No-reference image quality assessment through the von Mises distribution
An innovative way of calculating the von Mises distribution (VMD) of image entropy is introduced in this paper. The VMD’s concentration parameter and some fitness parameter that will be later defined, have been analyzed in the experimental part for determining their suitability as a image quality assessment measure in some particular distortions such as Gaussian blur or additive Gaussian noise. To achieve such measure, the local Rényi entropy is calculated in four equally spaced orientations and used to determine the parameters of the von Mises distribution of the image entropy. Considering contextual images, experimental results after applying this model show that the best-in-focus noise-free images are associated with the highest values for the von Mises distribution concentration parameter and the highest approximation of image data to the von Mises distribution model. Our defined von Misses fitness parameter experimentally appears also as a suitable no-reference image quality assessment indicator for no-contextual images.
Keywords: Rényi entropy, directional entropy, Wigner distribution, von Mises distribution, contextual/no-contextual metric.
Instituto de Óptica (CSIC),
Serrano 121, 28006 Madrid, Spain \addressCorresponding author: firstname.lastname@example.org
110.3000, 100.2000, 330.6180, 100.6640
Digital images can suffer a number of different operations that take into account e.g. acquisition, coding, compression, transmission, and many others before their final use. In such processes, digital images might be affected by many kinds of degradations revealed by different types of visual distortions. A list of different possible types of degradations has been considered in the image database TID2008 by Ponomarenko et al. . A key feature about images is their quality. However, image quality has not a simple definition. In a semantic approximation, image quality is understood as the subjective impression of how well image content is rendered or reproduced . This definition yields to the concept of image quality assessment. In practice, such assessment can be achieved by two means: one is by psychophysical subjective experiments with human observers, and the other one is by objective metrics applied directly to digital images . Image quality assessment is a very active research area with many contributions, techniques and models . Typically, image quality can be measured by comparison with a reference, but unfortnately such reference is not available in many applications. Hence, no-reference image quality assessment methods are required and they have been the subject of a very active research during the last years. However, a universal method for estimating the overall image quality is still a challenging open issue. Customarily, existing no-reference methods deal with simple specific degradations such as Gaussian blur or additive white noise. A good survey about existing no-reference sharpness metrics may be found in Ferzli and Karan  where their metric is compared to others with good results. Ferzli and Karan introduced the notion of just noticeable blur (JNB). It is an edge-based sharpness metric based on a human visual system model. Other metrics have been developed including blur and noise simultaneously, such as the one by Zhu and Milanfar , introducing a new concept called true image content. Their measure is correlated with noise, sharpness, and intensity contrast, manifested in visually salient geometric features such as edges, showing that such measure correlates well with subjective quality evaluation for both blur and noise distortions. However, Zhu and Milanfar technique has been designed to compare images within the same context, while the sharpness metric of Ferzli and Karam has been developed to predict the relative amount of blurriness in images regardless of their context (note that we consider that images resulting from distorting a given original are contextual each other and we use the term no-contextual when the degraded images are the result of distorting different originals). According to Zhu, JNB technique fails to capture the trend of quality change in BM3D  denoising experiments, since it cannot handle noise well. Later on, Narvekar and Karam  proposed an improved algorithm based on the JNB paradigm for a no-reference objective image sharpness metric, introducing a technique they called the cumulative probability of blur detection (CPBD). In such work, the sharpness metric converges to a finite number of quality classes. They used the LIVE  database to validate the performance of their metric. A training-based method determines the centroids of the quality classes that represent the perceived quality levels. Classification is based on assigning the image to one of the quality classes and then using the index of the corresponding quality class as the metric value for that image. They include measuring experiments for Gaussian blur and JPEG2000-compressed images and they show that this metric performs better than other known metrics.
In summary, different reference and no-reference classifications of metrics can be considered. According to the type of distortion they measure, specific metrics (only one type) versus universal metrics (all types) could be considered. According to their contextual character, contextual metrics, if the measures have sense only when the images have the same context versus no-contextual metrics, if the measure have sense by itself, regardless of the image context could be considered. In addition to the previous schemes, supervised and unsupervised approaches may be considered in the image quality context.
