SIViP DOI 10.1007/s11760-015-0775-3
ORIGINAL PAPER
Image focus measure based on polynomial coefficients and spectral radius Vilas H. Gaidhane1 · Yogesh V. Hote2 · Vijander Singh3
Received: 9 June 2014 / Revised: 17 March 2015 / Accepted: 13 April 2015 © Springer-Verlag London 2015
Abstract In this paper, a new measure of image focus based on the statistical properties of polynomial coefficients and spectral radius is proposed. Spectral radius captures the dominant features and represents the important dynamics of an image. It is shown that the proposed focus measure is monotonic and unimodal with respect to the degree of defocusation, noise and blurring effects. Moreover, it is sufficiently invariant to contrast changes occur due to the variations in intensities of illumination. The noise studies show that the proposed focus measure is robust under the different noisy and blurring conditions. The performance of proposed focus measure is gauged by comparing with the existing image focus measures. Experimental results using synthetic as well as real-time images with known and unknown distortion conditions show the wider working capability and higher prediction consistency of the proposed focus measure. Moreover, the performance of the proposed approach is validated with most popular five image quality
B
Vilas H. Gaidhane
[email protected] Yogesh V. Hote
[email protected] Vijander Singh
[email protected]
1
Department of Electrical and Electronics Engineering, Birla Institute of Technology and Science (BITS) Pilani, Dubai Campus, Dubai International Academic City, P O Box 345055, Dubai, UAE
2
Department of Electrical Engineering, Indian Institute of Technology (IIT), Roorkee 247667, India
3
ICE Division, Netaji Subhas Institute of Technology, University of Delhi, New Delhi 110078, India
databases: TID2008, LIVE, CSIQ, IVC and Cornell-A57. Experimentation on the databases shows that the proposed metric provides the comparatively higher correlation with ideal mean observer score than the existing metrics. Keywords Focus measure · Polynomial coefficient · Companion matrix · Spectral radius · Mean observer score · Spearman’s correlation · Kendall’s tau correlation
1 Introduction A well-focused and clear image is very important in human and computer vision applications such as astronomical imaging, remote sensing, digital microscopy and digital photography [1–4]. A focusing is a process and focus measure is a quantity which measures blurring effect in an acquired image. Generally, image blurring is caused by atmospheric disturbances rather than the camera defocus. To resolve these problems, a reliable and robust focus measure is required. In the literature, various researchers have suggested two types of focusing methods to examine the degree of defocus of images taken with variation in different camera parameters. The first method is active approach where the focusing is carried out with the help of sensing equipments. Although active approach is fast, the extra sensing equipments increases the cost of the system. The second method is passive approach which is totally depends on the quality of the captured images. The quality of such images is derived from the measurement of image sharpness. This measurement is called as image focus measure. The value of such focus measure is found maximum for less blurring effect and reduces as blurring effect increases. Thus, a passive approach is totally relies on the quality of acquired images. However,
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this method increases the burden of computational complexity in the system. Any good-quality image focus measure should satisfy some basic requirements. First, it should be independent of image contents such as brightness and noise. Secondly, it should be unimodal and monotonic to blurring effects. Lastly, the focus measure should be less complex and robust to noise effects [5]. The blurring effect in the images can be modeled by the convolution of the original image R(x, y) and the point spread function (PSF) p (x, y). The relation between the original image R(x, y) and the acquired sequence of the blurred images I1 (x, y), I2 (x, y), I3 (x, y), . . . , I p (x, y) is given as I p (x, y) = R(x, y) × p (x, y), ∀ p = 1, 2, . . . , n.
