CN104392411A - Image processing method and device based on Shannon-Blackman wavelet sparse representation - Google Patents

Abstract

The invention discloses an image processing method and device based on Shannon-Blackman wavelet sparse representation. The method comprises steps as follows: a to-be-processed image is acquired; a multi-scale interpolation operator based on Shannon-Blackman interpolating wavelets is established, and the image is amplified; an image variation model for the amplified image under a wavelet frame is established; and the image variation model under the wavelet frame is solved to obtain a sharp image. According to the Shannon-Blackman wavelet sparse representation based image processing method and device, the image is amplified through the multi-scale interpolation operator, the image variation model under the wavelet frame is established, the image is solved through a sparse grid algorithm to obtain a super-resolution image, and the image processing efficiency and accuracy are improved.

Description

Based on image processing method and the device of Shannon-Blackman small echo sparse expression

Technical field

The present invention relates to technical field of image processing, particularly a kind of image processing method based on Shannon-Blackman small echo sparse expression and device.

Background technology

High resolving power Biomedical Image can provide more accurate and abundant visual information for medical diagnosis and pathological analysis.Although adopt video high density pick-up transducers can improve image resolution ratio to a certain extent, due to the restriction of imaging system self-sensor device arranging density, with high costs and the pollution of noise in image acquisition process cannot be avoided completely; Improve chip size and then can cause the decline of Charger transfer speed and the increase of electric capacity.Therefore, adopt Technique of Super-resolution Image Construction to improve image resolution ratio and there is important scientific meaning and practical value.

Verified, image procossing Variation Model is the effective tool improving Biomedical Image resolution.But when image super-resolution rebuilding, inevitably there is artificial artifact and distortion phenomenon in conventional method of difference and linear interpolation method; And in existing image super-resolution rebuilding Variation Model, the spread function of recognition image object boundary automatically can not have multiple dimensioned characteristic, cause the image local detailed structure after amplifying unintelligible, have impact on the raising of image resolution ratio.For solving this problem, partial differential equation denoising model experienced by from low order to high-order, vector diffusion is spread to tensor, real evolution of spreading to multiple diffusion, but can by the tiny boundary vague in local when the method is corrected the image after amplification, its image processing effect is always not obvious.

In addition, when traditional method of difference solves parabolic type nonlinear partial differential equation, the stability requirement of algorithm is higher, and computational accuracy and efficiency are not high, have impact on application and the popularization of this algorithm.

Summary of the invention

Based on the problems referred to above, the invention provides a kind of image processing method based on Shannon-Blackman small echo sparse expression and device, by multiple dimensioned interpolation operator, image is amplified, Image Variational model is set up under interpolating wavelet framework, and obtain super-resolution image by sparse grid Algorithm for Solving image, improve efficiency and the precision of image procossing.

For above-mentioned purpose, the invention provides a kind of image processing method based on Shannon-Blackman small echo sparse expression, it is characterized in that, comprising:

Obtain pending image;

Construct the multiple dimensioned interpolation operator based on Shannon-Blackman interpolating wavelet, described image is amplified;

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

Solve the Image Variational model under described wavelet frame, get a distinct image.

Wherein, the described structure detailed process of amplifying image based on the multiple dimensioned interpolation operator of Shannon-Blackman interpolation plait is as follows:

If the field of definition of image is (x _min, x _max) × (y _min, y _max), in image, the position of each pixel is defined as wherein j is scale parameter, k _jxand k _jyfor the location parameter on j yardstick;

Definition Shannon-Blackman scaling function is:

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

Shannon-Blackman scaling function is defined as by tensor product:

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

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

$\begin{array}{c}{I}_{n_1,n_2}(x,y)=\underset{k_{0x}=0}{\overset{2^{j_0}}{\mathrm{\Σ}}}\underset{k_{0y}=0}{\overset{2^{j_0}}{\mathrm{\Σ}}}{R}_{k_{0x},k_{0y},n_1,n_2}^{j_0,j_0,J,J}{w}_{k_{0x,}{k}_{0y}}^{j_0}(x,y)\\ +\underset{j=j_0}{\overset{J-1}{\mathrm{\Σ}}}\underset{k_{\mathrm{jx}}=0}{\overset{2^j}{\mathrm{\Σ}}}\underset{k_{\mathrm{jy}}=0}{\overset{2^j}{\mathrm{\Σ}}}({C1}_{k_{\mathrm{jx}},k_{\mathrm{jy}},n_1,n_2}^{j,j,J,J}{w}_{2k_{\mathrm{jx}}+\mathrm{1,2}{k}_{\mathrm{jy}}}^{j+1}(x,y)\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ {+C2}_{k_{\mathrm{jx}},k_{\mathrm{jy}},n_1,n_2}^{j,j,J,J}{w}_{2k_{\mathrm{jx}},2k_{\mathrm{jy}}+1}^{j+1}(x,y)\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ {+C3}_{k_{\mathrm{jx}},k_{\mathrm{jy}},n_1,n_2}^{j,j,J,J}{w}_{2k_{\mathrm{jx}}+\mathrm{1,2}{k}_{\mathrm{jy}}+1}^{j+1}(x,y)\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\end{array}$

