CN110880192A - Image DCT coefficient distribution fitting method based on probability density function dictionary - Google Patents

Image DCT coefficient distribution fitting method based on probability density function dictionary Download PDF

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CN110880192A
CN110880192A CN201911069046.8A CN201911069046A CN110880192A CN 110880192 A CN110880192 A CN 110880192A CN 201911069046 A CN201911069046 A CN 201911069046A CN 110880192 A CN110880192 A CN 110880192A
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宋传鸣
葛明博
王相海
刘丹
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Liaoning Normal University
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Abstract

The invention discloses an image DCT coefficient distribution fitting method based on a probability density function dictionary, which comprises the steps of firstly establishing a probability density function dictionary by utilizing Cauchy distribution, Laplace distribution, logistic distribution, extremum distribution and Gaussian distribution; secondly, measuring the precision of each single distribution fitting DCT coefficient in the probability density function dictionary by adopting a Kullback-Leibler (K-L) measure, selecting two single distributions with optimal fitting degrees from the DCT coefficients to form a mixed distribution function, and respectively setting a self-adaptive weight for the two distributions; then, the real probability density of the DCT coefficient of the input image is counted by adopting a kernel density estimation method, and a linear over-determined equation set which takes the self-adaptive weight as an unknown number is established by utilizing the discrete sampling value of the DCT coefficient in each interval (bin); and finally, solving the linear overdetermined equation set to obtain the self-adaptive weight of the mixed distribution function.

Description

Image DCT coefficient distribution fitting method based on probability density function dictionary
Technical Field
The invention relates to the field of digital image processing, in particular to a probability density function dictionary-based image DCT coefficient distribution fitting method which is stable, efficient, strong in adaptability and high in fitting precision.
Background
Images are a description or portrayal of the human being's objective world, and are one of the most commonly used information carriers in human social activities. It has been shown that about 75-80% of the information obtained by humans is from the visual sense. Therefore, image analysis, recognition and understanding are core problems in the field of digital image processing, and have important theoretical and practical significance for information acquisition in the multimedia era. However, with the rapid development of image acquisition technologies and devices, the types of digital images are becoming more and more diversified, and there are not only natural images and computer-generated images (also referred to as "screen content images") that reflect visible light information, but also radar images, sonar images, X-ray images, ultrasonic images, remote sensing images, hyperspectral images, and the like that depict invisible light information. The digital images are different in light source, acquisition means, shooting objects and the like, and the distribution rules of pixels and sparse coefficients of the digital images are often obviously different. In this case, image statistics should be generated. Image statistics is one of the research bases of various processing such as image denoising, segmentation, compression, content classification, texture analysis, quality evaluation and the like, has important theoretical and application values for deepening the macroscopic cognitive degree of people on the image essence and improving the efficiency of digital image and video processing, and has been paid attention and paid attention by more and more researchers.
Research on image statistics dates back to the middle of the last 50 th century, so far, researchers concluded that digital images have two typical distribution rules, including scale invariance and non-gaussian. The former means that the power spectrum of the image follows power law distribution in a spatial domain, and the latter means that the image has the characteristics of high peak and thick tail in the spatial domain and a transformation domain. After researching the distribution rule of the forest image, Ruderman et al find that the local statistics (such as contrast, contrast gradient, local variance and the like) of the forest image show scale invariance and tail index which are not possessed by Gaussian distribution; further, they indicate that natural images are composed of a series of statistically independent small regions and provide a theoretical basis for the power-law distribution of images. Meanwhile, Luo et al uniformly model the spatial distribution and intensity distribution of wavelet transform coefficients by means of a Bayesian estimation framework and a Gibbs random field. Later, with the continuous and intensive research on Random structures of images, Markov Random Fields (MRFs) have been widely used in the Field of image modeling. By expanding the traditional MRF model, Roth and Black establish an expert domain (Fields of Experts, FOE) model which can be used for learning image prior; yousefi et al introduces a bilateral Markov grid random field model, which can overcome the difficulty of classical MRF model and the resulting model asymmetry; zhai et al propose an image context model based on the minimum description length principle, which has a great application potential in the aspects of lossless predictive coding and image denoising; the operation of Zoran and Weiss further reveals the advantages of Gaussian Mixture Models (GMMs) in image statistical modeling by fusing components such as covariance structures, contrast changes, and complex textures.
