CN104392411B - Image processing method and device based on Shannon Blackman small echo sparse expressions - Google Patents

Abstract

The invention discloses a kind of image processing method and device based on Shannon Blackman small echo sparse expressions, methods described includes：Obtain pending image；Multiple dimensioned interpolation operator of the construction based on Shannon Blackman interpolating wavelets, is amplified to described image；To the Image Variational model under the picture construction wavelet frame after amplification；The Image Variational model under the wavelet frame is solved, is got a distinct image.The present invention is amplified by multiple dimensioned interpolation operator to image, and Image Variational model is set up under interpolating wavelet framework, and obtains super-resolution image by sparse grid Algorithm for Solving image, improves the efficiency and precision of image procossing.

Description

Image processing method and device based on Shannon-Blackman small echo sparse expressions

Technical field

It is more particularly to a kind of to be based on Shannon-Blackman small echo sparse tables the present invention relates to technical field of image processing The image processing method and device for reaching.

Background technology

High-resolution Biomedical Image can provide more accurate and abundant vision for medical diagnosis and pathological analysis Information.Although image resolution ratio can be improved to a certain extent using video high density collection sensor, because imaging is The limitation of system self-sensor device arranging density, pollution that is with high costs and cannot completely avoiding noise in image acquisition process；Carry Chip size high can then cause the decline of electric charge transfer speed and the increase of electric capacity.Therefore, using super-resolution image reconstruction skill Art improves image resolution ratio has important scientific meaning and practical value.

It has been proved that image procossing Variation Model is the effective tool for improving Biomedical Image resolution ratio.But conventional There is artificial artifact and distortion phenomenon in image super-resolution rebuilding, inevitably in calculus of finite differences and linear interpolation method； And in existing image super-resolution rebuilding Variation Model, can not have with the spread function on automatic identification image object border Standby multiple dimensioned characteristic, causes the image local detailed structure after amplifying unintelligible, have impact on the raising of image resolution ratio.To solve The problem, partial differential equation denoising model experienced from low order to high-order, vector spreads to tensor diffusion, spreads to multiple expansion in fact Scattered evolution, but the method can be by local tiny boundary vague, at its image when being corrected to the image after amplification Reason effect is not obvious always.

In addition, when traditional calculus of finite differences solves parabolic type nonlinear partial differential equation, the stability requirement of algorithm is higher, meter Calculate precision and efficiency is not high, have impact on the application and popularization of the algorithm.

The content of the invention

Based on above mentioned problem, the present invention provides a kind of image procossing based on Shannon-Blackman small echo sparse expressions Method and device, is amplified by multiple dimensioned interpolation operator to image, and Image Variational model is set up under interpolating wavelet framework, And super-resolution image is obtained by sparse grid Algorithm for Solving image, improve the efficiency and precision of image procossing.

For above-mentioned purpose, the present invention provides a kind of image procossing based on Shannon-Blackman small echo sparse expressions Method, it is characterised in that including：

Obtain pending image；

Multiple dimensioned interpolation operator of the construction based on Shannon-Blackman interpolating wavelets, is amplified to described image；

To the Image Variational model under the picture construction wavelet frame after amplification；

The Image Variational model under the wavelet frame is solved, is got a distinct image.

Wherein, the multiple dimensioned interpolation operator of the construction based on Shannon-Blackman interpolating wavelets is put to image Big detailed process is as follows：

If the domain of definition of image is (x_min,x_max)×(y_min,y_max), the position of each pixel is defined as in imageWherein j is scale parameter, k_jxAnd k_jyIt is the location parameter on j yardsticks；

Defining Shannon-Blackman scaling functions is：

Shannon-Blackman scaling functions are defined as by tensor product：

According to the definition of interpolating wavelet transform, derive that multi-scale wavelet interpolation operator is as follows：

Described image u (x, y) is expressed as by multiple dimensioned interpolating function：

Wherein, N is compact schemes constant, and C1, C2, C3 are Wavelet Interpolation transformation matrix, n₁,n₂It is the position of interpolation operator, R It is Restriction Operators, J is the maximum of scale parameter, j₀It is expressed as 0 layer of yardstick, j₁To be different from the scale parameter of j；

Wherein,

Wherein, in described image each pixel positionIt is defined as：

Wherein, the Restriction Operators R is defined as：

Wherein, l is the scale parameter for being different from j.

