CN110880192A - Image DCT Coefficient Distribution Fitting Method Based on Probability Density Function Dictionary - Google Patents

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 an image DCT coefficient distribution fitting method based on a probability density function dictionary, which is stable, efficient, strong in adaptability and high in fitting accuracy.

Background technique

Image is a description or portrait of the objective world by human beings, and it is one of the most commonly used information carriers in human social activities. Some studies have pointed out that about 75-80% of the information obtained by humans comes from vision. Therefore, image analysis, recognition and understanding are the core issues in the field of digital image processing, and have important theoretical and practical significance for information acquisition in the multimedia age. However, with the rapid development of image acquisition technology and equipment, the types of digital images are becoming more and more diverse, including natural images reflecting visible light information, computer-synthesized images (also known as "screen content images"), and characterizations. Radar images, sonar images, X-ray images, ultrasound images, remote sensing images, hyperspectral images, etc. of invisible light information. These digital images are different in terms of light sources, acquisition methods, and shooting objects, and there are often significant differences in the distribution laws of pixels and sparse coefficients. In this case, image statistics came into being. Image statistics is one of the research bases for various processing such as image denoising, segmentation, compression, content classification, texture analysis, and quality evaluation. Efficiency has important theoretical and applied value, and has been paid more and more attention by researchers.

The research on image statistics can be traced back to the mid-1950s. So far, researchers have concluded that digital images have two typical distribution laws, including scale invariance and non-Gaussian. Among them, the former means that the power spectrum of the image obeys a power-law distribution in the spatial domain, and the latter means that the image exhibits the characteristics of "high peaks and thick tails" in both the spatial and transform domains. After studying the distribution law of forest images, Ruderman et al. found that their local statistics (such as contrast, contrast gradient, local variance, etc.) showed scale invariance and tail exponential that Gaussian distribution did not have; further, they pointed out Natural images consist 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 modeled the spatial distribution and intensity distribution of wavelet transform coefficients with the help of Bayesian estimation framework and Gibbs random field. Later, with the deepening of the research on the random structure of images, the Markov Random Field (MRF) was widely used in the field of image modeling. By extending the traditional MRF model, Roth and Black established a Fields of Experts (FOE) model that can be used to learn image priors; Yousefi et al. introduced a bilateral Markov grid random field model that can overcome The intractability of the classical MRF model and the resulting model asymmetry; Zhai et al. proposed an image context model based on the principle of minimum description length, which showed great application potential in lossless predictive coding and image denoising The work of Zoran and Weiss further reveals the advantages of Gaussian Mixture Model (GMM) in image statistical modeling by fusing components such as covariance structure, contrast variation, and complex texture.

Considering that there is a lot of data redundancy in the image in the spatial domain, orthogonal sparse transforms such as Discrete Cosine Transform (DCT) can provide an excellent nonlinear approximation for the image, effectively remove the local correlation between pixels, and represent the image as The set of independent and identically distributed uncorrelated coefficients is more conducive to people's analysis of the statistical characteristics of images. Therefore, orthogonal sparse transform is widely used in image and video processing fields. Therefore, researchers have also carried out in-depth research on the distribution of DCT transform coefficients of digital images. Pratt believed that the AC DCT coefficient of the image obeyed the Gaussian distribution; Reininger et al. used the Kolmogorov-Smirnov goodness-of-fit test method to verify that the AC DCT coefficient of the image obeyed the Laplace distribution, and Lam et al. also characterized it as Laplace distribution and has been widely recognized; Muller et al. modeled the DCT transform coefficient of the image through the generalized Gaussian distribution, and Joshi et al. also believed that the DCT communication coefficient of the image obeyed the generalized Gaussian distribution with zero mean, and used the maximum likelihood estimation method. It is concluded that the shape parameters of the distribution are concentrated between 1 and 2, which means that the DCT AC coefficient does not obey the Gaussian distribution nor the Laplace distribution, but is between the Gaussian distribution and the Laplace distribution. while Kang further models the DCT coefficient distribution of the image as a mixed Cauchy distribution. Although the above works have effectively verified that the DCT coefficients of the image are non-Gaussian with "high peaks and thick tails", Zoran and Weiss proved the scale invariance of the kurtosis of the response distribution of the edge filter (DCT basis), and proposed a The conclusion that the noise existing in the image will change the kurtosis through the scale shows that due to the complexity of the image acquisition process and the diversity of image types, whether it is a single distribution such as Laplace distribution, generalized Gaussian distribution, or two fixed distributions. The mixed distribution composed of distributions has certain limitations in the statistical modeling of image DCT coefficients, and its fitting accuracy still needs to be further improved.

