Fuzzy Nonlinear Regression - Report Example

Student Name: Tutor: Subject: Date: Abstract Diverse picture ready applications, such as image/ video weight and picture entertainment rely heavily on the quality of a picture, more precisely, its quality concerning its perception by a human being. This is referred to as Subjective Image Quality Assessment (IQA). It gets identified as the indispensable component of picture quality assessment as high-quality images determined through objective methods, have been shown to correlate badly with ratings from human viewers. However, the lack of uniformity resulting from human judgement poses a significant discrepancy in the reliability and accuracy of the assessment. A need to develop accurate human perception models therefore exists. The approach of fuzzy regression methods using linear polynomials has various limitations. This research proposes the use of non- linear polynomials in the fuzzy regression models. This process is designed to provide more accurate results compared to the unequivocal showing procedure, in cases where (a) the number of individuals (viewers) are inadequate; (b) assumptions using linear regression is not applicable; (c) data collected contains vague characteristics; (d) variety fuzziness gets achieved by human judgement, and (e) the event in question is ambiguous [Arn05] and [Eng09]. Results would demonstrate that more fruitful data fitting and better hypothesis capacity can be achieved through fuzzy regression models. Introduction The number of digital pictures uploaded to the internet has risen exponentially over the past few years. In 2013, 208,300 images were uploaded to Facebook every minute, and 27,800 images were uploaded to Instagram every minute [Pop13]. This influx of photos can be attributed to various digital and social trends such as ‘The Selfie Craze’, as well as new mobile applications to capture and upload pictures, which actively encourage the uploading of digital images. Technological advancements have resulted in the invention of devices customized for capture, storage, compression, transmission and display of digital images, increasing the overall quality of the original image. The raw visual information (original image) passes through a series of steps in an imaging pipeline, each of which affects the quality of the picture. There is, therefore, need to develop prediction algorithms to evaluate automatically image quality and determine the effect of each of these steps on image quality. The creation of prediction models that accurately represent image quality as perceived by humans is very challenging. As humans are considered to be the observers and consumers of virtually all digital images, subjective image quality assessment (IQA) is commonly used as a ground truth to develop computational image quality prediction models [Eng09]. However, it is impossible to incorporate subjective IQA into the design and optimization of image processing algorithms to achieve enhanced image quality. There has been increasing interest, therefore, in correlating subjective IQA with quality assessment considered to be objective so as to predict or estimate perceived quality of the image. At the moment, there are no image quality prediction models that apply to a broad range of visual content and distortion types [Eng09]. Traditionally, image quality was determined by use of fidelity metrics. However, fidelity metrics has been shown to, more often, not correlate well with human perceived quality, and thus are not popular. There has been considerable research to devise other image quality estimators with higher correlation with human-perceived quality [Kee02]. Such perceived quality determiners are usually devised and ascertained based on outcomes of psychophysical experiments [SEM04]. The ratings by the research population are averaged into one parameter; Mean Opinion Scores (MOS), that are mostly accepted as a baseline for the design of image and video quality estimators. These scores are instrumental in the training and validation of computational image quality prediction models. Visual stimuli rating, however, is a laborious undertaking given that the quality is to be judged across various visual contents or distortions, further worsened in case there is mixing of various distortions or such discrepancies exhibit complex patterns [CTV08]. This results in a significant discrepancy in the scores of quality between the volunteers of the experiment, compromising the reliability of the MOS based on significant deviation in opinion among the observers [UEn11]. Historically, models for predicting subjective IQA have been developed based on neural networks. However, neural networks lack transparency as they are the black box in nature and the training time required is much longer compared with statistical regression when the network size is large. Fuzzy modelling- based approaches have also been applied to develop models for IQA, but more explicit information can be found in statistical regression models that are in polynomial form. Hence, variable significances and interactions can be determined by the polynomial of the regression models. Most of the time, people opt for statistical methods over fuzzy model base approaches or neural networks to generate an explicit model. Subjective image quality experiments are more inconsistent to changes over time because it involves human judgment, which may not perform accurately. The new image quality prediction model links objective IQA and subjective IQA. The main drawback with this method is it can function efficiently only within the range over which it is developed. Another constraint is that the experimental data should be normally distributed, which may not be possible every time. Another drawback is possibility of misconception due to the close relationship between subjective image quality measurements versus objective image quality metric [Bob]. To solve the above issues, the fuzzy regression method is used. The advantage of fuzzy regression method is that it can model the fuzziness in the subjective opinion scores of people. This method can be applied even on incomplete or small sets of data. Fuzzy regression method is done in three individual cases by varying fuzziness, data size, and some participants with MOS data sets. From the