This paper is a step forward to the approach introduced by Gabarda and Cristóbal  that links image directional entropy with image quality assessment. In , the variance of the directional entropy of the image was introduced as a quality assessment parameter, presenting the highest value when the image is free of Gaussian blur or Gaussian noise and decreases monotonously with blur an noise increments. Gabarda and Cristóbal’s method may be classified as a no-reference, specific, contextual and unsupervised metric. In this paper we extend this methodology to both contextual and no-contextual applications for image quality assessment by introducing the von Mises distribution of the directional image entropy as the cornerstone of a new image quality metric for blur and noise.
This paper is organized as follows. Section 2 gives the mathematical background required to understand our methodology. Section 3 presents some applications of the VMD for image quality measurement, along with the performances of this method against the Mean Opinion Score (MOS) of TID2008 image database and Differential Mean Opinion Score (DMOS) of LIVE image database. Results are compared with scores from other no-reference quality assessment existing methods. Finally, conclusions are drawn in Section 4.
2 Description of the method
2.1 Directional entropy
The information content of images can be measured in different ways. One of the most popular is the entropy. Generally speaking, the measurement of entropy was initially proposed by Shannon  as a measure of the information content per symbol, coming from a stochastic information source. It can be considered as a quantifier of the uncertainty or randomness of a signal or, in other words, a measure of the information content of the signal. Given a discrete random variable , the Shannon entropy of this discrete random variable can be computed as
where is the probability of event .
Later, Rényi  extended this notion to yield the generalized entropy, whose representation for discrete random variables is
Here is a real-positive number. It can be shown that the Rényi entropy, as a generalization of the Shannon entropy, reverts to the Shannon entropy when .
Measures of entropy may be global, affecting the whole image, or may be localized in each pixel by the use of an sliding window, centered in each pixel, that gathers the neighborhood of that pixel with the desired geometry.
Our proposal is to use a linear array of pixel values, centered in the interest pixel and oriented in a set of desired directions, in order to have a directional measure of entropy at pixel level. To do so, an operational sliding window L is defined, containing the pixels that better fit the ideal line centered in the interest pixel and directed by angle . N is required to be an even number to preserve the symmetry of the window.
Better than considering the gray-values of the image as input to the entropy calculation, we prefer considering the space-frequency information of the image. Space-frequency representations of a signal open up the possibility of exploring the image information taking into account the position and the local frequency content of the image. One of these space-frequency (SF) representations is the Wigner distribution . A common analytical framework of SF representations was introduced by Cohen . The Wigner distribution (WD) has been selected due to its excellent properties and because it can be regarded as a masterform distribution function from which the other existing representation can be derived as filtered versions of it . The WD approximation for discrete signals is better known as pseudo-Wigner distribution (PWD). To calculate a local PWD we will use the following equation
This approximation of the PWD is due to Claasen and Mecklembr uker  and is similar to Brenner’s expression . In Eq. (3) the variable represents the gray value of pixel in a given image . A directional PWD may be calculated by using Eq. (3) along with a pixel-by-pixel sliding window to pick up the values . The central position of the window matches exactly pixel on the image. Eq. (3) can be interpreted as the discrete Fourier transform (DFT) of the product . Here indicates the complex-conjugate of signal (note that for real valued images). The sum is limited to a spatial interval . In Eq. (3), and represent the space and frequency discrete variables respectively, and is a shifting parameter, which is also discrete. By shifting the window to all possible positions over the image, the full pixelwise PWD of the image is produced.
Diverse measures of entropy may be defined upon the kind of distribution and normalization used with the formulation due to Rényi. Although the Rényi measures in time-frequency distributions formally resemble the original entropies, they do not have the same properties, conclusions and results derived in classical information theory. The positivity, will not be always preserved, along with the unity energy condition for a given pixel . In order to reduce a distribution to the unity signal energy case, some kind of normalization should be applied. The normalization can be performed in various ways, leading to a variety of possible measurement definitions , , , , .
The Rényi entropy measure applied to a discrete space-frequency distribution, as the PWD denoted by , can be written as
Here and we have dropped the subscript . The summation is restricted to the 1-D neighborhood of pixel , as described above. Again is the frequency discrete variable. In order to reduce the distribution to the unity signal energy case, some kind of normalization must be done . To do so, the PWD given by Eq.(3) is normalized by applying the following normalizing law, as a previous step to the Rényi entropy measure:
Squaring of is justified due to the real character of the PWD, and the normalizing step affect exclusively to index , when the window operation in Eq. (4) is applied to pixel ; hence the condition in such position is fulfilled.