SUMMARY OF THE INVENTION

The present invention aims to solve the above-mentioned technical problems existing in the prior art, and provides a method for fitting the distribution of image DCT coefficients based on a probability density function dictionary, which is stable and efficient, has strong adaptability and high fitting accuracy.

The technical solution of the present invention is: an image DCT coefficient distribution fitting method based on a probability density function dictionary, which is characterized in that it is performed according to the following steps:

Step 1. Input an image I, divide it into a series of pixel blocks whose size is B×B and do not overlap, and perform block DCT transformation to obtain a set of DCT transformation coefficients;

Step 2. adopt the kernel density estimation method to count the distribution of image DCT transform coefficients, obtain the true probability density pdf _{I of the DCT transform coefficients of image I} ;

Step 3. Establish a probability density function dictionary D (pdf _Cauchy , pdf _Laplacian , pdf _Logistic , pdf _Gaussian , pdf _extreme ) composed of 5 probability density functions, the pdf _Cauchy represents the Cauchy probability density function, and pdf _Laplacian represents the Laplacian Plath probability density function, pdf _Logistic represents the logistic probability density function, pdf _Gaussian represents the Gaussian probability density function, and pdf _Extreme represents the extreme value probability density function;

Step 3.1 According to the definition of formula (1), establish the Cauchy probability density function with parameters θ ^C and λ ^C :

Step 3.2 According to the definition of formula (2), establish the Laplace probability density function with parameters α ^L and β ^L :

Step 3.3 According to the definition of formula (3), establish a logistic probability density function with parameters α ^O and β ^O :

Step 3.4 According to the definition of formula (4), establish the extreme value probability density function with parameters α ^E , β ^E :

Step 3.5 According to the definition of formula (5), establish a Gaussian probability density function with parameters μ and σ:

Step 4. Use each distribution in the probability density function dictionary to fit the true probability density pdf _I of the DCT transform coefficients of the input image I respectively;

Step 4.1 Use the Cauchy probability density function pdf _Cauchy to fit the image DCT coefficient distribution, and use the maximum likelihood estimation method to calculate the parameters θ ^C and λ ^C of the pdf _Cauchy ;

(a) According to the definition of formula (6), establish the likelihood function of the Cauchy probability density function:

The _xi represents the value of the ith interval (bin) of the true probability density pdf _I ;

(b) Calculate the partial derivatives of the parameters θ ^C and λ ^C by formula (7) respectively, and make their partial derivatives equal to 0, and then combine the obtained equations into a system of equations, the definition of which is given by formula (7):

以及相应的单一柯西分布拟合结果

(c) Solve the system of equations given by equation (7) to obtain the optimal fitting parameters of pdf _Cauchy

and the corresponding fitting results of a single Cauchy distribution

Step 4.2 uses the Laplacian probability density function pdf _Laplacian to fit the image DCT coefficient distribution, and adopts the maximum likelihood estimation method to calculate the parameters α ^L and β ^L of the pdf _Laplacian ;

(a) According to the definition of formula (8), establish the likelihood function of the Laplace probability density function:

(b) According to the definition of formula (9), establish the log-likelihood function of the Laplace probability density function:

(c) Calculate the partial derivatives of the parameters α ^L and β ^L by formula (9) respectively, and make their partial derivatives equal to 0, and then combine the obtained equations into a system of equations, the definition of which is given by formula (10):

以及相应的单一拉普拉斯分布拟合结果

(d) Solve the system of equations given by equation (10) to obtain the optimal fitting parameters of pdf _Laplacian

and the corresponding fitting results of a single Laplace distribution

Step 4.3 utilizes the logistic probability density function pdf _Logistic to fit the image DCT coefficient distribution, and adopts the maximum likelihood estimation method to calculate the parameters α ^O and β ^O of pdf _Logistic ;

(a) According to the definition of formula (11), establish the likelihood function of the logistic probability density function:

(b) Calculate the partial derivatives of the parameters α ^O and β ^O by formula (11) respectively, and set their partial derivatives equal to 0, and then combine the obtained equations into a system of equations, the definition of which is given by formula (12):