(1)
where p (x, y) has the characteristics of low-pass filter and reflects its effect of blurring on the focused image. When the object is located at desired position, the term p (x, y) = δ(x, y) = 1 and the well-focused and clear image is obtained, i.e., I p (x, y) = R(x, y)
(2)
Using the focus measure, one can understand and able to estimate the reflection of blurring effect on the plane of image. Thus, having the good focus measure, a best image can be obtained by maximizing the value of focus measure. In recent years, the great efforts have been given by the various researchers toward the development of objective quality assessment measures that take the advantage of known characteristics of the human visual system (HVS). There are basically two classes of objective assessment focus measures. The first are mathematically defined measures such as mean square (MSE), peak signal-to-noise ratio (PSNR), root-mean-square error (RMSE), and mean absolute error (MAE). The second class of methods considers the HVS characteristics. In last few decades, based on these methods the different objective assessment focus measures such as similarity PSNR human visual system measure (sPHVSM) [6], structural similarity (SSIM), mean structural similarity (MSSIM) [7], visual information fidelity (VIF) [8] and universal image quality index (UQI) [9] have been presented in the literature. However, none of the HVS-based measures have shown any clear advantages over the mathematical measures. Therefore, the mathematical measures are still attractive as they are easy to calculate and usually have less complexity. Moreover, they are independent of viewing conditions and individual observers. The motivation of the proposed focus measure came from the fact that the images acquired from the remote sensing, astronomical imaging or solar atmosphere are degraded due
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to the visible light band refractive index fluctuation generally caused by temperature variation. Since the atmospheric conditions are changes continuously, the sequence of acquired images contains the images of different quality. The selection of good-quality image from such image sequence is a challenging task. Therefore, in this paper, a mathematical approach for image focus measure based on polynomial coefficients and spectral radius is proposed. It is modeled as a higher polynomial coefficients problem and then solved by the spectral radius of reduced dimension companion matrix. The rest of the paper is organized as follows: A brief overview of existing focus measures is explained in Sect. 2. Section 3 describes the fundamentals of spectral radius and proposed approach. Section 4 presents the experimental results, and conclusions are drawn in Sect. 5.
2 Existing similarity measures In recent years, several mathematical image focus measures have been reported in the literature [10–13]. Most of the focus measures are based on the high frequency components of an image. The simple focus measure is the variance of image gray levels [14] and it is given as Mv =
2 I p (x, y) − μ dx dy
(3)
where I p (x, y) is the captured image and μ is the mean graylevel value of the image. The other focus measures are defined by the norm of the derivatives of an image [14]. ∂ I p (x, y) ∂ I p (x, y) dx dy Mg = + ∂x ∂y
(4)
The second derivative of the 1 -norm of the image [15] Md =
2 ∂ I p (x, y) ∂ 2 I p (x, y) + dx dy ∂x2 ∂ y2
(5)
The energy of the Laplacian of the image [15] Ml =
∂ 2 I p (x, y) ∂ 2 I p (x, y) + ∂x2 ∂ y2
2 dx dy
(6)
The Chebyshev moments-based focus measure is defined as the ratio of the norm of the high-order to the low-order Chebyshev moments which is given as [16]
H I˜; p
Mt = 1 − L I˜; p
(7)
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The eigenvalue-based focus measure is based on the trace of the m largest eigenvalues of the image which is given as [17] Me = Trace(m ), ∀ m = 1, 2, 3, . . . , 6.
(8)
where Trace(m ) is the sum of the first six eigenvalues. These existing focus measures are used for determining the best focused image among the series of captured images. In the literature, the comparison of these existing focus measures is carried out by normalizing the best focused image having the largest focus measure value M = 1. The most of the focus measures are based on the high-spatial-frequency components of the images without assuming the changes due to the noise effects. Therefore, the focus measures based on the image variance and its variant such as gradient, Laplacian energy, derivative and Chebyshev moments perform well only under the noiseless conditions. The working range of the image focus measures is an important issue and plays a vital role in image focusing. After detailed study of existing focus measures, it is observed that the working range is very narrow and dropping continuously with respect to noise variation. Therefore, it is very difficult to select a good image from the sequence of captured images if the noise in the images is marginally large. To overcome these drawbacks, Wee and Parmesran [17] introduced an image focus measure using the larger eigenvalues and shown that the larger eigenvalues are less affected by noise and the smaller eigenvalues are more sensitive to noise. However, the eigenvalue-based method needs the complex calculations. The motivation of companion matrixbased approach came from the fact that the eigenvalue-based focus measure requires the computation covariance matrix and singular value decomposition (SVD). These calculations are more complicated and contributes most of the computational complexity in the focus measure calculation. In this paper, a new image focus measure based on the polynomial coefficients and spectral radius is proposed. The companion matrix is constructed from the few polynomial coefficients (k) which reduces the dimension of N × N image matrix. Thus, the computational complexity of the proposed focus measure is of order O(k)3 , (k << N ), overcomes the computational complexity problem occurs in existing focus measure.
