CN101887576A - Image de-noising method based on partial differential equation filter - Google Patents

Image de-noising method based on partial differential equation filter Download PDF

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CN101887576A
CN101887576A CN 201010192760 CN201010192760A CN101887576A CN 101887576 A CN101887576 A CN 101887576A CN 201010192760 CN201010192760 CN 201010192760 CN 201010192760 A CN201010192760 A CN 201010192760A CN 101887576 A CN101887576 A CN 101887576A
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image
differential equation
partial differential
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gradient
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张小华
焦李成
王然
王爽
侯彪
马文萍
尚荣华
盖超
张强
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Xidian University
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Abstract

The invention discloses an image de-nosing method based on a partial differential equation filter, mainly solving the problems that the traditional de-nosing method has weaker de-nosing effect and performance. The method comprises the following steps of: (1) inputting a noising image u and calculating a partial derivative of the image u; (2) calculating a gradient modulus absolute Delta<u> of the noising image u; (3) establishing a partial differential equation according to the gradient Delta <u> and the gradient modulus absolute Delta <u>; (4) calculating a diffusion coefficient and Phi in the partial differential equation; (5) solving the partial differential equation to obtain a filter image by utilizing the coefficient and Phi; (6) calculating the PSNR (Peak Signal to Noise Ratio) of the filter image; and (7) repeating from the step 1 to the step 6. When the PSNR value of the filter image output at some iteration is less than that of the filter image iteratively output at the last time, the iteration is stopped, and the filter image at the last iteration is output. The invention can carry out filtering by utilizing the detailed structures of the image, has simple calculation and fast operation speed, can keep image texture details better at the same time of smoothening noise and can be used for the de-noising treatment of natural images.

