CN108596859A - A kind of image de-noising method based on partial differential equation of higher order - Google Patents
A kind of image de-noising method based on partial differential equation of higher order Download PDFInfo
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Abstract
The present invention proposes a kind of image de-noising method based on partial differential equation of higher order to solve existing alias when partial differential equation of second order handles noisy image, belongs to the technical field of image procossing.It includes the following steps:The spread function of fourth order PDEs denoising model is set as c (u)=e‑(u/k);Establish denoising model;Discretization is carried out to the continuous model in preceding step with finite difference scheme, obtains the corresponding iterative equation of model, the solution of iterative equation is the numerical solution of PDE model;And emulation experiment is carried out using MATLAB to denoising model, obtain denoising image, Y-PSNR (PSNR), signal-to-noise ratio (SNR), structural similarity (SSIM) are used as evaluation index and obtains its numerical result, its numerical value is directly proportional to image denoising effect, bigger to be worth, image denoising effect is better.The present invention can effectively remove multiplicative noise, while retain image edge detailss information, be compared with classical denoising model, denoising effect is more preferable.
Description
Technical field
The present invention relates to a kind of image de-noising methods based on partial differential equation of higher order, belong to the technology neck of image procossing
Domain.
Background technology
A kind of important way that image is transmitted as information has been dissolved into the various aspects of people's life, wherein clearly
Image can bring us more useful information.However durings image formation, image recording, image transmitting etc., all
It can be interfered by one or more factors so that observed image quality reduces, wherein most typical is exactly noise.In order to obtain
The clear image of high quality needs to carry out denoising to noisy image.
The partial differential equation (partial differential equations, PDE) being applied to earliest in image denoising
It is the equation of heat conduction, but it is to being damaged to a certain extent property of image boundary.Nineteen ninety, Perona and Malik are in the equation of heat conduction
On the basis of propose that the anisotropic diffusion equation on boundary, i.e. P-M models can be kept:
Wherein, u indicates the gray value of image,Indicate divergence,It is gradient operator, c (s) is spread function
Or
It is threshold value for keeping image border, k, the feature for judging image.P-M models eliminate to a certain extent
Noise improves the clarity of image, but the image after denoising has " alias ".
In order to overcome this problem, scholars propose numerous high order PD E models.Wherein more classical has broad sense P-M
Model and Y-K models.Broad sense P-M models are that G.W.Wei in 1999 proposes that model is as follows:
Y-K models are 2000, what Y-L.You and M.Kaveh were proposed, and model is as follows:
ut=-△ (c (| △ u |) △ u)
2014, Min proposed a kind of denoising model (being called M models in the present invention):
In addition, proposing " mixing " denoising model based on PDE there are many more scholar has reached certain denoising result,
But it can not effectively solve to be effectively maintained " ladder effect caused by image edge detailss information while removal multiplicative noise
Answer " problem.
Invention content
To solve the deficiencies in the prior art, the purpose of the present invention is to provide a kind of images based on partial differential equation of higher order
Denoising method solves the problems, such as existing for second order PDE models " alias ", realizes the more efficient denoising to image.
In order to realize that above-mentioned target, the present invention adopt the following technical scheme that:
A kind of image de-noising method based on partial differential equation of higher order, characterized in that include the following steps:
Step 1, set the spread function of fourth order PDEs denoising model as
C (u)=e-(u/k)
Wherein u indicates the gamma function of image.
Step 2 establishes denoising model
Wherein, △ is Laplace operator, constant k (k>0) it is threshold value, the feature for judging image.
Step 3 carries out discretization with finite difference scheme to the continuous model in step 2, obtains the corresponding iteration of model
Equation, the solution of iterative equation are the numerical solution of PDE model.
Step 4 carries out emulation experiment to the model in step 2 using MATLAB, obtains denoising image, uses Y-PSNR
(PSNR), signal-to-noise ratio (SNR), structural similarity (SSIM) as evaluation index and obtain its numerical result, numerical value and image
Denoising effect is directly proportional, and bigger to be worth, image denoising effect is better.
Further, the step 3 specifically includes the following contents:
For the image that a width size is M × N, method is dissected using grid, is enabled
If um,n=u (m, n), boundary are handled in the following way
u0,n=u1,n uM+1,n=uM,n um,0=um,1 um,N+1=um,N
M=1,2 ..., M n=1,2 ..., N
If time step is △ t, spatial mesh size h=1, p indicate iterations, then
T=p △ t p=0,1,2 ...
X=m m=0,1,2 ..., M
Y=n n=0,1,2 ..., N
Laplace operator is calculated using central difference schemes:
M=0,1,2 ..., M n=0,1,2 ..., N
According to the model in step 2, there is iterative equation
The result of pth time iteration is up+1, i.e.,
Further, evaluation index Y-PSNR (PSNR) is in step 4 described in evaluation index in the step 4
Further, evaluation index signal-to-noise ratio (SNR) is in the step 4
WhereinFor the mean value of the pixel value of image I.
