CN111311525A

CN111311525A - Image gradient field double-interval equalization algorithm based on histogram probability correction

Info

Publication number: CN111311525A
Application number: CN201911141281.1A
Authority: CN
Inventors: 赵辉; 徐先明; 方禄发
Original assignee: Chongqing University of Post and Telecommunications
Current assignee: Chongqing University of Post and Telecommunications
Priority date: 2019-11-20
Filing date: 2019-11-20
Publication date: 2020-06-19

Abstract

In order to effectively enhance the image lacking texture details, the invention expands the traditional gray level histogram to the gradient histogram and provides an image gradient field double-interval equalization algorithm based on histogram probability correction. The algorithm considers the over-enhancement effect of global equalization, improves the traditional global equalization, provides that the gradient histogram is divided into an edge part and a non-edge part of an image, and then the two parts are equalized respectively, and the method solves the over-enhancement of the traditional global equalization. Finally, according to the mapping rule of histogram equalization, the algorithm provides single threshold value method correction probability before the two intervals are equalized respectively, the operation effectively avoids the 'phagocytosis' effect of histogram equalization, and the information entropy of the enhanced image is greatly improved compared with that of the original image.

Description

Image gradient field double-interval equalization algorithm based on histogram probability correction

Technical Field

The invention belongs to the technical field of digital image processing, and particularly relates to an image gradient field double-interval equalization algorithm based on histogram probability correction.

Background

Histogram Equalization (HE) is a common method for image enhancement, and has the advantages of small time complexity, obvious enhancement effect and the like, but after Equalization, the method can cause the area with high probability of the original Histogram to be over-enhanced and the area with low probability to be reduced in contrast. In view of the above-mentioned disadvantages, many scholars have recently proposed an image enhancement algorithm based on histogram equalization, and many algorithms have been proposed to achieve a better effect after equalization by dividing an original histogram or original image into a plurality of sub-blocks by setting a threshold value and then equalizing the sub-blocks individually, or by directly changing the distribution of the histogram to achieve a predetermined shape.

The image gradient is an important index for the quality and the quality of image texture details, the gradient amplitude at a certain position of an image is large and represents that the gray value at the position is greatly changed, namely the texture contour is clear, if the gradient field of the image is well enhanced, particularly for the image lacking the texture details, the image can be effectively enhanced.

Disclosure of Invention

Aiming at the defects of the prior art, the invention provides an image gradient field double-interval equalization algorithm based on histogram probability correction. The algorithm firstly proves that gradient variables of the edge part of an original gradient field obey mixed Gaussian distribution, gradient variables of the non-edge part obey mixed Rayleigh distribution, then a proper threshold value is found according to the characteristic that the deviation of the mixed Rayleigh distribution is a fixed value, a gradient histogram is divided into an edge part (large gradient interval) and a non-edge part (small gradient interval), and finally the two parts are respectively balanced. Further to solve the problem of 'phagocytosis' of two-part equilibrium, the invention proposes a single threshold method to correct the probability before the two parts are equilibrated.

The technical scheme and the flow sequence of the invention are as follows:

(1) inputting an original image, determining a pixel point at the leftmost upper corner of the original image I as an origin of a plane rectangular coordinate system (a y-axis direction is set downwards and an X-axis direction is set to be a positive direction), calculating gradient amplitudes of all pixel points in the original image I, and counting the number of the gradient amplitudes of all the pixel points to obtain a gradient amplitude histogram X;

(2) size gradient bi-compartment equalization: PDFs of gradient variables of an edge part (large gradient interval) and a non-edge part (small gradient interval) are firstly solved, and the fact that the gradient variables of the edge part obey mixed Gaussian distribution and the gradient variables of the non-edge part obey mixed Rayleigh distribution is proved. And then testing a large number of pictures according to the characteristic that the deviation of the mixed Rayleigh distribution is a fixed value to obtain an optimal threshold value for segmenting the original gradient amplitude histogram X, wherein the deviation of the mixed Rayleigh distribution corresponding to the threshold value is 0.75. Finally, dividing the original gradient amplitude histogram X into small gradient intervals X according to the optimal threshold value_SAnd large gradient interval X_BAs shown in fig. 4;