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

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

Wherein, N is compact schemes constant, and C1, C2, C3 are Wavelet Interpolation transformation matrix, n ₁, n ₂for the position of interpolation operator, R is Restriction Operators, and J is the maximal value of scale parameter, j ₀be expressed as 0 layer of yardstick, j ₁for being different from the scale parameter of j;

Wherein,

$\begin{array}{c}{C1}_{k_{\mathrm{jx}},k_{\mathrm{jy}},n_1,n_2}^{j,j,J,J}=R_{2k_{\mathrm{jx}}+\mathrm{1,2}{k}_{\mathrm{jy}},n_1,n_2}^{j+1,j_{+2},J,J}\\ -[\underset{k_{0x}=0}{\overset{2^{j_0}}{\mathrm{\Σ}}}\underset{k_{0y}=0}{\overset{2^{j_0}}{\mathrm{\Σ}}}{R}_{k_{0x},k_{0y},n_1,n_2}^{j_0,j_0,J,J}\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)w_{k_{0x},k_{0y}}^{j_0}(x_{2k_{\mathrm{jx}}+1}^{j+1},y_{{2k}_{\mathrm{jy}}}^{j+1})\\ +\underset{j_1=j_0}{\overset{j-1}{\mathrm{\Σ}}}\underset{n_2=0}{\overset{2^J}{\mathrm{\Σ}}}\underset{k_{j_{1}x}=0}{\overset{2^{j_1}}{\mathrm{\Σ}}}({C1}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x}+\mathrm{1,2}{k}_{j_{1}y}}^{j_1+1}(x_{2k_{\mathrm{jx}}+1}^{j+1},y_{{2k}_{\mathrm{jy}}}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +{C2}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x},2k_{j_{1}y}+1}^{j_1+1}(x_{2k_{\mathrm{jx}}+1}^{j+1},y_{{2k}_{\mathrm{jy}}}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +{C3}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x}+\mathrm{1,2}{k}_{j_{1}y}+1}^{j_1+1}(x_{2k_{\mathrm{jx}}+1}^{j+1},y_{{2k}_{\mathrm{jy}}}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)]\end{array}$

$\begin{array}{c}{C2}_{k_{\mathrm{jx}},k_{\mathrm{jy}},n_1,n_2}^{j,j,J,J}=R_{2k_{\mathrm{jx}},2k_{\mathrm{jy}}+1,n_1,n_2}^{j+1,j+2,J,J}\\ -[\underset{k_{0x}=0}{\overset{2^{j_0}}{\mathrm{\Σ}}}\underset{k_{0y}=0}{\overset{2^{j_0}}{\mathrm{\Σ}}}{R}_{k_{0x},k_{0y},n_1,n_2}^{j_0,j_0,J,J}\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)w_{k_{0x},k_{0y}}^{j_0}(x_{2k_{\mathrm{jx}}}^{j+1},y_{{2k}_{\mathrm{jy}}+1}^{j+1})\\ +\underset{j_1=j_0}{\overset{j-1}{\mathrm{\Σ}}}\underset{n_2=0}{\overset{2^J}{\mathrm{\Σ}}}\underset{k_{j_{1}x}=0}{\overset{2^{j_1}}{\mathrm{\Σ}}}({C1}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x}+\mathrm{1,2}{k}_{j_{1}y}}^{j_1+1}(x_{2k_{\mathrm{jx}}}^{j+1},y_{{2k}_{\mathrm{jy}}+1}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +{C2}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x},2k_{j_{1}y}+1}^{j_1+1}(x_{2k_{\mathrm{jx}}}^{j+1},y_{{2k}_{\mathrm{jy}}+1}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +{C3}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x}+\mathrm{1,2}{k}_{j_{1}y}+1}^{j_1+1}(x_{2k_{\mathrm{jx}}}^{j+1},y_{{2k}_{\mathrm{jy}}+1}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)]\end{array}$

