CN103390265B - A kind of Denoising of Texture Images wave filter based on fractional order evolution equation - Google Patents

Abstract

A kind of Denoising of Texture Images wave filter based on fractional order evolution equation proposed by the invention realizes the fractional order to texture image, non-linear, multiple dimensioned, high precision denoising based on a kind of special fractional order evolution equation denoise algorithm.This wave filter adopts buffer, squared difference device one ,-ν ₃power power side device, multiplier one, λ ⁿgenerator, generator, multiplier two, multiplier three and totalizer one are formed in cascaded fashion.This wave filter has while removal texture image noise, the low frequency profile in smooth region can be retained as far as possible, simultaneously again can the relatively large high frequency edge of non-linear reservation gray-scale value transition amplitude, and can also the remarkable advantage of the relatively little grain details of non-linear reservation gray-scale value transition changes in amplitude.This wave filter is specially adapted to the application scenario of the image being rich in complex texture minutia being carried out to denoising.The invention belongs to the technical field of applied mathematics, Digital Image Processing and digital circuit cross discipline.

Description

A kind of Denoising of Texture Images wave filter based on fractional order evolution equation

Technical field

A kind of Denoising of Texture Images wave filter based on fractional order evolution equation proposed by the invention realizes the fractional order to texture image, non-linear, multiple dimensioned, high precision denoising based on a kind of special fractional order evolution equation denoise algorithm.The order v of the fractional calculus that the present invention relates to ₁, v ₂and v ₃not traditional integer rank, but non-integral order, generally get mark or rational fractional in engineer applied, and v ₃≠ 1,2,3.See Fig. 1, this wave filter adopts buffer 2, squared difference device 1 ,-v ₃power power side device 4, multiplier 1, λ ⁿgenerator 11, generator 10, multiplier 28, multiplier 39 and totalizer 1 are formed in cascaded fashion.This wave filter has while removal texture image noise, the low frequency profile in smooth region can be retained as far as possible, simultaneously again can the relatively large high frequency edge of non-linear reservation gray-scale value transition amplitude, and can also the remarkable advantage of the relatively little grain details of non-linear reservation gray-scale value transition changes in amplitude.This wave filter is specially adapted to the application scenario of the image being rich in complex texture minutia being carried out to denoising.The invention belongs to the technical field of applied mathematics, Digital Image Processing and digital circuit cross discipline.

Background technology

Digital Image Processing theory mainly comprises three major types method: stochastic modeling, wavelet theory and Partial Differential Equation method.Wherein, the image procossing based on partial differential equation belongs to a part important in mathematical analysis, is an important branch in image processing field.Partial Differential Equation method is together with it is closely connected with physical world.Famous wave equation and heat-conduction equation all belong to integer rank partial differential equation, also have Euler equation, Poisson equation and Laplace equation etc.Integer rank partial differential equation in physics is often applied to other field, as biological, financial etc., and has been applied to digital image processing field.About the digital image processing techniques based on integer rank partial differential equation, on the one hand, this image processing method belongs to the category of low layer pictures process, and its result is usually taken as intermediate result and is supplied to other image processing methods and uses further; On the other hand, along with the further investigation of this image processing method, people more and more profoundly excavate the essence of image and image procossing, and attempt to transform existing traditional images disposal route by strict mathematical theory, this is a kind of challenge for the traditional images disposal route based on practicality.