(3) single threshold method: after dividing the large and small gradient regions, the invention firstly finds the small gradient region X_SAnd large gradient interval X_BDistribution average probability of (T)_SAnd T_BAs shown in fig. 5. Statistics of X_SLi is greater than and less than T_SThe number of gradient stages of (1), two numbers being denoted as C_hS(k)＞TSAnd C_hS(k)＜TSComparison C_hS(k)＞TSAnd C_hS(k)＜TSThe gradient histogram of (1) is obtained by changing the probability of the gradient level corresponding to the larger one to T_S，X_BProbability correction of similar X_S；

(4) The corrected large and small gradient intervals X_SAnd X_BRespectively equalizing, and combining the two equalized gradient regions to obtain an equalized image gradient field

(5) According to

Reconstructing an enhanced image: let the enhanced image be u, the gradient field according to u should be maximally close to the gradient field

According to the requirement, the reconstruction problem is converted into an extreme value problem of a functional, and then the extreme value solving is converted into a solution of a Poisson equation according to a variational method;

(6) and performing experimental simulation by using MATLAB, objectively presenting an experimental result, and verifying the effectiveness of the algorithm.

Compared with the prior image gradient field equalization algorithm, the algorithm provided by the invention comprises the following steps: 1) the original gradient amplitude histogram is divided into an edge part and a non-edge part to be balanced respectively, and the divided threshold value is not selected randomly but is the boundary gradient value of the two parts, so that the two parts can effectively enhance the gradient field after being balanced respectively, the gradient value of each other can not occur, and the over-enhancement effect of the traditional global equalization is eliminated; 2) the probability of the two parts is corrected by a single threshold value method before the two parts are respectively balanced, so that the 'phagocytosis' problem of histogram balance is effectively avoided. Experimental results show that compared with the original image, the entropy of the image information enhanced by the algorithm is greatly improved and is superior to that of the traditional algorithm at present.

Drawings

FIG. 1 is a comparison graph of animal simulation effects in the example of the present invention

FIG. 2 is a comparison graph of simulation effect of a house according to an embodiment of the present invention

FIG. 3 is a diagram showing the comparison of forest simulation effects in the embodiment of the present invention

FIG. 4 is a diagram illustrating the effect of segmenting the gradient histogram by optimal threshold values in an embodiment of the present invention

FIG. 5 is a diagram illustrating a single threshold method according to an embodiment of the present invention

FIG. 6 is an abstract drawing of the present invention

Detailed Description

The following describes the practice of the present invention in detail with reference to the accompanying drawings.

ZHU proposes a global equalization algorithm of the gradient histogram, and the algorithm directly equalizes the original gradient histogram by keeping the direction of the original gradient field unchanged, so that the method has the advantages of obvious effect on enhancing image details, high efficiency and the like. However, the algorithm does not consider the problems of overlarge equilibrium range, uneven probability distribution and the like, so that the disadvantages of gradient field over-enhancement, gradient level phagocytosis and the like exist after equilibrium. The invention provides an image gradient field double-interval equalization algorithm based on histogram probability correction, firstly, a gradient histogram is divided into an edge part and a non-edge part of an image, and then the two parts are equalized respectively, so that the operation effectively weakens the problem of over-enhancement of global equalization. In order to solve the problem of gradient level phagocytosis, the invention provides a single threshold value method to correct the probability before the two parts are respectively balanced.

One, two interval equalization

Conventional gradient histogram equalization employs global equalization, which results in an over-enhancement of the histogram equalization. In order to eliminate the over-enhancement phenomenon, the gradient amplitude histogram is divided into large and small gradient intervals by an optimal threshold value and then the gradient intervals are respectively balanced according to the mapping rule of histogram balancing, the two intervals can be balanced in the original gradient range, and the gradient values cannot occur after the two intervals are balanced, so that the over-enhancement is avoided. The steps for finding the optimal threshold value provided by the invention are as follows:

1. proving that the gradient amplitude variable of the original gradient field obeys the Rice distribution