$\begin{array}{c}{C3}_{k_{\mathrm{jx}},k_{\mathrm{jy}},n_1,n_2}^{j,j,J,J}=R_{2k_{\mathrm{jx}}+\mathrm{1,2}{k}_{\mathrm{jy}}+1,n_1,n_2}^{j+1,j+1,J,J}\\ -[\underset{k_{0x}=0}{\overset{2^{j_0}}{\mathrm{\Σ}}}\underset{k_{0y}=0}{\overset{2^{j_0}}{\mathrm{\Σ}}}{R}_{k_{0x},k_{0y},n_1,n_2}^{j_0,j_0,J,J}\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)w_{k_{0x},k_{0y}}^{j_0}(x_{2k_{\mathrm{jx}}+1}^{j+1},y_{{2k}_{\mathrm{jy}}+1}^{j+1})\\ +\underset{j_1=j_0}{\overset{j-1}{\mathrm{\Σ}}}\underset{n_2=0}{\overset{2^J}{\mathrm{\Σ}}}\underset{k_{j_{1}x}=0}{\overset{2^{j_1}}{\mathrm{\Σ}}}({C1}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x}+\mathrm{1,2}{k}_{j_{1}y}}^{j_1+1}(x_{2k_{\mathrm{jx}}+1}^{j+1},y_{{2k}_{\mathrm{jy}}+1}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +{C2}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x},2k_{j_{1}y}+1}^{j_1+1}(x_{2k_{\mathrm{jx}}+1}^{j+1},y_{{2k}_{\mathrm{jy}}+1}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +{C3}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x}+\mathrm{1,2}{k}_{j_{1}y}+1}^{j_1+1}(x_{2k_{\mathrm{jx}}+1}^{j+1},y_{{2k}_{\mathrm{jy}}+1}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)]\end{array}.$

Wherein, the position of each pixel in described image be defined as:

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

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

Wherein, described Restriction Operators R is defined as:

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

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

The described detailed process to the Image Variational model under the picture construction wavelet frame after amplification is:

The calibration model defining the image after described 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 parameter diffusion time, and f (x, y) is original two dimensional image, and c is spread function, and u is the expression formula of the image of being expressed by multiple dimensioned interpolating function;

The expression formula of the image of being expressed by multiple dimensioned interpolating function is brought in described calibration model, obtains the Image Variational model under described wavelet frame.

Wherein, described spread function c is defined as:

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

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

Wherein, the detailed process solving the Image Variational model under described wavelet frame described in comprises:

Described Image Variational model is rewritten as:

$\begin{array}{c}\frac{{\mathrm{du}}^J(x,y,t)}{\mathrm{dt}}=F[t,x,y,u^J(x,y,t),u^{J\left(\mathrm{1,0}\right)}(x,y,t),u^{J\left(\mathrm{0,1}\right)}(x,y,t),\\ u^{J\left(\mathrm{2,0}\right)}(x,y,t),u^{J\left(\mathrm{1,1}\right)}(x,y,t),u^{J\left(\mathrm{0,2}\right)}(x,y,t)]\end{array},$

Wherein, and by u ^j(x, y, t _n) be set as u _n, t _nthe function F in moment is set as F _n;

Structure linear homotopy is: u ^j(x, y, t)=(1-ε) F _n+ ε F _n+1;

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

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

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

Wherein, described ε is homotopy parameter, and

According to another aspect of the present invention, a kind of image processing apparatus based on Shannon-Blackman small echo sparse expression is provided, it is characterized in that, comprising:

Image acquisition unit, for obtaining pending image;

Image enlarging unit, for constructing the multiple dimensioned interpolation operator based on Shannon-Blackman interpolating wavelet, amplifies described image;

Unit set up by Image Variational model, for amplify after picture construction wavelet frame under Image Variational model;

Solving unit, for solving the Image Variational model under described wavelet frame, getting a distinct image.

Image processing method based on Shannon-Blackman small echo sparse expression provided by the invention and device, by multiple dimensioned interpolation operator, image is amplified, achieve the adaptive configuration of image slices vegetarian refreshments, effectively can retain detailed structure simultaneously, improve and rebuild efficiency, avoid the phenomenon occurring artificial artifact in amplification process; In addition, adopt multiple dimensioned Blackman function to replace traditional spread function, effectively can avoid the obfuscation of local detail, thus the oversubscription not rate of image can be improved further; Finally, by structure sparse grid Algorithm for Solving Image Variational model, effectively improve efficiency and precision.

Accompanying drawing explanation

Fig. 1 shows the process flow diagram of the image processing method based on Shannon-Blackman small echo sparse expression of the present invention.

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

Fig. 3 shows the structured flowchart of the image processing apparatus based on Shannon-Blackman small echo sparse expression of the present invention.

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

Fig. 5 show an embodiment of reality of the present invention by the image of Nonlinear magnify 2.

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

The image that Fig. 7 shows one embodiment of the invention uses Shannon-Blackman scaling function to carry out the expression schematic diagram of multiple dimensioned interpolation.

Embodiment

Below in conjunction with drawings and Examples, the specific embodiment of the present invention is described in further detail.Following examples for illustration of the present invention, but are not used for limiting the scope of the invention.

An embodiment provides a kind of image processing method based on Shannon-Blackman small echo sparse expression.