At present, although partial differential equation has been applied to digital image processing field, but most correlative study is all also only confined to the application of integer rank partial differential equation, but all also study very few both at home and abroad for partial fractional differential equation being applied in digital image processing field.Integer rank partial differential equation itself is from continuous domain, so it can describe in essence is analog image, once its existence of solution and uniqueness have been proved to be, we just can utilize discrete numerical method to ask for its numerical solution to the integer rank partial differential equation for digital picture.Because can be associated with some physical processes based on the Digital Image Processing of integer rank partial differential equation, therefore they are all described with continuous domain usually.In general, Partial Differential Equation method calculated amount compared with common filtering method in integer rank is larger: need the solving equations that iterative or finite difference construct, the major advantage of integer rank partial differential equation class methods is: first, integer rank partial differential equation and corresponding curve (curved surface) stream provide the continuous model of analysis chart picture, discrete filtering shows as continuous print differentiating operator, thus make the division of grid, local nonlinearity analysis be easy to realize.On the other hand, when image is expressed as continuous signal, integer rank partial differential equation can be regarded as the iteration at small subneighborhood part filter, and this characteristic allows to be carried out synthesizing by existing filtering method and classify, and can form new filtering method.Second, utilize integer rank partial differential equation processing digital images to be easy to directly grasp and process the geometric properties that such as gradient, tangent line, curvature and level set etc. are visually important, effectively can also simulate the visually significant dynamic process that such as linear processes diffusion and information transmission mechanism are such.3rd, the unique analytical theory in partial differential equation field, integer rank is the better Digital Image Processing algorithm of research and significant notional result, as existence of solution, uniqueness etc., provides possibility.Especially, the most noticeable advantage is that integer rank Partial Differential Equation method can obtain good picture quality, and has certain stability.Flexile numerical scheme is that the numerical evaluation of image procossing equation provides larger help.The phase at the end of the eighties in last century can be traced back in the introducing of digital image processing field more effective integer rank partial differential equation, obtain very significant progress in the nineties.This research can trace back to Nagao, Rudin etc. about image smoothing and image enhaucament and the Koenderink exploration for picture structure.The integer rank partial differential equation of multiple classics has been applied in the middle of Digital Image Processing, such as heat-conduction equation, schrodinger equation, convect h equation etc.At present, the image processing techniques based on integer rank partial differential equation achieves some and applies preferably, and such as French NASA have employed AMSS method as standard method Aerial Images being carried out to image enhaucament.Integer rank Partial Differential Equation method itself is physical content, the heat-conduction equation calculated possibly in isotropic medium applied the earliest in Digital Image Processing.If gray level image is regarded as the temperature field in an isotropic medium, so the heat transfer process in this temperature field correspond to the Gaussian smoothing process of image just, and Gaussian filter variance parameter is relevant with conduction time.But because Gaussian smoothing is isotropic, so very large for the destruction at edge, its application is also restricted.1987, Kass etc. utilized the internal-external constraint definition required for image border one to characterize the energy function of contour curve quality, and wherein internal constraint mainly considers slickness and the curvature of profile, and external energy represents the attraction of image border profile.By optimizing (minimizing) this energy function, initially given profile can converge on contiguous image border.The explicit physical meaning of this method, but due to its consider object (profile) be [] ²one dimension target (suppose it is the profile in two dimensional image, if consider 3-D view, as 3 d medical images, what so this profile was corresponding is [] ³in a Two-dimensional Surfaces), its describing mode and discretize all receive certain restriction, and its describing mode directly limit the change in topology of contour curve, as division, merging etc.1989, Mumford and Shah proposed the Variation Model of Iamge Segmentation.1992, the minimum thought of the inter-object distance in the identification of Chan and Vese Land use models constructed unskirted movable contour model, and Yezzi etc. utilize the maximum thought of between class distance to construct a kind of new movable contour model equally afterwards.Nineteen ninety-five, the front propagation process that the proposition level sets such as Osher go description one relevant with curvature.The essence of these class methods image outline is regarded as the zero level collection of a two-dimensional function, so by the change behavior of this two-dimensional function of research, just can know the variation pattern of profile.Simultaneously due to this studied to as if a two-dimensional function, it is [] ²in be easy to describe and solve (for a dimensional object), and one-dimensional profile is not the object of direct solution, so by the change of two-dimensional function, make the process of the change in topology such as the division of profile, merging become relatively easy.From the later stage nineties in last century, integer rank partial differential equation starts to be applied to digital picture reparation, the part of namely losing in stuffing digit image or the barrier etc. removed in digital picture, to make result images appear to really, it is the very difficult task of of picture editting field.Image repair based on integer rank partial differential equation has two general orientation, Chan and Shen of UCLA etc. utilize and energy-optimisedly process this problem, mainly do to suppose to the properties at structural images edge (as little in simplicity, curvature etc.), then construct corresponding energy function to describe, change into integer rank partial differential equation by the integer rank variational method and solve; Take Bertalmio as the diffusion process that another school of representative then directly considers some character in image, directly provide integer rank partial differential equation and carry out evolution and ask Partial Differential Equations.The method of this two class all achieves larger success.In addition in picture editting field, Poisson equation also sticks at the seamless sticky of image and occupies leading position.

In the image procossing based on integer rank partial differential equation, image denoising is one of its most important research contents.Image denoising based on integer rank partial differential equation is divided into two classes: based on the method for nonlinear diffusion with based on the minimized variational method of energy model letter.Two kinds of corresponding with it basic models are: full variation (ROF) model that Anisotropic diffusion (PM) model proposed by Perona and Malik and Rudin, Osher and Fatemi propose.PM model uses the diffusion process of heat energy to carry out the denoising process of analog image, and the result of image denoising is exactly the state that heat energy spreads when reaching balance.Describing above-mentioned heat energy with full variation, is exactly ROF model.On this basis, scholar is had PM model and ROF model to be generalized among Color Image Processing respectively.There is the Selecting parameter in scholar's research model, and how to calculate the Optimal Stop point of iterative process.The people such as Rudin propose the rectangular method of a kind of variable time step to separate Euler-Lagrange equation.C.R.Vogel and M.E.Oman fixed point iteration method improves the stability of ROF model.D.C.Dobson and C.R.Vogel revises full variational form to ensure the convergence that ROF model value calculates.A.Chambolle proposes a kind of fast algorithm based on dual formula.J.Darbon and M.Sigelle utilizes Level Set Method primal problem to be decomposed into the optimization problem of separate markov random file, obtains globally optimal solution by rebuilding.Scholar is had to propose a kind of iteration weighted norm to solve full variation to improve counting yield.Original image first through a Gaussian smoothing, makes PM model have well-posedness by F.Catte etc.PM model and ROF model all have easily generation comparative information and lose, the significantly shortcoming such as texture information loss and staircase effect.For these shortcomings, there has been proposed many improved models.In order to keep comparative information and texture information, scholar is had to use L ¹norm replaces L ²norm.S.Osher etc. propose a kind of Iteration Regularized method.G.Gilboa, Y.Y.Zeevi and N.Sochen propose a kind of method of the self-adaptation numerical value fidelity item with spatial variations.S.Esedoglu and S.Osher proposes a kind of directional information keeping particular edge; In order to eliminate staircase effect, full P.Blomgren proposes the model of a kind of variation item with graded.There is scholar also higher derivative to be introduced in energy model letter, or higher derivative and original ROF model are combined, or propose two benches denoising etc. and improve one's methods.The above-mentioned image denoising based on integer rank partial differential equation is improved one's methods, and for the comparative information and the marginal information that keep image, and elimination staircase effect achieves certain effect.

