CN103914811B - A kind of algorithm for image enhancement based on gauss hybrid models - Google Patents

Abstract

A kind of algorithm for image enhancement based on gauss hybrid models relates to technical field of image processing, and the method is: first, and the luminance component of coloured image is added up into rectangular histogram, and rectangular histogram is carried out Gaussian modeling；Secondly, the EM algorithm of application enhancements, rectangular histogram is carried out gauss hybrid models estimation, finds the parameter of likelihood function expectation maximization, self adaptation determines optimal number of clusters simultaneously；Then, according to the intersection point of neighboring clusters by rectangular histogram subregion, it is thus achieved that many sub-rectangular histograms；Finally, find the cluster after mapping according to the sub-rectangular histogram area ratio having mapping relations is equal, and apply holding maximum entropy method to tend to human visual system inching mapping function, obtain final enhancing image.The image enhancement technique that the present invention uses, is effectively improved the contrast of image, and improves processing speed.In terms of subjective vision perceptible aspect or objective evaluation, good effect is either all achieved by the enhancing image acquired in the inventive method.

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 image based on gauss hybrid models and increase Strong algorithms.

Background technology

Image information is more and more used by people to identify and judge things, solves actual problem.But by In factors such as weather brightness, conditions of exposures, cause brightness of image dark the fuzzyyest, tend not to meet and answer Demand, this can badly influence the identification to target.This class image typically presents gray level and compares Concentrating, the characteristic that the contrast of image is low, therefore, the contrast improving image carries out later stage process to image It is very important.Histogram modification technology receives publicity because of its simple easily realization.Wherein, rectangular histogram Equalize and be used widely in terms of improving picture contrast, but be currently based on the enhancing algorithm of histogram equalization Luminance saturation, loss in detail or the phenomenon of amplification noise easily occur.

Summary of the invention

In order to improve the definition of image, the invention provides a kind of image enhancement technique, can effectively carry The contrast of hi-vision, can keep image detail and prevent the excessive tensile of gray level simultaneously.

It is as follows that the present invention solves the technical scheme that technical problem taked:

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

The first step, the luminance component of coloured image is added up into rectangular histogram, rectangular histogram is carried out mixed Gaussian and builds Mould；

Assuming that X is input picture, data are histogram data h (x)={ h (x₁),h(x₂),...,h(x_N), its gray level Probability distribution is p (x), then the rectangular histogram of image utilizes GMM to construct M Gaussian clustering linear hybrid Form, i.e.

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

In formula (1), and p (x | w_n) it is the probability density function of the n-th cluster, P (w_n) it is the weighting of the n-th cluster Coefficient；

The EM algorithm that second step, utilization improve carries out gauss hybrid models estimation to rectangular histogram, finds likelihood The parameter of function expectation maximization, self adaptation determines optimal number of clusters simultaneously；

The EM algorithm of above-mentioned improvement is as follows:

1) E step, by data X and current estimationCalculate the expected value of possibility predication, by formula (2) by the condition phase Prestige p (x | w_n) obtain p (w_n| x), then obtained 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 resultProbability density function as parameter；

2) M step, obtains satisfiedMaximized parameter 、With P (w_n) according to following equation Renewal is tried to achieve

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

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

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

In formula, h (x) is the rectangular histogram of statistical pixel number；p(w_n| x) by p (x | w_n) obtained by Bayesian formula； Application fitting function l (θ | x) state that tends to be steady judgement iteration stopping, choosing before and after's difference here is 10^-5For stopping Only condition, selects to make iteration most stable of cluster number gather as final in minimum and maximum number of clusters Class quantity；

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

Finally obtaining the rectangular histogram approached, N is image maximum gray scale, and the form of expression is as follows

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

3rd step, according to the intersection point of neighboring clusters by rectangular histogram subregion, it is thus achieved that many sub-rectangular histograms；

4th step, according to having that sub-rectangular histogram cumulative probability density CDF of mapping relations is equal and finding mapping after Gaussian clustering parameter, and then by the cumulative probability Density Weighted of Gaussian clustering and the mapping letter trying to achieve gray value Number, and apply holding maximum entropy method to tend to human visual system inching mapping function, obtain final increasing Strong image.

The medicine have the advantages that this algorithm can keep image detail effectively, also prevent gray level simultaneously The luminance saturation phenomenon that overstretching causes, is effectively improved the contrast of image, strengthens image and either exists Subjective vision perceptible aspect or objective evaluation aspect all achieves good effect.

Accompanying drawing explanation

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

Fig. 2 is the mapping curve as u=0.5 and u=0.2.

Fig. 3 is the experimental result picture before and after using the inventive method to strengthen.

Detailed description of the invention

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

Present invention algorithm for image enhancement based on gauss hybrid models, its step is as follows:

First, the luminance component of coloured image is added up into rectangular histogram, rectangular histogram is carried out Gaussian modeling, I.e. initialize Gaussian parameter.Gauss hybrid models (GaussianMixtureModeling, GMM) is to have not The Gauss distribution linear hybrid of same parameter, the corresponding class mean of each Gaussian clustering, variance and weight coefficient. Assuming that X is input picture, data are histogram data h (x)={ h (x₁),h(x₂),...,h(x_N), the probability of its gray level Be distributed as p (x), then the rectangular histogram of image can utilize GMM to construct M Gaussian clustering linear hybrid Form, i.e.