Firstly, the vertical gradient of an input image is set as a variable Y_ijRThe gradient in the horizontal direction being the variable Y_ijCConsidering the gradient distribution of the input image as a binary mixed Gaussian distribution, Y is obtained because the gradient value of the input image depends on various reasons_ijRAnd Y_ijCPDF obeying the following formula (1):

wherein r is_GAre the regions corresponding to the binary mixed gaussian distribution,

and

is Y in each region_ijRAnd Y_ijCIn the expectation that the position of the target is not changed,

and

is Y in each region_ijRAnd Y_ijCThe standard deviation of (a) is determined,

is Y_ijRAnd Y_ijCThe covariance of the regions in each of the regions,

is the weight of each region. To simplify equation (1), a series of proofs is required, the procedure is as follows:

(1) proving that Y in formula (1)_ijRAnd Y_ijCIndependent of each other

① finding Y_ijRAnd Y_ijCCorrelation coefficient of (1) ("rho")

Because of Y_ijRAnd Y_ijCThe correlation coefficient ρ of (d) is:

wherein I₁And I₂Is Y_ijRAnd Y_ijCFirst moment of (d). At the same time according to Y_ijRAnd Y_ijCAre mutually orthogonal to obtain Y_ijRAnd Y_ijCHas a cross-correlation function of 0, i.e. E (Y)_ijRY_ijC) When 0, then equation (2) can be reduced to equation (3):

wherein v is₁And v₂Is Y_ijRAnd Y_ijCStandard deviation of (1), M₁And M₂Is Y_ijRAnd Y_ijCSecond moment of (I)₁And I₂Is Y_ijRAnd Y_ijCFirst moment of (d).

② finding the gradient variable Y in the formula (3)_ijRAnd Y_ijCFirst moment of₁And I₂

Because the input image is composed of many non-edge and edge regions, I is obtained₁And I₂It becomes to find the first moment I of each region_1r1And I_2r2The weighted sum of (2) is given by the following equation (4)_ijRAnd Y_ijCThe second moment of (2) is the same.

Wherein w_1r1And w_2r2Is I_1r1And I_2r2The corresponding weight.

③ analysis of the magnitude relation of each variable in the formula (3) yields the value of ρ

For a certain non-edge area, the gray scale difference in the area is not large, namely the gradient value in the area is relatively small, and I is deduced_1r1And I_2r2Close to 0. And the ratio of the number of pixels in the edge area is small and the obtained I_1r1And I_2r2Has positive or negative values, so that the first moment I in the formulas (3) and (4) is obtained₁And I₂Close to 0. At the same time due to M₁And M₂Is a gradient variable Y_ijRAnd Y_ijCSo that they are much larger than I₁And I₂. From the above, it is found that p is a very small value, i.e. Y is proved_ijRAnd Y_ijCAre independent of each other.

(2) In analytical formula (1)

And

in relation to (2)

Due to the variable Y_ijRAnd Y_ijCAre obtained by operating on the same group of pixels of the input image, so they have the same standard deviation, i.e. v₁＝v₂. And the gradient variables of each region in the input image are subjected to Gaussian distribution, and all the gradient variables are obtained by using the same gradient operator action on each region of the input image, so that all the gradient variables are subjected to the same mixed Gaussian distribution, and the standard deviation of the mixed Gaussian distribution is set as

According to the above Y in each region of the input image_ijRAnd Y_ijCStandard deviation of (2)

And

are equal to, and have

Approximating equation (1) in two steps according to (1) (2) as follows:

f(Y_ijR,Y_ijC)≈f_k(Y_ijR,Y_ijC) (5)

wherein r is_GIs Y_ijRAnd Y_ijCThe corresponding respective areas are set to be in correspondence with each other,

is the weight of each of the regions and,

and

is Y in each region_ijRAnd Y_ijCIs markedTolerance, p_rGIs Y in each region_ijRAnd Y_ijCThe covariance of (a) of (b),

and

N₁，N₀，N_-1is f_k(Y_ijR,Y_ijC) Correction constants that make the curves uniform in three cases. According to formula (7) and formula (1), G is obtained_ijThe PDF of (1) is formula (8):

wherein Y is_ijRAnd Y_ijCAre gradient variables in the vertical and horizontal directions of the image. F of formula (6)_kSubstituting the value into formula (8) to obtain G_ijThe PDF of (a) can be approximated as a weighted sum rice PDFs, i.e. to prove that the gradient variables of the original gradient field obey a rice distribution.