Further, evaluation index structural similarity (SSIM) is in the step 4
Wherein, x, y are original image signal and testing image signal, μxIt is the average value of x, μyIt is the average value of y,It is
The variance of x,It is the variance of y, σxyIt is the covariance of x and y.
Further, the software used in the present invention is MATLAB.
Further, the present invention carries out testing required be configured to:Processor:Intel(R)Core(TM)i5-4210U
CPU@1.70GHz 2.40GHz;Memory:4.00GB;System type:64 bit manipulation systems, the processor based on x64.
Further, when carrying out MATLAB emulation experiments in the step 4, addition is multiplicative noise.
The present invention using the above scheme, can generate following effect:
The denoising model that this method proposes can effectively remove multiplicative noise, inhibit " alias ", be effectively maintained image
Edge detail information.
Description of the drawings
Fig. 1 is the flow diagram of the present invention;
Fig. 2 is original Lena image and its image that noise is added;
Fig. 3 is Lena figures in threshold value k=140, image when iteration 1000 times after each model denoising:
(a) it is Y-K model denoising results, is (b) M model denoising results, is (c) denoising result of model of the present invention;
Fig. 4 is Lena figures in threshold value k=140, image when iteration 2000 times after each model denoising:
(a) it is Y-K model denoising results, is (b) M model denoising results, is (c) denoising result of model of the present invention;
Fig. 5 is original Cameraman images and its image that noise is added;
Fig. 6 is Cameraman figures in threshold value k=92, image when iteration 1000 times after each model denoising:
(a) it is Y-K model denoising results, is (b) M model denoising results, is (c) denoising result of model of the present invention;
Fig. 7 is Cameraman figures in threshold value k=92, image when iteration 2000 times after each model denoising:
(a) it is Y-K model denoising results, is (b) M model denoising results, is (c) denoising result of model of the present invention;
Fig. 8 is original color Lena images and its image that noise is added;
Fig. 9 is colour Lena figures in threshold value k=110, image when iteration 2000 times after each model denoising:
(a) it is Y-K model denoising results, is (b) M model denoising results, is (c) denoising result of model of the present invention;
Figure 10 is colour Lena figures in threshold value k=110, image when iteration 3000 times after each model denoising:
(a) it is Y-K model denoising results, is (b) M model denoising results, is (c) denoising result of model of the present invention;
Figure 11 is original color Peppers images and its image that noise is added;
Figure 12 is colour Peppers figures in threshold value k=110, image when iteration 2000 times after each model denoising:
(a) it is Y-K model denoising results, is (b) M model denoising results, is (c) denoising result of model of the present invention;
Figure 13 is colour Peppers figures in threshold value k=110, image when iteration 3000 times after each model denoising:
(a) it is Y-K model denoising results, is (b) M model denoising results, is (c) denoising result of model of the present invention.
Specific implementation mode
The present invention is further described in detail below in conjunction with the accompanying drawings.Implement to be only used for that the present invention will be described in detail below
Embodiment, the present invention can not be limited with this.
As shown in Figure 1, the present invention devises a kind of method of the image denoising based on partial differential equation of higher order, this method packet
Contain:1) discretization carried out to continuous model with finite difference scheme, 2) choose Lena images, Cameraman images and colour
Lena images, colour Peppers images progress MATLAB emulation experiments, 3) select Y-PSNR (PSNR), signal-to-noise ratio
(SNR), structural similarity (SSIM) is used as evaluation index.
The present invention is as follows:
Step 1, set the spread function of quadravalence partial differential denoising model as
C (u)=e-(u/k)
Wherein u indicates the gamma function of image.
Step 2 establishes denoising model
Wherein, △ is Laplace operator, constant k (k>0) it is threshold value, the feature for judging image.
Step 3 carries out discretization with finite difference scheme to the continuous model in step 2, is M × N for a width size
Image, utilize grid dissect method, enable
If um,n=u (m, n), boundary are handled in the following way
u0,n=u1,n uM+1,n=uM,n um,0=um,1 um,N+1=um,N
M=1,2 ..., M n=1,2 ..., N
If time step is △ t, spatial mesh size h=1, p indicate iterations, then
T=p △ t p=0,1,2 ...
X=m m=0,1,2 ..., M
Y=n n=0,1,2 ..., N
Laplace operator is calculated using central difference schemes:
M=0,1,2 ..., M n=0,1,2 ..., N
According to the model in step 2, there is iterative equation
The result of the iteration is up+1, i.e.,
Step 4 carries out emulation experiment to the model in step 2 using MATLAB, denoising image is obtained, with peak value noise
Than, signal-to-noise ratio, structural similarity as evaluation index and obtain its numerical result.
Three evaluation indexes are specific as follows:
1) Y-PSNR (peak signal to noise ratio, PSNR)
2) signal-to-noise ratio (signal to noise ratio, SNR)
WhereinFor the mean value of the pixel value of image I.