I p (x, y), λ is the eigenvalue and I is the identity matrix. The determinant of (λI − A) can be calculated as [18] λ − a11 a12 a13 a21 λ − a a23 22 a31 a λ − a33 32 det (λI − A) = . . . .. .. .. a aN 2 aN 3 N1
... ... ... .. .
The proposed focus measure is based on the polynomial coefficients of image matrix and its spectral radius. Suppose I p (x, y) is 2D image such that I p (x, y) ∈ N ×N , then its characteristics polynomial can be obtained by det (λI − A) where A is N × N square matrix obtained from 2D image
. . . λ − aN N (9)
The characteristics polynomial can be obtained as p (λ) = λ N − a1 λ N −1 − a2 λ N −2 − · · · − a N
(10)
where a1 , a2 , a3 , . . . , a N are the polynomial coefficients which can be calculated as follows [18]. a1 =
p i=1
aii aii , a2 = a ji
aii a ji a3 = i< j
i< j
ai j ajj ak j
ai j , ajj aik a jk , a N = det ( A) . akk
(11)
The companion matrix [19] is a simple form of the reduced dimension matrix. It can be obtained using the polynomial coefficients. From Eq. (11), the companion matrix can be written as ⎡ ⎢ ⎢ ⎢ C=⎢ ⎢ ⎣
0 0 0 .. .
1 0 0 .. .
a1 a2
⎤ 0 ... 0 1 ... 0 ⎥ ⎥ 0 ... 0 ⎥ ⎥ ⎥ .. . . . . 1 ⎦ a3 . . . a N
(12)
Here, it is important to note that the first few polynomial coefficients are finite and rest of the coefficients are very small or zero. Therefore, the companion matrix can be constructed using only few polynomial coefficients k, where k << N , represents the main feature of the 2D image. Let C be the n-dimensional companion matrix having λ, v pair of eigenvalue–eigenvector. Then it can be represented as [20,21] Cv = λv
3 Proposed image focus measure
a1N a2N a3N .. .
(13)
Using the sub-multicative property of the norm, Eq. (13) can be written as n λ v ∀ n = 1, 2, 3, . . . , ∞. n C · v , |λ|n v = C n · v |λ|n = C n as v = 0
|λ|n v = n C v ≤
(14) (15) (16)
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Therefore, 1/n ρ (C) = C n , ∀ sup {|λ| : λ ∈ ρ (C)}
(17)
Thus, the spectral radius of a square matrix is the supremum among the absolute values. The direction indicated by this supreme value is same as the direction indicated by the highest variance of 2D image matrix. Therefore, most of the dynamics of the images are indicated by the spectral radius [22]. Using these properties of spectral radius, an efficient image focus measure can be represented as 1 Ms = lim C nk n = ρ (C) , ∀ k = 1, 2, 3, . . . , N . (18) n→∞
Based on the above mathematical preliminaries, a new focus measure is proposed which is summarized in Algorithm 1. Algorithm 1 Proposed focus measure 1: Given an acquired image I p (x, y) of size N × N pixels normalized by its energy defining as 2 N −1 ˜p (x, y) = 1. ˜I p (x, y) = I p (x,y) I such that N −1 2 x,y=0 [ I p (x,y)] x,y=0
2: Calculate the polynomial p(λ) using Eq. (10). 3: Compute the companion matrix C using Eq. (12). 1 4: Calculate lim C nk n = |λk | , ∀ k = 0, 1, 2, 3, . . . , N . n→∞ 1 5: Compute the proposed focus measure as ρ (C) = C nk n = |λk | Thus, the proposed image focus measure, Ms = ρ(C).
The order of companion matrix very small (k ×k) as compared to 2D image matrix (N × N ). Thus, the computational complexity reduced from O(N )3 to O(k)3 , (k << N ). In proposed approach (Ms ), the polynomial coefficients of the image matrix are used to obtain the companion matrix. In this, only few polynomial coefficients are finite and remaining coefficients are zero. Therefore, the size of the companion matrix is very small as compared to the 2D image matrix. Figure 1 shows the variation of magnitude of eigenvalues and polynomial coefficients. It is observed from Fig. 1 that the convergence of magnitude of polynomial coefficients is faster than the eigenvalues. Moreover, the polynomial coefficients have larger magnitude as compared to the eigenvalues. Therefore, most of the features of an image can be represented efficiently using few nonzero polynomial coefficients than the eigenvalues.