Unfortunately, when traditional image de-noising method based on integer rank partial differential equation is directly applied to Denoising of Texture Images by us, be generally difficult to obtain good treatment effect.Because on the one hand, traditional image de-noising method based on integer rank partial differential equation is based on the infinitesimal analysis computing of integer rank in itself.It is difficult to process the additive noise etc. that some are non-linear, non-causal, non-minimum phase system, non-gaussian, non-stationary, non-integer tie up (fractal) signal and non-white preferably.If when the image de-noising method based on integer rank partial differential equation is directly applied to Denoising of Texture Images by us, following limitation can be there is: first, integer rank partial differential equation is only according to local message process image, therefore can not the textural characteristics of hold period, can not Recovery image global characteristics; The second, based on the integer rank PDE model of integer rank variational principle, realize image procossing by optimization energy functional.What energy functional realized is optimization in local neighborhood, therefore there is blocky effect in image after process; 3rd, the integer rank partial differential equation processing power only comprising forward direction or backward diffusion is limited, and the equation of directional diffusion there will be the phenomenon that both direction diffuse information is offset in diffusion process, affects final process result.On the other hand, for being rich in the texture image of complex texture detailed information, the accuracy of grain details information to its interpretation of image seems valuable.Denoising of Texture Images method has the particular/special requirements such as contrast is constant, textural characteristics is constant.Because the integer rank differential value of constant or DC component is zero, the integer rank differential value of the AC compounent of Slight undulations is after specific threshold current limliting, and its value is also zero, so differentiating in integer rank can cause extreme loss to image complex texture detailed information.When traditional image de-noising method based on integer rank partial differential equation directly applies to Denoising of Texture Images, under low resolution condition, because the grain details information in original image is natively abundant and clear not, its result can also be accepted reluctantly; But when resolution increases gradually, the complex texture detailed information in its result image will, by extreme loss, cause us to be difficult to carry out accurate interpretation to the grain details of result image.Therefore, for being rich in the Denoising of Texture Images of complex texture detailed information, in order to more effectively keep while denoising and utilize complex texture detailed information, this with regard to an urgent demand we a class is proposed can fractional order, non-linear, multiple dimensioned the Denoising of Texture Images method based on partial fractional differential equation processing image complex texture minutia.

Summary of the invention

A kind of Denoising of Texture Images wave filter based on fractional order evolution equation proposed by the invention realizes the fractional order to texture image, non-linear, multiple dimensioned, high precision denoising based on a kind of special fractional order evolution equation denoise algorithm.The order v of the fractional calculus that the present invention relates to ₁, v ₂and v ₃not traditional integer rank, but non-integral order, generally get mark or rational fractional in engineer applied, and v ₃≠ 1,2,3.See Fig. 1, this wave filter adopts buffer 2, squared difference device 1 ,-v ₃power power side device 4, multiplier 1, λ ⁿgenerator 11, generator 10, multiplier 28, multiplier 39 and totalizer 1 are formed in cascaded fashion.This wave filter has while removal texture image noise, the low frequency profile in smooth region can be retained as far as possible, simultaneously again can the relatively large high frequency edge of non-linear reservation gray-scale value transition amplitude, and can also the remarkable advantage of the relatively little grain details of non-linear reservation gray-scale value transition changes in amplitude.This wave filter is specially adapted to the application scenario of the image being rich in complex texture minutia being carried out to denoising.

See Fig. 1, in order to the circuit clearly demonstrating a kind of Denoising of Texture Images wave filter based on fractional order evolution equation of the present invention is formed, be necessary first to carry out following brief description to this deduce mathematical being suitable for the high-precision filter of Denoising of Texture Images based on partial fractional differential equation and numerical operation rule:

As everyone knows, fractal date theory creates the transformation estimating sight, and fractal geometry have negated the existence of newton-Leibnitz's derivative.Fractal theory based on Hausdorff estimates, although the research going through for more than the 90 years still very incomplete mathematical theory of one so far.The structure of the infinitesimal analysis mathematical theory under Hausdorff estimates not yet can complete so far.Current development comparative maturity be estimate undefined fractional calculus at Euclidean, it mathematically requires to use Euclidean to estimate.Under Euclidean is estimated, what fractional calculus was the most frequently used is that Gr ü mwald-Letnikov defines and Riemann-Liouville defines two kinds.