Described pair amplify after picture construction wavelet frame under the detailed process of Image Variational model be：

The calibration model for defining the image after the amplification is：

Wherein, (x, y) represents the position of pixel, and t is diffusion time parameter, and f (x, y) is original two dimensional image, and c is expansion Function is dissipated, u is the expression formula of the image expressed by multiple dimensioned interpolating function；

The expression formula of the image expressed by multiple dimensioned interpolating function is brought into the calibration model, obtains described small Image Variational model under ripple framework.

Wherein, the spread function c is defined as：

Wherein, N is compact schemes constant,It is gradient operator.

Wherein, the detailed process for solving the Image Variational model under the wavelet frame includes：

Described image Variation Model is rewritten as：

Wherein,And by u^J(x, y, t_n) it is set as u_n, t_nThe function F at moment is set as F_n；

Constructing linear homotopy is：u^J(x, y, t)=(1- ε) F_n+εF_n+1；

According to perturbation theory, the linear homotopy is expanded into following formula：

According to the expression formula and the revised Image Variational model, one group of ordinary differential system is obtained, and solve Obtain the correction result of described image；

Wherein, the ε is homotopy parameter, and

According to another aspect of the present invention, there is provided a kind of image based on Shannon-Blackman small echo sparse expressions Processing unit, it is characterised in that including：

Image acquisition unit, for obtaining pending image；

Image enlarging unit, it is right for constructing the multiple dimensioned interpolation operator based on Shannon-Blackman interpolating wavelets Described image is amplified；

Image Variational model sets up unit, for the Image Variational model under the picture construction wavelet frame after amplification；

Unit is solved, for solving the Image Variational model under the wavelet frame, is got a distinct image.

Image processing method and device based on Shannon-Blackman small echo sparse expressions that the present invention is provided, pass through Multiple dimensioned interpolation operator is amplified to image, realizes the adaptive configuration of image slices vegetarian refreshments, while can be effectively retained thin Nodule structure, improves and rebuilds efficiency, it is to avoid occur the phenomenon of artificial artifact in amplification process；In addition, using multiple dimensioned Blackman Function replaces traditional spread function, can be effectively prevented from the obfuscation of local detail, such that it is able to further improve image Super-resolution；Finally, by constructing sparse grid Algorithm for Solving Image Variational model, efficiency and precision are effectively improved.

Brief description of the drawings

Fig. 1 shows the flow of the image processing method based on Shannon-Blackman small echo sparse expressions of the invention Figure.

Fig. 2 (a) and (b) show the Shannon- of existing Shannon wavelet scaling functions and embodiments of the invention The comparison diagram of Blackman scaling functions.

Fig. 3 shows the structure of the image processing apparatus based on Shannon-Blackman small echo sparse expressions of the invention Block diagram.

Fig. 4 shows the original image of one embodiment of the present of invention.

Fig. 5 shows the image that image is amplified 2 times of one embodiment of the present of invention.

Fig. 6 shows the image after the correction of one embodiment of the present of invention.

It is multiple dimensioned that Fig. 7 shows that the image of one embodiment of the invention is carried out using Shannon-Blackman scaling functions The expression schematic diagram of interpolation.

Specific embodiment

With reference to the accompanying drawings and examples, specific embodiment of the invention is described in further detail.Hereinafter implement Example is not limited to the scope of the present invention for illustrating the present invention.

At a kind of image based on Shannon-Blackman small echo sparse expressions Reason method.

Fig. 1 shows the flow of the image processing method based on Shannon-Blackman small echo sparse expressions of the invention Figure.Fig. 2 shows the Shannon-Blackman yardstick letters of existing Shannon wavelet scaling functions and embodiments of the invention Several comparison diagrams.

Reference picture 1, the image processing method based on Shannon-Blackman small echo sparse expressions of the invention, specific mistake Journey includes：

The pending image of S1, acquisition；

The multiple dimensioned interpolation operator of S2, construction based on Shannon-Blackman interpolating wavelets, puts to described image Greatly.