Fig. 1 shows the process flow diagram of the image processing method based on Shannon-Blackman small echo sparse expression of the present invention.Fig. 2 shows the comparison diagram of the Shannon-Blackman scaling function of existing Shannon wavelet scaling function and embodiments of the invention.

With reference to Fig. 1, the image processing method based on Shannon-Blackman small echo sparse expression of the present invention, detailed process comprises:

S1, obtain pending image;

S2, construct multiple dimensioned interpolation operator based on Shannon-Blackman interpolating wavelet, described image is amplified.

Detailed process is as follows:

If the field of definition of image is (x _min, x _max) × (y _min, y _max), in image, the position of each pixel is defined as wherein j is scale parameter, k _jxand k _jyfor the location parameter on j yardstick;

The position of each pixel in image be defined as:

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

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

First, defining Shannon-Blackman scaling function is:

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

According to above-mentioned scaling function, N is compact schemes constant, as shown in Figure 2, relative to Shannon wavelet scaling function, the scaling function that the interpolating wavelet of the present embodiment is corresponding remains interpolation characteristic and orthogonal property, also add compactly supported, the scaling function of compact schemes, can simultaneously innovatory algorithm numerical precision and speed of convergence as the odd function solving partial differential equation simultaneously.

Then Shannon-Blackman scaling function is defined as by tensor product:

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

According to the definition of interpolating wavelet transform, multi-scale wavelet interpolation operator can be derived as follows:

$\begin{array}{c}{I}_{n_1,n_2}(x,y)=\underset{k_{0x}=0}{\overset{2^{j_0}}{\mathrm{\Σ}}}\underset{k_{0y}=0}{\overset{2^{j_0}}{\mathrm{\Σ}}}{R}_{k_{0x},k_{0y},n_1,n_2}^{j_0,j_0,J,J}{w}_{k_{0x,}{k}_{0y}}^{j_0}(x,y)\\ +\underset{j=j_0}{\overset{J-1}{\mathrm{\Σ}}}\underset{k_{\mathrm{jx}}=0}{\overset{2^j}{\mathrm{\Σ}}}\underset{k_{\mathrm{jy}}=0}{\overset{2^j}{\mathrm{\Σ}}}({C1}_{k_{\mathrm{jx}},k_{\mathrm{jy}},n_1,n_2}^{j,j,J,J}{w}_{2k_{\mathrm{jx}}+\mathrm{1,2}{k}_{\mathrm{jy}}}^{j+1}(x,y)\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ {+C2}_{k_{\mathrm{jx}},k_{\mathrm{jy}},n_1,n_2}^{j,j,J,J}{w}_{2k_{\mathrm{jx}},2k_{\mathrm{jy}}+1}^{j+1}(x,y)\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ {+C3}_{k_{\mathrm{jx}},k_{\mathrm{jy}},n_1,n_2}^{j,j,J,J}{w}_{2k_{\mathrm{jx}}+\mathrm{1,2}{k}_{\mathrm{jy}}+1}^{j+1}(x,y)\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\end{array}$

Wherein, C1, C2, C3 are Wavelet Interpolation transformation matrix, n ₁, n ₂for the position of interpolation operator, R is Restriction Operators, and J is the maximal value of scale parameter, j ₀be expressed as 0 layer of yardstick, j ₁for being different from the scale parameter of j, and

$\begin{array}{c}{C1}_{k_{\mathrm{jx}},k_{\mathrm{jy}},n_1,n_2}^{j,j,J,J}=R_{2k_{\mathrm{jx}}+\mathrm{1,2}{k}_{\mathrm{jy}},n_1,n_2}^{j+1,j_{+2},J,J}\\ -[\underset{k_{0x}=0}{\overset{2^{j_0}}{\mathrm{\Σ}}}\underset{k_{0y}=0}{\overset{2^{j_0}}{\mathrm{\Σ}}}{R}_{k_{0x},k_{0y},n_1,n_2}^{j_0,j_0,J,J}\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)w_{k_{0x},k_{0y}}^{j_0}(x_{2k_{\mathrm{jx}}+1}^{j+1},y_{{2k}_{\mathrm{jy}}}^{j+1})\\ +\underset{j_1=j_0}{\overset{j-1}{\mathrm{\Σ}}}\underset{n_2=0}{\overset{2^J}{\mathrm{\Σ}}}\underset{k_{j_{1}x}=0}{\overset{2^{j_1}}{\mathrm{\Σ}}}({C1}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x}+\mathrm{1,2}{k}_{j_{1}y}}^{j_1+1}(x_{2k_{\mathrm{jx}}+1}^{j+1},y_{{2k}_{\mathrm{jy}}}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +{C2}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x},2k_{j_{1}y}+1}^{j_1+1}(x_{2k_{\mathrm{jx}}+1}^{j+1},y_{{2k}_{\mathrm{jy}}}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +{C3}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x}+\mathrm{1,2}{k}_{j_{1}y}+1}^{j_1+1}(x_{2k_{\mathrm{jx}}+1}^{j+1},y_{{2k}_{\mathrm{jy}}}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)]\end{array}$