Considering that a large amount of data redundancy exists in the image in a spatial domain, orthogonal sparse transform such as Discrete Cosine Transform (DCT) can provide excellent nonlinear approximation for the image, effectively remove local correlation among pixels, and represent the image as a set of independent and uniformly distributed uncorrelated coefficients, which is more beneficial to the analysis of image statistical characteristics by people. As such, orthogonal sparse transforms are widely used in the field of image and video processing. Therefore, researchers have also conducted intensive research on the distribution of DCT transform coefficients in digital images. Pratt considers that the alternating current DCT coefficients of the image follow Gaussian distribution; reininger et al verified that the alternating current DCT coefficient of the image obeys Laplace distribution by adopting a Kolmogorov-Smirnov goodness-of-fit test method, and Lam et al also described the alternating current DCT coefficient as Laplace distribution and obtained wide acceptance; muller et al models DCT transform coefficients of an image through generalized Gaussian distribution, Joshi et al also considers that DCT alternating current coefficients of the image obey generalized Gaussian distribution of zero mean, and adopts a maximum likelihood estimation method to obtain a conclusion that shape parameters of the distribution are concentrated between 1 and 2, which indicates that the DCT alternating current coefficients do not obey neither Gaussian distribution nor Laplace distribution but are between Gaussian distribution and Laplace distribution; while Kang further models the DCT coefficient distribution of the image as a hybrid cauchy distribution. Although the above works all effectively verify that the DCT coefficient of the image has non-gaussian properties of "high peak, thick tail", Zoran and Weiss prove the scale invariance of the kurtosis of the response distribution of the edge filter (DCT basis), and propose the conclusion that the noise existing in the image can change the kurtosis through the scale, which shows that due to the complexity of the image acquisition process and the diversity of the image types, no matter single distribution such as laplacian distribution, generalized gaussian distribution, etc., or mixed distribution composed of some two fixed distributions, the statistical modeling of the DCT coefficient of the image has certain limitation, and the fitting accuracy of the statistical modeling of the DCT coefficient of the image still needs to be further improved.
Disclosure of Invention
The invention aims to solve the technical problems in the prior art and provides a method for fitting the DCT coefficient distribution of an image based on a probability density function dictionary, which is stable and efficient, strong in self-adaptability and high in fitting precision.
The technical solution of the invention is as follows: a probability density function dictionary-based image DCT coefficient distribution fitting method is characterized by comprising the following steps:
step 1, inputting an image I, dividing the image I into a series of pixel blocks with the size of BxB and without overlapping, and performing block DCT (discrete cosine transformation) to obtain a DCT coefficient set;
step 2, counting the distribution of the DCT transform coefficients of the image by adopting a kernel density estimation method to obtain the true probability density pdf of the DCT transform coefficients of the image II；
Step 3, establishing a summary composed of 5 probability density functionsRate Density function dictionary D (pdf)Cauchy,pdfLaplacian,pdfLogistic,pdfGaussian,pdfextreme) The pdf ofCauchyRepresenting the Cauchy probability density function, pdfLaplacianRepresenting the Laplace probability density function, pdfLogisticRepresenting the logistic probability density function, pdfGaussianRepresenting a Gaussian probability density function, pdfExtremeRepresenting an extreme probability density function;
step 3.1 establishing a parameter theta according to the definition of the formula (1)C、λCCauchy probability density function of (c):
Figure BDA0002260360360000031
step 3.2 establishing a parameter of α according to the definition of equation (2)L、βLLaplacian probability density function of:
Figure BDA0002260360360000032
step 3.3 according to the definition of formula (3) a parameter is established of αO、βOLogistic probability density function of (1):
Figure BDA0002260360360000033
step 3.4 establishing a parameter of α according to the definition of equation (4)E、βEThe extreme probability density function of:
Figure BDA0002260360360000034
step 3.5, according to the definition of the formula (5), establishing a Gaussian probability density function with the parameters of mu and sigma:
Figure BDA0002260360360000035
step 4. use the outlineFitting each distribution in the dictionary of rate density functions to the true probability density pdf of the DCT transform coefficients of the input image I, respectivelyI；