The v rank infinitesimal analysis of Gr ü mwald-Letnikov definition signal s (x) is $D_{G-L}^{v}s\left(x\right)=\frac{d^v}{{\&lsqb;d\left(x-a\right)\&rsqb;}^v}s\left(x\right){|}_{G-L}=\underset{N\&RightArrow;\mathrm{\&infin;}}{\lim }\left\{\frac{{\left(\frac{x-a}{N}\right)}^{-v}}{\mathrm{\&Gamma;}\left(-v\right)}\underset{k=0}{\overset{N-1}{\mathrm{\&Sigma;}}}\frac{\mathrm{\&Gamma;}\left(k-v\right)}{\mathrm{\&Gamma;}\left(k+v\right)}s\left(x-k\left(\frac{x-a}{N}\right)\right)\right\}.$ Wherein, the extended period of signal s (x) is [a, x], and v is any real number (comprising mark), represent the fractional order differential operator defined based on Gr ü mwald-Letnikov, Г is Gamma function.From the Gr ü mwald-Lemikov definition of fractional calculus, Gr ü mwald-Letnikov is defined in Euclidean to be estimated the lower integer step by integer rank infinitesimal analysis and is generalized to mark step-length, thus the integer rank of infinitesimal analysis are generalized to fractional order.The calculating that the Gr ü mwald-Lemikov of fractional calculus defines is simple and easy to do, and it only needs relevant to signal s (x) self dis-crete sample values, and the derivative of undesired signal s (x) and integrated value.The v rank integration of Riemann-Liouville definition signal s (x) for $D_{R-L}^{v}s\left(x\right)=\frac{d^v}{{\&lsqb;d\left(x-a\right)\&rsqb;}^v}s\left(x\right){|}_{R-L}=\frac{1}{\mathrm{\&Gamma;}\left(-v\right)}\underset{a}{\overset{x}{\&Integral;}}{\left(x-\mathrm{\&eta;}\right)}^{-v-1}s\left(\mathrm{\&eta;}\right)d\mathrm{\&eta;}=\frac{-1}{\mathrm{\&Gamma;}\left(-v\right)}\underset{a}{\overset{x}{\&Integral;}}s\left(\mathrm{\&eta;}\right)d{\left(x-\mathrm{\&eta;}\right)}^{-v},v<0.$ Wherein, represent the fractional order differential operator based on Riemann-Liouville definition.For v rank differential (v >=0) of signal s (x), n meets n-1<v≤n.So by Riemann-Liouville Definitions On Integration formula, the Riemann-Liouville that the present invention can derive the v rank differential of signal s (x) is defined as $D_{R-L}^{v}s\left(x\right)=\frac{d^v}{{\&lsqb;d\left(x-a\right)\&rsqb;}^v}s\left(x\right){|}_{R-L}=\frac{d^n}{{\mathrm{dx}}^n}\frac{d^{v-n}}{{\&lsqb;d\left(x-a\right)\&rsqb;}^{v-n}}s\left(x\right){|}_{R-L}=\underset{k=0}{\overset{n-1}{\mathrm{\&Sigma;}}}\frac{{\left(x-a\right)}^{k-v}{s}^{\left(k\right)}\left(a\right)}{\mathrm{\&Gamma;}\left(k-v+1\right)}+\frac{1}{\mathrm{\&Gamma;}\left(n-v\right)}\underset{a}{\overset{x}{\&Integral;}}\frac{s^{\left(n\right)}\left(\mathrm{\&eta;}\right)}{{\left(x-\mathrm{\&eta;}\right)}^{v-n+1}}d\mathrm{\&eta;},0\&le;v<n.$ By the Riemann-Liouville definition of fractional order differential, can the derive Fourier of signal s (x) of the present invention is transformed to $FT\&lsqb;D^{v}s\left(x\right)\&rsqb;={\left(i\mathrm{\&omega;}\right)}^{v}FT\&lsqb;s\left(x\right)\&rsqb;-\underset{k=0}{\overset{n-1}{\&Sigma;}}{\left(i\mathrm{\&omega;}\right)}^k\frac{d^{v-1-k}}{{\mathrm{dx}}^{v-1-k}}s\left(0\right).$ Wherein, i is imaginary unit, and ω is numerical frequency.When signal s (x) is causal signal, above formula can be reduced to FT [D ^vs (x)]=(i ω) ^vfT [s (x)].

The present invention will construct the partial fractional differential equation being suitable for Denoising of Texture Images, just must construct fractional order Euler-Lagrange equation, and constructs the prerequisite that fractional order Green formula is structure fractional order Euler-Lagrange equation.Therefore, first traditional integer rank Green formula must be generalized to fractional order by the present invention, derives and constructs fractional order Green formula.

See Fig. 2, the present invention makes Ω be the plane simply connected region that is border with piecewise smooth curve C, can micro-Product function P (x, y) and Q (x, y) on Ω and C continuously, and there is the fractional order continuous offset derivative to x and y; Make D ¹represent 1 rank differentiating operator, D ^vrepresent v rank fractional order differential operator, I ¹=D ^-1represent 1 rank integral operator, I ^v=D ^-vrepresent v > 0 rank fractional order integration operator, represent the v rank fractional order curve surface integral operator (the Riemann-Liouville definition of fractional order integration is generalized to two dimension from one dimension) on plane Ω, represent the AC in curve C ₁edge in B section the v rank fractional order curvilinear integral operator in direction, represent v rank fractional order closed curve integral operator in the counterclockwise direction on closed curve C; The border C of region Ω is made to be by two curve y=φ ₁(x), y=φ ₂(x), a≤x≤b or x=ψ ₁(y), x=ψ ₂y (), c≤y≤d surrounded.

