CN103914811A - Image enhancement algorithm based on gauss hybrid model - Google Patents

Abstract

The invention relates to the technical field of image processing and provides an image enhancement algorithm based on a gauss hybrid model. According to the method, at first, luminance components of a color image are counted into a histogram, and mixture gauss modeling is carried out on the histogram; secondly, an improved EM algorithm is used for carrying out gauss hybrid model estimation on the histogram, a parameter of expectation maximization of a likelihood function is found out, and the optimum cluster quantity is determined through self-adaptation; thirdly, partition is carried out on the histogram according to an intersection point of adjacent clusters, and a plurality of sub-histograms are obtained; finally, the mapped clusters are found out according to the fact that area proportions of the sub-histograms with mapping relations are equal, the mapping function is adjusted in a micro mode according to application of the characteristic that the maximum entropy method tends to the human vision, and the final enhanced image is obtained. By the adoption of the image enhancement technology, the algorithm effectively improves the contrast ratio of the image, and increases the processing speed. The enhanced image obtained through the method achieves good effects in subjective visual perception aspect and objective evaluation aspect.

Description

A kind of algorithm for image enhancement based on gauss hybrid models

Technical field

The present invention relates to technical field of image processing, be specifically related to a kind of algorithm for image enhancement based on gauss hybrid models.

Background technology

Image information is more and more used for identifying and judging things by people, solves actual problem.But due to factors such as weather brightness, conditions of exposures, cause brightness of image fuzzyyer, often can not meet the demand of application, this can badly influence the identification to target.It is more concentrated that this class image generally presents gray level, the low characteristic of contrast of image, and therefore, it is very important that the contrast of raising image is carried out post-processed to image.Histogram modification technology receives publicity because it is simple and easy to realize.Wherein, histogram equalization is being used widely aspect raising picture contrast, but luminance saturation easily appears in the enhancing algorithm based on histogram equalization at present, loss in detail or the phenomenon of amplifying noise.

Summary of the invention

In order to improve the sharpness of image, the invention provides a kind of image enhancement technique, can effectively improve the contrast of image, the while can keep image detail and prevent the excessive stretching of gray level.

The technical scheme that technical solution problem of the present invention is taked is as follows:

A kind of algorithm for image enhancement based on gauss hybrid models comprises:

The first step, the luminance component of coloured image is added up into histogram, histogram is carried out to Gaussian modeling;

Suppose that X is input picture, data are histogram data h (x)={ h (x ₁), h (x ₂) ..., h (x _n), the probability distribution of its gray level is p (x), the histogram of image utilizes GMM to construct the form of M Gaussian clustering linear hybrid,

$p\left(x\right)=\underset{n=1}{\overset{M}{\mathrm{\Σ}}}P\left(w_n\right)p\left(x\right|w_n)---\left(1\right)$

In formula (1), p (x|w _n) be the probability density function of n cluster, P (w _n) be the weighting coefficient of n cluster;

Second step, utilize improved EM algorithm to carry out gauss hybrid models estimation to histogram, find the parameter of likelihood function expectation maximization, self-adaptation is determined best number of clusters simultaneously;

Above-mentioned improved EM algorithm is as follows:

1) E step, by data X and current estimation calculate the expectation value that likelihood is estimated, pass through conditional expectation p (x|w by formula (2) _n) obtain p (w _n| x), then obtain final expectation function by formula (3):

$p\left(w_n\right|x)=\frac{p\left(x\right|w_n)P\left(w_n\right)}{\underset{n=1}{\overset{M}{\mathrm{\Σ}}}p\left(x\right|w_n)P\left(w_n\right)}---\left(2\right)$

$\begin{array}{c}Q(\mathrm{\θ},\widehat{{\mathrm{\θ}}_t}))=E[l\left(\mathrm{\θ}\right)\left|\widehat{\mathrm{\θ}}\right]=\\ \underset{x\∈L}{\mathrm{\Σ}}\underset{n=1}{\overset{M}{\mathrm{\Σ}}}p\left(w_n\right|x)h\left(x\right)\log \left[P\left(w_n\right)p\left(x\right|w_n)\right]\end{array}---\left(3\right)$