$\begin{array}{c}{C2}_{k_{\mathrm{jx}},k_{\mathrm{jy}},n_1,n_2}^{j,j,J,J}=R_{2k_{\mathrm{jx}},2k_{\mathrm{jy}}+1,n_1,n_2}^{j+1,j+2,J,J}\\ -[\underset{k_{0x}=0}{\overset{2^{j_0}}{\mathrm{\Σ}}}\underset{k_{0y}=0}{\overset{2^{j_0}}{\mathrm{\Σ}}}{R}_{k_{0x},k_{0y},n_1,n_2}^{j_0,j_0,J,J}\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)w_{k_{0x},k_{0y}}^{j_0}(x_{2k_{\mathrm{jx}}}^{j+1},y_{{2k}_{\mathrm{jy}}+1}^{j+1})\\ +\underset{j_1=j_0}{\overset{j-1}{\mathrm{\Σ}}}\underset{n_2=0}{\overset{2^J}{\mathrm{\Σ}}}\underset{k_{j_{1}x}=0}{\overset{2^{j_1}}{\mathrm{\Σ}}}({C1}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x}+\mathrm{1,2}{k}_{j_{1}y}}^{j_1+1}(x_{2k_{\mathrm{jx}}}^{j+1},y_{{2k}_{\mathrm{jy}}+1}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +{C2}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x},2k_{j_{1}y}+1}^{j_1+1}(x_{2k_{\mathrm{jx}}}^{j+1},y_{{2k}_{\mathrm{jy}}+1}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +{C3}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x}+\mathrm{1,2}{k}_{j_{1}y}+1}^{j_1+1}(x_{2k_{\mathrm{jx}}}^{j+1},y_{{2k}_{\mathrm{jy}}+1}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)]\end{array}$

$\begin{array}{c}{C3}_{k_{\mathrm{jx}},k_{\mathrm{jy}},n_1,n_2}^{j,j,J,J}=R_{2k_{\mathrm{jx}}+\mathrm{1,2}{k}_{\mathrm{jy}}+1,n_1,n_2}^{j+1,j+1,J,J}\\ -[\underset{k_{0x}=0}{\overset{2^{j_0}}{\mathrm{\Σ}}}\underset{k_{0y}=0}{\overset{2^{j_0}}{\mathrm{\Σ}}}{R}_{k_{0x},k_{0y},n_1,n_2}^{j_0,j_0,J,J}\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)w_{k_{0x},k_{0y}}^{j_0}(x_{2k_{\mathrm{jx}}+1}^{j+1},y_{{2k}_{\mathrm{jy}}+1}^{j+1})\\ +\underset{j_1=j_0}{\overset{j-1}{\mathrm{\Σ}}}\underset{n_2=0}{\overset{2^J}{\mathrm{\Σ}}}\underset{k_{j_{1}x}=0}{\overset{2^{j_1}}{\mathrm{\Σ}}}({C1}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x}+\mathrm{1,2}{k}_{j_{1}y}}^{j_1+1}(x_{2k_{\mathrm{jx}}+1}^{j+1},y_{{2k}_{\mathrm{jy}}+1}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +{C2}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x},2k_{j_{1}y}+1}^{j_1+1}(x_{2k_{\mathrm{jx}}+1}^{j+1},y_{{2k}_{\mathrm{jy}}+1}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +{C3}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{2k_{j_{1}x}+\mathrm{1,2}{k}_{j_{1}y}+1}^{j_1+1}(x_{2k_{\mathrm{jx}}+1}^{j+1},y_{{2k}_{\mathrm{jy}}+1}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)]\end{array}.$

In addition, described Restriction Operators R is defined as:

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

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

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

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

By above-mentioned formula, the multiple dimensioned interpolation amplification of image can be realized.

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

Its detailed process is:

The calibration model of the image after definition 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 parameter diffusion time, and f (x, y) is original two dimensional image, and c is spread function, and u is the expression formula of the image of being expressed by multiple dimensioned interpolating function;

Spread function c is defined as:

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

Wherein, for gradient operator.The present embodiment uses above-mentioned spread function, can guarantee that the spread function in calculating has strict compactly supported.