Step 4.1 utilizing the Cauchy probability density function pdfCauchyFitting DCT coefficient distribution of the image, and calculating pdf by adopting a maximum likelihood estimation methodCauchyParameter theta ofC、λC；
(a) Establishing a likelihood function of the Cauchy probability density function according to the definition of equation (6):
Figure BDA0002260360360000041
said xiPdf representing true probability densityIThe value of the ith interval (bin);
(b) the equation (7) is respectively applied to the parameter thetaCAnd λCThe partial derivatives are calculated and made equal to 0, and the resulting equations are combined into a system of equations, the definition of which is given by equation (7):
Figure BDA0002260360360000042
(c) solving the system of equations given in equation (7) to obtain pdfCauchyOf (2) optimal fitting parameters
Figure BDA0002260360360000043
And corresponding single Cauchy distribution fitting result
Figure BDA0002260360360000044
Step 4.2 Using Laplace probability Density function pdfLaplacianFitting DCT coefficient distribution of the image, and calculating pdf by adopting a maximum likelihood estimation methodLaplacianParameter αL、βL；
(a) Establishing a likelihood function of the laplacian probability density function according to the definition of equation (8):
Figure BDA0002260360360000045
(b) establishing a log-likelihood function of the laplacian probability density function according to the definition of equation (9):
Figure BDA0002260360360000046
(c) the formula (9) is respectively matched with the parameter αLAnd βLThe partial derivatives are calculated and made equal to 0, and the resulting equations are combined into a system of equations, the definition of which is given by equation (10):
Figure BDA0002260360360000047
(d) solving the system of equations given in equation (10) to obtain pdfLaplacianOf (2) optimal fitting parameters
Figure BDA0002260360360000048
And corresponding single Laplace distribution fitting result
Figure BDA0002260360360000049
Step 4.3 PDF Using logistic probability Density functionLogisticFitting DCT coefficient distribution of the image, and calculating pdf by adopting a maximum likelihood estimation methodLogisticParameter αO、βO；
(a) The likelihood function of the logistic probability density function is established according to the definition of equation (11):
Figure BDA0002260360360000051
(b) the formula (11) is respectively matched with the parameter αOAnd βOThe partial derivatives are calculated and made equal to 0, and the resulting equations are combined into a system of equations, the definition of which is given by equation (12):
Figure BDA0002260360360000052
(c) solving the system of equations given in equation (12) to obtain pdfLogisticOf (2) optimal fitting parameters
Figure BDA0002260360360000053
And the corresponding single logistic distribution fitting results
Figure BDA0002260360360000054
Step 4.4 use of the extreme probability density function pdfExtremeFitting DCT coefficient distribution of the image, and calculating pdf by adopting a maximum likelihood estimation methodExtremeParameter αE、βE；
(a) Establishing a likelihood function of the extreme probability density function according to the definition of equation (13):
Figure BDA0002260360360000055
(b) establishing a log-likelihood function of the extreme probability density function according to the definition of equation (14):
Figure BDA0002260360360000056
(c) the formula (14) is respectively matched with the parameter αEAnd βEThe partial derivatives are calculated and made equal to 0, and the resulting equations are combined into a system of equations, the definition of which is given by equation (15):
Figure BDA0002260360360000057
(d) solving the system of equations given in equation (15) to obtain pdfExtremeOf (2) optimal fitting parameters
Figure BDA0002260360360000058
And corresponding single extremum distribution fitting results
Figure BDA0002260360360000059
Step 4.5 using the Gaussian probability density function pdfGaussianFitting DCT coefficient distribution of the image, and calculating pdf by adopting a maximum likelihood estimation methodGaussianParameters μ, σ of (d);
(a) establishing a likelihood function of the gaussian probability density function according to the definition of equation (16):
Figure BDA0002260360360000061
(b) establishing a log-likelihood function of the gaussian probability density function according to the definition of equation (17):
Figure BDA0002260360360000062
(c) the parameters μ and σ are separately subjected to partial derivatives by equation (17) and the partial derivatives are made equal to 0, and the resulting equations are combined into a system of equations, the definition of which is given by equation (18):
Figure BDA0002260360360000063
(d) solving the system of equations given in equation (18) to obtain pdfGaussianOf (d) is the best fit parameter muopt、σoptAnd corresponding single Gaussian distribution fitting result pdfGaussian(x；μopt,σopt)；
And 5, respectively calculating probability density functions by utilizing Kullback-Leibler measure (KLD) according to the definitions of the formula (19) and the formula (20)