4 Experimental results and discussion In this section, the performance of the proposed focus measure is tested by performing a series of experiments on various synthetic as well as real-time images.
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Magnitude of Plolynomial coefficients/Eigenvalues
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6 5 Polynomial Coefficients Eigenvalues
4 3 2 1 0
5
10
15
20
25
30
Number of Plolynomial coefficients/Eigenvalues
Fig. 1 variation of magnitude of polynomial coefficients and eigenvalues
4.1 Blurred images without noise The efficacy of proposed focus measure is tested using the blurring effects such as motion point spread function (MPSF) and Gaussian point spread function (GPSF). These two PSFs are considered in this experiment because they are most commonly introduced PSFs during the imaging process. For motion PSF, the motion filter is used with its linear motion pixels η = 1, 2, 3, . . . , 20. For Gaussian PSF, Gaussian low-pass filter is used with its standard deviation σ = 1, 2, . . . , 10. The set of three grayscale test images [23] of size 256 × 256 pixel is used for experimentations. The original reference images are shown in Fig. 2a–c. Similarly, the noiseless blurred images with motion PSF (η = 15) and Gaussian PSF (σ = 6) are shown in Fig. 2d–f and g–i, respectively. A new focus measure |Ms | is obtained using the proposed algorithm. Similarly, the focus measure based on eigenvalues (Me ) is also obtained using first six largest eigenvalues as reported in [17]. Moreover, the parameter p = 2 is selected to calculate the six Chebyshev moments in the case of Mt focus measure [16]. Then the performance of the proposed focus measure (Ms ) is compared with the various reported focus measures Mv , Mg , Md , Ml , Mt and Me for motion PSF as well as Gaussian PSF. The comparison of various focus measures for motion PSF and Gaussian PSF is shown in Fig. 3a, b, respectively. A good focus measure declines slowly when there is increase in blurring effect (PSF). The performance of the focus measures is approximately similar for both PSFs. It is observed from Fig. 3 that the proposed measure (Ms ) decline slowly with respect to increase in blurring effect. This implies that Ms performs better under varying and larger blurring condi-
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(a)
(b)
(c)
tions. However, the focus measures Mg , Md and Ml shows the saturation as the degree of burring effect increases. This behavior indicates that they are unable to differentiate the change of PSFs in the images. 4.2 Blurred images with noise
(d)
(e)
(f)
(g)
(h)
(i)
Fig. 2 Effect of PSF. a–c reference images, d–f with motion PSF (η = 15), g–i with Gaussian PSF (σ = 6)
(a) 1.1 1
Focus Measure Value
0.9 0.8 0.7
Ms
0.6
Me
0.5
Mv
0.4
Mt Mg
0.3
Md
0.2 0.1
Focus Measure Value
(b)