In formula (3), p (w _n| x) represent with the t time iteration result as the probability density function of parameter;

2) M step, obtains satisfied maximized parameter , and P (w _n) upgrade and try to achieve according to following formula

${{\mathrm{\μ}}_w}_n=\frac{\underset{x\∈L}{\mathrm{\Σ}}h\left(x\right)p\left(w_n\right|x)x}{\underset{x\∈L}{\mathrm{\Σ}}h\left(x\right)p\left(w_n\right|x)}---\left(4\right)$

${\mathrm{\σ}}_{w_n}^2=\frac{\underset{x\∈L}{\mathrm{\Σ}}h\left(x\right)p\left(w_n\right|x){(x-{{\mathrm{\μ}}_w}_n)}^2}{\underset{x\∈L}{\mathrm{\Σ}}h\left(x\right)p\left(w_n\right|x)}---\left(5\right)$

$P\left(w_n\right)=\frac{\underset{x\∈L}{\mathrm{\Σ}}h\left(x\right)p\left(w_n\right|x)}{\underset{x\∈L}{\mathrm{\Σ}}h\left(x\right)}---\left(6\right)$

In formula, h (x) is the histogram of statistical pixel number; P (w _n| x) by p (x|w _n) obtain by Bayesian formula; Application fitting function l (θ | x) the condition judgement iteration stopping that tends to be steady, before and after choosing here, difference is 10 ^-5for stop condition, in minimum and maximum number of clusters, select to make cluster number that iteration is the most stable as final number of clusters;

$l\left(\mathrm{\θ}\right|x)=\frac{1}{2}\underset{n=1}{\overset{M}{\mathrm{\Σ}}}\log \left(\frac{\mathrm{nP}\left(w_n\right)}{12}\right)+\frac{M}{2}\log \frac{n}{12}+M-\log p\left(w_n\right|x)---\left(7\right)$

Finally obtain the histogram approaching, N is image maximum gray scale, and the form of expression is as follows

$\hat{h}\left(x\right)=N\×\underset{n=1}{\overset{M}{\mathrm{\Σ}}}p\left(x\right|w_n)P\left(w_n\right)---\left(8\right)$

The 3rd step, according to the intersection point of adjacent cluster by histogram subregion, obtain multiple sub-histograms;

The 4th step, basis have the sub-histogram cumulative probability density CDF of mapping relations to equate to find the Gaussian clustering parameter after mapping, and then by the cumulative probability Density Weighted and the mapping function of trying to achieve gray-scale value of Gaussian clustering, and application keep maximum entropy method be tending towards human visual system inching mapping function, obtain final enhancing image.

Beneficial effect of the present invention is: this algorithm can keep image detail effectively, also prevented the luminance saturation phenomenon that gray level overstretching causes simultaneously, effectively improved the contrast of image, strengthening image is all to have obtained good effect at aspect subjective vision perception or objective evaluation aspect.

Accompanying drawing explanation

Gauss hybrid models (GMM) when Fig. 1 is k=3 approaches histogram.

Fig. 2 is the mapping curve in the time of u=0.5 and u=0.2.

Fig. 3 adopts the inventive method to strengthen the experimental result picture of front and back.

Embodiment

Below in conjunction with accompanying drawing, the present invention is described in further details.

The present invention is based on the algorithm for image enhancement of gauss hybrid models, its step is as follows:

First, the luminance component of coloured image is added up into histogram, histogram is carried out to Gaussian modeling, be i.e. initialization Gaussian parameter.Gauss hybrid models (GaussianMixtureModeling, GMM) is the Gaussian distribution linear hybrid with different parameters, the corresponding class mean of each Gaussian clustering, variance and weighting coefficient.Suppose that X is input picture, data are histogram data h (x)={ h (x ₁), h (x ₂) ..., h (x _n), the probability distribution of its gray level is p (x), the histogram of image can utilize GMM to construct the form of M Gaussian clustering linear hybrid,

$p\left(x\right)=\underset{n=1}{\overset{M}{\mathrm{\Σ}}}P\left(w_n\right)p\left(x\right|w_n)---\left(1\right)$

In formula (1), p (x|w _n) be the probability density function of n cluster, P (w _n) be the weighting coefficient of n cluster.