Figure BDA0002260360360000064
Figure BDA0002260360360000065
And pdfGaussian(x；μopt,σopt) Real probability density pdf of DCT transform coefficient of image I to be fittedIThe distance of (c):
Figure BDA0002260360360000066
Figure BDA0002260360360000067
said p is1(x) And p2(x) Representing two given probability density functions;
step 5.1 order
Figure BDA0002260360360000068
p2(x)＝pdfIAnd p is1(x) And p2(x) Substituting equation (19) and equation (20) into the calculation, fitting the pdf with a single Cauchy distributionIDistance d ofCauchy；
Step 5.2 order
Figure BDA0002260360360000069
p2(x)＝pdfIAnd p is1(x) And p2(x) Substituting equation (19) and equation (20) into the calculation, fit pdf with a single laplacian distributionIDistance d ofLaplacian；
Step 5.3 order
Figure BDA00022603603600000610
p2(x)＝pdfIAnd p is1(x) And p2(x) Substituting equation (19) and equation (20), calculate the pdf fit using a single logistic distributionIDistance d ofLogistic；
Step 5.4 order
Figure BDA00022603603600000611
p2(x)＝pdfIAnd p is1(x) And p2(x) Substituting equation (19) and equation (20) into the calculation, the pdf is fitted with a single extreme value distributionIDistance d ofExtreme；
Step 5.5 let p1(x)＝pdfGaussian(x；μopt,σopt)，p2(x)＝pdfIAnd p is1(x) And p2(x) Substituting equation (19) and equation (20) into the calculation, fit pdf with a single gaussian distributionIDistance d ofGaussian；
Step 6, fitting distance d in 5 single distributionsCauchy、dLaplacian、dLogistic、dExtremeAnd dGaussianIn (2), the 2 smallest distances are selected, and the single distributions corresponding to the 2 smallest distances are respectively made to be F1And F2；
Step 7. from F1And F2A mixture distribution F is composed, the definition of which probability density function is given by equation (21):
F(x)＝AF1(x)+BF2(x) (21)
the A, B represents the adaptive weight to be determined, and then the true probability density pdf is obtainedIOf each interval xi(i ∈ {1,2,3, …, n }) and its corresponding probability density pdfI(xi) Substituting equation (21) to obtain a linear overdetermined system of equations, the definition of which is given by equation (22):
Figure BDA0002260360360000071
and 8, converting the formula (22) into a matrix form according to the definition of the formula (23):
Figure BDA0002260360360000072
and solving the least squares solution of the linear overdetermined system of equations using left division to obtain the adaptive weights A, B for the mixed distribution F, which is defined by equation (24):
Figure BDA0002260360360000073
the "/" denotes the left divide operator;
step 9, substituting A, B into the formula (21) to obtain a probability density function of the mixed distribution F, and taking the probability density function as a fitting result of the DCT coefficient distribution of the input image I;
and 10, outputting a probability density function of the mixed distribution F.
Compared with the prior art, the method ensures the fitting precision of the image DCT transformation coefficient from three aspects: firstly, aiming at the non-Gaussian characteristic of image DCT (discrete cosine transform) coefficient distribution, the complexity of an image acquisition process and the diversity of image types, a probability density function dictionary is constructed by utilizing 5 types of thick tail distributions, so that the limitation of the traditional method in statistical modeling of the image DCT coefficient distribution by adopting single distribution or certain two fixed distributions is favorably broken through, and the distribution fitting precision is improved; secondly, the Kullback-Leibler measure is used as a standard for evaluating the fitting accuracy, so that the defect that the fitting accuracy of the Laplace distribution cannot be effectively evaluated by the traditional Kolmogorov-Smirnov goodness-of-fit test method can be overcome; thirdly, two single distributions which best accord with the DCT coefficient distribution characteristics of the input image are selected in a probability density function dictionary in a self-adaptive mode by utilizing the Kullback-Leibler measure, the self-adaptive weight of the two single distributions is determined by solving the least square solution of a linear overdetermined equation set, and then self-adaptive mixed thick tail distribution is formed, and richer thick tail distribution and probability density function forms thereof can be obtained theoretically, so that the category limit of the existing thick tail distribution is broken through, and greater flexibility and more degrees of freedom can be provided for DCT transformation coefficient fitting. Therefore, the method has the advantages of stability, high efficiency, strong adaptability and high fitting precision.
Drawings
FIG. 1 is a comparison of the DCT coefficients of images separately fitted using the present invention and a single distribution.
Table 1 shows the distance comparison results of fitting the DCT coefficients of the image separately using the present invention and a single distribution.