以及相应的单一逻辑斯谛分布拟合结果

(c) Solve the equation system given by formula (12) to obtain the optimal fitting parameters of pdf _Logistic

and the corresponding fitting result of a single logistic distribution

Step 4.4 utilizes extreme value probability density function pdf _Extreme to fit the image DCT coefficient distribution, and adopts the maximum likelihood estimation method to calculate the parameters α ^E and β ^E of pdf _Extreme ;

(a) According to the definition of formula (13), establish the likelihood function of extreme value probability density function:

(b) According to the definition of formula (14), establish the log-likelihood function of the extreme probability density function:

(c) Calculate the partial derivatives of the parameters α ^E and β ^E by formula (14) respectively, and make their partial derivatives equal to 0, and then combine the obtained equations into a system of equations, the definition of which is given by formula (15):

以及相应的单一极值分布拟合结果

(d) Solve the system of equations given by equation (15) to obtain the optimal fitting parameters of pdf _Extreme

and the corresponding single extreme value distribution fitting results

Step 4.5 Use the Gaussian probability density function pdf _Gaussian to fit the image DCT coefficient distribution, and use the maximum likelihood estimation method to calculate the parameters μ and σ of the pdf _Gaussian ;

(a) According to the definition of formula (16), establish the likelihood function of the Gaussian probability density function:

(b) According to the definition of formula (17), establish the log-likelihood function of the Gaussian probability density function:

(c) Calculate the partial derivatives of the parameters μ and σ by formula (17) respectively, and make their partial derivatives equal to 0, and then combine the obtained equations into a system of equations, the definition of which is given by formula (18):

(d) Solving the equation system given by formula (18) to obtain the optimal fitting parameters μ _opt , σ _opt of pdf _Gaussian and the corresponding fitting result of single Gaussian distribution pdf _Gaussian (x; μ _opt ,σ _opt );

和pdf_Gaussian(x；μ_opt,σ_opt)与待拟合图像I的DCT变换系数真实概率密度pdf_I的距离：Step 5. According to the definition of formula (19) and formula (20), use the Kullback-Leibler measure (KLD) to calculate the probability density function respectively

and the distance between pdf _Gaussian (x; μ _opt, σ _opt ) and the true probability density pdf _I of the DCT transform coefficients of the image I to be fitted:

The p ₁ (x) and p ₂ (x) represent two given probability density functions;

p₂(x)＝pdf_I，并将p₁(x)和p₂(x)代入公式(19)和公式(20)，计算利用单一柯西分布拟合pdf_I的距离d_Cauchy；Step 5.1 Order

p ₂ (x)=pdf _I , and substituting p ₁ (x) and p ₂ (x) into formula (19) and formula (20), calculate the distance d _Cauchy fitting pdf _I with a single Cauchy distribution;

p₂(x)＝pdf_I，并将p₁(x)和p₂(x)代入公式(19)和公式(20)，计算利用单一拉普拉斯分布拟合pdf_I的距离d_Laplacian；Step 5.2 Order

p ₂ (x)=pdf _I , and substituting p ₁ (x) and p ₂ (x) into formula (19) and formula (20), calculate the distance d _Laplacian fitting pdf _I with a single Laplace distribution;

p₂(x)＝pdf_I，并将p₁(x)和p₂(x)代入公式(19)和公式(20)，计算利用单一逻辑斯谛分布拟合pdf_I的距离d_Logistic；Step 5.3 Order

p ₂ (x)=pdf _I , and substituting p ₁ (x) and p ₂ (x) into formula (19) and formula (20), calculate the distance d _Logistic fitting pdf _I with a single logistic distribution;

p₂(x)＝pdf_I，并将p₁(x)和p₂(x)代入公式(19)和公式(20)，计算利用单一极值分布拟合pdf_I的距离d_Extreme；Step 5.4 Order

p ₂ (x)=pdf _I , and substituting p ₁ (x) and p ₂ (x) into formula (19) and formula (20), calculate the distance d _Extreme that fits pdf _I with a single extreme value distribution;

Step 5.5 Let p ₁ (x)=pdf _Gaussian (x; μ _opt ,σ _opt ), p ₂ (x)=pdf _I , and substitute p ₁ (x) and p ₂ (x) into equation (19) and equation (20), calculate the distance d _Gaussian that utilizes a single Gaussian distribution to fit pdf _I ;

Step 6. Among the fitting distances d _Cauchy , d _Laplacian , d _Logistic , d _Extreme and d _Gaussian of the five single distributions, select the two smallest distances, and make the single distribution corresponding to the two smallest distances respectively. are F ₁ and F ₂ ;

