimum settings are compared, and their differences are discussed.

Introduction

In many different fields, image quality measurement is important for various image processing tasks. Traditional tasks as image enhancement [1, 2], sharpening, and smoothing are solved by digital filters of various types and parameter settings. Filter performance can be compared by different image quality assessment techniques [3, 4]. Image quality measure signal-to-noise ratio (SNR) [5] or its modifications are the most commonly used to compare filter performance [68]. SNR measure is based on the knowledge of referential image which is a kind of Full-Reference Image Quality Assessment. However, the original image is not available in real-world tasks. Therefore, No-Reference Image Quality Assessment (NR-IQA) technique [9, 10] must be used to measure image quality. Our approach is focused on relationship between SNR and Hartley entropy. In this paper, a novel NR-IQA method based on image entropy is introduced and verified on image dataset. Alternative approach focused on motion estimation and parallel computing is included in [11, 12].

Methods

Quality measures

A digital image is a 2D discrete signal obtained by a sampling process of analogous 2D signal. A digital image will be denoted by real function x(n 1,n 2) which describes image amplitude at an integer coordinate position (n 1,n 2). Image quality can be measured by standard measures as mean squared error or SNR. However, both mentioned measures are based on the knowledge of original image. Other measures must be used when original image is not known. Founding a relationship between SNR and entropy allows us to use also entropy as image quality measure.

Signal-to-noise ratio

The SNR is an image property comparing the ratio of signal power to noise power. The SNR measure can be used to analyze image quality. The estimation of SNR is based on knowledge of original undegraded image s(n 1,n 2). The SNR of input noisy image x(n 1,n 2) is calculated in the spatial domain as

$$ {SNR}_{x} = 10 \log_{10}\frac{\mathrm{E}\left[\mathrm{s}(n_{1}, n_{2})^{2}\right]}{\mathrm{E}\left[\left(\mathrm{x}(n_{1}, n_{2}) - \mathrm{s}(n_{1}, n_{2})\right)^{2}\right]} $$
(1)

with E[ ·] standing for an expected value. The SNR of improved image y(n 1,n 2) is

$$ {SNR}_{y}= 10 \log_{10}\frac{\mathrm{E}\left[\mathrm{s}(n_{1}, n_{2})^{2}\right]}{\mathrm{E}\left[\left(\mathrm{y}(n_{1}, n_{2}) - \mathrm{s}(n_{1}, n_{2})\right)^{2}\right]}. $$
(2)

The traditional improving measure Δ SNR is defined as a difference between SNR y and SNR x , and it allows to compare filter performances.

$$ \Delta {SNR} = 10 \log_{10}\frac{\mathrm{E}\left[\left(\mathrm{x}(n_{1}, n_{2}) - \mathrm{s}(n_{1}, n_{2})\right)^{2}\right]}{\mathrm{E}\left[\left(\mathrm{y}(n_{1}, n_{2}) - \mathrm{s}(n_{1}, n_{2})\right)^{2}\right]}. $$
(3)

Positive Δ SNR value expresses the improvement of noisy image after its reconstruction. On the other hand, negative value expresses noisy image degradation.

Robust signal-to-noise ratio

Measure Δ SNR compares squared error of image intensities. Image filtering can cause intensities shifting or scaling, which will automatically decrease image quality measure Δ SNR. Therefore, we introduce robust version of Δ SNR designated as Δ R. Definition of Δ R is similar to Δ SNR but Δ R compares squared errors of statical rank

$$ \Delta R = 10 \log_{10}\frac{\mathrm{E}\left[\left(\mathrm{R}(\mathrm{x}(n_{1}, n_{2})) - \mathrm{R}(\mathrm{s}(n_{1}, n_{2}))\right)^{2}\right]}{\mathrm{E}\left[\left(\mathrm{R}(\mathrm{y}(n_{1}, n_{2})) - \mathrm{R}(\mathrm{s}(n_{1}, n_{2}))\right)^{2}\right]}, $$
(4)

where R(·) is a rank function [13] returning the rank of a pixel intensity inside an image. This measure is shift and scale invariant, but its time complexity is greater than time complexity of Δ SNR due to embedded sorting.

Hartley entropy

Entropy is well known as a measure in statistical thermodynamics and information theory. We use entropy as a measure for image quality. To estimate image entropy, we use entropy estimation algorithm described in [14, 15]. Let \(n \in \mathbb {N}\) be the number of image pixels, x k [0,1] be the intensity of kth pixel for k=1,,n, and ε(0,0.5] be the width parameter. Hartley entropy [16] (in nats) can be estimated as

$$ \hat{H}(\varepsilon) = \ln\frac{\mu(\mathcal{C})}{2\varepsilon}, $$
(5)

where \(\mu (\mathcal {C})\) is a measure of a set

$$ \mathcal{C} = \left(\bigcup\limits_{k=1}^{n} (x_{k} - \varepsilon, x_{k} +\varepsilon)\right) \cap (0,1). $$
(6)

The measure of a set \(\mu (\mathcal {C})\) can be calculated as

$$ \mu(\mathcal{C}) = x_{(1)} + x_{(n)} -1 + \sum\limits_{k = 1}^{n-1} x_{(k+1)} - x_{(k)} - 2\varepsilon, $$
(7)

where x (1)x (2)x (n). Supposing, a reconstructed image is the result of any filter application with parameters \({\bold {p}} \in \left (\mathbb {R}_{0}^{+}\right)^{q}\) where \(q \in \mathbb {N}\) is a number of parameters.