Detailed Description
A probability density function dictionary-based image DCT coefficient distribution fitting method is characterized by comprising the following steps:
step 1, inputting an image I, dividing the image I into a series of non-overlapping pixel blocks with the size of bxb, and performing block DCT transformation to obtain a DCT transformation coefficient set, where in this embodiment, B is set to 16;
step 2, counting the distribution of the DCT transform coefficients of the image by adopting a kernel density estimation method to obtain the true probability density pdf of the DCT transform coefficients of the image II；
Step 3, establishing a probability density function dictionary D (pdf) consisting of 5 probability density functionsCauchy,pdfLaplacian,pdfLogistic,pdfGaussian,pdfextreme) The pdf ofCauchyRepresenting the Cauchy probability density function, pdfLaplacianRepresenting the Laplace probability density function, pdfLogisticRepresenting the logistic probability density function, pdfGaussianRepresenting a Gaussian probability density function, pdfExtremeRepresenting an extreme probability density function;
step 3.1 establishing a parameter theta according to the definition of the formula (1)C、λCCauchy probability density function of (c):
Figure BDA0002260360360000091
step 3.2 establishing a parameter of α according to the definition of equation (2)L、βLLaplacian probability density function of:
Figure BDA0002260360360000092
step 3.3 according to the definition of formula (3) a parameter is established of αO、βOLogistic probability density function of (1):
Figure BDA0002260360360000093
step 3.4 establishing a parameter of α according to the definition of equation (4)E、βEThe extreme probability density function of:
Figure BDA0002260360360000094
step 3.5, according to the definition of the formula (5), establishing a Gaussian probability density function with the parameters of mu and sigma:
Figure BDA0002260360360000095
step 4, fitting the true probability density pdf of the DCT transformation coefficient of the input image I by using each distribution in the probability density function dictionaryI；
Step 4.1 utilizing the Cauchy probability density function pdfCauchyFitting DCT coefficient distribution of the image, and calculating pdf by adopting a maximum likelihood estimation methodCauchyParameter theta ofC、λC；
(a) Establishing a likelihood function of the Cauchy probability density function according to the definition of equation (6):
Figure BDA0002260360360000096
said xiPdf representing true probability densityIThe value of the ith interval (bin);
(b) the equation (7) is respectively applied to the parameter thetaCAnd λCThe partial derivatives are calculated and made equal to 0, and the resulting equations are combined into a system of equations, the definition of which is given by equation (7):
Figure BDA0002260360360000101
(c) solving the system of equations given in equation (7) to obtain pdfCauchyOf (2) optimal fitting parameters
Figure BDA0002260360360000102
And corresponding single Cauchy distribution fitting result
Figure BDA0002260360360000103
Step 4.2 Using Laplace probability Density function pdfLaplacianFitting image DCT coefficient distribution and using maximumLikelihood estimation method calculating pdfLaplacianParameter αL、βL；
(a) Establishing a likelihood function of the laplacian probability density function according to the definition of equation (8):
Figure BDA0002260360360000104
(b) establishing a log-likelihood function of the laplacian probability density function according to the definition of equation (9):
Figure BDA0002260360360000105
(c) the formula (9) is respectively matched with the parameter αLAnd βLThe partial derivatives are calculated and made equal to 0, and the resulting equations are combined into a system of equations, the definition of which is given by equation (10):
Figure BDA0002260360360000106
(d) solving the system of equations given in equation (10) to obtain pdfLaplacianOf (2) optimal fitting parameters
Figure BDA0002260360360000107
And corresponding single Laplace distribution fitting result
Figure BDA0002260360360000108
Step 4.3 PDF Using logistic probability Density functionLogisticFitting DCT coefficient distribution of the image, and calculating pdf by adopting a maximum likelihood estimation methodLogisticParameter αO、βO；
(a) The likelihood function of the logistic probability density function is established according to the definition of equation (11):
Figure BDA0002260360360000109
(b) the formula (11) is respectively matched with the parameter αOAnd βOThe partial derivatives are calculated and made equal to 0, and the resulting equations are combined into a system of equations, the definition of which is given by equation (12):
Figure BDA00022603603600001010
(c) solving the system of equations given in equation (12) to obtain pdfLogisticOf (2) optimal fitting parameters
Figure BDA00022603603600001011
And the corresponding single logistic distribution fitting results
Figure BDA0002260360360000111
Step 4.4 use of the extreme probability density function pdfExtremeFitting DCT coefficient distribution of the image, and calculating pdf by adopting a maximum likelihood estimation methodExtremeParameter αE、βE；
(a) Establishing a likelihood function of the extreme probability density function according to the definition of equation (13):
Figure BDA0002260360360000112
(b) establishing a log-likelihood function of the extreme probability density function according to the definition of equation (14):
Figure BDA0002260360360000113
(c) the formula (14) is respectively matched with the parameter αEAnd βEThe partial derivatives are calculated and made equal to 0, and the resulting equations are combined into a system of equations, the definition of which is given by equation (15):
Figure BDA0002260360360000114
(d) solving the system of equations given in equation (15) to obtainpdfExtremeOf (2) optimal fitting parameters
Figure BDA0002260360360000115
And corresponding single extremum distribution fitting results
Figure BDA0002260360360000116
Step 4.5 using the Gaussian probability density function pdfGaussianFitting DCT coefficient distribution of the image, and calculating pdf by adopting a maximum likelihood estimation methodGaussianParameters μ, σ of (d);