Step 7. The mixture distribution F is composed of F ₁ and F ₂ , and the definition of its probability density function is given by formula (21):

F(x)=AF ₁ (x)+BF ₂ (x) (21)

The A and B represent the undetermined adaptive weights, and then the value x _i (i∈{1,2,3,...,n}) of each interval of the true probability density pdf _I and its corresponding probability density pdf _I (x _i ) is substituted into equation (21) to obtain a linear overdetermined system of equations whose definition is given by equation (22):

Step 8. According to the definition of formula (23), convert formula (22) into matrix form:

And use the left division method to solve the least squares solution of the linear overdetermined equation system, and obtain the adaptive weights A and B of the mixture distribution F, whose definition is given by formula (24):

The "/" represents the left division operator;

Step 9. Substitute A and B into formula (21), obtain the probability density function of mixed distribution F, and use it as the fitting result of the DCT transform coefficient distribution of input image I;

Step 10. Output the probability density function of the mixture distribution F.

Compared with the prior art, the present invention guarantees the fitting accuracy of the image DCT transform coefficients from three aspects: first, for the non-Gaussian characteristic of the image DCT transform coefficient distribution, the complexity of the image acquisition process and the diversity of image types, A probability density function dictionary is constructed by using five "thick-tailed" distributions, which is beneficial to break through the limitations of traditional methods in statistical modeling of image DCT coefficient distributions by using a single distribution or two fixed distributions, and improve distribution fitting. Second, using the Kullback-Leibler measure as the standard for evaluating the fitting accuracy can overcome the inability of the traditional Kolmogorov-Smirnov goodness-of-fit test method to effectively evaluate the fitting accuracy of the Laplace distribution; third, using The Kullback-Leibler measure adaptively selects two single distributions in the probability density function dictionary that best match the distribution characteristics of the DCT coefficients of the input image, and determines the difference between the two single distributions by solving the least squares solution of the linear overdetermined equation system. Adaptive weights, and then form an adaptive hybrid "thick-tailed" distribution, in theory, a richer "thick-tailed" distribution and its probability density function form can be obtained, so as to break through the existing "thick-tailed" distribution type limitations, can be Provides greater flexibility and more degrees of freedom for DCT transform coefficient fitting. Therefore, the present invention has the advantages of stability and efficiency, strong adaptability and high fitting precision.

Description of drawings

Fig. 1 is the comparison result of fitting the DCT coefficients of the images using the present invention and a single distribution respectively.

Table 1 shows the distance comparison results of the DCT coefficients of images fitted by the present invention and a single distribution respectively.

Detailed ways

A method for fitting an image DCT coefficient distribution based on a probability density function dictionary, characterized in that it is performed according to the following steps:

Step 1. Input an image I, divide it into a series of pixel blocks whose size is B×B and do not overlap and carry out block DCT transformation to obtain a set of DCT transformation coefficients, in this embodiment, let B=16;

Step 2. adopt the kernel density estimation method to count the distribution of image DCT transform coefficients, obtain the true probability density pdf _{I of the DCT transform coefficients of image I} ;

Step 3. Establish a probability density function dictionary D (pdf _Cauchy , pdf _Laplacian , pdf _Logistic , pdf _Gaussian , pdf _extreme ) composed of 5 probability density functions, the pdf _Cauchy represents the Cauchy probability density function, and pdf _Laplacian represents the Laplacian Plath probability density function, pdf _Logistic represents the logistic probability density function, pdf _Gaussian represents the Gaussian probability density function, and pdf _Extreme represents the extreme value probability density function;

Step 3.1 According to the definition of formula (1), establish the Cauchy probability density function with parameters θ ^C and λ ^C :

Step 3.2 According to the definition of formula (2), establish the Laplace probability density function with parameters α ^L and β ^L :

Step 3.3 According to the definition of formula (3), establish a logistic probability density function with parameters α ^O and β ^O :

Step 3.4 According to the definition of formula (4), establish the extreme value probability density function with parameters α ^E , β ^E :

Step 3.5 According to the definition of formula (5), establish a Gaussian probability density function with parameters μ and σ:

Step 4. Use each distribution in the probability density function dictionary to fit the true probability density pdf _I of the DCT transform coefficients of the input image I respectively;