Now consider the blurred images with noise. In this experiment, the original test images are convolved with the Gaussian PSF (σ = 4). Further, these blurred images are convolved with the most common additive random Gaussian noise with the variance v = 0.001 to 0.01 and the saltand-pepper noise with the noise density δ = 0.10–0.20, respectively. It is reported in [17] that the variation of the focus measure for motion PSF and Gaussian PSF are approximately similar and therefore, only Gaussian PSF along with the Gaussian noise and salt-and-pepper noise are considered. The set of three grayscale test images of size 256 × 256 pixel is used for experimentations as shown in Fig. 4a– c [23]. The various blurred images with Gaussian noise are shown in Fig. 4d–f and salt-and-pepper noise are shown in Fig. 4g–i. The variation of mean value of all focus measures in the presence of Gaussian noise and salt-and-pepper noise is shown in Fig. 5a, b, respectively. The focus measure is said to be robust if its mean value of focus measure saturates slowly for Gaussian as well as salt-and-paper noise. It is observed from Fig. 5a, b, that the proposed focus measure (Ms ) and eigenvalue-based measure (Me ) performs better in the presence of noise and preserve the monotonicity and unimodality properties with slower saturation. Thus, in the presence of additive random noise, the proposed focus measure (Ms ) provides the wider working range. However, the focus measure Mg , Md and Ml saturates abruptly and therefore, they are
Ml
0
2
4
6
8
10
η
12
14
16
18
20
(a)
(b)
(c)
1
(d) σ = 5, v = 0.01 (e) σ = 5, v = 0.01 (f) σ = 5, v = 0.01
0.8 0.6
M M
0.4
M M
0.2
M
s e v
(g) σ = 8,δ = 0.2
t
(h) σ = 8, δ = 0.2 (i) σ = 8, δ = 0.2
g
Md
0
M
0
1
l
2
3
4
5
σ
6
7
8
9
10
Fig. 3 The variation of mean value of focus measures. a motion PSF, b Gaussian PSF
Fig. 4 Noisy blurred images. a–c reference images, d–f blurred images with Gaussian PSF and Gaussian noise, g–i blurred images with Gaussian PSF and salt-and-pepper noise
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(a)
1
Focus Measure Value
0.98 0.96 0.94
M M
0.92
M s = 0.90
M s = 0.907
M s = 0.889
M s = 0.807
M s = 1.00
M s = 0.91 M s = 0.909
M s = 0.892
M s = 0.822
M s = 0.82
e
Mv
0.9
Mt
0.88
Mg
0.86
M M
0.84
M s = 1.00 s
0
d l
0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01
v
(b)
1
M s = 1.00
M s = 0.911
M s = 0.856
M s = 0.8
M s = 1.00
M s = 0.97
M s = 0.976
M s = 0.969
0.98
Focus Measure value
0.96 0.94 Ms
0.92 0.9
Fig. 6 Real images from camera (M × N = 256 × 256)
Mv Mt
0.88
Mg Md
0.86
(a)
Ml
0.84
0.1
0.12
0.14
δ
0.16
0.18
(b)
(c)
(d)
0.2
Fig. 5 The variation of mean value of focus measures for blurred images. a Gaussian PSF with Gaussian noise. b Gaussian PSF with salt-and-pepper noise
unable to preserve the properties of monotonicity and unimodality for Gaussian as well as salt-and-pepper noise. 4.3 Real-time images In order to verify the accuracy of the proposed focus measure in real situation, two experiments are carried out on real images captured under the degraded conditions. The realtime images are shown in Fig. 6. These images are taken with a Nikon Power Shot digital camera at different focal length, cropped to size of 256 × 256 pixels. It is observed from Fig. 6 that the value of proposed focus measure (Ms ) decreases with the increase in the degree of blurring effect in the image. Thus, the proposed focus measure provides the different focus measure values under different degraded conditions which are compatible with the human vision perception. In second experiment, the Ms is tested with the set of four real-time images shown in Fig. 7. These images are also taken with a Nikon Power Shot digital camera at different focal length. In this experiment, the blur windows of pixel size 2 × 2, 4 × 4, 8 × 8, 16 × 16 and 24 × 24 are considered.