(a) establishing a likelihood function of the gaussian probability density function according to the definition of equation (16):
Figure BDA0002260360360000117
(b) establishing a log-likelihood function of the gaussian probability density function according to the definition of equation (17):
Figure BDA0002260360360000118
(c) the parameters μ and σ are separately subjected to partial derivatives by equation (17) and the partial derivatives are made equal to 0, and the resulting equations are combined into a system of equations, the definition of which is given by equation (18):
Figure BDA0002260360360000119
(d) solving the system of equations given in equation (18) to obtain pdfGaussianOf (d) is the best fit parameter muopt、σoptAnd corresponding single Gaussian distribution fitting result pdfGaussian(x；μopt,σopt)；
And 5, respectively calculating probability density functions by utilizing Kullback-Leibler measure (KLD) according to the definitions of the formula (19) and the formula (20)
Figure BDA0002260360360000121
Figure BDA0002260360360000122
And pdfGaussian(x；μopt,σopt) Real probability density pdf of DCT transform coefficient of image I to be fittedIThe distance of (c):
Figure BDA0002260360360000123
Figure BDA0002260360360000124
said p is1(x) And p2(x) Representing two given probability density functions;
step 5.1 order
Figure BDA0002260360360000125
p2(x)＝pdfIAnd p is1(x) And p2(x) Substituting equation (19) and equation (20) into the calculation, fitting the pdf with a single Cauchy distributionIDistance d ofCauchy；
Step 5.2 order
Figure BDA0002260360360000126
p2(x)＝pdfIAnd p is1(x) And p2(x) Substituting equation (19) and equation (20) into the calculation, fit pdf with a single laplacian distributionIDistance d ofLaplacian；
Step 5.3 order
Figure BDA0002260360360000127
p2(x)＝pdfIAnd p is1(x) And p2(x) Substituting equation (19) and equation (20), calculate the pdf fit using a single logistic distributionIDistance d ofLogistic；
Step 5.4 order
Figure BDA0002260360360000128
p2(x)＝pdfIAnd p is1(x) And p2(x) Substituting equation (19) and equation (20) into the calculation, the pdf is fitted with a single extreme value distributionIDistance d ofExtreme；
Step 5.5 let p1(x)＝pdfGaussian(x；μopt,σopt)，p2(x)＝pdfIAnd p is1(x) And p2(x) Substituting equation (19) and equation (20) into the calculation, fit pdf with a single gaussian distributionIDistance d ofGaussian；
Step 6, fitting distance d in 5 single distributionsCauchy、dLaplacian、dLogistic、dExtremeAnd dGaussianIn (2), the 2 smallest distances are selected, and the single distributions corresponding to the 2 smallest distances are respectively made to be F1And F2；
Step 7. from F1And F2A mixture distribution F is composed, the definition of which probability density function is given by equation (21):
F(x)＝AF1(x)+BF2(x) (21)
the A, B represents the adaptive weight to be determined, and then the true probability density pdf is obtainedIOf each interval xi(i ∈ {1,2,3, …, n }) and its corresponding probability density pdfI(xi) Substituting equation (21) to obtain a linear overdetermined system of equations, the definition of which is given by equation (22):
Figure BDA0002260360360000131
and 8, converting the formula (22) into a matrix form according to the definition of the formula (23):
Figure BDA0002260360360000132
and solving the least squares solution of the linear overdetermined system of equations using left division to obtain the adaptive weights A, B for the mixed distribution F, which is defined by equation (24):
Figure BDA0002260360360000133
the "/" denotes the left divide operator;
step 9, substituting A, B into the formula (21) to obtain a probability density function of the mixed distribution F, and taking the probability density function as a fitting result of the DCT coefficient distribution of the input image I;
and 10, outputting a probability density function of the mixed distribution F.
The present invention is used to fit pairs of image DCT coefficients separately to a single distribution as shown in figure 1. Wherein, (a) is an original image; (b) real probability density of DCT coefficient of the original image; (c) local amplification results at the peak portion when a single Cauchy distribution is used to fit the probability density distribution of the image DCT coefficients; (d) local amplification results at the tail when a single Laplace distribution is used to fit the probability density distribution of image DCT coefficients; (e) fitting the results of the probability density distribution of image DCT coefficients for a single logistic distribution; (f) fitting the result of the probability density distribution of the image DCT coefficients using a single extreme value distribution; (g) fitting the result of the probability density distribution of the image DCT coefficients using a single Gaussian distribution; (h) is the result of fitting the probability density distribution of the image DCT coefficients using the present invention. As can be seen from fig. 1, the DCT coefficients of the original image basically exhibit the distribution characteristics of "high peak, long tail" (as shown in fig. (b)), which reflects the non-gaussian characteristics of the DCT coefficient distribution of the image; the single cauchy distribution of graph (c) does not fit well to the peak portion of the probability density distribution of the DCT coefficients of the image, with the peak being significantly higher than the true distribution of the original image; the single laplacian distribution of graph (d) does not fit exactly the tail of the DCT coefficient probability density distribution of the image; the single logistic distribution of plot (e), neither in the peak nor in the tail, has an ideal fitting accuracy; the fit accuracy of the single extremum distribution of plot (f) and the single gaussian distribution of plot (g) is lower; in contrast, the fit result of graph (h) is closest to the true probability density distribution of the original image DCT coefficients.
The distance comparison table for fitting the image DCT coefficients respectively by using the invention and the single distribution is shown in the following table.