Step 4.1 Use the Cauchy probability density function pdf _Cauchy to fit the image DCT coefficient distribution, and use the maximum likelihood estimation method to calculate the parameters θ ^C and λ ^C of the pdf _Cauchy ;

(a) According to the definition of formula (6), establish the likelihood function of the Cauchy probability density function:

The _xi represents the value of the ith interval (bin) of the true probability density pdf _I ;

(b) Calculate the partial derivatives of the parameters θ ^C and λ ^C by formula (7) respectively, and make their partial derivatives equal to 0, and then combine the obtained equations into a system of equations, the definition of which is given by formula (7):

以及相应的单一柯西分布拟合结果

(c) Solve the system of equations given by equation (7) to obtain the optimal fitting parameters of pdf _Cauchy

and the corresponding fitting results of a single Cauchy distribution

Step 4.2 uses the Laplacian probability density function pdf _Laplacian to fit the image DCT coefficient distribution, and adopts the maximum likelihood estimation method to calculate the parameters α ^L and β ^L of the pdf _Laplacian ;

(a) According to the definition of formula (8), establish the likelihood function of the Laplace probability density function:

(b) According to the definition of formula (9), establish the log-likelihood function of the Laplace probability density function:

(c) Calculate the partial derivatives of the parameters α ^L and β ^L by formula (9) respectively, and make their partial derivatives equal to 0, and then combine the obtained equations into a system of equations, the definition of which is given by formula (10):

以及相应的单一拉普拉斯分布拟合结果

(d) Solve the system of equations given by equation (10) to obtain the optimal fitting parameters of pdf _Laplacian

and the corresponding fitting results of a single Laplace distribution

Step 4.3 utilizes the logistic probability density function pdf _Logistic to fit the image DCT coefficient distribution, and adopts the maximum likelihood estimation method to calculate the parameters α ^O and β ^O of pdf _Logistic ;

(a) According to the definition of formula (11), establish the likelihood function of the logistic probability density function:

(b) Calculate the partial derivatives of the parameters α ^O and β ^O by formula (11) respectively, and set their partial derivatives equal to 0, and then combine the obtained equations into a system of equations, the definition of which is given by formula (12):

以及相应的单一逻辑斯谛分布拟合结果

(c) Solve the equation system given by formula (12) to obtain the optimal fitting parameters of pdf _Logistic

and the corresponding fitting result of a single logistic distribution

Step 4.4 utilizes extreme value probability density function pdf _Extreme to fit the image DCT coefficient distribution, and adopts the maximum likelihood estimation method to calculate the parameters α ^E and β ^E of pdf _Extreme ;

(a) According to the definition of formula (13), establish the likelihood function of extreme value probability density function:

(b) According to the definition of formula (14), establish the log-likelihood function of the extreme probability density function:

(c) Calculate the partial derivatives of the parameters α ^E and β ^E by formula (14) respectively, and make their partial derivatives equal to 0, and then combine the obtained equations into a system of equations, the definition of which is given by formula (15):

以及相应的单一极值分布拟合结果

(d) Solve the system of equations given by equation (15) to obtain the optimal fitting parameters of pdf _Extreme

and the corresponding single extreme value distribution fitting results

Step 4.5 Use the Gaussian probability density function pdf _Gaussian to fit the image DCT coefficient distribution, and use the maximum likelihood estimation method to calculate the parameters μ and σ of the pdf _Gaussian ;

(a) According to the definition of formula (16), establish the likelihood function of the Gaussian probability density function:

(b) According to the definition of formula (17), establish the log-likelihood function of the Gaussian probability density function:

(c) Calculate the partial derivatives of the parameters μ and σ by formula (17) respectively, and make their partial derivatives equal to 0, and then combine the obtained equations into a system of equations, the definition of which is given by formula (18):

(d) Solving the equation system given by formula (18) to obtain the optimal fitting parameters μ _opt , σ _opt of pdf _Gaussian and the corresponding fitting result of single Gaussian distribution pdf _Gaussian (x; μ _opt ,σ _opt );

和pdf_Gaussian(x；μ_opt,σ_opt)与待拟合图像I的DCT变换系数真实概率密度pdf_I的距离：Step 5. According to the definition of formula (19) and formula (20), use the Kullback-Leibler measure (KLD) to calculate the probability density function respectively

and the distance between pdf _Gaussian (x; μ _opt ,σ _opt ) and the true probability density pdf _I of the DCT transform coefficients of the image I to be fitted:

The p ₁ (x) and p ₂ (x) represent two given probability density functions;

p₂(x)＝pdf_I，并将p₁(x)和p₂(x)代入公式(19)和公式(20)，计算利用单一柯西分布拟合pdf_I的距离d_Cauchy；Step 5.1 Order

p ₂ (x)=pdf _I , and substituting p ₁ (x) and p ₂ (x) into formula (19) and formula (20), calculate the distance d _Cauchy fitting pdf _I with a single Cauchy distribution;

p₂(x)＝pdf_I，并将p₁(x)和p₂(x)代入公式(19)和公式(20)，计算利用单一拉普拉斯分布拟合pdf_I的距离d_Laplacian；Step 5.2 Order

p ₂ (x)=pdf _I , and substituting p ₁ (x) and p ₂ (x) into formula (19) and formula (20), calculate the distance d _Laplacian fitting pdf _I with a single Laplace distribution;

p₂(x)＝pdf_I，并将p₁(x)和p₂(x)代入公式(19)和公式(20)，计算利用单一逻辑斯谛分布拟合pdf_I的距离d_Logistic；Step 5.3 Order

p ₂ (x)=pdf _I , and substituting p ₁ (x) and p ₂ (x) into formula (19) and formula (20), calculate the distance d _Logistic fitting pdf _I with a single logistic distribution;

p₂(x)＝pdf_I，并将p₁(x)和p₂(x)代入公式(19)和公式(20)，计算利用单一极值分布拟合pdf_I的距离d_Extreme；Step 5.4 Order

p ₂ (x)=pdf _I , and substituting p ₁ (x) and p ₂ (x) into formula (19) and formula (20), calculate the distance d _Extreme that fits pdf _I with a single extreme value distribution;

Step 5.5 Let p ₁ (x)=pdf _Gaussian (x; μ _opt ,σ _opt ), p ₂ (x)=pdf _I , and substitute p ₁ (x) and p ₂ (x) into equation (19) and equation (20), calculate the distance d _Gaussian of fitting pdf _I using a single Gaussian distribution;

Step 6. Among the fitting distances d _Cauchy , d _Laplacian , d _Logistic , d _Extreme and d _Gaussian of the five single distributions, select the two smallest distances, and make the single distribution corresponding to the two smallest distances respectively. are F ₁ and F ₂ ;

Step 7. The mixture distribution F is composed of F ₁ and F ₂ , and the definition of its probability density function is given by formula (21):

F(x)=AF ₁ (x)+BF ₂ (x) (21)

The A and B represent the undetermined adaptive weights, and then the value x _i (i∈{1,2,3,...,n}) of each interval of the true probability density pdf _I and its corresponding probability density pdf _I (x _i ) is substituted into equation (21) to obtain a linear overdetermined system of equations whose definition is given by equation (22):

Step 8. According to the definition of formula (23), convert formula (22) into matrix form:

And use the left division method to solve the least squares solution of the linear overdetermined equation system, and obtain the adaptive weights A and B of the mixture distribution F, whose definition is given by formula (24):

The "/" represents the left division operator;

Step 9. Substitute A and B into formula (21), obtain the probability density function of mixed distribution F, and use it as the fitting result of the DCT transform coefficient distribution of input image I;

Step 10. Output the probability density function of the mixture distribution F.

Figure 1 shows a comparison of the DCT coefficients of the images fitted with the present invention and a single distribution respectively. Among them, (a) is the original image; (b) is the true probability density of the DCT coefficients of the original image; (c) is the local amplification result of the peak part when using a single Cauchy distribution to fit the probability density distribution of the image DCT coefficients; (d) is the partially enlarged result at the tail when using a single Laplace distribution to fit the probability density distribution of the image DCT coefficients; (e) is the result of using a single logistic distribution to fit the probability density distribution of the image DCT coefficients (f) is the result of using a single extreme value distribution to fit the probability density distribution of the image DCT coefficients; (g) is the result of using a single Gaussian distribution to fit the probability density distribution of the image DCT coefficients; The result of the probability density distribution of the combined image DCT coefficients. It can be seen from Figure 1 that the DCT coefficients of the original image basically show the distribution characteristics of "high peaks and long tails" (Figure (b)), which reflects the non-Gaussian distribution of the DCT coefficients of the image; The western distribution cannot well fit the peak part of the DCT coefficient probability density distribution of the image, and its peak value is significantly higher than the true distribution of the original image; the single Laplace distribution of Figure (d) cannot accurately fit the image DCT coefficient probability density distribution. The tail of ; the single logistic distribution of Figure (e), whether in the peak part or the tail, its fitting accuracy is not ideal; the single extreme value distribution of Figure (f) and the single Gaussian distribution of Figure (g) The fitting accuracy is lower; in contrast, the fitting result of Figure (h) is closest to the true probability density distribution of the DCT coefficients of the original image.