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

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

Its detailed process comprises:

Described Image Variational model is rewritten as:

$\begin{array}{c}\frac{{\mathrm{du}}^J(x,y,t)}{\mathrm{dt}}=F[t,x,y,u^J(x,y,t),u^{J\left(\mathrm{1,0}\right)}(x,y,t),u^{J\left(\mathrm{0,1}\right)}(x,y,t),\\ u^{J\left(\mathrm{2,0}\right)}(x,y,t),u^{J\left(\mathrm{1,1}\right)}(x,y,t),u^{J\left(\mathrm{0,2}\right)}(x,y,t)]\end{array}$

Wherein, and by u ^j(x, y, t _n) be set as u _n, t _nthe function F in moment is set as F _n;

Structure linear homotopy is: u ^j(x, y, t)=(1-ε) F _n+ ε F _n+1;

Wherein, described ε is homotopy parameter, and

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

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

According to described expression formula and described revised Image Variational model, obtain one group of ordinary differential system, this solving equations can be obtained the correction result of described image, thus the super-resolution rebuilding result of image can be obtained.

In an alternative embodiment of the invention, provide a kind of image processing apparatus based on Shannon-Blackman small echo sparse expression.

Fig. 3 shows the structured flowchart of the image processing apparatus based on Shannon-Blackman small echo sparse expression of the present invention.

With reference to Fig. 3, the image processing apparatus based on Shannon-Blackman small echo sparse expression of the present embodiment comprises:

Image acquisition unit 10, for obtaining pending image;

Image enlarging unit 20, for constructing the multiple dimensioned interpolation operator based on Shannon-Blackman interpolating wavelet, amplifies described image;

Unit 30 set up by Image Variational model, for amplify after picture construction wavelet frame under Image Variational model;

Solving unit 40, for solving the Image Variational model under described wavelet frame, getting a distinct image.

Below by way of specific embodiment, said method is described.

Fig. 4 shows the original image of an alternative embodiment of the invention.Fig. 5 show another embodiment of reality of the present invention by the image of Nonlinear magnify 2.Fig. 6 shows the image after the correction of an alternative embodiment of the invention.The image that Fig. 7 shows one embodiment of the invention uses Shannon-Blackman scaling function to carry out the expression schematic diagram of multiple dimensioned interpolation.

The present embodiment extracts an image as shown in Figure 4 and uses said method to correct.

First the multiple dimensioned interpolation operator of Shannon-Blackman is adopted to amplify image, shown in figure Fig. 5;

Then adopt the Variation Model under interpolating wavelet framework to correct to the image after amplification, obtain the image after correcting, as shown in Figure 6, thus complete the super-resolution rebuilding of image.

By image processing method of the present invention, multiple dimensioned interpolation operator is used to amplify image at timing, the adaptive configuration of image slices vegetarian refreshments can be realized, as shown in Figure 7, image pixel is 300*300, the adaptive configuration point number of its pixel is 23488, the border annex pixel point density of image object thing is large, and other equal value parts are relatively sparse, when adopting this scheme to carry out Nonlinear magnify, effectively can retain detailed structure, improve simultaneously and rebuild efficiency, avoid in amplification process, occurring artificial artifact phenomenon.

In addition, the present invention adopts multiple dimensioned employing multiple dimensioned Blackman function to replace traditional spread function, effectively can avoid the obfuscation of local detail, thus can improve the oversubscription not rate of image further.

Finally, adopt the multiple dimensioned operator of Shannon-Blackman Construction of Wavelets, and by structure sparse grid Algorithm for Solving Image Variational model, effectively improve efficiency and precision.

Above embodiment is only for illustration of the present invention; and be not limitation of the present invention; the those of ordinary skill of relevant technical field; without departing from the spirit and scope of the present invention; can also make a variety of changes and modification; therefore all equivalent technical schemes also belong to category of the present invention, and scope of patent protection of the present invention should be defined by the claims.

Claims (8)

1. based on an image processing method for Shannon-Blackman small echo sparse expression, it is characterized in that, comprising:

Obtain pending image;

Construct the multiple dimensioned interpolation operator based on Shannon-Blackman interpolating wavelet, described image is amplified;

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

Solve the Image Variational model under described wavelet frame, get a distinct image.

2. image processing method as claimed in claim 1, is characterized in that, the detailed process that described structure amplifies image based on the multiple dimensioned interpolation operator of Shannon-Blackman interpolation plait is as follows:

If the field of definition of image is (x _min, x _max) × (y _min, y _max), in image, the position of each pixel is defined as wherein j is scale parameter, k _jxand k _jyfor the location parameter on j yardstick;

Definition Shannon-Blackman scaling function is:

$\begin{array}{c}\mathrm{\φ}\left(x\right)\\ =\begin{array}{}\end{array}\left\{\begin{array}{c}\frac{\sin \left(\mathrm{\πx}\right)}{\mathrm{\πx}}{(0.12+0.5\cos \left(\frac{2\mathrm{\πx}}{N}\right)+0.08\cos \left(\frac{4\mathrm{\πx}}{N}\right))}^2,-N/2\≤x\≤N/2\\ 0,\mathrm{otherwise}\end{array}\right.\end{array};$