Figure BDA0002260360360000141
As can be seen from FIG. 1 and Table 1, the subjective quality and objective quality of the DCT coefficients of the images fitted by the present invention are both higher than the fitting accuracy of a single distribution.

Claims (1)

1. A method for fitting image DCT coefficient distribution based on a probability density function dictionary is characterized by comprising the following steps:
step 1, inputting an image I, dividing the image I into a series of pixel blocks with the size of BxB and without overlapping, and performing block DCT (discrete cosine transformation) to obtain a DCT coefficient set;
step 2, counting the distribution of the DCT transform coefficients of the image by adopting a kernel density estimation method to obtain the true probability density pdf of the DCT transform coefficients of the image II；
Step 3, establishing a probability density function dictionary D (pdf) consisting of 5 probability density functionsCauchy,pdfLaplacian,pdfLogistic,pdfGaussian,pdfextreme) The pdf ofCauchyRepresenting the Cauchy probability density function, pdfLaplacianRepresenting the Laplace probability density function, pdfLogisticRepresenting the logistic probability density function, pdfGaussianRepresenting a Gaussian probability density function, pdfExtremeRepresenting an extreme probability density function;
step 3.1 establishing a parameter theta according to the definition of the formula (1)C、λCCauchy probability density function of (c):
Figure FDA0002260360350000011
step 3.2 establishing a parameter of α according to the definition of equation (2)L、βLLaplacian probability density function of:
Figure FDA0002260360350000012
step 3.3 according to the definition of formula (3) a parameter is established of αO、βOLogistic probability density function of (1):
Figure FDA0002260360350000013
step 3.4 establishing a parameter of α according to the definition of equation (4)E、βEThe extreme probability density function of:
Figure FDA0002260360350000014
step 3.5, according to the definition of the formula (5), establishing a Gaussian probability density function with the parameters of mu and sigma:
Figure FDA0002260360350000021
step 4, fitting the true probability density pdf of the DCT transformation coefficient of the input image I by using each distribution in the probability density function dictionaryI；
Step 4.1 utilizing the Cauchy probability density function pdfCauchyFitting DCT coefficient distribution of the image, and calculating pdf by adopting a maximum likelihood estimation methodCauchyParameter theta ofC、λC；
(a) Establishing a likelihood function of the Cauchy probability density function according to the definition of equation (6):
Figure FDA0002260360350000022
said xiPdf representing true probability densityIThe value of the ith interval (bin);
(b) the equation (7) is respectively applied to the parameter thetaCAnd λCThe partial derivatives are calculated and made equal to 0, and the resulting equations are combined into a system of equations, the definition of which is given by equation (7):
Figure FDA0002260360350000023
(c) solving the system of equations given in equation (7) to obtain pdfCauchyOf (2) optimal fitting parameters
Figure FDA0002260360350000024
And corresponding single Cauchy distribution fitting result
Figure FDA0002260360350000025
Step 4.2 Using Laplace probability Density function pdfLaplacianFitting DCT coefficient distribution of the image, and calculating pdf by adopting a maximum likelihood estimation methodLaplacianParameter αL、βL；
(a) Establishing a likelihood function of the laplacian probability density function according to the definition of equation (8):
Figure FDA0002260360350000026
(b) establishing a log-likelihood function of the laplacian probability density function according to the definition of equation (9):
Figure FDA0002260360350000027
(c) the formula (9) is respectively matched with the parameter αLAnd βLThe partial derivatives are calculated and made equal to 0, and the resulting equations are combined into a system of equations, the definition of which is given by equation (10):
Figure FDA0002260360350000031
(d) solving the system of equations given in equation (10) to obtain pdfLaplacianOf (2) optimal fitting parameters
Figure FDA0002260360350000032
And corresponding single Laplace distribution fitting result
Figure FDA0002260360350000033
Step 4.3 PDF Using logistic probability Density functionLogisticFitting DCT coefficient distribution of the image, and calculating pdf by adopting a maximum likelihood estimation methodLogisticParameter αO、βO；
(a) The likelihood function of the logistic probability density function is established according to the definition of equation (11):
Figure FDA0002260360350000034
(b) the formula (11) is respectively matched with the parameter αOAnd βOThe partial derivatives are calculated and made equal to 0, and the resulting equations are combined into a system of equations, the definition of which is given by equation (12):
Figure FDA0002260360350000035
(c) solving the system of equations given in equation (12) to obtain pdfLogisticOf (2) optimal fitting parameters
Figure FDA0002260360350000036
And the corresponding single logistic distribution fitting results