Detailed process is as follows：

If the domain of definition of image is (x_min,x_max)×(y_min,y_max), the position of each pixel is defined as in imageWherein j is scale parameter, k_jxAnd k_jyIt is the location parameter on j yardsticks；

The position of each pixel in imageIt is defined as：

First, defining Shannon-Blackman scaling functions is：

According to above-mentioned scaling function, N is compact schemes constant, as shown in Fig. 2 relative to Shannon wavelet scaling functions, this The corresponding scaling function of interpolating wavelet of embodiment remains interpolation characteristic and orthogonal property, while compactly supported is also add, The scaling function of compact schemes, can be while innovatory algorithm numerical precision and convergence speed used as the odd function for solving partial differential equation Degree.

Then Shannon-Blackman scaling functions are defined as by tensor product：

According to the definition of interpolating wavelet transform, can derive that multi-scale wavelet interpolation operator is as follows：

Wherein, C1, C2, C3 are Wavelet Interpolation transformation matrix, n₁,n₂It is the position of interpolation operator, R is Restriction Operators, and J is The maximum of scale parameter, j₀It is expressed as 0 layer of yardstick, j₁To be different from the scale parameter of j, and

In addition, the Restriction Operators R is defined as：

Wherein, l is the scale parameter for being different from j.

According to above-mentioned derivation, image u (x, y) is expressed as by multiple dimensioned interpolating function：

By above-mentioned formula, you can realize the multiple dimensioned interpolation amplification of image.

S3, to the Image Variational model under the picture construction wavelet frame after amplification；

Its detailed process is：

The calibration model of image after definition is amplified is：

Wherein, (x, y) represents the position of pixel, and t is diffusion time parameter, and f (x, y) is original two dimensional image, and c is expansion Function is dissipated, u is the expression formula of the image expressed by multiple dimensioned interpolating function；

Spread function c is defined as：

Wherein,It is gradient operator.The present embodiment uses above-mentioned spread function, it can be ensured that the spread function tool in calculating There is strict compactly supported.

The expression formula of the image expressed by multiple dimensioned interpolating function in step S2 is brought into the calibration model, is obtained Image Variational model under to the wavelet frame.

S4, the Image Variational model solved under the wavelet frame, get a distinct image.

Its detailed process includes：

Described image Variation Model is rewritten as：

u^{J (2,0)}(x, y, t), u^{J (1,1)}(x, y, t), u^{J (0,2)}(x, y, t)]

Wherein,And by u^J(x, y, t_n) it is set as u_n, t_nThe function F at moment is set as F_n；

Constructing linear homotopy is：u^J(x, y, t)=(1- ε) F_n+εF_n+1；

Wherein, the ε is homotopy parameter, and

According to perturbation theory, the linear homotopy is expanded into following formula：

According to the expression formula and the revised Image Variational model, one group of ordinary differential system is obtained, by the party Journey group solves the correction result that can obtain described image, such that it is able to obtain the super-resolution rebuilding result of image.

In an alternative embodiment of the invention, there is provided a kind of image based on Shannon-Blackman small echo sparse expressions Processing unit.

Fig. 3 shows the structure of the image processing apparatus based on Shannon-Blackman small echo sparse expressions of the invention Block diagram.

Reference picture 3, the image processing apparatus based on Shannon-Blackman small echo sparse expressions of the present embodiment include：

Image acquisition unit 10, for obtaining pending image；

Image enlarging unit 20, for constructing the multiple dimensioned interpolation operator based on Shannon-Blackman interpolating wavelets, Described image is amplified；

Image Variational model sets up unit 30, for the Image Variational mould under the picture construction wavelet frame after amplification Type；

Unit 40 is solved, for solving the Image Variational model under the wavelet frame, is got a distinct image.

The above method is illustrated below by way of specific embodiment.

Fig. 4 shows the original image of an alternative embodiment of the invention.Fig. 5 shows one embodiment of the present of invention Image is amplified 2 times of image.Fig. 6 shows the image after the correction of an alternative embodiment of the invention.Fig. 7 shows this hair The image of bright one embodiment carries out the expression schematic diagram of multiple dimensioned interpolation using Shannon-Blackman scaling functions.