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

In formula (1), and p (x | w_n) it is the probability density function of the n-th cluster, P (w_n) it is the weight coefficient of the n-th cluster.

Secondly, EM algorithm (Expectation-maximizationalgorithm, the greatest hope of application enhancements Algorithm), rectangular histogram is carried out gauss hybrid models (GMM) and estimates, find likelihood function expectation maximization Parameter, self adaptation determines optimal number of clusters (need not arrange parameter for different images) simultaneously.Change The EM algorithm entered can be directly applied on histogram data, saves storage sky compared to processing image array Between, and improve processing speed.Improve EM algorithm be mainly reflected in M step, add rectangular histogram with Gray-scale information, decreases amount of calculation, constantly updates average, variance and weight coefficient by iteration.EM calculates Method detailed process is as follows:

1) E step, by data X and current estimationCalculate the expected value of possibility predication, by formula (2) by the condition phase Prestige p (x | w_n) obtain p (w_n| x), then obtained 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 resultProbability density function as parameter.

2) M step, obtains satisfiedMaximized parameter 、With P (w_n) according to following public affairs Formula updates tries to achieve

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

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

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

In formula, h (x) is the rectangular histogram of statistical pixel number.p(w_n| x) by p (x | w_n) obtained by Bayesian formula. Application fitting function l (θ | x) state that tends to be steady judgement iteration stopping, choosing before and after's difference here is 10^-5For stopping Only condition, selects to make iteration most stable of cluster number gather as final in minimum and maximum number of clusters Class quantity.

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

Finally obtaining the rectangular histogram approached, 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, according to the intersection point of neighboring clusters by rectangular histogram subregion, it is thus achieved that many sub-rectangular histograms.Such as Fig. 1 institute Showing, optimum cluster number M=3, the hollow small circle of Lycoperdon polymorphum Vitt represents the significant intersection point of Gaussian clustering, i.e. subregion Point.The solid small circle of Lycoperdon polymorphum Vitt represents the end points of rectangular histogram dynamic range.According to intersection point and end points, rectangular histogram is divided It is four parts, it is ensured that each subinterval has a Gaussian clustering to occupy an leading position.

Finally, gathering after finding mapping according to sub-rectangular histogram accumulated probability density CDF having mapping relations is equal Gaussian parameter corresponding in class, i.e. output image, wherein, weight 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 cluster weighted sums all in GMM, will not be answered by original equalization algorithm here Crossing enhancing phenomenon with what accumulated probability density caused, 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 holding maximum entropy method tends to human visual system and is adjusted, and is increased Image after Qiang.To asking entropy formula derivation, obtain extreme point, i.e. maximum entropy point.Application Lagrange multiplier Method finds brightness of image u_ySolution, i.e.

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

It is thus known that u_yAfter, there is a unique λ the most corresponding, it is possible to found finally by formula (10) Mapping relations.

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

According to the visual characteristic of human eye, human eye is relatively strong, to high grade grey level identification energy to low gray level identification ability Power is more weak.As in figure 2 it is shown, during u=0.5, c (y) function is linearly；During u=0.2, c (y) is a concave function. By being adjusted function after mapping, low gray level is suitably compressed, and high grade grey level suitably stretches so that it is more Level off to human visual system, and then improve the recognizable ability of image.

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

Claims (1)

1. an algorithm for image enhancement based on gauss hybrid models, it is characterised in that this algorithm includes as follows Step:

The first step, the luminance component of coloured image is added up into rectangular histogram, rectangular histogram is carried out mixed Gaussian and builds Mould；

Assuming that X is input picture, data are histogram data h (x)={ h (x₁),h(x₂),...,h(x_N), its gray level Probability distribution is p (x), then the rectangular histogram of image utilizes GMM to construct M Gaussian clustering linear hybrid Form, i.e.

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

In formula (1), and p (x | w_n) it is the probability density function of the n-th cluster, P (w_n) it is the weighting of the n-th cluster Coefficient；

The EM algorithm that second step, utilization improve carries out gauss hybrid models estimation to rectangular histogram, finds likelihood The parameter of function expectation maximization, self adaptation determines optimal number of clusters simultaneously；

The EM algorithm of above-mentioned improvement is as follows:

1) E step, by data X and current estimationCalculate the expected value of possibility predication, by formula (2) by the condition phase Prestige p (x | w_n) obtain p (w_n| x), then obtained 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 resultProbability density function as parameter；

2) M step, obtains satisfiedMaximized parameter 、With P (w_n) according to following equation Renewal is tried to achieve

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

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

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

In formula, h (x) is the rectangular histogram of statistical pixel number；p(w_n| x) by p (x | w_n) obtained by Bayesian formula； Application fitting function l (θ | x) state that tends to be steady judgement iteration stopping, choosing before and after's difference here is 10^-5For stopping Only condition, selects to make iteration most stable of cluster number gather as final in minimum and maximum number of clusters Class quantity；

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

Finally obtaining the rectangular histogram approached, N is image maximum gray scale, and the form of expression is as follows

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

3rd step, according to the intersection point of neighboring clusters by rectangular histogram subregion, it is thus achieved that many sub-rectangular histograms；

4th step, according to having that sub-rectangular histogram cumulative probability density CDF of mapping relations is equal and finding mapping after Gaussian clustering parameter, and then by the cumulative probability Density Weighted of Gaussian clustering and the mapping letter trying to achieve gray value Number, and apply holding maximum entropy method to tend to human visual system inching mapping function, obtain final increasing Strong image.