Figure FDA0002260360350000037
Step 4.4 use of the extreme probability density function pdfExtremeFitting DCT coefficient distribution of the image, and calculating pdf by adopting a maximum likelihood estimation methodExtremeParameter αE、βE；
(a) Establishing a likelihood function of the extreme probability density function according to the definition of equation (13):
Figure FDA0002260360350000038
(b) establishing a log-likelihood function of the extreme probability density function according to the definition of equation (14):
Figure FDA0002260360350000039
(c) the formula (14) is respectively matched with the parameter αEAnd βEThe partial derivatives are calculated and made equal to 0, and the resulting equations are combined into a system of equations, the definition of which is given by equation (15):
Figure FDA00022603603500000310
(d) solving the system of equations given in equation (15) to obtain pdfExtremeOf (2) optimal fitting parameters
Figure FDA0002260360350000041
And corresponding single extremum distribution fitting results
Figure FDA0002260360350000042
Step 4.5 using the Gaussian probability density function pdfGaussianFitting DCT coefficient distribution of the image, and calculating pdf by adopting a maximum likelihood estimation methodGaussianParameters μ, σ of (d);
(a) establishing a likelihood function of the gaussian probability density function according to the definition of equation (16):
Figure FDA0002260360350000043
(b) establishing a log-likelihood function of the gaussian probability density function according to the definition of equation (17):
Figure FDA0002260360350000044
(c) the parameters μ and σ are separately subjected to partial derivatives by equation (17) and the partial derivatives are made equal to 0, and the resulting equations are combined into a system of equations, the definition of which is given by equation (18):
Figure FDA0002260360350000045
(d) solving the system of equations given in equation (18) to obtain pdfGaussianOf (d) is the best fit parameter muopt、σoptAnd corresponding single Gaussian distribution fitting result pdfGaussian(x；μopt,σopt)；
And 5, respectively calculating probability density functions by utilizing Kullback-Leibler measure (KLD) according to the definitions of the formula (19) and the formula (20)
Figure FDA0002260360350000046
Figure FDA0002260360350000047
And
Figure FDA0002260360350000048
real probability density pdf of DCT transform coefficient of image I to be fittedIThe distance of (c):
Figure FDA0002260360350000049
Figure FDA00022603603500000410
said p is1(x) And p2(x) Representing two given probability density functions;
step 5.1 order
Figure FDA00022603603500000411
p2(x)＝pdfIAnd p is1(x) And p2(x) Substituting into formula (19) and formula (20), the calculation utilizes a singleOne Cauchy distribution fitting pdfIDistance d ofCauchy；
Step 5.2 order
Figure FDA00022603603500000412
p2(x)＝pdfIAnd p is1(x) And p2(x) Substituting equation (19) and equation (20) into the calculation, fit pdf with a single laplacian distributionIDistance d ofLaplacian；
Step 5.3 order
Figure FDA0002260360350000051
p2(x)＝pdfIAnd p is1(x) And p2(x) Substituting equation (19) and equation (20), calculate the pdf fit using a single logistic distributionIDistance d ofLogistic；
Step 5.4 order
Figure FDA0002260360350000052
p2(x)＝pdfIAnd p is1(x) And p2(x) Substituting equation (19) and equation (20) into the calculation, the pdf is fitted with a single extreme value distributionIDistance d ofExtreme；
Step 5.5 let p1(x)＝pdfGaussian(x；μopt,σopt)，p2(x)＝pdfIAnd p is1(x) And p2(x) Substituting equation (19) and equation (20) into the calculation, fit pdf with a single gaussian distributionIDistance d ofGaussian；
Step 6, fitting distance d in 5 single distributionsCauchy、dLaplacian、dLogistic、dExtremeAnd dGaussianIn (2), the 2 smallest distances are selected, and the single distributions corresponding to the 2 smallest distances are respectively made to be F1And F2；
Step 7. from F1And F2A mixture distribution F is composed, the definition of which probability density function is given by equation (21):
F(x)＝AF1(x)+BF2(x) (21)
the A, B represents the adaptive weight to be determined, and then the true probability density pdf is obtainedIOf each interval xi(i ∈ {1,2,3, …, n }) and its corresponding probability density pdfI(xi) Substituting equation (21) to obtain a linear overdetermined system of equations, the definition of which is given by equation (22):
Figure FDA0002260360350000053
and 8, converting the formula (22) into a matrix form according to the definition of the formula (23):
Figure FDA0002260360350000054
and solving the least squares solution of the linear overdetermined system of equations using left division to obtain the adaptive weights A, B for the mixed distribution F, which is defined by equation (24):
Figure FDA0002260360350000061
the "/" denotes the left divide operator;
step 9, substituting A, B into the formula (21) to obtain a probability density function of the mixed distribution F, and taking the probability density function as a fitting result of the DCT coefficient distribution of the input image I;
and 10, outputting a probability density function of the mixed distribution F.
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