2. Proving the gradient amplitude of the non-edge region of the original gradient map

And edge region gradient magnitude

Obeying Rayleigh and Gaussian distributions

Due to the non-edge area A_HIs/are as follows

And

about 0, will

And

substitution of 0 into f in formula (6)_kThen f of formula (6)_kSubstituting the value into the formula (8) to obtain

The probability density of (a) is as follows (9):

wherein r is_GIs that

The corresponding respective areas are set to be in correspondence with each other,

is the weight of each of the regions and,

is in each region

The standard deviation of (a) is determined,

is in each region

Of (a) autocovariance, N₁，N₀，N_-1Is f_k(Y_ijR,Y_ijC) Correction constants that make the curves uniform in three cases.

For the edge area A_EIs provided with

And substituting into formula (8) to obtain (G)_ij)_AEThe probability density of (a) is as follows (10):

wherein r is_GIs that

is the weight of each of the regions and,

is in each region

Standard deviation of (1), ρ_rGIs in each region

The gradient amplitude variable of the non-edge region obtained by the derivation approximately follows mixed Rayleigh distribution, and the gradient amplitude variable of the edge region approximately follows mixed Gaussian distribution.

3. Selection of optimal segmentation threshold

In the gradient histogram of the lack of detail image, when the original gradient histogram is divided by different threshold values, the distribution of the gradient amplitudes of small gradient intervals is different, and the skewness coefficients of the small gradient intervals calculated by taking the gradient amplitudes as sample values are also different, so that the optimal threshold value can be determined by measuring the skewness coefficients of the small gradient intervals under different division threshold values and then utilizing the corresponding relation between the skewness coefficients and the threshold values.

When the mixed rayleigh distribution includes only one rayleigh distribution, the skewness coefficient thereof is a constant value of 0.63, and when the mixed rayleigh distribution includes a large number of rayleigh distributions, the skewness coefficient thereof is a constant value of 1.25. Because gradient amplitude variables of non-edge areas of different images approximately obey mixed Rayleigh distribution, and the number of Rayleigh distributions contained in the mixed Rayleigh distribution is uncertain, the optimal segmentation threshold is determined by the following two steps:

(1) testing the skewness coefficient of small gradient intervals of a certain input image under different segmentation threshold values to obtain the following conclusion: when a certain segmentation threshold value is taken, the skewness coefficient can be monotonically increased from 0.63 to 1.25 correspondingly;

(2) testing a large amount of gradient histograms lacking texture details to obtain a corresponding threshold value x when the skewness coefficient is 0.75_tIt is reasonable to segment the gradient histograms of different images.

Setting the gradient histogram X according to the above by the optimal threshold value X_tDivided into small gradient regions X_SAnd large gradient interval X_BThere are then the formulae (11) to (13)

X＝X_S∪X_B(11)

Wherein

X_SIs a partial gradient level of X {0,1,2_tComposition X_BIs a partial gradient order by X { X_t+1,X_t+ 2.., L-1} as in fig. 4.

Two, one threshold method

After the large and small gradient intervals are segmented, because the probability of gradient level distribution of each interval is uneven, phagocytosis can occur when the two intervals are directly balanced, and a part of gradient can be lost in a reconstructed image, the probability in a gradient histogram is adjusted by adopting a single threshold method before the two intervals are respectively balanced, and the specific process of the single threshold method is as follows:

(1) average probability of two intervals of gradient histogram is calculated

The above small gradient interval X_SThe original probability (number of pixel points) of the inner gradient level k is h_s(k) Denotes, the large gradient region X_BThe original probability of the inner gradient level k is h_B(k) And (4) showing. Then, the average probability T of each of the two intervals is calculated_SAnd T_B，T_SAnd T_BAdaptive calculation as follows (14) and (15):

(2) correcting the probability of two intervals by taking the average probability as a reference

Set small gradient interval h_S(k)＞T_SHas a gradient number of

h_S(k)＜T_SNumber of gradient stages of

To correct X_SIs performed by the following formulas (16) and (17):

wherein h is_S(k) Is X_SThe original probability of the pre-gradient level k, h, is corrected_s(k) After correction, use h_mS(k) Denotes, T_SIs X_SAverage probability of. The formulae (16) and (17) are represented by X_SInterior with T_SFor reference, the summary of all gradient levelsRate is greater than and less than T_SDividing the data into an upper part and a lower part, comparing the number of gradient levels of the two parts, and changing the more probability into T_S，X_BThe probability correction method of (3) is similar.