Secondly, EM algorithm (the Expectation-maximizationalgorithm of application enhancements, greatest hope algorithm), histogram is carried out to gauss hybrid models (GMM) to be estimated, find the parameter of likelihood function expectation maximization, simultaneously self-adaptation is determined best number of clusters (not need to for different images parameters).Improved EM algorithm can directly be applied on histogram data, has saved storage space, and improved processing speed than processing image array.Improved EM algorithm is mainly reflected in M step, adds histogram and gray-scale information, has reduced calculated amount, constantly updates average, variance and weighting coefficient by iteration.EM algorithm detailed process is as follows:

1) E step, by data X and current estimation calculate the expectation value that likelihood is estimated, pass through conditional expectation p (x|w by formula (2) _n) obtain p (w _n| x), then obtain final expectation function by formula (3):

$p\left(w_n\right|x)=\frac{p\left(x\right|w_n)P\left(w_n\right)}{\underset{n=1}{\overset{M}{\mathrm{\Σ}}}p\left(x\right|w_n)P\left(w_n\right)}---\left(2\right)$

$\begin{array}{c}Q(\mathrm{\θ},\widehat{{\mathrm{\θ}}_t}))=E[l\left(\mathrm{\θ}\right)\left|\widehat{\mathrm{\θ}}\right]=\\ \underset{x\∈L}{\mathrm{\Σ}}\underset{n=1}{\overset{M}{\mathrm{\Σ}}}p\left(w_n\right|x)h\left(x\right)\log \left[P\left(w_n\right)p\left(x\right|w_n)\right]\end{array}---\left(3\right)$

In formula (3), p (w _n| x) represent with the t time iteration result as the probability density function of parameter.

2) M step, obtains satisfied maximized parameter , and P (w _n) upgrade and try to achieve according to following formula

${{\mathrm{\μ}}_w}_n=\frac{\underset{x\∈L}{\mathrm{\Σ}}h\left(x\right)p\left(w_n\right|x)x}{\underset{x\∈L}{\mathrm{\Σ}}h\left(x\right)p\left(w_n\right|x)}---\left(4\right)$

${\mathrm{\σ}}_{w_n}^2=\frac{\underset{x\∈L}{\mathrm{\Σ}}h\left(x\right)p\left(w_n\right|x){(x-{{\mathrm{\μ}}_w}_n)}^2}{\underset{x\∈L}{\mathrm{\Σ}}h\left(x\right)p\left(w_n\right|x)}---\left(5\right)$

$P\left(w_n\right)=\frac{\underset{x\∈L}{\mathrm{\Σ}}h\left(x\right)p\left(w_n\right|x)}{\underset{x\∈L}{\mathrm{\Σ}}h\left(x\right)}---\left(6\right)$

In formula, h (x) is the histogram of statistical pixel number.P (w _n| x) by p (x|w _n) obtain by Bayesian formula.Application fitting function l (θ | x) the condition judgement iteration stopping that tends to be steady, before and after choosing here, difference is 10 ^-5for stop condition, in minimum and maximum number of clusters, select to make cluster number that iteration is the most stable as final number of clusters.

$l\left(\mathrm{\θ}\right|x)=\frac{1}{2}\underset{n=1}{\overset{M}{\mathrm{\Σ}}}\log \left(\frac{\mathrm{nP}\left(w_n\right)}{12}\right)+\frac{M}{2}\log \frac{n}{12}+M-\log p\left(w_n\right|x)---\left(7\right)$

Finally obtain the histogram approaching, N is image maximum gray scale, and the form of expression is as follows

$\hat{h}\left(x\right)=N\×\underset{n=1}{\overset{M}{\mathrm{\Σ}}}p\left(x\right|w_n)P\left(w_n\right)---\left(8\right)$

Then, by histogram subregion, obtain multiple sub-histograms according to the intersection point of adjacent cluster.As shown in Figure 1, optimum cluster number M=3, the hollow small circle of grey represents the significant intersection point of Gaussian clustering, i.e. subregion point.The solid small circle of grey represents the end points of histogram dynamic range.According to intersection point and end points, histogram is divided into four parts, guarantees that each sub-range has a Gaussian clustering to occupy an leading position.