Description

Image de-noising method based on partial differential equation filter
Technical field
The invention belongs to technical field of image processing, relate to image de-noising method, be applicable to the noise remove of SAR image and natural image.
Background technology
Image denoising is intended to handle to reduce noise the influence of original useful information is restored more approaching Utopian image as much as possible undertaken certain by the image of noise pollution by algorithm; it is to carry out the preconditioning technique that often can use in the field Flame Image Process such as forest inventory investigation, soil utilization, covering variation research, environmental hazard assessment, city planning, the monitoring of national defence military situation, medical image and uranology image, has exigence and application prospects.Synthetic-aperture radar SAR image and natural image all can need denoising, and research SAR image and natural image denoising technology have boundless application prospect.
In order to satisfy, very many denoising methods have been emerged at present, as wavelet method, beamlet, shearlet, countlet, non-local mean method or the like to pressing for that image denoising is used.Though the purpose that these denoising methods can reasonable realization denoising, for the profuse image of image detail, the denoising result of these methods is all not ideal enough, can not reach the specific (special) requirements of denoising effect.
In order to solve the problem of said method, become the hot issue of image denoising area research based on the denoising method of partial differential equation filter, many scholars classify to existing denoising method from different angles, analyze and improve, but for the different characteristic area of image, the effect of denoising is not very good.Particularly require than higher image for some real-times, the speed of denoising is very slow, influences the operate as normal of follow-up system.
Summary of the invention
The objective of the invention is to overcome the shortcoming of above-mentioned prior art, proposed a kind of image de-noising method,, improve the effect and the speed of denoising with the image denoising of automatic realization based on SAR image and natural image feature based on partial differential equation filter.
Realize that technical scheme of the present invention is: the gradient-norm value and the partial derivative that calculate pending noise image earlier
Figure BSA00000147955400011
With in the Partial Differential Equation method denoising model of noise image substitution based on characteristics of image, calculate the recovery value of each pixel again, thereby obtain final filtering result images by the partial differential equation in the solving model.Concrete steps comprise as follows:
(1) the input size is M * N single width noise image u, calculates the partial derivative of this image u in the x direction
Figure BSA00000147955400021
Partial derivative with the y direction
Figure BSA00000147955400022
(2) utilize the gradient-norm value of gradient formula calculating noise image u | ▽ u|;
▽u＝(u x，u y)，
| &dtri; u | = u x 2 + u y 2 ,
Wherein, ▽ u is the gradient of noise image u, u xExpression
Figure BSA00000147955400024
u yExpression
Figure BSA00000147955400025
(3) according to gradient ▽ u that calculates in the step (2) and gradient-norm value | ▽ u|, it is as follows to set up partial differential equation:
Figure BSA00000147955400026
Wherein,
Figure BSA00000147955400027
Expression noise image u is about the partial derivative of time t;
Figure BSA00000147955400028
Main diffusion coefficient for flat site;
ψ is the main diffusion coefficient of fringe region;
Div () is a divergence;
G (| ▽ u|) for the diffusion adjustment function, be used to control the diffusion of noise image u on certain direction,
Figure BSA00000147955400029
Wherein k is used to judge that certain pixel is the image border or the threshold value of flat site, k=k 0e -t, k wherein 0Be initial value, t is an iteration time, t=Δ t (n-1), and wherein Δ t is an iteration step length, n is an iterations;
U (0) expression zero image constantly, u (0)=u 0Expression zero initial input image constantly is u 0
(4) main diffusion coefficient of the flat site in the calculating partial differential equation
Figure BSA000001479554000210
Main diffusion coefficient ψ with fringe region:
Figure BSA000001479554000211
Figure BSA000001479554000212
Wherein, h is an empirical value, gets 0.5～0.9;
(5) utilize coefficient in the partial differential equation calculate
Figure BSA00000147955400031
And ψ, obtaining each gray values of pixel points by the partial differential equation in the solution procedure (3), these pixels are formed filtering image;
(6) Y-PSNR of calculation of filtered image: PSNR=20log 10(255/RMSE),
Wherein, the 255th, maximum gray scale,
Figure BSA00000147955400032
(i j) is the grey scale pixel value of the filtering image that obtains in the step (5) to f, and (i be the grey scale pixel value of the noise image u that imports in the step (1) j) to F, and i and j are the pixel coordinate in the image;
(7) repeating step 1 is to step 6, during the PSNR value of the filtering image of exporting less than last iteration when the PSNR value of the filtering image of certain iteration output, and termination of iterations, the filtering image of exporting last iteration is a denoising result.
The present invention has the following advantages compared with prior art:
1. the present invention is because the model that proposes has utilized the diffusion adjustment factor
Figure BSA00000147955400033
And ψ, take different diffusion smoothing strategies at different details area, have stronger adaptivity;
2. in case the stopping criterion for iteration of the present invention's employing is effect assessment indices P SNR to occur to descend, the denoising process stops immediately, has improved the travelling speed of denoising like this;
3. the present invention adopts the threshold k of bringing in constant renewal in iterations, makes the accuracy of denoising be greatly improved, and has improved denoising effect.
Description of drawings
Fig. 1 is realization flow figure of the present invention;
Fig. 2 is to the denoising result comparison diagram of lena figure among the present invention;
Fig. 3 is to the denoising result comparison diagram of Barbara figure among the present invention;
Fig. 4 is to the denoising result comparison diagram of camera figure among the present invention.
Embodiment
With reference to Fig. 1, concrete implementation step of the present invention is as follows:
Step 1. input size is M * N single width noise image u, calculates the partial derivative of this image u in the x direction
Figure BSA00000147955400034
Partial derivative with the y direction
Figure BSA00000147955400035
Step 2. is according to the partial derivative that calculates
Figure BSA00000147955400036
With Utilize gradient ▽ u and the gradient-norm value of gradient formula calculating noise image u | ▽ u|.
Gradient ▽ u=(u x, u y), the gradient-norm value
Figure BSA00000147955400038
Wherein, u xExpression
Figure BSA00000147955400039
u yExpression
Figure BSA000001479554000310
In the image denoising process, require the detailed information of reservation image as much as possible itself and remove The noise.The region gradient mould value more at image detail is bigger, carries out less diffusion smoothing; Less in the flat site gradient-norm value that details is less, carry out more diffusion smoothing.
Step 3. is according to gradient ▽ u that calculates in the step 2 and gradient-norm value | and ▽ u|, set up partial differential equation.
It is as follows to set up partial differential equation:
Figure BSA00000147955400041
Wherein,
Figure BSA00000147955400042
Expression noise image u is about the partial derivative of time t;
Figure BSA00000147955400043
Be the main diffusion coefficient of flat site, When value was big, flat site mainly spread according to the first half div in the partial differential equation (▽ u/| ▽ u|);
ψ is the main diffusion coefficient of fringe region, and when the ψ value was big, fringe region was mainly according to the latter half div in the partial differential equation (g (| ▽ u|) ▽ u) spread;
Div () is a divergence;