results in [Kit15], the fuzzy regression method outperforms the traditional statistical method. However, this method uses linear polynomials. So this method only captures the linear relationship. In this project, we propose to use nonlinear polynomials in the fuzzy regression method. Problem In this digital age, many applications demand accurate picture quality assessment methods, for instance, biomedical image and communication, to facilitate their proper functioning. In this fast growing world, we need an appropriate algorithm that can evaluate the picture quality automatically and inconsistent manner so as not to compromise on speed and efficiency. Not only that, these algorithms must be in agreement with human judgments and perception. Several IQA methods that have been proposed in last decade to produce quality results, some of which have succeeded. Subjective image quality assessment has for a while now been taken as the basic truth for IQA. Since, this method is directly related to human judgment, it is a bit time- consuming and expensive as well, besides being impractical in real world applications. Furthermore, many other factors, including lighting conditions, image viewing distance, human’s vision ability and the observer’s mood, are to be considered. Hence, an appropriate method is needed; that can average human observing to evaluate image quality. The primary objective of IQA is to develop a model that can estimate the quality of an image entirely and automatically for use in many applications in real time. Traditionally, statistical linear regression methods have been utilized in every field of engineering. The primary goal of the regression method is to express the changes of a dependent variable Y regarding the variation of the independent variable X. However, this basic model do not accurately represent the data. Fuzzy regression functions were developed to come up with models that could more accurately describe the image quality. The general form for these models is; The fuzzy components are assumed to be triangular fuzzy numbers [Arn05]. These models were deemed to be adequate. However, when the fuzzy polynomial only consists of the independent variables, the fuzziness estimated by the regression is only linear to the value of [Jin09]. This is the main limitation of the fuzzy regression. The resultant fuzzy regression model, therefore, bears some resemblance to the statistically linear model. Nonetheless, this linear characteristic may not always be true particularly for image quality assessment. The fuzziness would decrease with increasing magnitude of x. Therefore, we may need to introduce the term or lower terms to the equations, to achieve the characteristic of decreasing fuzziness with increasing x. in this early stage, we need to see the result of adding the term . We may consider adding other terms later. Method Subjective IQA model: 1. Statistical regression method As per the statistical regression model developed in the paper[Kit15], the author defined equations for collected mean opinion score of Ni images. Where, the error term consider normally distributed with zero mean. M+1 coefficient is βj and with, M is normalized objective IQA matrix for the following There are two primary drawbacks associated with this method (Kim et. al, 1996). In conventional MOS, the fuzziness of human judgments can be addressed, whereas, in statistical regression method, it is not possible. To come up with an estimate, the following assumptions should be made; 1. and relationship should be continue over the data ray. 2. Deviations of are normally distributed. 3. All are normally distributed on regression. 2. Fuzzy non- linear Regression Method Fuzzy regression method supersedes the statistical regression method in every aspect, such as a small error rate even with little data, with the most significant advantage being it considers the fuzziness of human judgment. Fuzzy non- linear regression models extend above fuzzy regression models by introducing non- linear polynomials. First, we plan to add the term to the fuzzy regression equation and observe the performance of the resulting non- linear method. Fuzzy non-linear regression can therefore be stated in a general equation as follows; The fuzzy coefficients are set as constants in a symmetric and triangular format, represented as whereandre the center and spread values of thefuzzy coefficientrespectively. Through basic fuzzy arithmetic, the center and spread values of can be calculated as; The term is introduced to reduce the fuzziness of the model with increasing values of. This, therefore, makes the above equation non- linear. According to [Tan82] and [Wat88], the centre and spread values can then get established by obtaining the solutions to the following analogous FnLR model; The assumption here is that the method used to solve the FLR model can be applied to the FnLR model to come up with accurate results. The above equations aim to reduce the fuzziness of the system Δ within the constraints that the membership degree of each crisp output should be greater than or equal to the interval of the fuzzy outputs. As a solution to the above equations, the optimal centre values of remain constant viz a vi the changes of the value of, denoted by ; according to [Mos93]. The resulting optimal coefficients of the fuzzy model through the FnLR system can be denoted with the setting as and the corresponding fuzzy outputs denoted as. It was proved that the optimal fuzzy coefficients and the related fuzzy outputs with respect to can be deduced through the corresponding results, with respect to as; (Mat lab) Forming the inequalities; The lower and upper bonds are given by; The general form of the equations is therefore, given as; Database Databases based on subjective IQA have an imperative role in developing and putting new image quality measures to test. There are several such databases used to establish the correlation with objective measures, but this study focuses on the ‘Video Communications Laboratory @ FER (VCL@FER)’ database. The database includes of 575 images, 4% of which are original images, having not undergone any degradation. All other images have experienced four different types of degradation, each divided into six quality levels of increasing magnitude, with the first level representing the mildest distortion while the sixth represents the most severe deterioration. The database contains the following degradations; i. Average White Gaussian Noise (AWGN) A sum of the original image and normally distributed pseudorandom numbers was used to calculate the AWGN degradation, at six different standard deviations for the degradation levels. This was done in Matlab. ii. Gaussian Blur It is calculated as filtering of an image with Gaussian function with different size pixels. Using the Irfanview software, six different sizes of Gaussian function were used to calculate blur degradation [Irf]. Without normalization, it can be expressed by; iii. JPEG2000 This was performed so that the final size was and finally bits per pixel, using ‘kdu_compress’.