Equalization of large and small gradient intervals

Correcting the probability of the large and small gradient regions, balancing the two regions, and setting the threshold for dividing the original gradient histogram as x_tDividing the original gradient histogram X into small gradient intervals X_SAnd large gradient interval X_BInterval X_SHas a gradient range of [0, x ]_t]Interval X_SHas a gradient range of

Is the maximum gradient level in the original gradient histogram. The formula for calculating the magnitude of the equalized gradient field is shown in (18):

wherein I₁And I₂Are respectively the interval X_SAnd X_BThe original gradient amplitude of (a).

And

respectively, the equalized gradient amplitudes. n (I)₁) Is the interval X_SThe value of the interior gradient is less than or equal to I₁All pixel numbers of (n) (I)₂) Is the interval X_BThe value of the internal gradient being greater than x_tAnd is less than or equal to I₂All pixel numbers of (N)₁And N₂Are respectively the interval X_SAnd X_BThe total number of pixels in.

Will be provided with

And

gradient magnitudes combined together to form a target gradient field

Maintaining the direction of the original gradient field before the balance, and establishing a target gradient field

For a vector field containing direction and magnitude, it is expressed by equation (19):

wherein

The original gradient vector is used as the vector of the original gradient,

is the modulus of the original gradient vector,

representing the original unit vector.

Image reconstruction based on target gradient field

After the double-interval equalization, the target gradient field is obtained

When the gradient field of image u is maximally close

When u is then according to

And (4) reconstructing an enhanced image. The solution for u can be found by finding the minimum of the functional e (u) of the following formula:

in the formula

Is a gradient operator. The minimization of E (u) is equivalent to the Poisson equation of equation (21) according to the variational method:

where Δ is the laplacian, u is the enhanced image, div is the sign of the divergence,

representing the target gradient field. The solution to the poisson equation of formula (21) is many, and the invention adopts a matrix transformation method with small time complexity. The method is characterized in that the effect of the Laplace operator on an image U is equivalent to that two matrixes are multiplied by an image matrix U and then added, as shown in a formula (22), the effect of the divergence operator on a target gradient field G is equivalent to a formula (23), and the formula (21) is equivalent to a formula (24):

Δu＝A_m*mU_m*n+U_m*nB_n*n(22)

A_m*mU_m*n+U_m*nB_n*n＝C_m*n(24)

wherein A and B in the formulas (22) and (24) are shown in the formula (25), U is a matrix of the enhanced image U, and the size is m x n.

Because the eigenvalues of both A and B matrices are distinct and negative, both matrices can be diagonalized. The matrix U is obtained according to the above formula as follows:

(1) solving a similar diagonalized matrix I of A and B_AAnd I_BSimultaneously obtaining A and B similarity transformation matrixes P and Q;

(2) rewriting formula (24) to P^-1APP^-1UQ+P^-1UQQ^-1BQ＝P^-1CQ, Dei_AP^-1UQ+P^-1UQI_B＝P^-1CQ；

(3) Let Y be P^-1UQ, based on the property of the product of diagonal matrix and general matrix

(4) According to U-PYQ^-1Obtaining a matrix U, namely an enhanced image U;

the method adopts matrix form to solve.

In order to evaluate the performance of the algorithm provided by the invention, three original images of a forest, an animal and a house are input for simulation. The algorithm chosen for the performance comparison was: HE. AHE and gradient field global equalization which are histogram equalization algorithms of the current mainstream are used for evaluating the algorithm provided by the invention from both subjective and objective aspects.