Finally, equate to find the cluster after mapping according to the sub-histogram accumulated probability density CDF that has mapping relations, i.e. the Gaussian parameter of correspondence in output image, wherein, weighting coefficient is constant, and formula is as follows

${{\mathrm{\μ}}_w}_{k^{\′}}=\frac{(\frac{x_s^{\left(k\right)}-{{\mathrm{\μ}}_w}_k}{x_s^{(k+1)}-{{\mathrm{\μ}}_w}_k}{y}^{(k+1)}-y^{\left(k\right)})}{(\frac{x_s^{\left(k\right)}-{{\mathrm{\μ}}_w}_k}{x_s^{(k+1)}-{{\mathrm{\μ}}_w}_k}-1)}---\left(9\right)$

${{\mathrm{\σ}}_w}_{k^{\′}}=\frac{(y^{\left(k\right)}-{{\mathrm{\μ}}_w}_{k^{\′}})}{x_s^{(k+1)}-{{\mathrm{\μ}}_w}_k}{\mathrm{\σ}}_{w_k}---\left(10\right)$

Mapping function is tried to achieve by all cluster weighted sums in GMM, can not be subject to here to apply the enhancing phenomenon excessively that accumulated probability density causes in original equalization algorithm, and formula is as follows

$y=\underset{i=1}{\overset{N}{\mathrm{\Σ}}}(\left(\frac{x-{{\mathrm{\μ}}_w}_k}{{{\mathrm{\σ}}_w}_k}\right){{\mathrm{\σ}}_w}_{k^{\′}}+{{\mathrm{\μ}}_w}_{k^{\′}})P\left(w_i\right)---\left(11\right)$

Final mapping function application keeps maximum entropy method to be tending towards human visual system adjustment, the image after being enhanced.To asking the differentiate of entropy formula, obtain extreme point, i.e. maximum entropy point.Application method of Lagrange multipliers finds brightness of image u _ysolution,

$u_y=\frac{\mathrm{\λ}{e}^{\mathrm{\λ}}-e^{\mathrm{\λ}}+1}{\mathrm{\λ}(e^{\mathrm{\λ}}-1)}---\left(12\right)$

Therefore, known u _yafter, exist a unique λ corresponding with it, just can find final mapping relations by through type (10).

$c\left(y\right)={\∫}_0^{y}f\left(t\right)=\left\{\begin{array}{cc}y,\mathrm{if}& u_y=0.5\\ \frac{e^{\mathrm{\λy}}-1}{e^{\mathrm{\λ}}-1},\mathrm{if}& u_y\∈\left(\mathrm{0,0.5}\right)\∪\left(\mathrm{0.5,1}\right)\end{array}\right.---\left(13\right)$

According to the visual characteristic of human eye, human eye is stronger to low gray level recognition capability, to high grade grey level recognition capability a little less than.As shown in Figure 2, when u=0.5, it is linear that c (y) function is; When u=0.2, c (y) is a concave function.By adjusting shining upon rear function, low gray level is suitably compressed, and high grade grey level suitably stretches, and makes it more level off to human visual system, so improved image can recognition capability.

Fig. 3 is three groups of experimental result pictures that adopt after the algorithm for image enhancement that the present invention is based on gauss hybrid models.