[htt] iv. JPEG Using Matlab, JPEG degradation was performed using six different qualities in the range . The subjective experiment got accomplished based on a group of 188 volunteers between the ages of 20 – 30, who were non- experts. Each subject had to grade about 96 images, but with no prior knowledge of the degradation that the picture had experienced. Each image got rated between 16 and 36 times. The method of Single Stimulus (SS), which uses a numeric criterion with 100 grades, was employed in the experiment. The external factors for perception were controlled by providing artificial lighting and the monitors calibrated appropriately. Experimental outcomes were collected, with mean ratings for each picture being obtained. The results from all observers on a single image are compared, and if the result from either of the observers differs significantly from the average, the result is discarded [ITU02]. In this test configuration, there was only one test condition. To make up for this, the single test configuration had one iteration and single window per mix of test conditions and linear alignment. It allowed for the second step to be discarded. There were 118 observers and 575 test images. The results were checked using kurtosis β to determine whether their distribution was normal or not. is defined as the 4th primary moment of the variable parameter Y. In this case described as; The process can be mathematically expressed as; For every observer, the values of P and Q got established and in case any value was greater than the number of tested images by 2% the observer got eliminated; In this case, 2 observers were eliminated. Recommendation[ITU02] proposes 0.2% that would lead to 55 observers being discarded. For a 1% ratio, 17 volunteers would be eliminated but correlation results between OIQA and SIQA measures would subsequently be lower. Results for every observer were later rescaled to the same full range of, using the equations; represents the grade that theviewer gave for the image (including reference images) represents rescaled grades of the same viewer represents all grades of subject Average Mean Opinion Score (MOS) finally got determined for each of the distorted images as an arithmetic mean of all grades for each image. Results The data can be illustrated using a line of best fit whose characteristics depend on the type of regression being used and the distribution of the data. The different methods of distortion used were; a. Average White Gaussian Noise (AWGN): Statistical regression does not accurately represent the various levels of AWGN distortion. This can be seen as the line of best fit does not cut the data cluster at each distortion level at the center, and in some cases (such as 0.4 and 0.7 distortion levels), the line of best fit is way off the mark. The below figure represents a statistical regression of 138 samples. The fuzzy linear regression graph does not accurately represent the data distribution, resulting in even larger error compared to statistical regression. The line of best fit entirely misses the data cluster at the 0.7 distortion level and is not close to the average value of MOS at the other distortion levels. The range between the data limits is also an important aspect, as it shows the data range represented by the line of best fit provided. This broad range indicates that the line doesn’t accurately represent the data presented in this case The below figure represents linear regression of 138 samples. The fuzzy non- linear regression graph presents a more accurate representation of the data distribution. The line of best fit is close to the average value of MOS at a majority of the distortion levels. The reduced range between the data limits also shows that the data is more concentrated (clustered together). However, it can be seen that the most of the data at the 0.2 distortion level is below the line of best fit. It shows a small error in this method. The below figure represents a non-linear regression of 138 samples. b. Gaussian Blur (BLU): The statistical regression method represents the Gaussian Blur distortion more accurately than the AWGN. The line of best fit cuts the data clusters close to the average MOS values, except at the 0.4 distortion level where it gets slightly skewed. However, the broad distribution of the data clusters at each distortion level could be a source of error. The below figure represents a statistical regression of 138 BLU samples. The fuzzy linear regression graph does not accurately represent the Gaussian Blur distortion. The line of best fit chosen crosses the data clusters close the average value of MOS at only one of the distortion levels, being significantly off the mark at the other distortion levels. The range between the data limits is also significantly large, showing that the line of best fit does not efficiently represent the data provided. The below figure represents linear regression of 138 BLU samples. The fuzzy non- linear regression method provides a more accurate representation of the Gaussian Blur distortion than the linear regression method. The line of best fit crosses the data clusters close to the average MOS value at all but two points (0.6 and 0.8 distortion levels). However, the range between the data limits means that the line of best fit represents a data distribution larger than that which is provided. The below figure represents a non-linear regression of 138 samples. c. JPEG2000 (J2K): The statistical linear regression graph provides a more or less accurate representation of the JPEG2000 