For can for micro-Product function P (x, y), when then so, $I_x^{v_2}{I}_y^{v_2}{D}_y^{v_1}P\left(x,y\right)=I_x^{v_2}\left\{D_y^{v_1-v_2}P\left(x,y\right)-D_y^{v_1-v_2}\&lsqb;P\left(x,y\right)-D_y^{-v_1}{D}_y^{v1}P\left(x,y\right)\&rsqb;\right\},$ The present invention can derive $\begin{array}{c}\underset{\mathrm{\&Omega;}}{I_x^{v_2}{I}_y^{v_2}}{D}_y^{v_1}P\left(x,y\right)=I_x^{v_2}{}_a{}^{b}I_y^{v_2}{}_{{\mathrm{\&phi;}}_1\left(x\right)}{}^{{\mathrm{\&phi;}}_2\left(x\right)}{D}_y^{v_1}P\left(x,y\right)=I_x^{v_2}{}_a{}^b\left\{D_y^{v_1-v_2}P\left(x,y\right)-D_y^{v_1-v_2}\&lsqb;P\left(x,y\right)-D_y^{-v_1}{D}_y^{v_1}P\left(x,y\right)\&rsqb;\right\}{|}_{{\mathrm{\&phi;}}_1\left(x\right)}^{{\mathrm{\&phi;}}_2\left(x\right)}\\ =-\underset{C\left({\mathrm{BC}}_{2}A\right)}{I_x^{v_2}}\left\{D_y^{v_1-v_2}P\left(x,y\right)-D_y^{v_1-v_2}\&lsqb;P\left(x,y\right)-D_y^{-v_1}{D}_y^{v_1}P\left(x,y\right)\&rsqb;\right\}-\underset{C\left({\mathrm{AC}}_{1}B\right)}{I_x^{v_2}}\left\{D_y^{v_1-v_2}P\left(x,y\right)-D_y^{v_1-v_2}\&lsqb;P\left(x,y\right)-D_y^{-v_1}{D}_y^{v_1}P\left(x,y\right)\&rsqb;\right\}\\ =-\underset{C-}{I_x^{v_2}}\left\{D_y^{v_1-v_2}P\left(x,y\right)-D_y^{v_1-v_2}\&lsqb;P\left(x,y\right)-D_y^{-v_1}{D}_y^{v_1}P\left(x,y\right)\&rsqb;\right\}\end{array}$ In like manner can obtain $\underset{\mathrm{\&Omega;}}{I_x^{v_2}{I}_y^{v_2}}{D}_x^{v_1}Q\left(x,y\right)=\underset{C-}{I_y^{v_2}}\left\{D_x^{v_1-v_2}Q\left(x,y\right)-D_x^{v_1-v_2}\&lsqb;Q\left(x,y\right)-D_x^{-v_1}{D}_x^{v_1}Q\left(x,y\right)\&rsqb;\right\}.$ By above-mentioned two formulas, the present invention's goals for rank Green formula of can deriving is $\underset{\mathrm{\&Omega;}}{I_x^{v_2}{I}_y^{v_2}}\left(D_x^{v_1}Q\left(x,y\right)-D_y^{v_1}P\left(x,y\right)\right)=\underset{C-}{I_x^{v_2}}\left\{D_y^{v_1-v_2}P\left(x,y\right)-D_y^{v_1-v_2}\&lsqb;P\left(x,y\right)-D_y^{-v_1}{D}_y^{v_1}P\left(x,y\right)\&rsqb;\right\}+\underset{C-}{I_y^{v_2}}\left\{D_x^{v_1-v_2}Q\left(x,y\right)-D_x^{v_1-v_2}\&lsqb;Q\left(x,y\right)-D_x^{-v_1}{D}_x^{v_1}Q\left(x,y\right)\&rsqb;\right\}.$ Especially, when with time reciprocal (this conditional request is higher, and general being difficult to meets), namely then can be obtained fom the above equation the simplification under given conditions of fractional order Green formula show out for $\underset{\mathrm{\&Omega;}}{I_x^{v_2}{I}_y^{v_2}}\left(D_x^{v_1}Q\left(x,y\right)-D_y^{v_1}P\left(x,y\right)\right)=\underset{C-}{I_x^{v_2}}{D}_y^{v_1-v_2}P\left(x,y\right)+\underset{C-}{I_y^{v_2}}{D}_x^{v_1-v_2}Q\left(x,y\right).$ From above formula, work as v ₁=v ₂when=1, can be derived from $\underset{\mathrm{\&Omega;}}{I_x^1{I}_y^1}\left(D_x^{1}Q\left(x,y\right)-D_y^{1}P\left(x,y\right)\right)=\underset{C-}{I_x^1}P\left(x,y\right)+\underset{C-}{I_y^1}Q\left(x,y\right),$ Traditional integer rank Green formula is the special case of fractional order Green formula; Work as v ₁=v ₂during=v, can be derived from this is fractional order Green formula applied widely.

Based on the fractional order Green formula of above-mentioned derivation gained, the present invention can derive further for the general Euler-Lagrange equation of fractional order of two dimensional image process.

See Fig. 3, the present invention makes can be u (x, y) and can micro-long-pending phasor function be by micro-product flow function in two-dimensional space v rank fractional order differential operator $D^v=i\frac{{\&part;}^v}{\&part;x^v}+j\frac{{\&part;}^v}{\&part;y^v}={\mathrm{iD}}_x^v+{\mathrm{jD}}_y^v=(D_x^v,D_y^v),$ D ^vbe a linear operator (as v=0, D ⁰representing neither differential also not integration, is an identity operator), wherein i and j is illustrated respectively in the unit vector on x and y direction.Generally speaking, two dimensional image region Ω is a rectangular simply connected region, and therefore the sectionally smooth border C of Ω is a closed rectangular lines.

Can be obtained by the Riemann-Liouville definition of fractional order integration $I_y^{v}s\left(x,y\right)=\frac{1}{\mathrm{\&Gamma;}\left(v\right)}\underset{a_y}{\overset{y}{\&Integral;}}{\left(y-\mathrm{\&zeta;}\right)}^{v-1}s\left(x,\mathrm{\&zeta;}\right)d\mathrm{\&zeta;}$ And $I_x^v{I}_y^{v}s\left(x,y\right)=\frac{1}{{\mathrm{\&Gamma;}}^2\left(v\right)}\underset{a_x}{\overset{x}{\&Integral;}}\underset{a_y}{\overset{y}{\&Integral;}}{\left(x-\mathrm{\&eta;}\right)}^{v-1}{\left(y-\mathrm{\&zeta;}\right)}^{v-1}s\left(\mathrm{\&eta;},\mathrm{\&zeta;}\right)d\mathrm{\&eta;}d\mathrm{\&zeta;}.$ So be can be derived from by above-mentioned fractional order Green formula $\begin{array}{c}{I}_x^{v_2}{I}_y^{v_2}\left(D_x^{v_1}Q\left(x,y\right)-D_y^{v_1}P\left(x,y\right)\right)=\underset{C-}{I_x^{v_2}}\left\{D_y^{v_1-v_2}P\left(x,y\right)-D_y^{v_1-v_2}\&lsqb;P\left(x,y\right)-D_y^{-v_1}{D}_y^{v_1}P\left(x,y\right)\&rsqb;\right\}+\underset{C-}{I_y^{v_2}}\left\{D_x^{v_1-v_2}Q\left(x,y\right)-D_x^{v_1-v_2}\&lsqb;Q\left(x,y\right)-D_x^{-v_1}{D}_x^{v_1}Q\left(x,y\right)\&rsqb;\right\}\\ =I_x^{v_2}{}_{x_0}{}^{x_1}\left\{D_y^{v_1-v_2}P-D_y^{v_1-v_2}\&lsqb;P-D_y^{-v_1}{D}_y^{v_1}P\&rsqb;\right\}+I_x^{v_2}{}_{x_1}{}^{x_0}\left\{D_y^{v_1-v_2}P-D_y^{v_1-v_2}\&lsqb;P-D_y^{-v_1}{D}_y^{v_1}P\&rsqb;\right\}+I_y^{v_2}{}_{y_0}{}^{y_1}\left\{D_x^{v_1-v_2}Q-D_x^{v_1-v_2}\&lsqb;Q-D_x^{-v_1}{D}_x^{v_1}Q\&rsqb;\right\}+I_y^{v_2}{}_{y_1}{}^{y_0}\left\{D_x^{v_1-v_2}Q-D_x^{v_1-v_2}\&lsqb;Q-D_x^{-v_1}{D}_x^{v_1}Q\&rsqb;\right\}\&equiv;0\end{array}$ Permanent establishment.Due to and $\left(\begin{array}{c}v\\ r+n\end{array}\right)\left(\begin{array}{c}r+n\\ n\end{array}\right)\&equiv;\left(\begin{array}{c}v\\ n\end{array}\right)\left(\begin{array}{c}v-n\\ r\end{array}\right),$ So the present invention can derive $D_{x-a}^v\left(\mathrm{fg}\right)=\underset{n=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\left[\left(\begin{array}{c}v\\ n\end{array}\right)\left(D_{x-a}^{v-n}\right)D_{x-a}^{n}g\right].$ By fractional calculus homogeneous so that can obtain so the present invention can derive wherein, symbol represents inner product.With fractional order divergence operator definition similar, the present invention makes v rank fractional order class differentiating operator v rank fractional order subhuman robot operator divP ^vand P ^vbe all linear operator, so theoretical according to Hilbert adjoint operator, the present invention can derive