Fig. 1 to 3 are comparison graphs of simulation results of the algorithms, and in subjective view, the HE algorithm has an obvious over-enhancement effect although the HE algorithm improves the contrast of the image as a whole and has a certain effect on enhancing details of the image. The AHE algorithm is a self-adaptive HE algorithm, has obvious effect on enhancing images lacking texture details, but has poor image authenticity after enhancement. The gradient field global equalization algorithm has over-enhancement and 'phagocytosis' effects on the equalized gradient field, the gradient of a part of regions of an output image is overlarge, and the gradient of the other part of regions is lost. The algorithm of the invention has obvious effect on enhancing the detail texture of the original image, and because the edge and the non-edge are respectively balanced, the gradient values of the edge and the non-edge can not appear after the balance, so that the enhanced image has no distortion of overlarge or undersize gradient values.

Objectively, the invention uses the information entropy to evaluate the result of each algorithm after enhancement, as shown in table 1, and uses the average gradient to evaluate the result, as shown in table 2. As can be seen from the observation of the two tables, the algorithm provided by the invention can better improve the information entropy and the average gradient of the original image. Meanwhile, in contrast, although the average gradient after processing is larger, the gradient field global equalization algorithm is established on the premise that the gradient field is over-enhanced. The average gradient processed by the HE algorithm and the AHE algorithm is good in performance, but the output image has an over-enhancement phenomenon, and the effectiveness of the algorithm provided by the invention is demonstrated. For the comparison of effectiveness of the single threshold method provided by the invention, as shown in table 3, the entropy of the output image information is larger with the single threshold method than without the single threshold method as shown in table 3.

TABLE 1

TABLE 2

Serial number	Original drawing	Correction for removing single threshold value by algorithm	Algorithm of the invention
				FIG. 1 shows a schematic view of a	6.59	7.00	14.78
FIG. 2	6.15	7.63	21.03
				FIG. 3	6.77	7.38	19.21

Table 3.

Claims

1. An image gradient field double-interval equalization algorithm based on histogram probability correction is characterized by comprising the following steps:

firstly, solving a gradient amplitude value of each pixel point of an original image, counting the gradient amplitude values of all the pixel points to obtain a gradient histogram, using the gradient value of a small gradient interval as a corresponding threshold value when the sample deviation degree is 0.75, and dividing the gradient histogram into large and small gradient intervals by using the threshold value;

then, before the two intervals are respectively balanced, respectively solving the average probability of the distribution of the two intervals, dividing the probability of the gradient level in the corresponding interval into an upper part and a lower part according to the reference by taking the average probability as the reference, then changing all the probability of the part with more gradient levels into the average probability, keeping the probability of the part with less gradient levels unchanged, and finally respectively balancing the two intervals;

and finally, reconstructing an enhanced image according to the equalized image gradient field, converting the problem into a functional extreme value solving problem, and solving an extreme value of the functional to obtain the enhanced image.

2. The histogram probability modification-based image gradient field dual-interval equalization algorithm as claimed in claim 1, wherein: after the gradient histogram of the original image is solved, the traditional global gradient domain equalization is improved, and the original gradient histogram is divided into an edge interval and a non-edge interval by adopting a proper threshold value; the specific threshold solving process comprises the following steps:

(1) proving the gradient amplitude of the non-edge part of the original image gradient domain

Obeying a mixed rayleigh distribution;

(2) proving the gradient amplitude of the edge part of the original image gradient domain

Obeying a mixed Gaussian distribution;

(3) according to the skewness characteristics of Rayleigh distribution, a large number of gradient histograms lacking texture details are tested; finding out an optimal threshold value, and dividing the gradient histogram into an edge part and a non-edge part to obtain an optimal threshold value x_tThe corresponding skewness factor is 0.75.

3. The histogram probability modification-based image gradient field dual-interval equalization algorithm as claimed in claim 1, wherein: before the double-interval equalization, the average probability of the double-interval distribution is solved, the number of gradient levels which are larger than and smaller than the reference is compared by taking the average probability as the reference, the probability of the larger gradient level is changed into the average probability, and the modified model is as follows:

the single threshold method provided by the invention is adopted to modify and balance, so that the information entropy of the output image can be greatly improved.