3) structural similarity (structural similarity, SSIM)
Wherein, x, y are original image signal and testing image signal, μxIt is the average value of x, μyIt is the average value of y,It is x
Variance,It is the variance of y, σxyIt is the covariance of x and y.
The numerical value of three evaluation indexes is bigger, illustrates that denoising effect is better.
The effect of the present invention is illustrated with reference to embodiment:
It is 0 that mean value, which is added, to Lena images and Cameraman images in the present invention, the multiplicative noise that variance is 0.4, to colour
It is 0 that mean value, which is added, in Lena images and colour Peppers images, and the multiplicative noise that variance is 0.099 carries out MATLAB emulation experiments.
Experimental situation is MATLAB r2016a, tests computer used and is configured to:Processor:Intel(R)Core(TM)i5-4210U
CPU@1.70GHz 2.40GHz;Memory:4.00GB;System type:64 bit manipulation systems, the processor based on x64.In experiment
It is 0.01 that time step, which is arranged, spatial mesh size 1.Model of the present invention is compared with Y-K models, M models, with peak value noise
Than, signal-to-noise ratio, structural similarity carry out denoising effect evaluation.As a result as shown in table 1- tables 4.
1 Lena figure denoising effects of table compare
2 Cameraman figure denoising effects of table compare
3 colour Lena figure denoising effects of table compare
4 colour Peppers figure denoising effects of table compare
From the point of view of the data result in table:When choosing different parameters to different images respectively, model proposed by the present invention
Compared with Y-K models, M models, Y-PSNR, signal-to-noise ratio, the structural similarity of the image after denoising significantly improve, explanation
The carried model of the present invention has good denoising effect really.
In conclusion method proposed by the present invention is an effective denoising model, while removing multiplicative noise very
Good remains image edge detailss information.
The above is the detailed description done to embodiment of the present invention, for the ordinary skill people of the art
Member, under the premise of not departing from the technology of the present invention, can also make several improvement, belong to protection scope of the present invention.
Claims (8)
1. a kind of image de-noising method based on partial differential equation of higher order, which is characterized in that include the following steps:
Step 1, set the spread function of fourth order PDEs denoising model as
C (u)=e-(u/k)
Wherein u indicates the gamma function of image;
Step 2 establishes denoising model
Wherein, △ is Laplace operator, constant k (k>0) it is threshold value, the feature for judging image;
Step 3 carries out discretization with finite difference scheme to the continuous model in step 2, obtains the corresponding iterative equation of model,
The solution of iterative equation is the numerical solution of PDE model;
Step 4 carries out emulation experiment to the model in step 2 using MATLAB, obtains denoising image, uses Y-PSNR
(PSNR), signal-to-noise ratio (SNR), structural similarity (SSIM) as evaluation index and obtain its numerical result, numerical value and image
Denoising effect is directly proportional, and bigger to be worth, image denoising effect is better.
2. the image de-noising method based on partial differential equation of higher order according to claim 1, which is characterized in that the step 3
Specifically include following content:
For the image that a width size is M × N, method is dissected using grid, is enabled
If um,n=u (m, n), boundary are handled in the following way
u0,n=u1,n uM+1,n=uM,n um,0=um,1 um,N+1=um,N
M=1,2 ..., M n=1,2 ..., N
If time step is △ t, spatial mesh size h=1, p indicate iterations, then
T=p △ t p=0,1,2 ...
X=m m=0,1,2 ..., M
Y=n n=0,1,2 ..., N
Laplace operator is calculated using central difference schemes:
According to the model in step 2, there is iterative equation
The result of pth time iteration is up+1, i.e.,
3. the image de-noising method according to claim 1 or claim 2 based on partial differential equation of higher order, which is characterized in that the step
Evaluation index Y-PSNR (PSNR) is in rapid 4
4. the image de-noising method according to claim 1 or claim 2 based on partial differential equation of higher order, which is characterized in that the step
Evaluation index signal-to-noise ratio (SNR) is in rapid 4
WhereinFor the mean value of the pixel value of image I.
5. the image de-noising method according to claim 1 or claim 2 based on partial differential equation of higher order, which is characterized in that the step
Evaluation index structural similarity (SSIM) is in rapid 4
Wherein, x, y are original image signal and testing image signal, μxIt is the average value of x, μyIt is the average value of y,It is the side of x
Difference,It is the variance of y, σxyIt is the covariance of x and y.
6. the image de-noising method based on partial differential equation of higher order according to claim 1, which is characterized in that used in the present invention
Software be MATLAB.
7. the image de-noising method based on partial differential equation of higher order according to claim 1, which is characterized in that the present invention carries out
It is configured to needed for experiment:Processor:Intel(R)Core(TM)i5-4210U CPU@1.70GHz 2.40GHz;Memory:
4.00GB;System type:64 bit manipulation systems, the processor based on x64.
8. the image de-noising method according to claim 1 or claim 2 based on partial differential equation of higher order, which is characterized in that the step
When carrying out MATLAB emulation experiments in rapid 4, addition is multiplicative noise.
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