It can be shown that the Rényi entropy meet the constraint . Hence, the measure can be normalized to the interval by applying in Eq. (4).
The 1-D window used to measure entropy by this method may be oriented in any desired direction , as indicated above, and entropy is now denoted by . This local pixel-wise directional entropy can be constrained to a scalar value for the whole image by averaging all the values by means of
provided that is the number of pixels in the image.
2.2 The von Mises distribution
The von Mises (VM) distribution belongs to probability theory and is used to handle directional statistics for continuous probability distribution on a circular basis. It appears, in many respects, analogous to the normal distributions for a scalar variable. It was proposed by von Mises  to study deviations of atomic weights from integer values. This distribution has also been applied to diverse applications in many fields and has become an important tool in the statistical theory of directional data .
The von Mises probability density function for the angular variable is given by:
where is the modified Bessel function of order , and is defined in the interval . The parameter is responsible for how concentrated the distribution is around the mean direction . Larger values of the concentration parameter indicate that the distribution is more closely grouped around the mean direction. When , this distribution is equivalent to the uniform distribution. This fact happens when directions in the image are endowed with equal probability. The uniform circular distribution is, for example, a good model for Gaussian noise.
Different authors have dedicated some research to the link between images and the von Mises distribution. For instance, Vo et al.  proposed a new statistical modeling of natural images in the wavelet transform domain. They claimed that the von Mises distribution fits accurately the behaviors of relative phases in the complex directional wavelet sub-band from different natural images, and introduced a new image feature based on the VM model for image texture retrieval applications. Palacios et al.  presented what they called new tools for color image processing, based on the circularity of the hue variable of a color image. They give a definition of the median and the range of angular data and apply their results to detect hue edges. Feng  presented a local image orientation estimation method using the image gradient, using a combination of the principal component analysis (PCA) and the multiscale pyramid decomposition. Grana et al.  described a new approach to texture characterization for document analysis. By considering the autocorrelation matrix, they described image texture through a mixture of VM distributions.
2.3 Von Mises distribution of image entropy
As a further contribution to the applications cited in the previous section, we propose to use the VM distribution for modeling the directional distribution of the image entropy, calculated by means of Eq. (6). The measure defined for the image by Eq. (6) requires the use of a bimodal von Mises distribution, provided that . This requirement is fulfilled when we take , and in the following bimodal expression of the von Mises distribution
from which we arrive to 
Eq. (9) can be considered as the basis for modeling directional statistics of image entropy. One of the features to take into account in the discrete directional model for image entropy starting from Eq. (6) concerns the number of axis that can be defined in the image. Two requirements must to be fulfilled in order to consider that the directional measures, based on Eq. (6) are completely comparable. First all directions must be measured by windows consisting in the same number of pixels, and the second one is that the span of the windows in all possible directions must have the same length. These two constrains are only possible when using four directions along the axes of a regular octagon, i.e.: . Under this requirements, we are going to restrict our calculations upon 1D windows directed by the axes along these angles. The arrangement of the pixels in these four operational windows may be visualized in Fig. 1.
The four directionalities that we have defined are used to calculate four values of entropy by means of Eq. (6) that it can be referred as vectors . These directional entropies can be represented also by means of Cartesian vectors by .
Some definitions for a resultant vector and a mean angular direction in VM distribution have been given by Jammalamadaka and SenGupta  and one analysis for modeling circular data may be found in Bentley .
In our approximation we first perform a preliminary estimation of by a single value decomposition (SVD) of the directional vectors , following to Feng , that deals the image orientation estimation problem through the local image gradient. The singular vector of the matrix
corresponding to its largest singular value, gives an estimation of the parameter .
Also, an initial estimation of can be achieved by the following approximation due to Dhillon : , where .
The second step is a gradient descent algorithm to find a better estimation for . The set of values , for , with respectively, are matched by a minimum square error estimation method to find the coefficients and that fit the equation . An exact fitting is attained if and . An error function is defined to control the increments, in order to find a minimum for this error function. The algorithm runs by means of a recursive law , for updating the values of . The algorithm stops when the error value reaches a minimum. A Matlab implementation of this algorithm may be found in .