Shannon-Blackman scaling function is defined as by tensor product:

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

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

$\begin{array}{c}{I}_{n_1,n_2}(x,y)=\underset{k_{0x}=0}{\overset{2^{j_0}}{\mathrm{\Σ}}}\underset{k_{0y}=0}{\overset{2^{j_0}}{\mathrm{\Σ}}}{R}_{k_{0x},k_{0y},n_1,n_2}^{j_0,j_0,J,J}{w}_{k_{0x},k_{0y}}^{j_{0}j}(x,y)\\ +\underset{j=j_0}{\overset{J-1}{\mathrm{\Σ}}}\underset{k_{\mathrm{jx}}=0}{\overset{2^j}{\mathrm{\Σ}}}\underset{k_{\mathrm{jy}}=0}{\overset{2^j}{\mathrm{\Σ}}}\left({C1}_{k_{\mathrm{jx}},k_{\mathrm{jy}},n_1,n_2}^{j,j,J,J}{w}_{{2k}_{\mathrm{jx}}+1,{2k}_{\mathrm{jy}}}^{j+1}\right)(x,y)\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +{C2}_{k_{\mathrm{jx}},k_{\mathrm{jy}},n_1,n_2}^{j,j,J,J}{w}_{{2k}_{\mathrm{jx}},{2k}_{\mathrm{jy}}+1}^{j+1}(x,y)\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +{C3}_{k_{\mathrm{jx}},k_{\mathrm{jy}},n_1,n_2}^{j,j,J,J}{w}_{{2k}_{\mathrm{jx}}+1,{2k}_{\mathrm{jy}}+1}^{j+1}(x,y)\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\end{array}$

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

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

Wherein, N is compact schemes constant, and C1, C2, C3 are Wavelet Interpolation transformation matrix, n ₁, n ₂for the position of interpolation operator, R is Restriction Operators, and J is the maximal value of scale parameter, j ₀be expressed as 0 layer of yardstick, j ₁for being different from the scale parameter of j;

Wherein,

$\begin{array}{c}{C1}_{k_{\mathrm{jx}},k_{\mathrm{jy}},n_1,n_2}^{j,j,J,J}=R_{{2k}_{\mathrm{jx}}+1,{2k}_{\mathrm{jy}},n_1,n_2}^{j+1,j_{+2},J,J}\\ -[\underset{k_{0x}=0}{\overset{2^{j_0}}{\mathrm{\Σ}}}\underset{k_{0x}=0}{\overset{2^{j_0}}{\mathrm{\Σ}}}\underset{k_{0y}=0}{\overset{2^{j_0}}{\mathrm{\Σ}}}{R}_{k_{0x},k_{0y},n_1,n_2}^{j_0,j_0,J,J}\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)w_{k_{0x},k_{0y}}^{j_0}(x_{{2k}_{\mathrm{jx}}+1}^{j+1},y_{{2k}_{\mathrm{jy}}}^{j+1})\\ +{C2}_{k_{j_{1}x},k_{j1y},n_1,n_2}^{j_1,j_1,J,J}{w}_{{2k}_{j_{1}x},{2k}_{j_{1}y}+1}^{j_1+1}(x_{{2k}_{\mathrm{jx}}+1}^{j+1},y_{{2k}_{\mathrm{jy}}}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +{C3}_{k_{j_{1}x},k_{j1y},n_1,n_2}^{j_1,j_1,J,J}{w}_{{2k}_{j_{1}x}+1,{2k}_{j_{1}y}+1}^{j_1+1}(x_{{2k}_{\mathrm{jx}}+1}^{j+1},y_{{2k}_{\mathrm{jy}}}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)]\end{array}$

$\begin{array}{c}{C2}_{k_{\mathrm{jx}},k_{\mathrm{jy}},n_1,n2}^{j,j,J,J}=R_{{2k}_{\mathrm{jx}},{2k}_{\mathrm{jy}},n_1,n_2}^{j+1,j+2,J,J}\\ -[\underset{k_{0x}=0}{\overset{2^{j_0}}{\mathrm{\Σ}}}\underset{k_{0y}=0}{\overset{2^{j_0}}{\mathrm{\Σ}}}{R}_{k_{0x},k_{0y},n_1,n_2}^{j_0,j_0,J,J}\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)w_{k_{0x},k_{0y}}^{j_0}(x_{{2k}_{\mathrm{jx}}}^{j+1},y_{{2k}_{\mathrm{jy}}+1}^{j+1})\\ +{C2}_{k_{1_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{{2k}_{j_{1}x},{2k}_{j_{1}y}+1}^{j_1+1}(x_{{2k}_{\mathrm{jx}}}^{j+1},y_{{2k}_{\mathrm{jy}}+1}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +{C3}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{{2k}_{j_{1}x}+1,{2k}_{j_{1}y}+1}^{j_1+1}(x_{{2k}_{\mathrm{jx}}}^{j+1},y_{{2k}_{\mathrm{jy}}}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)]\end{array}$