distortion. The line of best fit crosses the data clusters close to the average value of MOS at each distortion level. The relatively wide distribution of the data clusters, however, could be construed as a slight cause of error in the method due to loss of some information. The below figure represents a statistical regression of 138 J2K samples. The fuzzy linear regression method provides a relatively accurate representation with the line of best fit passing close to the average value of MOS at each of the distortion levels. The range between the data limits is, however, significantly large, compromising the accuracy of the distribution. The below figure represents linear regression of 138 J2K samples. Fuzzy non- linear regression graph provides a relatively accurate representation of the distortion. The line of best-fit cuts through the data cluster at each distortion level at a point, more or less close to the average value of MOS. It also describes the changes in the slope of data with different levels of distortion. However, the range between the data limits means that the line of best fit represents a data distribution larger than that which is provided. The below figure represents non-linear regression of 138 J2K samples. d. JPG: Statistical linear regression provides an accurate representation of the JPG distortion as the line of best fit crosses the data clusters at points relatively close to the average value of MOS. There is, however, a relative amount of data skewed away from the line, which could compromise on the level of accuracy represented. The below figure represents a statistical regression of 138 JPG samples. The fuzzy linear regression provides a slightly less accurate representation of the distribution as the line of best fit does not cross the data clusters at a point closest to the average value of MOS for all levels of distortion (such as at 0.7 the distortion level). The small range between the data limits, however, increases the level of accuracy, as the data is more concentrated together. The below figure represents linear regression of 138 JPG samples. Fuzzy non- linear regression graph provides a relatively accurate representation of the distortion. The line of best-fit cuts through the data cluster at each distortion level at a point, more or less close to the average value of MOS. It also describes the changes in the slope of data with different levels of distortion. The small range between the data limits, however, increases the level of accuracy, as the data is more concentrated together. The below figure represents nonlinear regression of 138 JPG samples. Conclusion The following observations have been made from the results collected and analyzed; 1. Fuzzy non- linear regression, on average, provides a more relatively accurate representation of the data distribution, as compared to the other forms of regression 2. Fuzzy non- linear regression is more accurate at representing specific distortions than it is at others. The level of accuracy in the JPEG2000 and Gaussian Blur distortions was comparably lower than that in the AWGN and JPEG distortions 3. Fuzzy linear regression is accurate at representing some distortions, such as the JPEG2000 distortion, rivaling the fuzzy non- linear regression method 4. Statistical linear regression is accurate at describing some distortions, such as the Gaussian Blur, rivalling both the linear and non- linear fuzzy regression methods. Therefore, the fuzzy non- linear regression method provides a, more or less, accurate representation of data that is not specifically statistical. It is instrumental in describing data distributions with an aspect of human perception, which cannot be efficiently described using statistical methods. Future Work The research conducted was interested in finding the effect of adding the term to form a fuzzy regression equation with non- linear polynomials. The study found out that this equation provided more accurate representations of various image distortions over existing statistical methods, such as statistical linear regression and fuzzy linear regression. However, this form of non- linear polynomial has some limitations, resulting in inaccurate representations for some kinds of distortion. Researchers strive to achieve more predictable level of observer confidence for a wide range of observable image parameters. To achieve this, the use of finer grained and medium level image quality. U Engelke in his paper suggests the use of pilot test whose confidence or reaction times are measured to help identify such distorted images. Repetitive analysis of difficult to assess images could also be employed to improve the level of confidence. Automated assessment of observer confidence can be allowed by forecasted models based on quality ratings and the duration of reaction (recorded unobtrusively with no burden on the observers), neglecting the active recording of confidence scores. In this paper, still images with different kinds of distortions are focused on. It is crucial to ascertain whether the conclusions arrived at in this document also hold for videos with temporary discrepancies. Some of the distortions that can be studied are spatially complex packet decline distortion patterns and their effect on the confidence of the observer in quality rating. Work done in this research paper is taken to be only an initial step towards broader subjective IQA methods resulting in deeper insight into human behavior and preference. Future work that can be in this area involves the addition of lower levels of the term to the fuzzy non- linear polynomial, in order to further decrease the fuzziness of the distribution with increasing values of x. More research will be required for both internal and external contributory factors to observer confidence and thus understand the causes of observer disagreements. References Arn05: , (Shapiro), Eng09: , (Engelke U.), Pop13: , (PopPhoto), Kee02: , (Keelan), SEM04: , (S.E. Maxwell), CTV08: , (C.T. Vu), UEn11: , (U. Engelke), Bob: , (Boberg), Kit15: , (Kit Yan Chan), Jin09: , (Jingli Lu), Kit15: , (Kit Yan Chan), Tan82: , (Tanaka H.), Wat88: , (Watada J.), Mos93: , (Moskowitz H.), Irf: , (http://www.irfanview.com/), htt: , (http://www.kakadusoftware.com), ITU02: , (ITU-R BT.500- 11), ITU02: , (ITU-R BT.500- 11), Read More