The present embodiment is extracted an image as shown in Figure 4 and is corrected using the above method.

Image is amplified using Shannon-Blackman multiple dimensioned interpolation operators first, as shown in Figure 5；

Then the image after amplification is corrected using the Variation Model under interpolating wavelet framework, the figure after being corrected Picture, as shown in fig. 6, the super-resolution rebuilding so as to complete image.

By image processing method of the invention, image is amplified using multiple dimensioned interpolation operator in timing, can To realize the adaptive configuration of image slices vegetarian refreshments, as shown in fig. 7, image pixel is 300*300, the adaptive configuration of its pixel Point number is 23488, and the border annex pixel point density of image object thing is big, and other equal value parts are relatively sparse, using this When the scheme of kind carries out image amplification, detailed structure can be effectively retained, while improve rebuilding efficiency, it is to avoid occur in amplification process Artificial artifact phenomenon.

In addition, the present invention replaces traditional spread function using multiple dimensioned Blackman functions, office can be effectively prevented from The obfuscation of portion's details, such that it is able to further improve the super-resolution of image.

Finally, the multiple dimensioned operator using Shannon-Blackman Construction of Wavelets, and by constructing sparse grid algorithm Image Variational model is solved, efficiency and precision is effectively improved.

Embodiment of above is merely to illustrate the present invention, and not limitation of the present invention, about the common of technical field Technical staff, without departing from the spirit and scope of the present invention, can also make a variety of changes and modification, therefore all Equivalent technical scheme falls within scope of the invention, and scope of patent protection of the invention should be defined by the claims.

Claims (7)

1. a kind of image processing method based on Shannon-Blackman small echo sparse expressions, it is characterised in that including：

Obtain pending image；

Multiple dimensioned interpolation operator of the construction based on Shannon-Blackman interpolating wavelets, is amplified to described image；

To the Image Variational model under the picture construction wavelet frame after amplification；

The Image Variational model under the wavelet frame is solved, is got a distinct image；

It is specific that the multiple dimensioned interpolation operator of the construction based on Shannon-Blackman interpolating wavelets is amplified to image Process is as follows：

If the domain of definition of image is (x_min,x_max)×(y_min,y_max), the position of each pixel is defined as in imageWherein j is scale parameter, k_jxAnd k_jyIt is the location parameter on j yardsticks；

Defining Shannon-Blackman scaling functions is：

$\begin{array}{c}\mathrm{\φ}\left(x\right)\\ =\left\{\begin{array}{c}\frac{sin\left(\mathrm{\π}x\right)}{\mathrm{\π}x}{(0.42+0.5cos(\frac{2\mathrm{\π}x}{N})+0.08cos(\frac{4\mathrm{\π}x}{N}\left)\right)}^2,-N/2\≤x\≤N/2\\ 0,otherwise\end{array}\right.\end{array};$

Shannon-Blackman scaling functions are defined as by tensor product：

$w_{k_{jx},k_{jy}}^j(x,y)={\mathrm{\Φ}}_{k_{jx}}^j\left(x\right){\mathrm{\Φ}}_{k_{jy}}^j\left(y\right)=\mathrm{\Φ}(2^{j}x-k_{jx})\mathrm{\Φ}(2^{j}y-k_{jy})$

According to the definition of interpolating wavelet transform, derive that multi-scale wavelet interpolation operator is as follows：

$\begin{array}{c}{I}_{n_1,n_2}(x,y)=\underset{k_{0x}=0}{\overset{2^{j_0}}{\Σ}}\underset{k_{0y}=0}{\overset{2^{j_0}}{\Σ}}{R}_{k_{0x},k_{0y},n_1,n_2}^{j_0,j_0,J,J}{w}_{k_{0x},k_{0y}}^{j_0}(x,y)\\ -\underset{j=j_0}{\overset{J-1}{\Σ}}\underset{k_{jx}=0}{\overset{2^j}{\Σ}}\underset{k_{jy}=0}{\overset{2^j}{\Σ}}\left(C{1}_{k_{jx},k_{jy},n_1,n_2}^{j,j,J,J}{w}_{2k_{jx}+1,2k_{jy}}^{j+1}\right(x,y)\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ -C{2}_{k_{jx},k_{jy},n_1,n_2}^{j,j,J,J}{w}_{2k_{jx},2k_{jy}+1}^{j+1}(x,y)\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +C{3}_{k_{jx},k_{jy},n_1,n_2}^{j,j,J,J}{w}_{2k_{jx}+1,2k_{jy}+1}^{j+1}(x,y)\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\end{array}$