G (| ▽ u|) for the diffusion adjustment function, be used to control the diffusion of noise image u on certain direction,
Figure BSA00000147955400045
Wherein k is used to judge that certain pixel is the image border or the threshold value of flat site, k=k 0e -t, k wherein 0Be initial value, t is an iteration time, t=Δ t (n-1), and wherein Δ t is an iteration step length, n is an iterations;
U (0) expression zero image constantly, u (0)=u 0Expression zero initial input image constantly is u 0
In the smoothing process of image u, the gradient of image u | ▽ u| constantly changes along with iteration develops, so judging certain pixel is that the image border or the threshold value k of flat site can not be set to constant again, and should be a decreasing function k=k who constantly changes with iterations 0e -t, wherein e is an exponential constant, and threshold value is constantly upgraded, and the effect of denoising just can be significantly improved.
Step 4. is calculated the main diffusion coefficient of the flat site in the partial differential equation
Figure BSA00000147955400046
Main diffusion coefficient ψ with fringe region.
The main diffusion coefficient of flat site
Figure BSA00000147955400051
Wherein, h is an empirical value, gets 0.5～0.9; At flat site, the gradient-norm value | ▽ u| is less, so
Figure BSA00000147955400052
Be worth lessly, then the ψ value is bigger, at div (g (| ▽ u|) ▽ u) the bigger situation of coefficient of diffusion ψ under, image is mainly according to div (g (| ▽ u|) ▽ u) carry out diffusion smoothing, can reasonable removal high gradient noise;
The main diffusion coefficient of fringe region
Figure BSA00000147955400053
In the more rich zone of image detail, the gradient-norm value | ▽ u| is bigger, so
Figure BSA00000147955400054
Be worth greatlyyer, then the ψ value is less, at the coefficient of diffusion of div (▽ u/| ▽ u|)
Figure BSA00000147955400055
Under the bigger situation, image mainly carries out diffusion smoothing according to div (▽ u/| ▽ u|), can keep the original detailed information of image when removing partial noise preferably.
Step 5. is according to the coefficient in the partial differential equation that calculates
Figure BSA00000147955400056
And ψ, obtaining each gray values of pixel points by the partial differential equation in the solution procedure 3, these pixels are formed filtering image.
The diffusion adjustment factor of partial differential equation With ψ is to take different diffusion smoothing strategies at different details area, with coefficient
Figure BSA00000147955400058
Bring in the partial differential equation with ψ.In finding the solution the process of partial differential equation, can adaptive adjusting dispersal direction go to each zone of smoothed image, less diffusion smoothing effect is carried out in more zone in the image border, and less flat site carries out more diffusion smoothing effect at the edge.
The Y-PSNR of step 6. calculation of filtered image.
Y-PSNR is a main quantizating index of estimating image denoising effect, and it is in order to determine next step termination of iterations time that each iteration all will be calculated Y-PSNR, and the Y-PSNR computing formula is as follows:
PSNR＝20log 10(255/RMSE)，
Wherein, the 255th, maximum gray scale,
Figure BSA00000147955400059
(i j) is the grey scale pixel value of the filtering image that obtains in the step (5) to f, and (i be the grey scale pixel value of the noise image u that imports in the step 1 j) to F, and i and j are the pixel coordinate in the image.
Step 7. repeating step 1 is to step 6, during the PSNR value of the filtering image of exporting less than last iteration when the PSNR value of the filtering image of certain iteration output, and termination of iterations, the filtering image of exporting last iteration is a denoising result.
In iterative process, if PSNR descends, occurred smoothly with regard to the key diagram picture, iteration has reached optimum efficiency in previous step, therefore, termination of iterations, the filtering image of output previous step iteration, promptly final denoising result image.
Effect of the present invention can further confirm by following experiment:
One. experiment condition and content
Experiment condition: adopt as Fig. 2 (a), Fig. 3 (a) and the described original noise-free picture of Fig. 4 (a), as experiment effect with reference to image.Test used input picture shown in Fig. 2 (b), Fig. 3 (b) and Fig. 4 (b).Fig. 2 (b) is that Fig. 2 (a) adding noise criteria difference is 10 noise image, and Fig. 3 (b) is that Fig. 3 (a) adding noise criteria difference is 10 noise image, and Fig. 4 (b) is 20 noise image for Fig. 4 (a) adding noise criteria difference.In the experiment, k 0Get 20, Δ t gets 0.1, and h gets 0.5.In the experiment, various filtering methods all are to use the MATLAB Programming with Pascal Language to realize.
Experiment content: under above-mentioned experiment condition, utilize PM method, TV method respectively and carry out the denoising emulation experiment, and provide experimental result and comparison based on the image de-noising method of partial differential equation filter.
Two. experimental result
A. Fig. 2 (b) is carried out the filtering emulation experiment with PM method, TV method with based on the image de-noising method of partial differential equation filter respectively, the filtering result who wherein uses the PM method is shown in Fig. 2 (c), shown in Fig. 2 (d), use filtering result based on the image de-noising method of partial differential equation filter with the filtering result of TV method shown in Fig. 2 (e).Figure is as can be seen as a result from these, the present invention is based on partial differential equation filter image de-noising method filtering as a result its minutia all obtained better reservation, visual effect is more near original image Fig. 2 (a), Y-PSNR PSNR is 34.18 simultaneously, also is higher than the PSNR of existing P M and two kinds of methods of TV.
B. Fig. 3 (b) is carried out the filtering emulation experiment with PM method, TV method with based on the image de-noising method of partial differential equation filter respectively, the filtering result who wherein uses the PM method is shown in Fig. 3 (c), shown in Fig. 3 (d), use filtering result based on the image de-noising method of partial differential equation filter with the filtering result of TV method shown in Fig. 3 (e).From the result of Fig. 3 as can be seen, what the marginal portion of Fig. 3 (e) kept is Fig. 3 (a) near original image more, keeps better such as the grid of clothes among Fig. 3 (e), and its Y-PSNR PSNR=31.16 also will be higher than the PSNR of Fig. 3 (c) and Fig. 3 (d).
C. Fig. 4 (b) is carried out the filtering emulation experiment with PM method, TV method and the image de-noising method that the present invention is based on partial differential equation filter respectively, the filtering result who wherein uses the PM method is shown in Fig. 4 (c), with the filtering result of TV method shown in Fig. 4 (d), with the filtering result of the image de-noising method that the present invention is based on partial differential equation filter shown in Fig. 4 (e).Singular point occurred the figure as can be seen from the result of Fig. 4, illustrated that the PM filtering method lost efficacy substantially under the very noisy level.Fig. 4 (d) is not though singular point occurs, and the filtering of flat site is not ideal enough.The visual effect of Fig. 4 (e) is compared the two kinds of methods in front and is significantly improved, and Y-PSNR also will be higher than the former.
Table 1 is among the present invention Fig. 2 (b), Fig. 3 (b) and the filtering result of Fig. 4 (b) under different noise levels to be quantized contrast.Wherein, Sigma is that noise criteria is poor, and Time is for testing working time, and unit is second.
The contrast of table 1 experimental result
Figure BSA00000147955400071
Table 1 is the result show, the present invention is based on the effect that the image de-noising method of partial differential equation filter carries out filtering to above-mentioned three kinds of images under different noise levels effect all is better than PM and TV method.Simultaneously, the algorithm speed that the present invention is based on the image de-noising method of partial differential equation filter obviously is better than the TV method.