Wherein, represent v ₂the integrated form of rank fractional order inner product, (D ^v) ^*represent D ^vv rank fractional order Hilbert adjoint operator.(D can be obtained by above formula the present invention ^v) ^*=-divP ^v.From above formula, fractional order Hilbert adjoint operator (D ^v) ^*it is a linear operator.Work as v ₁=v ₂when=1, the present invention can derive wherein, <, > ¹represent the integrated form of single order inner product, represent single order divergence operator, (D ¹) ^*represent D ¹single order Hilbert adjoint operator.For digital picture, the present invention can derive (D ¹) ^*=-div ¹.So known, single order Hilbert adjoint operator is the special case of fractional order Hilbert adjoint operator.When time, so can obtain in above formula, due to the orthogonality in x direction and y direction, for random two-dimensional quantity function u (trial function), accordingly with there is randomness, by variation fundamental lemma, above formula be made to set up, only need because n is the positive integer of 1 → ∞, make only need to make the Equivalent Form of above formula is so make above formula set up, and if only if set up. namely be with corresponding fractional order Euler-Lagrange equation.

Further, if it is phasor function quantity function; can micro-long-pending phasor function quantity function.In like manner, when time, the present invention can obtain

So make above formula set up, and if only if set up. namely be with corresponding fractional order Euler-Lagrange equation.

Because fractional calculus all exists for all v reason, so fractional order Euler-Lagrange equation and fractional order Line Integral integration order v ₂irrelevant, all do not adopt fractional order Line Integral when therefore the present invention constructs the energy functional of the partial fractional differential equation model being suitable for Denoising of Texture Images below and only adopt single order Line Integral form.

As described below, the present invention adopts single order extreme value to construct the energy functional based on fractional order total variation, thus constructs a kind of Denoising of Texture Images wave filter based on fractional order evolution equation.Family of surfaces s+ (β-1) ξ of the present invention's order containing small parameter β is v when β=1 ₃rank extremal surface s.First, the present invention's order ${\mathrm{\&Psi;}}_1\left(\mathrm{\&beta;}\right)=\underset{\mathrm{\&Omega;}}{I_x^1{I}_y^1}\&lsqb;f\left(\left|\right|\stackrel{\&RightArrow;}{\mathrm{\&phi;}}|{|}^{v_2}\right)\&rsqb;=\underset{\mathrm{\&Omega;}}{\&Integral;\&Integral;}f\left(\left|\right|\stackrel{\&RightArrow;}{\mathrm{\&phi;}}|{|}^{v_2}\right)dxdy.$ Meanwhile, the present invention makes vector $\stackrel{\&RightArrow;}{\mathrm{\&phi;}}\&lsqb;D^{v_1}s,\left(\mathrm{\&beta;}-1\right)D^{v_1}\mathrm{\&xi;}\&rsqb;=(\mathrm{\&beta;}-1)D^{v_1}\mathrm{\&xi;}-\frac{2\mathrm{\&Gamma;}(2-v_3)}{\mathrm{\&Gamma;}(3-v_3)}{D}^{v_1}s,$ And definition vector norm be $\left|\right|\stackrel{\&RightArrow;}{\mathrm{\&phi;}}\left|\right|=\sqrt{\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}{\left(D^{v_1}s\right)}^2-\frac{4(\mathrm{\&beta;}-1)\mathrm{\&Gamma;}(2-v_3)}{\mathrm{\&Gamma;}(3-v_3)}{D}^{v_1}s\&CenterDot;D^{v_1}\mathrm{\&xi;}+{\left(\mathrm{\&beta;}-1\right)}^2{\left(D^{v_1}\mathrm{\&xi;}\right)}^2}=\sqrt{{\left(\stackrel{\&RightArrow;}{\mathrm{\&phi;}}\right)}^2+C},$