Described image u (x, y) is expressed as by multiple dimensioned interpolating function：

$u^J(x,y)=\underset{n_1=0}{\overset{2J}{\Σ}}\underset{n_2=0}{\overset{2J}{\Σ}}{I}_{n_1,n_2}(x,y)u^J(x_{n1}^J,y_{n2}^J);$

Wherein, N is compact schemes constant, and C1, C2, C3 are Wavelet Interpolation transformation matrix, n₁,n₂It is the position of interpolation operator, R is limited Operator processed, J is the maximum of scale parameter, j₀It is expressed as 0 layer of yardstick, j₁To be different from the scale parameter of j；

Wherein,

$\begin{array}{c}C{1}_{k_{jx},k_{jy},n_1,n_2}^{j,j,J,J}=R_{2k_{jx}+1,2k_{jy},n_1,n_2}^{j+1,j_{+2},J,J}\\ -\[\underset{k_{0x}=0}{\overset{2^{j_0}}{\Σ}}\underset{k_{0y}=0}{\overset{2^{j_0}}{\Σ}}{R}_{k_{0x},k_{0y},n_1,n_2}^{j_0,j_0,J,J}\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)w_{k_{0x},k_{0y}}^{j_0}(x_{2k_{jx}+1}^{j+1},y_{2k_{jy}}^{j+1})\\ +\underset{j_1=j_0}{\overset{j-1}{\Σ}}\underset{n_2=0}{\overset{2^J}{\Σ}}\underset{k_{j_{1}x}=0}{\overset{2^{j_1}}{\Σ}}\left(C{1}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x}+1,2k_{j_{1}y}}^{j_1+1}\right(x_{2k_{jx}+1}^{j+1},y_{2k_{jy}}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +C{2}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x},2k_{j_y}+1}^{j_1+1}(x_{2k_{jx}+1}^{j+1},y_{2k_{jy}}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +C{3}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x}+1,2k_{j_{1}y}+1}^{j_1+1}(x_{2k_{jx}+1}^{j+1},y_{2k_{jy}}^{j+1})\mathrm{\Φ}(x_{n_1}^J,x_{n_2}^J)\]\end{array}$

$\begin{array}{c}C{2}_{k_{jx},k_{jy},n_1,n_2}^{j,j,J,J}=R_{2k_{jx},2k_{jy}+1,n_1,n_2}^{j+1,j+2,J,J}\\ -\[\underset{k_{0x}=0}{\overset{2^{j_0}}{\Σ}}\underset{k_{0y}=0}{\overset{2^{j_0}}{\Σ}}{R}_{k_{0x},k_{0y},n_1,n_2}^{j_0,j_0,J,J}\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)w_{k_{0x},k_{0y}}^{j_0}(x_{2k_{jx}+1}^{j+1},y_{2k_{jy}+1}^{j+1})\\ +\underset{j_1=j_0}{\overset{j-1}{\Σ}}\underset{n_2=0}{\overset{2^J}{\Σ}}\underset{k_{j_{1}x}=0}{\overset{2^{j_1}}{\Σ}}\left(C{1}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x}+1,2k_{j_{1}y}}^{j_1+1}\right(x_{2k_{jx}}^{j+1},y_{2k_{jy}+1}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +C{2}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x},2k_{j_{1}y}+1}^{j_1+1}(x_{2k_{jx}}^{j+1},y_{2k_{jy}+1}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +C{3}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x}+1,2k_{j_{1}y}+1}^{j_1+1}(x_{2k_{jx}}^{j+1},y_{2k_{jy}+1}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\]\end{array}$