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M s = 0.918
Me
Fig. 7 Real images from camera (M × N = 256 × 256)
In first case, the blurred images with different blurring windows of size 2 × 2, 4 × 4, 8 × 8, 16 × 16 and 24 × 24 are considered. Similarly, in second case, all the blurred images convolved with the Gaussian noise of variance v = 0.05 and in third case the blurred images convolved with salt-andpepper noise of density δ = 0.15. The different values of proposed focus measure (Ms ) for all three cases are summarized in Table 1. It is observed from Table 1 that the proposed focus measure follows the monotonicity and unimodality property. 4.4 Contrast invariance The image focus measure should be invariant to the variation in the intensity of illumination. Table 2 shows the point spread value (σ/μ) for the different focus measures for Gaussian noise and salt-and-pepper noise added with PSFs. The contrast changes are simulated by considering the modified image Fig. 7a. The point spread values of proposed focus measure (Ms ) are 0.0512 and 0.0346 for salt-and-pepper and Gaussian noise, respectively. These values are smaller than the point spread values of the other focus measures. This
SIViP Table 1 Values of proposed focus measure for blurred images
Image
2×2
4×4
8×8
16 × 16
24 × 24
(a) Noise free with blur effect Figure 7a
0.9801
0.9768
0.8801
0.8699
0.8626
Figure 7b
0.9991
0.9191
0.9163
0.9282
0.9052
Figure 7c
0.9622
0.9406
0.9386
0.9327
0.9073
Figure 7d
0.9822
0.9054
0.8811
0.8624
0.8428
(b) Images degraded with blur and Gaussian noise (v = 0.05) Figure 7a
0.9453
0.9145
0.8626
0.8432
0.8399
Figure 7b
0.9143
0.9129
0.9118
0.8805
0.8674
Figure 7c
0.9100
0.8872
0.8856
0.8801
0.8791
Figure 7d
0.9223
0.8998
0.8675
0.8544
0.8500
(c) Images with blur and salt-and-peppers noise (δ = 0.15) Figure 7a
0.9112
0.8571
0.8550
0.8463
0.8372
Figure 7b
0.9189
0.9172
0.8677
0.8418
0.7684
Figure 7c
0.9135
0.8931
0.8908
0.8822
0.8696
Figure 7d
0.9235
0.8995
0.8725
0.8647
0.8598
Table 2 Contrast invariance of reported and proposed focus measure
Table 3 Spearman’s correlations on LIVE, TID, CSIQ, IVC and CORNEL-A57 full image dataset
Focus measure
Salt-and-pepper noise
Gaussian noise
Mv
0.2069
0.2501
Mg
0.2196
0.2677
Proposed
0.958
0.895
0.931
0.928
0.934
0.935
0.844
0.898
0.886
0.906
Metric
LIVE
TID
CSIQ
IVC
Cornel
Md
0.3115
0.3621
sPHVSM
Ml
0.3155
0.6315
SSIM [7]
0.911
0.774
0.875
0.766
0.804
MSSIM [7]
0.945
0.852
0.907
0.880
0.839
Mt
0.1360
0.3711
Me
0.0717
0.0432
VIF [8]
0.950
0.750
0.913
0.897
0.622
0.0346
UQI [9]
0.892
0.600
0.801
0.819
0.425
NQM [29]
0.908
0.624
0.732
0.827
0.793
Ms
0.0512
Bold indicates best Spearman’s correlation values
indicates that the proposed focus measure (Ms ) perform well than the existing focus measures. 4.5 Test on image datasets In this experiment, five most popular image datasets: LIVE [24], TID2008 [25], CSIQ [26], IVC [27], and COENELA57 [28] are used. In the literature, Jin et al. [6] and Ponomarenko et al. [25] carried out the rigorous experimentations on these image quality datasets and compare the results with various image quality focus measures such as sPHVSM [6], SSIM, MSSIM [7], VIF [8], UQI [9], and NQM [29]. Therefore, for a fair comparison, the proposed approach is also tested on the MATLAB-based framework and the results are compared with the various existing image quality focus metrics. The comparison of Spearman’s correlations for proposed metric and the various existing image quality metrics for well known five full image quality databases is shown in Table 3. It is observed from Table 3 that the proposed metric performs better and it is more stable as compared to various
existing HVS-based methods as well as the similarity metric methods. Thus, the presented image focus measure provides comparatively good response to the contrast change, noise variation and illumination effect. Among all five image quality databases, TID2008 [25] is the largest dataset of distorted images specially intended for quality metrics performance evaluation. It consists of 1700 test images which includes 25 reference images, 17 types of distortion for each reference image, and 4 different levels of each type of distortion. It also includes an array of MOS values obtained for each distorted image. This database satisfies the main requirements and contains different types of distortions that relates to the peculiarities of the human vision system (HVS). Moreover, it allows to determine the drawbacks of the new as well as existing metrics. However, the scaling problem occurs during these experimentation. To avoid this problem, the rank correlations such as Spearman’s and Kendall’s tau is generally used in practice [30]. The values of these correlations are ideally unity which indicates the better quality of an image assessed by the metric.