$\left[\begin{array}{c}{C3}_{k_{\mathrm{jx}},k_{\mathrm{jy}},n_1,n_2}^{j,j,J,J}=R_{{2k}_{\mathrm{jx}}+1,{2k}_{\mathrm{jy}}+1,n_1,n_2}^{j+1,j+1,J,J}\\ -[\underset{k_{0x}=0}{\overset{2^{j_0}}{\mathrm{\Σ}}}\underset{k_{0y}=0}{\overset{2^{j_0}}{\mathrm{\Σ}}}{R}_{k_{0x},k_{0y},n_1,n_2}^{j_0,j_0,J,J}\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)w_{k_{0x},k_{0y}}^{j_0}(x_{{2k}_{\mathrm{jx}}+1}^{j+1},y_{{2k}_{\mathrm{jy}}+1}^{j+1})\\ +\underset{j_1=j_0}{\overset{j-1}{\mathrm{\Σ}}}\underset{n_2=0}{\overset{2^J}{\mathrm{\Σ}}}\underset{k_{j_{1}x}=0}{\overset{2^{j_1}}{\mathrm{\Σ}}}\left({C1}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{{2k}_{j_{1}x}+1,{2k}_{j_{1}y}}^{j_1+1}\right)(x_{{2k}_{\mathrm{jx}}+1}^{j+1},y_{{2k}_{\mathrm{jy}}+1}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +{C2}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{{2k}_{j_{1}x},{2k}_{j_{1}y}+1}^{j_1+1}(x_{{2k}_{j_{1}x}+1}^{j+1},y_{{2k}_{j_{1}y}+1}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)\\ +{C3}_{k_{j_{1}x},k_{j_{1}y},n_1,n_2}^{j_1,j_1,J,J}{w}_{{2k}_{j_{1}x}+1,{2k}_{j_{1}y}+1}^{j_1+1}(x_{{2k}_{j_{1}x}+1}^{j+1},y_{{2k}_{j_{1}y}+1}^{j+1})\mathrm{\Φ}(x_{n_1}^J,y_{n_2}^J)]\end{array}\right]$

3. image processing method as claimed in claim 2, is characterized in that, the position of each pixel in described image be defined as:

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

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

4. image processing method as claimed in claim 2, it is characterized in that, described Restriction Operators R is defined as:

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

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

5. image processing method as claimed in claim 2, is characterized in that, the described detailed process to the Image Variational model under the picture construction wavelet frame after amplification is:

The calibration model defining the image after described 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 parameter diffusion time, and f (x, y) is original two dimensional image, and c is spread function, and u is the expression formula of the image of being expressed by multiple dimensioned interpolating function;

The expression formula of the image of being expressed by multiple dimensioned interpolating function is brought in described calibration model, obtains the Image Variational model under described wavelet frame.

6. image processing method as claimed in claim 5, it is characterized in that, described spread function c is defined as:

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

Wherein, for gradient operator.

7. image processing method as claimed in claim 1, is characterized in that, described in the detailed process of Image Variational model that solves under described wavelet frame comprise:

Described Image Variational model is rewritten as:

$\begin{array}{c}\frac{{\mathrm{du}}^J(x,y,t)}{\mathrm{dt}}=F[t,x,y,u^J(x,y,t),u^{J\left(\mathrm{1,0}\right)}(x,y,t),u^{J\left(\mathrm{0,1}\right)}(x,y,t),\\ u^{J\left(\mathrm{2,0}\right)}(x,y,t),u^{J\left(\mathrm{1,1}\right)}(x,y,t),u^{J\left(\mathrm{0,2}\right)}(x,y,t)]\end{array},$

Wherein, and by u ^j(x, y, t _n) be set as u _n, t _nthe function F in moment is set as F _n;

Structure linear homotopy is: u ^j(x, y, t)=(1-ε) F _n+ ε F _n+1;

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

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

The expression formula launched according to described linear homotopy and described revised Image Variational model, obtain one group of ordinary differential system, and solve the correction result obtaining described pending image;

Wherein, described ε is homotopy parameter, and

8. based on an image processing apparatus for Shannon-Blackman small echo sparse expression, it is characterized in that, comprising:

Image acquisition unit, for obtaining pending image;

Image enlarging unit, for constructing the multiple dimensioned interpolation operator based on Shannon-Blackman interpolating wavelet, amplifies described image;

Unit set up by Image Variational model, for amplify after picture construction wavelet frame under Image Variational model;

Solving unit, for solving the Image Variational model under described wavelet frame, getting a distinct image.