Wherein $C=\frac{4\mathrm{\&Gamma;}(2-v_3)\mathrm{\&Gamma;}(3-v_3)-{\mathrm{\&Gamma;}}^2(3-v_3)-4{\mathrm{\&Gamma;}}^2(2-v_3)}{{\mathrm{\&Gamma;}}^2(3-v_3)}{\left(D^{v_1}s\right)}^2.$ Wherein, symbol represents inner product.For vector with , ${\left(D^{v_1}s\right)}^2={\left|D^{v_1}s\right|}^2={\left(\sqrt{{\left(D^{v_1}s\right)}^2}\right)}^2=D^{v_1}s\&CenterDot;D^{v_1}s$ And $\left(D^{v_1}\mathrm{\&xi;}\right)={\left|D^{v_1}\mathrm{\&xi;}\right|}^2={\left(\sqrt{{\left(D^{v_1}\mathrm{\&xi;}\right)}^2}\right)}^2=D^{v_1}\mathrm{\&xi;}\&CenterDot;D^{v_1}\mathrm{\&xi;}.$ Further, the present invention's order solve ψ ₂(β) namely minimizing process is solve noise and without the minimum process of the similarity of noise cancellation signal.ψ ₂(β) in denoising process, play non-linear fidelity, λ is regularization parameter.So be defined in the upper fractional order energy functional based on fractional order total variation of family of surfaces s+ (β-1) ξ be $\mathrm{\&Psi;}\left(\mathrm{\&beta;}\right)={\mathrm{\&Psi;}}_1\left(\mathrm{\&beta;}\right)+{\mathrm{\&Psi;}}_2\left(\mathrm{\&beta;}\right)=\underset{\mathrm{\&Omega;}}{I_x^1{I}_y^1}\&lsqb;f\left(\left|\right|\stackrel{\&RightArrow;}{\mathrm{\&phi;}}|{|}^{v_2}\right)+\mathrm{\&lambda;}\&lsqb;s+\left(\mathrm{\&beta;}-1\right)\mathrm{\&xi;}-s_0\&rsqb;\left(s-s_0\right)\&rsqb;=\underset{\mathrm{\&Omega;}}{\&Integral;\&Integral;}\&lsqb;f\left(\left|\right|\stackrel{\&RightArrow;}{\mathrm{\&phi;}}|{|}^{v_2}\right)+\mathrm{\&lambda;}\&lsqb;s+\left(\mathrm{\&beta;}-1\right)\mathrm{\&xi;}-s_0\&rsqb;\left(s-s_0\right)\&rsqb;dxdy.$ So, can derive $D_{\mathrm{\&beta;}}^{v_3}{\mathrm{\&psi;}}_1\left(\mathrm{\&beta;}\right){|}_{\mathrm{\&beta;}=1}=\frac{{\&PartialD;}^{v_3}}{\&PartialD;{\mathrm{\&beta;}}^{v_3}}\underset{\mathrm{\&Omega;}}{\&Integral;\&Integral;}f\left({\left|\right|\stackrel{\&RightArrow;}{\mathrm{\&phi;}}\left|\right|}^{v_2}\right)\mathrm{dxdy}{|}_{\mathrm{\&beta;}=1}=\underset{\mathrm{\&Omega;}}{\&Integral;\&Integral;}\frac{{\&PartialD;}^{v_3}}{\&PartialD;{\mathrm{\&beta;}}^{v_3}}f\left({\left|\right|\stackrel{\&RightArrow;}{\mathrm{\&phi;}}\left|\right|}^{v_2}\right)\mathrm{dxdy}{|}_{\mathrm{\&beta;}=1}=0.$ Different from the conventional first order variational method based on single order extreme value, above formula is ψ ₁(β) about the v of β ₃rank fractional order extreme value, its object is to utilize the special nature of image score rank infinitesimal analysis (the low frequency contour feature in image smoothing region can be retained as far as possible, simultaneously again can fractional order and the high frequency edge feature that in multi-scale enhancement image, gray-scale value transition amplitude is relatively large, and can also fractional order and gray-scale value transition amplitude little high frequency texture minutia relative to frequency change in multi-scale enhancement image), make texture image retain its complex texture minutia while denoising, with trying one's best fractional order nonlinear.And then, can derive $\underset{\mathrm{\&Omega;}}{\&Integral;\&Integral;}\frac{{\&lsqb;\sqrt{\left|\frac{4\mathrm{\&Gamma;}\left(2-v_3\right)-\mathrm{\&Gamma;}\left(3-v_3\right)}{\mathrm{\&Gamma;}\left(3-v_3\right)}\right|}\&rsqb;}^{v_2}{\left|D^{v_1}s\right|}^{v_2}}{\mathrm{\&Gamma;}\left(1-v_3\right)}+\underset{n=1}{\overset{\mathrm{\&infin;}}{\mathrm{\&Sigma;}}}\frac{{\left(-1\right)}^n\mathrm{\&Gamma;}\left(n-v_3\right)}{\mathrm{\&Gamma;}\left(-v_3\right)\mathrm{\&Gamma;}\left(n-v_3+1\right)}\frac{D_{\mathrm{\&beta;}}^n\left(\left|\right|\stackrel{\&RightArrow;}{\mathrm{\&phi;}}|{|}^{v_2}\right){|}_{\mathrm{\&beta;}=1}}{n!}dxdy=0.$ Wherein, in order to ensure be always positive number, to make meaningful, the present invention gets its absolute value when n gets odd number (n=2k+1, k=0,1,2,3 ...) get even number (n=2k, k=1,2,3 with as n ...) time, there is different expression formulas.So, the present invention can, to derive respectively $D_{\mathrm{\&beta;}}^n\left(\left|\right|\stackrel{\&RightArrow;}{\mathrm{\&phi;}}|{|}^{v_2}\right){|}_{\mathrm{\&beta;}=1}^{n=2k+1}=\frac{-2\mathrm{\&Gamma;}\left(2-v_3\right)\underset{\mathrm{\&pi;}=1}{\overset{n}{\mathrm{\&Pi;}}}\left(v_2-\mathrm{\&tau;}+1\right)}{\mathrm{\&Gamma;}\left(3-v_3\right)}{\&lsqb;\sqrt{\left|\frac{4\mathrm{\&Gamma;}\left(2-v_3\right)-\mathrm{\&Gamma;}\left(3-v_3\right)}{\mathrm{\&Gamma;}\left(3-v_3\right)}\right|}\&rsqb;}^{v_2-n-1}{\left|D^{v_1}s\right|}^{v_2-n-1}{\left|D^{v_1}\mathrm{\&xi;}\right|}^{n-1}\left(D^{v_1}\mathrm{\&xi;}\right)\&CenterDot;\left(D^{v_1}s\right)$ With $D_{\mathrm{\&beta;}}^n\left(\left|\right|\stackrel{\&RightArrow;}{\mathrm{\&phi;}}|{|}^{v_2}\right){|}_{\mathrm{\&beta;}=1}^{n=2k}=\underset{\mathrm{\&tau;}=1}{\overset{n}{\&Pi;}}\left(v_2-\mathrm{\&tau;}+1\right){\&lsqb;\sqrt{\left|\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}\right|}\&rsqb;}^{v_2-n}\left|D^{v_1}s{|}^{v_2-n}\right|D^{v_1}\mathrm{\&xi;}{|}^n.$ Further, the present invention can derive $\underset{\mathrm{\&Omega;}}{\&Integral;\&Integral;}\underset{k=0}{\overset{\mathrm{\&infin;}}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}\left(v_2-\mathrm{\&tau;}+1\right){\&lsqb;\sqrt{\left|\frac{4\mathrm{\&Gamma;}\left(2-v_3\right)-\mathrm{\&Gamma;}\left(3-v_3\right)}{\mathrm{\&Gamma;}\left(3-v_3\right)}\right|}\left|D^{v_1}s\right|\&rsqb;}^{v_2-2k-2}{\left|D^{v_1}\mathrm{\&xi;}\right|}^{2k}}{\mathrm{\&Gamma;}\left(-v_3\right)\mathrm{\&Gamma;}\left(3-v_3\right)\left(2k\right)!}\left(D^{v_1}s\right)\&CenterDot;D^{v_1}\left[\begin{array}{c}\frac{\mathrm{\&Gamma;}\left(2k-v_3\right)\&lsqb;4\mathrm{\&Gamma;}\left(2-v_3\right)-\mathrm{\&Gamma;}\left(3-v_3\right)\&rsqb;}{\mathrm{\&Gamma;}\left(2k-v_3+1\right)}s\\ +\frac{2\left(v_2-2k\right)\mathrm{\&Gamma;}\left(2k-v_3+1\right)\mathrm{\&Gamma;}\left(2-v_3\right)}{\left(2k+1\right)\mathrm{\&Gamma;}\left(2k-v_3+2\right)}\mathrm{\&xi;}\end{array}\right]dxdy=0.$