The accuracy of the estimated distribution to a true von Mises distribution is measured by defining a fitness parameter , whose value will be for the best fitting and for the worst. An example of estimation of the von Mises distribution for a given image is shown in Fig. 2. Triangles indicate the directional entropy values used as input to the algorithm, along with the estimated von Mises distribution, , of the image.
To characterize the behavior of and , the algorithm has been tested with the images in the image database TID2008 due to Ponomarenko et al. . This database contains 25 reference color images and 1700 distorted images (25 reference images 17 types of distortion 4 levels of distortion) in bitmap format without any compression. Images are pixels in size. The test has been performed taking the 25 reference images, excluding image 25th, provided that this last one is artificial and has been considered inappropriate for our study.
Reference images have been first degraded by adding blur in iterative manner by means of a rotationally symmetric Gaussian lowpass filter of size pixels with standard deviation , and in a second instance by adding Gaussian noise, also iteratively, with standard deviation . Considering the 24 originals and nine iterations for each original image, we produced a test set of 240 images. Figures 3 and 4 show the average values of and for all these images, having the degradation level as abscissa. Measurements indicated that as the expected value of the fitness parameter for the 24 originals (see abscissa=0 in Fig. 4), indicating that the fitness parameter of the VM distribution converges to a common value for good quality images. In general, when more blur or Gaussian noise is added, fitness decreases, and the variance of increases, indicating that, for severe degradation, the von Mises entropy paradigm for image modeling does not hold any more.
Figures 3 to 6 show the behavior of the parameters and by averaging the values from the outcome of our algorithm for the 24 images in TID2008 database. A hybrid blur-noise situation has been experimented by adding blur and noise simultaneously to the images in TID2008 database (see Fig. 7 and 8.
Two main consequences are derived from the previous observations. First, the fitness parameter has a low value () for blurred images or very noisy images, and a high value () for undegraded images. The second consequence is that quality of the images is maximum when reaches a maximum in the series of diversely degraded images from a given original.
3 Application of VM distribution to image quality assessment
3.1 VM distribution and image quality assessment of contextual images
The behavior of the parameter of the von Mises distribution suggest that it can be considered as an image quality assessmente index, when dealing with Gaussian blur or Gausssian noise. In order to have a quantitative evaluation of this statement, we compare the classification after this parameter vs. the Mean Opinion Score (MOS) for two specified distortion types in database TID2008.
The image database TID2008 includes 17 different degradation types with one to four strength levels. The image database includes information about the MOS for each distorted image. The degraded images labeled as Gaussian noise, and Gaussian blur have been scored by means the value determined by our VM algorithm. Later on, Kendall, Spearman and Pearson correlation coefficients have been evaluated by comparing our scores with the MOS of the images in TID2008 image database. Results are shown in Table 1. Figures in Table 1 must be understood as measures in a contextual, specialized, no-reference and no-learning metric.
In order to put figures in Table 1 in line with other existing no-reference methods, we include Table 2, where our measures are compared to the results published by Zhu and Milanfar  using the same image database and the Spearman correlation coefficient.
Figures in Table 2 reveal that the parameter of VM distribution has a strong correlation with Zhu and Milanfar measures for Gaussian blur and Gaussian noise distortions.
3.2 Autofocusing example
A short depth of field is an inherent limitation in optical microscopy. Basically, the focal plane of the microscope needs to be critically located to observe the sharper image. Hence, auto-focusing techniques must be included in automatized optical microscope systems. Relevant examples of images from bright field microscopy may be found in Valdecasas .
The behavior of the parameter of the von Mises distribution suggest that it can be considered as an image quality assessmente index, when dealing with Gaussian blur or Gausssian noise. After the above quantitative evaluation of this parameter through the TID2008 database, we present a real life example to illustrate one of the possible applications of this no-reference image quality assessment method for contextual images. In this example, a sequence of 100 images from an optical microscope have been processed by our algorithm to determine their values. Images are pixels in size. Images 10, 30, 50 and 70 of the stack are shown in Fig. 9. The best image according to our algorithm is number 57 in the sequence, that can be seen in Fig. 10. Fig. 11 plots the values of for the whole sequence of images.