$\begin{array}{c}C{3}_{k_{jx},k_{jy},n_1,n_2}^{j,j,J,J}=R_{2k_{jx}+1,2k_{jy}+1,n_1,n_2}^{j+1,j+1,J,J}\\ -\[\underset{k_{0x}=0}{\overset{2^{j_0}}{\Σ}}\underset{k_{0y}=0}{\overset{2^{j_0}}{\Σ}}{R}_{k_{0x},k_{0y},n_1,n_2}^{j_0,j_0,J,J}\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)w_{k_{0x},k_{0y}}^{j_0}(x_{2k_{jx}+1}^{j+1},y_{2k_{jy}+1}^{j+1})\\ +\underset{j_1=j_0}{\overset{j-1}{\Σ}}\underset{n_2=0}{\overset{2^J}{\Σ}}\underset{k_{j_{1}x}=0}{\overset{2^{j_1}}{\Σ}}\left(C{1}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x}+1,2k_{j_{1}y}}^{j_1+1}\right(x_{2k_{jx}+1}^{j+1},y_{2k_{jy}+1}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +C{2}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x},2k_{j_{1}y}+1}^{j_1+1}(x_{2k_{jx}+1}^{j+1},y_{2k_{j_{1}y}+1}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +C{3}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x}+1,2k_{j_{1}y}+1}^{j_1+1}(x_{2k_{j_{1}x}+1}^{j+1},y_{2k_{j_{1}y}+1}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\]\end{array}.$

2. image processing method as claimed in claim 1, it is characterised in that the position of each pixel in described imageIt is defined as：

$x_{k_{jx}}^j=x_{min}+k_{jx}\frac{x_{max}-x_{min}}{2^j}$

$y_{k_{jy}}^j=y_{min}+k_{jy}\frac{y_{max}-y_{min}}{2^j}.$

3. image processing method as claimed in claim 1, it is characterised in that the Restriction Operators R is defined as：

$R_{k_{lx},k_{ly},k_{jx},k_{jy}}^{l,l,j,j}=\left\{\begin{array}{cc}1,& {\mathrm{xl}}_{klx}^l=x_{jx}^{j}and{y}_{ly}^i=y_{jy}^j\\ 0,& otherwise\end{array}\right.,$

Wherein, l is the scale parameter for being different from j.

4. image processing method as claimed in claim 1, it is characterised in that described pair amplify after picture construction wavelet frame Under the detailed process of Image Variational model be：

The calibration model for defining the image after the amplification is：

$\left\{\begin{array}{c}\frac{\∂u(x,y,t)}{\∂t}=\frac{\∂\[c(x,y,t).\frac{\∂u(x,y,t)}{\∂x}\]}{\∂x}+\frac{\∂\[c(x,y,t).\frac{\∂u(x,y,t)}{\∂y}\]}{\∂y}\\ u(x,y,0)=f(x,y)\end{array}\right.,$

Wherein, (x, y) represents the position of pixel, and t is diffusion time parameter, and f (x, y) is original two dimensional image, and c is diffusion letter Number, u is the expression formula of the image expressed by multiple dimensioned interpolating function；

The expression formula of the image expressed by multiple dimensioned interpolating function is brought into the calibration model, the small echo frame is obtained Image Variational model under frame.

5. image processing method as claimed in claim 4, it is characterised in that the spread function c is defined as：

$c\left(\right|\▿u\left|\right)=\left\{\begin{array}{c}{\left(cos\right(\frac{\mathrm{\π}\▿u}{N}\left)\right)}^2,-N/2\≤\▿u\≤N/2\\ 0,otherwise\end{array}\right.,$

Wherein,It is gradient operator.

6. image processing method as claimed in claim 1, it is characterised in that the image under the solution wavelet frame becomes The detailed process of sub-model includes：

Described image Variation Model is rewritten as：

$\frac{{\mathrm{du}}^J(x,y,t)}{dt}=F\[t,x,y,u^J(x,y,t),u^{J(1,0)}(x,y,t),u^{J(0,1)}(x,y,t),$

u^J(2·⁰⁾(x, y, t), u^{J (1,1)}(x, y, t), u^{J (0,2)}(x, y, t)],

Wherein,And by u^J(x, y, t_n) it is set as u_n, t_nThe function F at moment is set as F_n；

Constructing linear homotopy is：u^J(x, y, t)=(1- ε) F_n+εF_n+1；

According to perturbation theory, the linear homotopy is expanded into following formula：