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SIViP Table 4 Comparison of Spearman’s correlation and Kendall’s tau correlation
Table 5 Computational speed and computational complexity of proposed and reported focus measure
Metric
Focus measure
Computational time (second)
Computational complexity (order)
Actual
Full
Noise
JPEG
Exotic
Spearman’s correlation Proposed
0.965
0.948
0.969
0.962
0.857
sPHVS [6]
0.844
0.862
0.923
0.962
0.760
Mv
0.400
O(N )2
sPHVSM [6]
0.934
0.844
0.930
0.975
0.750
Mg
0.560
O(N )2
SSIM [7]
0.882
0.808
0.856
0.964
0.468
Md
1.361
O(N )2
MSSIM [7]
0.868
0.853
0.813
0.957
0.673
Ml
1.615
O(N )2
VIF [8]
0.841
0.750
0.820
0.956
0.045
Mt
0.811
O(N )2
UQI [9]
0.677
0.600
0.526
0.860
0.156
Me
1.621
O(N )3
NQM [29]
0.874
0.624
0.865
0.932
0.517
Ms
0.540
O(k)3
PSNRHVS
0.920
0.594
0.917
0.966
0.541
0.803
0.798
0.819
0.812
0.653
sPHVS [6]
0.747
0.682
0.766
0.830
0.564
sPHVSM [6]
0.769
0.659
0.764
0.858
0.552
SSIM [7]
0.691
0.605
0.658
0.828
0.311
MSSIM [7]
0.675
0.654
0.609
0818
0.478
Kendall’s tau correlation Proposed
VIF [8]
0.657
0.586
0.634
0814
0.092
UQI [9]
0.489
0.435
0.363
0.666
0.115
NQM [29]
0.678
0.461
0.673
0.766
0.349
PSNRHVS
0.750
0.476
0.751
0.837
0.385
Best three values of Spearman’s correlation and Kendall’s tau correlation are indicated in bold
Here, the proposed metric is compared with existing metrics such as sPHVS, sPHVSM [6], SSIM, MSSIM [7], VIF [8], UQI [9], and NQM [29]. All these metrics are used the grayscale images or luminance component of the reference and distorted images. Table 4 shows the comparison of Spearman’s and Kendall’s tau correlation for the proposed metric as well as existing metrics with respect to subsets (Actual, Full, Noise, JPEG and Exotic) of TID2008 database. The best three focus metrics producing the largest Spearman and Kendall’s tau correlation are marked as bold in Table 4. It is observed from Table 4 that the proposed approach perform comparatively better than the existing metrics. Moreover, Spearman’s and Kendall’s tau correlation are close to the ideal value (unity) of the HVS. 4.6 Computational complexity All experiments are carried out on Intel CoreTM i3-370M 2.4 GHZ and 4 GB RAM processor, MATLAB platform of version 7.0.4. The average computation time and computational complexity of proposed as well as high frequency moments-based focus measures is summarized in Table 5. However, the various objective image quality focus measures [6–9] are based on the block matching approach which are more complex and needs more time duration as compared
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to moment-based approaches. Therefore, objective image quality measures are not considered in complexity and speed comparison. It is observed from Table 5 that the Ms has the higher computational speed as compared to Md , Ml and Me . However, the computational time for Ms is slightly more than Mv , Mg and Mt due to the polynomial coefficients computation. Although the computational time of proposed focus measure (Ms ) is slightly greater than Mv , it is acceptable due to its overall better performance and robustness in the presence of blurred and noisy conditions. The computational complexity of most of the focus measure is of order O(N )2 . However, in proposed approach, the finite polynomial coefficients form a companion matrix of order k × k = 12 × 12. Therefore, the computational complexity reduced to order of O(k)3 . Thus, the computational complexity reduced significantly which results into less memory storage space and computational time.
5 Conclusions In this paper, a new image focus measure based on the spectral radius is proposed. The spectral radius captures the dominant features and represents the important dynamics of an image. It is shown that the proposed focus measure fulfills the basic requirements usually imposed on an ideal focus measure. The new focus measure overcomes the drawback of conventional focus measures. It has high discrimination power in the presence of blurring effect and noises. The computational performance and contrast invariance of the proposed focus measure is compared with the various reported focus measures. The experimental results using the synthetic and real-time images under blurred and noisy conditions show that the proposed focus measure performs significantly better in terms of noise robustness, discrimination capability, contrast invariance and computational complexity. Moreover, experimentations on the image quality databases also show that the proposed measure performs comparatively better than the existing image quality measures.
SIViP
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