The novel characteristic which helps to optimize digital filter design is the component wise maximum of Hartley entropy gradient

$$ G = \max\limits_{i=1,\cdots, q} \frac{\partial \hat{H}({\bold{p}})}{\partial p_{i}} $$
(8)

that should be minimum possible which is the main supposition and matter of novel approach. The G criterion design is motivated as follows. When the filter has only one parameter (q=1), we minimize \(\partial \hat {H}/\partial p < 0\). Therefore, we obtain inflection point of \(\hat {H}(p_{1})\) for value of p 1 where the Hartley entropy rapidly decreases. The generalization for \(q \in \mathbb {N}\) is based on minimax approach when we minimize the maximal parameter sensitivity \(\partial \hat {H} / \partial p_{i} < 0\) over all tuning parameters. Whenever any \(\partial \hat {H}/\partial p_{i} \geq 0\), we set G=0. The partial derivative of Hartley entropy \(\hat {H}({\bold {p}})\) with respect to the variable p i can be approximated by finite differences

$$ \frac{\partial \hat{H}({\bold{p}})}{\partial p_{i}} \approx \frac{\hat{H}(\ldots,p_{i} + h,\ldots) - \hat{H}(\ldots,p_{i} - h,\ldots)}{2h}, $$
(9)

where spacing h>0 approaches zero and i{1,,q}.

Linear filter primer

We have to introduce sharpening filters that will be used for studying relationship between SNR and entropy changes in image enhancement. Our interest [15] is focused only on linear infinite impulse response (IIR) filters [17] with radial symmetry in frequency domain whose response can be easily calculated by the Discrete Fourier Transform [18] (DFT). Their advantage is in the side effect suppression of a rectangular grid.

Let \(\omega = \lVert \boldsymbol {\omega } \rVert _{2}, {\boldsymbol \omega }\in \mathbb {R}^{2}\) be the angular frequency and ρ>0 be the radius. The radial filter has transfer function F(ω)=Φ (ω). Useful low-pass (LP) filter is a Gaussian filter [19] as traditional one

$$ \varPhi_{1}(\boldsymbol\omega) = \text{exp}(-\rho^{2} \omega^{2}/2). $$
(10)

The simplest sharpening filter based on Gaussian filter with sharpening parameter α>0 and its generalization include

$$ \varPhi_{2}(\boldsymbol\omega) = \text{LP}(\boldsymbol\omega) + \alpha(1 -\text{LP}(\boldsymbol\omega)), $$
(11)
$$ \varPhi_{3}(\boldsymbol\omega) = \text{LP}_{0}(\boldsymbol\omega) + \alpha(\text{LP}_{1}(\boldsymbol\omega)-\text{LP}_{2}(\boldsymbol\omega)), $$
(12)

where LP(ω), LP0(ω), LP1(ω), and LP2(ω) are four realizations of low-pass filter Φ1(ω). In the case of Φ2, only fundamental low-pass filter is used, but in Φ3, the difference between two low-pass filters (LP1,LP2) is used as high-pass filter added to the fundamental LP0 filter in accordance with conventions of image processing. The filters Φ1 (smoothing), Φ2, and Φ3 (sharpening) will be subject of parameter optimization in the next section. The filter Φ1 has only one parameter ρ which is an advantage for its optimization. The filters Φ2 and Φ3 have two (α,ρ) and four (α,ρ 0,ρ 1,ρ 2) parameters, respectively. Their tuning can be performed by any heuristics for multimodal function optimization. Both Φ1 and Φ2 quality measures (Δ SNR,G) can be easily visualized.

Results and discussion

The novel characteristic G was tested on real images with an additive noise. The role of filter parameters was investigated for log10ρ [−1,1], log10α [−4,0], and log10ρ k [−1,0] where k{0,1,2} in the case of Φ12, and Φ3. The Fast Simulated Annealing (FSA) [20] was used for Δ SNR maximization and G minimization inside given logarithmic ranges.

Test data

Four gray scale images (THISTLE, HOUSE, MAP, WINDMILL) of size 450×400 and two gray scale images (BRIDGE, BALCONY) of size 375×282 pixels were chosen to demonstrate the relationship between Δ SNR and G criteria. All image intensities were transformed from their original range to the interval [0,1] and were degraded by a box filter with squared mask of size 3×3 and then by Gaussian additive noise with σ=0.01. The original images and results of their degradation are depicted in Figs. 1 and 2.