There has been considerable research to devise other image quality estimators with higher correlation with human-perceived quality [Kee02]. Such perceived quality determiners are usually devised and ascertained based on outcomes of psychophysical experiments [SEM04]. The ratings by the research population are averaged into one parameter; Mean Opinion Scores (MOS), that are mostly accepted as a baseline for the design of image and video quality estimators. These scores are instrumental in the training and validation of computational image quality prediction models.

Visual stimuli rating, however, is a laborious undertaking given that the quality is to be judged across various visual contents or distortions, further worsened in case there is mixing of various distortions or such discrepancies exhibit complex patterns [CTV08]. This results in a significant discrepancy in the scores of quality between the volunteers of the experiment, compromising the reliability of the MOS based on significant deviation in opinion among the observers [UEn11]. Historically, models for predicting subjective IQA have been developed based on neural networks.

However, neural networks lack transparency as they are the black box in nature and the training time required is much longer compared with statistical regression when the network size is large. Fuzzy modelling- based approaches have also been applied to develop models for IQA, but more explicit information can be found in statistical regression models that are in polynomial form. Hence, variable significances and interactions can be determined by the polynomial of the regression models. Most of the time, people opt for statistical methods over fuzzy model base approaches or neural networks to generate an explicit model.

Subjective image quality experiments are more inconsistent to changes over time because it involves human judgment, which may not perform accurately. The new image quality prediction model links objective IQA and subjective IQA. The main drawback with this method is it can function efficiently only within the range over which it is developed. Another constraint is that the experimental data should be normally distributed, which may not be possible every time. Another drawback is possibility of misconception due to the close relationship between subjective image quality measurements versus objective image quality metric [Bob].

To solve the above issues, the fuzzy regression method is used. The advantage of fuzzy regression method is that it can model the fuzziness in the subjective opinion scores of people. This method can be applied even on incomplete or small sets of data. Fuzzy regression method is done in three individual cases by varying fuzziness, data size, and some participants with MOS data sets. From the results in [Kit15], the fuzzy regression method outperforms the traditional statistical method. However, this method uses linear polynomials.

So this method only captures the linear relationship. In this project, we propose to use nonlinear polynomials in the fuzzy regression method. Problem In this digital age, many applications demand accurate picture quality assessment methods, for instance, biomedical image and communication, to facilitate their proper functioning. In this fast growing world, we need an appropriate algorithm that can evaluate the picture quality automatically and inconsistent manner so as not to compromise on speed and efficiency.

Not only that, these algorithms must be in agreement with human judgments and perception. Several IQA methods that have been proposed in last decade to produce quality results, some of which have succeeded. Subjective image quality assessment has for a while now been taken as the basic truth for IQA. Since, this method is directly related to human judgment, it is a bit time- consuming and expensive as well, besides being impractical in real world applications. Furthermore, many other factors, including lighting conditions, image viewing distance, human’s vision ability and the observer’s mood, are to be considered.

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