3.3 VM distribution and image quality assessment of no-contextual images
One of the objectives of this paper consists in determining whether the VM distribution can be used to construct a no-reference quality metric in the no-contextual case. The definition of a metric for image quality assessment based on the VM distribution for no-contextual images requires some kind of normalization, to assure independence of the measure from the image context. To do so, we have designed two possible measures taking into account the results in the previous section. Following these results, we postulate that both, the concentration parameter, , and the fitness parameter, , decrease exponentially with increasing amount of degradation (e.g.: Gaussian blur), that can be modelled by expressions like or , where is the degradation and is a shape factor exclusive for each image. Empirically, we have found convergence at the origin for to 0.88, although there is not a similar convergence for . The second of these expressions is simpler to calculate, provided that is an unknown parameter, while has been experimentally determined as (for Gaussian blur). The exponential variation of can be written in a logarithmic way as
The derivative of Eq. (11) gives , and can be approximated by
for small increments, .
Now, let us suppose that an arbitrary image produces an outcome parameter . By convolving the image with an appropriate kernel, , (i.e.: a PSF of pixels and ) a controlled amount of blur may be added to the image. Then, is a little more degraded image than . Supposing that the fitness parameter for is , and making for a given measuring scale, we can say, from Eq. (12), that and introducing this value in (11)
Eq. (13) defines in this way a new no-contextual measure for image quality assessment for the Gaussian blur case. The measure requires the estimation of the decay constant for each specific image as indicated in Eq. (12). This new measure will be referred here after as von Mises degradation measure (VMDM) and will be expressed in VM degradation units.
In order to evaluate the performance of the VMDM, we have applied our algorithm to the Gaussian blur images in the LIVE image database. This group is composed by 144 degraded images plus 29 originals. For our validation test, we have selected a group of 29 images (20 percent of the degraded subgroup), i.e.: from image number 117 to 145. Then we have correlated our quality scores with the DMOS scores of this group. Considering that the observations are positive quantities varying over many orders of magnitude, it is plausible to assume that the noise will be Gaussian, and the data will be well modeled as a Gaussian process. In this context, it is standard practice in the statistics literature to take the log of the data . Then, we have used instead of for correlation. Results for the Pearson and Spearman coefficients are shown in Table 3.
As expected, Table 3 presents good peformance results, but still far from a perfect matching with the DMOS scores. An objective measure must necessarily have some differences when compared with recordings from subjective human observers. Even more, DMOS values will change if the database is created using different methodologies and environmental settings; hence, subjective scores are not comparable , while the objective test will remain the same. However, the scores in Table 3 can be improved by applying some weighting transformation to the VMDM scores by incorporating to these scores some learning process that mimic the specific human preferences existing in the DMOS recordings accompanying the LIVE database. To do so, the logarithmic VMDM measures and the transformed VMDM measures may be related, for example, through a function  given by
Taking , the values for the coefficients have been determined by a Monte Carlo process that maximizes the Pearson+Spearman correlation with the DMOS scores. We have used images 1 to 116 (the 80 percent of the 145 blurred images) in the LIVE database as the learning set and we have used the former set of 29 images for crossvalidating the measures. Our results are shown in Table 4. The table includes also the results from Narvekar and Karam’s method for the same group of distortion.
Comparative results indicate that this method based on the parameter of the VM distribution is highly matching the DMOS scores for the Gaussian blur set of images in the LIVE database. Our results are comparable with the results given by Narvekar and Karam’s method which is based on a sharpness metric for Gaussian blurring when applied to no-contextual images. Our method has the advantage to be self-contained and its performance can be even improved by adding a learning step.
In this paper, we have introduced a new way of determining the VM distribution of the image information. The possible applications of the VM distribution for image quality assessment have been experimentally tested. The parameter of the VM distribution has experimentally shown that can be considered as a suitable no-reference quality indicator when dealing with contextual images. Significative results have been presented for the Gaussian blur and Gaussian noise cases. Also, the defined VM distribution fitness parameter has shown its suitability as a no-reference quality indicator in the no-contextual case for blur images. At the same time as the relationship of the VM distribution to image quality assessment has been shown in this paper, this new methodology suggest new applications of the VM distribution of image entropy for image processing, that will be the subject of future work.
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|Type of noise||Kendall||Pearson||Spearman|
|Method||Gaussian blur||Gaussian noise||Blur + noise|
|Zhu and Milanfar||1.0000||0.9760||0.9210|
|Narvekar and Karam||0.9211||0.9449|