Claims (1)

1. the algorithm for image enhancement based on gauss hybrid models, is characterized in that, this algorithm comprises the steps:

The first step, the luminance component of coloured image is added up into histogram, histogram is carried out to Gaussian modeling;

Suppose that X is input picture, data are histogram data h (x)={ h (x ₁), h (x ₂) ..., h (x _n), the probability distribution of its gray level is p (x), the histogram of image utilizes GMM to construct the form of M Gaussian clustering linear hybrid,

$p\left(x\right)=\underset{n=1}{\overset{M}{\mathrm{\Σ}}}P\left(w_n\right)p\left(x\right|w_n)---\left(1\right)$

In formula (1), p (x|w _n) be the probability density function of n cluster, P (w _n) be the weighting coefficient of n cluster;

Second step, utilize improved EM algorithm to carry out gauss hybrid models estimation to histogram, find the parameter of likelihood function expectation maximization, self-adaptation is determined best number of clusters simultaneously;

Above-mentioned improved EM algorithm is as follows:

1) E step, by data X and current estimation calculate the expectation value that likelihood is estimated, pass through conditional expectation p (x|w by formula (2) _n) obtain p (w _n| x), then obtain final expectation function by formula (3):

$p\left(w_n\right|x)=\frac{p\left(x\right|w_n)P\left(w_n\right)}{\underset{n=1}{\overset{M}{\mathrm{\Σ}}}p\left(x\right|w_n)P\left(w_n\right)}---\left(2\right)$

$\begin{array}{c}Q(\mathrm{\θ},\widehat{{\mathrm{\θ}}_t}))=E[l\left(\mathrm{\θ}\right)\left|\widehat{\mathrm{\θ}}\right]=\\ \underset{x\∈L}{\mathrm{\Σ}}\underset{n=1}{\overset{M}{\mathrm{\Σ}}}p\left(w_n\right|x)h\left(x\right)\log \left[P\left(w_n\right)p\left(x\right|w_n)\right]\end{array}---\left(3\right)$

In formula (3), p (w _n| x) represent with the t time iteration result as the probability density function of parameter;

2) M step, obtains satisfied maximized parameter , and P (w _n) upgrade and try to achieve according to following formula

${{\mathrm{\μ}}_w}_n=\frac{\underset{x\∈L}{\mathrm{\Σ}}h\left(x\right)p\left(w_n\right|x)x}{\underset{x\∈L}{\mathrm{\Σ}}h\left(x\right)p\left(w_n\right|x)}---\left(4\right)$

${\mathrm{\σ}}_{w_n}^2=\frac{\underset{x\∈L}{\mathrm{\Σ}}h\left(x\right)p\left(w_n\right|x){(x-{{\mathrm{\μ}}_w}_n)}^2}{\underset{x\∈L}{\mathrm{\Σ}}h\left(x\right)p\left(w_n\right|x)}---\left(5\right)$

$P\left(w_n\right)=\frac{\underset{x\∈L}{\mathrm{\Σ}}h\left(x\right)p\left(w_n\right|x)}{\underset{x\∈L}{\mathrm{\Σ}}h\left(x\right)}---\left(6\right)$

In formula, h (x) is the histogram of statistical pixel number; P (w _n| x) by p (x|w _n) obtain by Bayesian formula; Application fitting function l (θ | x) the condition judgement iteration stopping that tends to be steady, before and after choosing here, difference is 10 ^-5for stop condition, in minimum and maximum number of clusters, select to make cluster number that iteration is the most stable as final number of clusters;

$l\left(\mathrm{\θ}\right|x)=\frac{1}{2}\underset{n=1}{\overset{M}{\mathrm{\Σ}}}\log \left(\frac{\mathrm{nP}\left(w_n\right)}{12}\right)+\frac{M}{2}\log \frac{n}{12}+M-\log p\left(w_n\right|x)---\left(7\right)$

Finally obtain the histogram approaching, N is image maximum gray scale, and the form of expression is as follows

$\hat{h}\left(x\right)=N\×\underset{n=1}{\overset{M}{\mathrm{\Σ}}}p\left(x\right|w_n)P\left(w_n\right)---\left(8\right)$

The 3rd step, according to the intersection point of adjacent cluster by histogram subregion, obtain multiple sub-histograms;

The 4th step, basis have the sub-histogram cumulative probability density CDF of mapping relations to equate to find the Gaussian clustering parameter after mapping, and then by the cumulative probability Density Weighted and the mapping function of trying to achieve gray-scale value of Gaussian clustering, and application keep maximum entropy method be tending towards human visual system inching mapping function, obtain final enhancing image.