Fig. 1
figure1

Portfolio of original images

Fig. 2
figure2

Input images after degradation

Image enhancement based on Δ SNR

The Δ SNR criterion was used for the optimization of filters Φ12, and Φ3 as a reference. The dependency of Δ SNR on ρ is demonstrated on Fig. 3 for smoother Φ1 and MAP image. The dependency of Δ SNR on α and ρ is depicted in Fig. 4 for sharpening filter Φ2 and the same image. Similarly, Fig. 5 is showing the dependency of Δ SNR on α and ρ 0 of filter Φ3 with ρ 1=0.9 and ρ 2=0.2. The numerical results of heuristics maximization are included in Tables 1, 2, and 3 for both traditional and referential approaches. Reconstructed images via Φ3 with maximal Δ SNR are shown in Fig. 6.

Fig. 3
figure3

Quality of Φ1 sharpening as Δ SNR and G for MAP image

Fig. 4
figure4

Quality of Φ2 sharpening as Δ SNR for MAP image

Fig. 5
figure5

Quality of Φ3 sharpening with ρ 1=0.9,ρ 2=0.2 as Δ SNR for MAP image

Fig. 6
figure6

Sharpened images via Φ3 with maximum Δ SNR

Table 1 Optimal low-pass smoothing Φ1 via Δ SNR maximization
Table 2 Optimal sharpening Φ2 via Δ SNR maximization
Table 3 Optimal sharpening Φ3 via Δ SNR maximization

Image enhancement based on G

The novel G measure was used for the optimization of filters mentioned above. The measure G was approximated by Eq. (9) with spacing h=10−12. Width parameter ε was set to value 0.01. The dependency of G on ρ is demonstrated on Fig. 3 for smoother Φ1 and MAP image. The dependency of G on α and ρ is depicted in Fig. 7 for sharpening filter Φ2 and the same image. For the last filter Φ3 with ρ 1=0.9 and ρ 2=0.2, the dependency of G on α and ρ 0 is depicted in Fig. 8. The numerical results of heuristic minimization via FSA are included in Tables 4, 5, and 6 with adequate values of Δ SNR and Δ R. Reconstructed images via Φ3 with minimal G are shown in Fig. 9.

Fig. 7
figure7

Quality of Φ2 sharpening as G for MAP image

Fig. 8
figure8

Quality of Φ3 sharpening with ρ 1=0.9,ρ 2=0.2 as G for MAP image

Fig. 9
figure9

Sharpened images via Φ3 with minimum G

Table 4 Optimal low-pass smoothing Φ1 via G minimization
Table 5 Optimal sharpening Φ2 via G minimization
Table 6 Optimal sharpening Φ3 via G minimization

Discussion

The proposed novel criterion G was minimized to obtain the optimal parameters of the three different filters tested on the real images. The quality of the optimal reconstruction was evaluated by the classical Δ SNR measure and our robust version Δ R. For a comparison, the same images were reconstructed by the filters whose optimal parameters were obtained by maximization of Δ SNR. The relative changes RC between the qualities of the optimal reconstruction according to the filters Φ12, and Φ3 evaluated for the criterion Δ SNR and G are summarized in the Table 7. When comparing the results, it can be seen that the achieved results are similar. The most considerable changes in the quality measures were obtained for the filter Φ2 settings providing significantly lower qualities but still improving image enhancement. The image intensities reconstructed by the filter Φ2 and proposed criterion G are shifted or scaled which results from the large values of the relative changes with respect to the quality measure Δ SNR.

Table 7 The relative changes [%] between quality measures

Conclusions

The novel No-Reference Image Quality Assessment method and adequate criterion were introduced in this paper. It is based on the Hartley entropy estimation from gray-level densities and the optimization of its changes during tuning of filter parameters. Three types of linear image filters with various number of parameters were optimized by using traditional SNR criterion as a reference, first. Using novel criterion G and its minimization, similar results of comparable SNR quality were obtained without prior knowledge of ideal image. The novel procedure is directly applicable to real image enhancement.

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Acknowledgements

Not applicable.

Funding

This study has been supported financially from specific university research MSMT no. 20-SVV/2016, CTU grant SGS 17/196/OHK4/3T/14 and grant GA17-05840S “Multicriteria Optimization of Shift-Variant Imaging System Models” of the Czech Science Foundation.

Availability of data and materials

The dataset supporting the conclusions of this article is included within the article.

Author information

JK suggested the main ideas of the research and realized a part of algorithms. ZK realized a part of algorithms and performed the comparison between entropy and SNR. ZK and JK took part in writing and approved the final version of the manuscript.

Correspondence to Zuzana Krbcova.

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Keywords

  • Entropy
  • Signal-to-noise ratio
  • Robust signal-to-noise ratio
  • Filter design optimization