Claims (1)

1. the image de-noising method based on partial differential equation filter comprises the steps:
(1) the input size is M * N single width noise image u, calculates the partial derivative of this image u in the x direction Partial derivative with the y direction
Figure FSA00000147955300012
(2) utilize the gradient-norm value of gradient formula calculating noise image u | ▽ u|:
▽u＝(u x，u y)，
| &dtri; u | = u x 2 + u y 2 ,
Wherein, ▽ u is the gradient of noise image u, u xExpression u yExpression
Figure FSA00000147955300015
(3) according to gradient ▽ u that calculates in the step (2) and gradient-norm value | ▽ u|, it is as follows to set up partial differential equation:
Figure FSA00000147955300016
Wherein, Expression noise image u is about the partial derivative of time t;
Figure FSA00000147955300018
Main diffusion coefficient for flat site;
ψ is the main diffusion coefficient of fringe region;
Div () is a divergence;
G (| ▽ u|) for the diffusion adjustment function, be used to control the diffusion of noise image u on certain direction,
Figure FSA00000147955300019
Wherein k is used to judge that certain pixel is the image border or the threshold value of flat site, k=k 0e -t, k wherein 0Be initial value, t is an iteration time, t=Δ t (n-1), and wherein Δ t is an iteration step length, n is an iterations;
U (0) expression zero image constantly, u (0)=u 0Expression zero initial input image constantly is u 0
(4) main diffusion coefficient of the flat site in the calculating partial differential equation
Figure FSA000001479553000110
Main diffusion coefficient ψ with fringe region:
Figure FSA00000147955300021
Figure FSA00000147955300022
Wherein, h is an empirical value, gets 0.5～0.9;
(5) utilize coefficient in the partial differential equation calculate
Figure FSA00000147955300023
And ψ, obtaining each gray values of pixel points by the partial differential equation in the solution procedure (3), these pixels are formed filtering image;
(6) Y-PSNR of calculation of filtered image: PSNR=20log 10(255/RMSE),
Wherein, the 255th, maximum gray scale,
Figure FSA00000147955300024
(i j) is the grey scale pixel value of the filtering image that obtains in the step (5) to f, and (i be the grey scale pixel value of the noise image u that imports in the step (1) j) to F, and i and j are the pixel coordinate in the image;
(7) repeating step 1 is to step 6, during the PSNR value of the filtering image of exporting less than last iteration when the PSNR value of the filtering image of certain iteration output, and termination of iterations, the filtering image of exporting last iteration is a denoising result.
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Application publication date: 20101117