The following table shows the distance comparison of the DCT coefficients of the images fitted by the present invention and the single distribution respectively.

It can be seen from FIG. 1 and Table 1 that the subjective quality and objective quality of the DCT coefficients of the fitted image using the present invention are both higher than the fitting accuracy of a single distribution.

Claims (1)

1. an image DCT coefficient distribution fitting method based on a probability density function dictionary, is characterized in that carrying out as follows: Step 1. Input an image I, divide it into a series of pixel blocks whose size is B×B and do not overlap, and perform block DCT transformation to obtain a set of DCT transformation coefficients; Step 2. adopt the kernel density estimation method to count the distribution of image DCT transform coefficients, obtain the true probability density pdf _{I of the DCT transform coefficients of image I} ; Step 3. Establish a probability density function dictionary D (pdf _Cauchy , pdf _Laplacian , pdf _Logistic , pdf _Gaussian , pdf _extreme ) composed of 5 probability density functions, the pdf _Cauchy represents the Cauchy probability density function, and pdf _Laplacian represents the Laplacian Plath probability density function, pdf _Logistic represents the logistic probability density function, pdf _Gaussian represents the Gaussian probability density function, and pdf _Extreme represents the extreme value probability density function; Step 3.1 According to the definition of formula (1), establish the Cauchy probability density function with parameters θ ^C and λ ^C :

Step 3.2 According to the definition of formula (2), establish the Laplace probability density function with parameters α ^L and β ^L :

Step 3.3 According to the definition of formula (3), establish a logistic probability density function with parameters α ^O and β ^O :

Step 3.4 According to the definition of formula (4), establish the extreme value probability density function with parameters α ^E , β ^E :

Step 3.5 According to the definition of formula (5), establish a Gaussian probability density function with parameters μ and σ:

Step 4. Use each distribution in the probability density function dictionary to fit the true probability density pdf _I of the DCT transform coefficients of the input image I respectively; Step 4.1 Use the Cauchy probability density function pdf _Cauchy to fit the image DCT coefficient distribution, and use the maximum likelihood estimation method to calculate the parameters θ ^C and λ ^C of the pdf _Cauchy ; (a) According to the definition of formula (6), establish the likelihood function of the Cauchy probability density function:

The _xi represents the value of the ith interval (bin) of the true probability density pdf _I ; (b) Calculate the partial derivatives of the parameters θ ^C and λ ^C by formula (7) respectively, and make their partial derivatives equal to 0, and then combine the obtained equations into a system of equations, the definition of which is given by formula (7):

(c) Solve the system of equations given by equation (7) to obtain the optimal fitting parameters of pdf _Cauchy

and the corresponding fitting results of a single Cauchy distribution

Step 4.2 uses the Laplacian probability density function pdf _Laplacian to fit the image DCT coefficient distribution, and adopts the maximum likelihood estimation method to calculate the parameters α ^L and β ^L of the pdf _Laplacian ; (a) According to the definition of formula (8), establish the likelihood function of the Laplace probability density function:

(b) According to the definition of formula (9), establish the log-likelihood function of the Laplace probability density function:

(c) Calculate the partial derivatives of the parameters α ^L and β ^L by formula (9) respectively, and make their partial derivatives equal to 0, and then combine the obtained equations into a system of equations, the definition of which is given by formula (10):

(d) Solve the system of equations given by equation (10) to obtain the optimal fitting parameters of pdf _Laplacian

and the corresponding fitting results of a single Laplace distribution

Step 4.3 utilizes the logistic probability density function pdf _Logistic to fit the image DCT coefficient distribution, and adopts the maximum likelihood estimation method to calculate the parameters α ^O and β ^O of pdf _Logistic ; (a) According to the definition of formula (11), establish the likelihood function of the logistic probability density function:

(b) Calculate the partial derivatives of the parameters α ^O and β ^O by formula (11) respectively, and set their partial derivatives equal to 0, and then combine the obtained equations into a system of equations, the definition of which is given by formula (12):