$u^J=u_0^J+{\mathrm{\ϵu}}_1^J+{\mathrm{\ϵ}}^2{u}_2^J+\mathrm{...},$

The expression formula launched according to the linear homotopy and the revised Image Variational model, obtain one group of ODE Group, and solve the correction result for obtaining the pending image；

Wherein, the ε is homotopy parameter, and

7. a kind of image processing apparatus based on Shannon-Blackman small echo sparse expressions, it is characterised in that including：

Image acquisition unit, for obtaining pending image；

Image enlarging unit, for constructing the multiple dimensioned interpolation operator based on Shannon-Blackman interpolating wavelets, to described Image is amplified；

Wherein, the multiple dimensioned interpolation operator of the construction based on Shannon-Blackman interpolating wavelets is amplified to image Detailed process is as follows：

If the domain of definition of image is (x_min,x_max)×(y_min,y_max), the position of each pixel is defined as in imageWherein j is scale parameter, k_jxAnd k_jyIt is the location parameter on j yardsticks；

Defining Shannon-Blackman scaling functions is：

$\begin{array}{c}\mathrm{\φ}\left(x\right)\\ =\left\{\begin{array}{c}\frac{sin\left(\mathrm{\π}x\right)}{\mathrm{\π}x}{(0.42+0.5cos(\frac{2\mathrm{\π}x}{N})+0.08cos(\frac{4\mathrm{\π}x}{N}\left)\right)}^2,-N/2\≤x\≤N/2\\ 0,otherwise\end{array}\right.\end{array};$

Shannon-Blackman scaling functions are defined as by tensor product：

$w_{k_{jx},k_{jy}}^j(x,y)={\mathrm{\Φ}}_{k_{jx}}^j\left(x\right){\mathrm{\Φ}}_{k_{jy}}^j\left(y\right)=\mathrm{\Φ}(2^{j}x-k_{jx})\mathrm{\Φ}(2^{j}y-k_{jy})$

According to the definition of interpolating wavelet transform, derive that multi-scale wavelet interpolation operator is as follows：

$\begin{array}{c}{I}_{n_1,n_2}(x,y)=\underset{k_{0x}=0}{\overset{2^{j_0}}{\Σ}}\underset{k_{0y}=0}{\overset{2^{j_0}}{\Σ}}{R}_{k_{0x},k_{0y},n_1,n_2}^{j_0,j_0,J,J}{w}_{k_{0x},k_{0y}}^{j_0}(x,y)\\ -\underset{j=j_0}{\overset{J-1}{\Σ}}\underset{k_{jx}=0}{\overset{2^j}{\Σ}}\underset{k_{jy}=0}{\overset{2^j}{\Σ}}\left(C{1}_{k_{jx},k_{jy},n_1,n_2}^{j,j,J,J}{w}_{2k_{jx}+1,2k_{jy}}^{j+1}\right(x,y)\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ -C{2}_{k_{jx},k_{jy},n_1,n_2}^{j,j,J,J}{w}_{2k_{jx},2k_{jy}+1}^{j+1}(x,y)\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +C{3}_{k_{jx},k_{jy},n_1,n_2}^{j,j,J,J}{w}_{2k_{jx}+1,2k_{jy}+1}^{j+1}(x,y)\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\end{array}$

Described image u (x, y) is expressed as by multiple dimensioned interpolating function：

$u^J(x,y)=\underset{n_1=0}{\overset{2J}{\Σ}}\underset{n_2=0}{\overset{2J}{\Σ}}{I}_{n_1,n_2}(x,y)u^J(x_{n1}^J,y_{n2}^J);$

Wherein, N is compact schemes constant, and C1, C2, C3 are Wavelet Interpolation transformation matrix, n₁,n₂It is the position of interpolation operator, R is limited Operator processed, J is the maximum of scale parameter, j₀It is expressed as 0 layer of yardstick, j₁To be different from the scale parameter of j；

