Relationship between entropy and SNR changes in image enhancement
- Zuzana Krbcova^{1}Email author and
- Jaromir Kukal^{1, 2}
https://doi.org/10.1186/s13640-017-0232-z
© The Author(s) 2017
Received: 14 June 2017
Accepted: 20 November 2017
Published: 19 December 2017
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.
Keywords
1 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 [6–8]. 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].
2 Methods
2.1 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.
2.1.1 Signal-to-noise ratio
Positive Δ SNR value expresses the improvement of noisy image after its reconstruction. On the other hand, negative value expresses noisy image degradation.
2.1.2 Robust signal-to-noise ratio
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.
2.1.3 Hartley entropy
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.
where spacing h>0 approaches zero and i∈{1,⋯,q}.
2.2 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.
The simplest sharpening filter based on Gaussian filter with sharpening parameter α>0 and its generalization include
where LP(ω), LP_{0}(ω), LP_{1}(ω), and LP_{2}(ω) are four realizations of low-pass filter Φ_{1}(ω). In the case of Φ_{2}, only fundamental low-pass filter is used, but in Φ_{3}, the difference between two low-pass filters (LP_{1},LP_{2}) is used as high-pass filter added to the fundamental LP_{0} filter in accordance with conventions of image processing. The filters Φ_{1} (smoothing), Φ_{2}, and Φ_{3} (sharpening) will be subject of parameter optimization in the next section. The filter Φ_{1} has only one parameter ρ which is an advantage for its optimization. The filters Φ_{2} and Φ_{3} 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 Φ_{1} and Φ_{2} quality measures (Δ SNR,G) can be easily visualized.
3 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 Φ_{1},Φ _{2}, and Φ_{3}. The Fast Simulated Annealing (FSA) [20] was used for Δ SNR maximization and G minimization inside given logarithmic ranges.
3.1 Test data
3.2 Image enhancement based on Δ SNR
Optimal low-pass smoothing Φ_{1} via Δ SNR maximization
Image | Quality measures | Parameter | |
---|---|---|---|
Δ SNR | Δ R | log10ρ | |
THISTLE | 3.897 | 3.990 | −0.059 |
HOUSE | 3.308 | 3.604 | 0.131 |
MAP | 4.706 | 3.324 | −0.053 |
WINDMILL | 9.014 | 7.632 | 0.184 |
BRIDGE | 4.578 | 3.839 | −0.140 |
BALCONY | 5.916 | 4.398 | −0.032 |
Optimal sharpening Φ_{2} via Δ SNR maximization
Image | Quality measures | Parameters | ||
---|---|---|---|---|
Δ SNR | Δ R | log10α | log10ρ | |
THISTLE | 3.881 | 3.968 | −2.203 | −0.074 |
HOUSE | 3.284 | 3.582 | −2.159 | 0.102 |
MAP | 4.705 | 3.306 | −2.520 | −0.062 |
WINDMILL | 8.818 | 7.054 | −2.124 | 0.089 |
BRIDGE | 3.943 | 3.603 | −1.417 | −0.238 |
BALCONY | 5.835 | 4.361 | −1.802 | −0.054 |
Optimal sharpening Φ_{3} via Δ SNR maximization
Image | Quality measures | Parameters | ||||
---|---|---|---|---|---|---|
Δ SNR | Δ R | log10α | log10ρ _{0} | log10ρ _{1} | log10ρ _{2} | |
THISTLE | 3.935 | 4.035 | −0.137 | −0.099 | −0.217 | −0.313 |
HOUSE | 3.273 | 3.501 | −0.077 | −0.005 | −0.100 | −0.161 |
MAP | 4.739 | 3.343 | −0.148 | −0.167 | −0.079 | −0.250 |
WINDMILL | 8.706 | 6.833 | −0.075 | −0.012 | −0.037 | −0.162 |
BRIDGE | 4.811 | 4.100 | −0.230 | −0.001 | −0.491 | −0.223 |
BALCONY | 6.029 | 4.484 | −0.621 | −0.012 | −0.481 | −0.216 |
3.3 Image enhancement based on G
Optimal low-pass smoothing Φ_{1} via G minimization
Image | Quality measures | Parameter | ||
---|---|---|---|---|
G | Δ SNR | Δ R | log10ρ | |
THISTLE | −0.126 | 3.862 | 3.898 | −0.105 |
HOUSE | −0.140 | 3.197 | 3.377 | −0.020 |
MAP | −0.360 | 4.613 | 3.133 | −0.122 |
WINDMILL | −0.106 | 8.471 | 6.464 | 0.021 |
BRIDGE | −0.265 | 4.341 | 3.584 | −0.242 |
BALCONY | −0.308 | 5.764 | 4.159 | −0.120 |
Optimal sharpening Φ_{2} via G minimization
Image | Quality measures | Parameters | |||
---|---|---|---|---|---|
G | Δ SNR | Δ R | log10α | log10ρ | |
THISTLE | −0.126 | 2.563 | 3.882 | −1.121 | −0.111 |
HOUSE | −0.408 | −1.710 | 1.275 | −0.652 | −0.472 |
MAP | −0.358 | 3.841 | 3.145 | −1.203 | −0.119 |
WINDMILL | −0.154 | −5.113 | 3.567 | −0.503 | −0.237 |
BRIDGE | −0.259 | −9.055 | 3.580 | −0.308 | −0.243 |
BALCONY | −0.315 | 5.405 | 4.197 | −1.343 | −0.111 |
Optimal sharpening Φ_{3} via G minimization
Image | Quality measures | Parameters | |||||
---|---|---|---|---|---|---|---|
G | Δ SNR | Δ R | log10α | log10ρ _{0} | log10ρ _{1} | log10ρ _{2} | |
THISTLE | −0.025 | 3.869 | 3.900 | −0.368 | −0.284 | −0.055 | −0.895 |
HOUSE | −0.005 | 3.274 | 3.502 | −0.348 | −0.027 | −0.006 | −0.168 |
MAP | −0.077 | 4.608 | 3.199 | −0.212 | −0.310 | −0.042 | −0.666 |
WINDMILL | −0.008 | 7.309 | 5.130 | −0.065 | −0.257 | −0.176 | −0.412 |
BRIDGE | −0.005 | 3.596 | 2.859 | −0.080 | −0.329 | −0.323 | −0.927 |
BALCONY | −0.044 | 5.728 | 4.164 | −0.196 | −0.390 | −0.009 | −0.630 |
3.4 Discussion
The relative changes [%] between quality measures
Image | Φ_{1} | Φ_{2} | Φ_{3} | |||
---|---|---|---|---|---|---|
RC _{ ΔSNR} | RC _{ Δ R } | RC _{ Δ SNR } | RC _{ Δ R } | RC _{ Δ SNR } | RC _{ Δ R } | |
THISTLE | 0.90 | 0.03 | 33.96 | 2.17 | 1.68 | 3.35 |
HOUSE | 3.36 | 6.30 | 152.07 | 64.41 | 0.03 | 0.03 |
MAP | 1.98 | 5.75 | 18.36 | 4.87 | 2.76 | 4.31 |
WINDMILL | 6.02 | 15.30 | 157.98 | 49.43 | 16.05 | 24.92 |
BRIDGE | 5.18 | 6.64 | 329.65 | 0.64 | 25.25 | 30.27 |
BALCONY | 2.57 | 5.43 | 7.37 | 3.76 | 4.99 | 7.14 |
Mean | 3.33 | 6.58 | 116.57 | 20.88 | 8.46 | 11.67 |
4 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.
Declarations
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.
Authors’ contributions
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.
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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