(c) Solve the equation system given by formula (12) to obtain the optimal fitting parameters of pdf _Logistic

and the corresponding fitting result of a single logistic distribution

Step 4.4 utilizes extreme value probability density function pdf _Extreme to fit the image DCT coefficient distribution, and adopts the maximum likelihood estimation method to calculate the parameters α ^E and β ^E of pdf _Extreme ; (a) According to the definition of formula (13), establish the likelihood function of extreme value probability density function:

(b) According to the definition of formula (14), establish the log-likelihood function of the extreme probability density function:

(c) Calculate the partial derivatives of the parameters α ^E and β ^E by formula (14) respectively, and make their partial derivatives equal to 0, and then combine the obtained equations into a system of equations, the definition of which is given by formula (15):

(d) Solve the system of equations given by Equation (15) to obtain the optimal fitting parameters of pdf _Extreme

and the corresponding single extreme value distribution fitting results

Step 4.5 Use the Gaussian probability density function pdf _Gaussian to fit the image DCT coefficient distribution, and use the maximum likelihood estimation method to calculate the parameters μ and σ of the pdf _Gaussian ; (a) According to the definition of formula (16), establish the likelihood function of the Gaussian probability density function:

(b) According to the definition of formula (17), establish the log-likelihood function of the Gaussian probability density function:

(c) Calculate the partial derivatives of the parameters μ and σ by formula (17) respectively, and make their partial derivatives equal to 0, and then combine the obtained equations into a system of equations, the definition of which is given by formula (18):

(d) Solving the equation system given by formula (18) to obtain the optimal fitting parameters μ _opt , σ _opt of pdf _Gaussian and the corresponding fitting result of single Gaussian distribution pdf _Gaussian (x; μ _opt ,σ _opt ); Step 5. According to the definition of formula (19) and formula (20), use the Kullback-Leibler measure (KLD) to calculate the probability density function respectively

and

The distance from the true probability density pdf _I of the DCT transform coefficient of the image I to be fitted:

The p ₁ (x) and p ₂ (x) represent two given probability density functions; Step 5.1 Order

p ₂ (x)=pdf _I , and substituting p ₁ (x) and p ₂ (x) into formula (19) and formula (20), calculate the distance d _Cauchy fitting pdf _I with a single Cauchy distribution; Step 5.2 Order

p ₂ (x)=pdf _I , and substituting p ₁ (x) and p ₂ (x) into formula (19) and formula (20), calculate the distance d _Laplacian fitting pdf _I with a single Laplace distribution; Step 5.3 Order

p ₂ (x)=pdf _I , and substituting p ₁ (x) and p ₂ (x) into formula (19) and formula (20) to calculate the distance d _Logistic fitting pdf _I with a single logistic distribution; Step 5.4 Order

p ₂ (x)=pdf _I , and substituting p ₁ (x) and p ₂ (x) into formula (19) and formula (20), calculate the distance d _Extreme that fits pdf _I with a single extreme value distribution; Step 5.5 Let p ₁ (x)=pdf _Gaussian (x; μ _opt ,σ _opt ), p ₂ (x)=pdf _I , and substitute p ₁ (x) and p ₂ (x) into equation (19) and equation (20), calculate the distance d _Gaussian that utilizes a single Gaussian distribution to fit pdf _I ; Step 6. Among the fitting distances d _Cauchy , d _Laplacian , d _Logistic , d _Extreme and d _Gaussian of the five single distributions, select the two smallest distances, and make the single distribution corresponding to the two smallest distances respectively. are F ₁ and F ₂ ; Step 7. The mixture distribution F is composed of F ₁ and F ₂ , and the definition of its probability density function is given by formula (21): F(x)=AF ₁ (x)+BF ₂ (x) (21) The A and B represent the undetermined adaptive weights, and then the value x _i (i∈{1,2,3,...,n}) of each interval of the true probability density pdf _I and its corresponding probability density pdf _I (xi) is substituted into Equation (21) to obtain a linear overdetermined system of equations whose definition is given by Equation (22):

Step 8. According to the definition of formula (23), convert formula (22) into matrix form:

And use the left division method to solve the least squares solution of the linear overdetermined equation system, and obtain the adaptive weights A and B of the mixture distribution F, whose definition is given by formula (24):

The "/" represents the left division operator; Step 9. Substitute A and B into formula (21), obtain the probability density function of mixed distribution F, and use it as the fitting result of the DCT transform coefficient distribution of input image I; Step 10. Output the probability density function of the mixture distribution F.

Images