Wherein,

$\begin{array}{c}C{1}_{k_{jx},k_{jy},n_1,n_2}^{j,j,J,J}=R_{2k_{jx}+1,2k_{jy},n_1,n_2}^{j+1,j_{+2},J,J}\\ -\[\underset{k_{0x}=0}{\overset{2^{j_0}}{\Σ}}\underset{k_{0y}=0}{\overset{2^{j_0}}{\Σ}}{R}_{k_{0x},k_{0y},n_1,n_2}^{j_0,j_0,J,J}\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)w_{k_{0x},k_{0y}}^{j_0}(x_{2k_{jx}+1}^{j+1},y_{2k_{jy}+1}^{j+1})\\ +\underset{j_1=j_0}{\overset{j-1}{\Σ}}\underset{n_2=0}{\overset{2^J}{\Σ}}\underset{k_{j_{1}x}=0}{\overset{2^{j_1}}{\Σ}}\left(C{1}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x}+1,2k_{j_{1}y}}^{j_1+1}\right(x_{2k_{jx}+1}^{j+1},y_{2k_{jy}}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +C{2}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x},2k_{j_{1}y}+1}^{j_1+1}(x_{2k_{jx}+1}^{j+1},y_{2k_{jy}}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +C{3}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x}+1,2k_{j_{1}y}+1}^{j_1+1}(x_{2k_{jx}+1}^{j+1},y_{2k_{jy}}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\]\end{array}$

$\begin{array}{c}C{2}_{k_{jx},k_{jy},n_1,n_2}^{j,j,J,J}=R_{2k_{jx},2k_{jy}+1,n_1,n_2}^{j+1,j+2,J,J}\\ -\[\underset{k_{0x}=0}{\overset{2^{j_0}}{\Σ}}\underset{k_{0y}=0}{\overset{2^{j_0}}{\Σ}}{R}_{k_{0x},k_{0y},n_1,n_2}^{j_0,j_0,J,J}\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)w_{k_{0x},k_{0y}}^{j_0}(x_{2k_{jx}}^{j+1},y_{2k_{jy}+1}^{j+1})\\ +\underset{j_1=j_0}{\overset{j-1}{\Σ}}\underset{n_2=0}{\overset{2^J}{\Σ}}\underset{k_{j_{1}x}=0}{\overset{2^{j_1}}{\Σ}}\left(C{1}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x}+1,2k_{j_{1}y}}^{j_1+1}\right(x_{2k_{jx}}^{j+1},y_{2k_{jy}+1}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +C{2}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x},2k_{j_{1}y}+1}^{j_1+1}(x_{2k_{jx}}^{j+1},y_{2k_{jy}+1}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +C{3}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x}+1,2k_{j_{1}y}+1}^{j_1+1}(x_{2k_{jx}}^{j+1},y_{2k_{jy}+1}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\]\end{array}$

$\begin{array}{c}C{3}_{k_{jx},k_{jy},n_1,n_2}^{j,j,J,J}=R_{2k_{jx}+1,2k_{jy}+1,n_1,n_2}^{j+1,j+1,J,J}\\ -\[\underset{k_{0x}=0}{\overset{2^{j_0}}{\Σ}}\underset{k_{0y}=0}{\overset{2^{j_0}}{\Σ}}{R}_{k_{0x},k_{0y},n_1,n_2}^{j_0,j_0,J,J}\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)w_{k_{0x},k_{0y}}^{j_0}(x_{2k_{jx}+1}^{j+1},y_{2k_{jy}+1}^{j+1})\\ +\underset{j_1=j_0}{\overset{j-1}{\Σ}}\underset{n_2=0}{\overset{2^J}{\Σ}}\underset{k_{j_{1}x}=0}{\overset{2^{j_1}}{\Σ}}\left(C{1}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x}+1,2k_{j_{1}y}}^{j_1+1}\right(x_{2k_{jx}+1}^{j+1},y_{2k_{jy}+1}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +C{2}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x},2k_{j_{1}y}+1}^{j_1+1}(x_{2k_{jx}+1}^{j+1},y_{2k_{j_{1}y}+1}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +C{3}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x}+1,2k_{j_{1}y}+1}^{j_1+1}(x_{2k_{j_{1}x}+1}^{j+1},y_{2k_{j_{1}y}+1}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\]\end{array}$

Image Variational model sets up unit, for the Image Variational model under the picture construction wavelet frame after amplification；

Unit is solved, for solving the Image Variational model under the wavelet frame, is got a distinct image.