Wherein, the present invention's order for random two-dimensional quantity function ξ (trial function), accordingly there is randomness, so $D^{v_1}\&lsqb;\frac{\mathrm{\&Gamma;}\left(2k-v_3\right)\&lsqb;4\mathrm{\&Gamma;}\left(2-v_3\right)-\mathrm{\&Gamma;}\left(3-v_3\right)\&rsqb;}{\mathrm{\&Gamma;}\left(2k-v_3+1\right)}s+\frac{2\left(v_2-2k\right)\mathrm{\&Gamma;}\left(2k-v_3+1\right)\mathrm{\&Gamma;}\left(2-v_3\right)}{\left(2k+1\right)\mathrm{\&Gamma;}\left(2k-v_3+2\right)}\mathrm{\&xi;}\&rsqb;$ Also there is randomness.So the present invention's general Euler-Lagrange equation of corresponding fractional order of can deriving is

$\frac{\mathrm{\&Gamma;}\left(1-v_1\right)}{\mathrm{\&Gamma;}\left(-v_1\right)\mathrm{\&Gamma;}\left(-v_3\right)\mathrm{\&Gamma;}\left(3-v_3\right)}\underset{k=0}{\overset{\mathrm{\&infin;}}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}\left(v_2-\mathrm{\&tau;}+1\right){\&lsqb;\sqrt{\left|\frac{4\mathrm{\&Gamma;}\left(2-v_3\right)-\mathrm{\&Gamma;}\left(3-v_3\right)}{\mathrm{\&Gamma;}\left(3-v_3\right)}\right|}\&rsqb;}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left({\left|D^{v_1}s\right|}^{v_2-2k-2}{D}_x^{v_1}s\right)\\ +D_y^1\left({\left|D^{v_1}s\right|}^{v_2-2k-2}{D}_y^{v_1}s\right)\end{array}\right]=0.$

Wherein, so the present invention gets v=v ₃≠ 1,2,3, can derive

$\frac{{\&part;}^{v_3}s}{\&part;t^{v_3}}=\frac{-\mathrm{\&Gamma;}\left(1-v_1\right)}{\mathrm{\&Gamma;}\left(-v_1\right)\mathrm{\&Gamma;}\left(-v_3\right)\mathrm{\&Gamma;}\left(3-v_3\right)}\underset{k=0}{\overset{\mathrm{\&infin;}}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}\left(v_2-\mathrm{\&tau;}+1\right){\&lsqb;\sqrt{\left|\frac{4\mathrm{\&Gamma;}\left(2-v_3\right)-\mathrm{\&Gamma;}\left(3-v_3\right)}{\mathrm{\&Gamma;}\left(3-v_3\right)}\right|}\&rsqb;}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left({\left|D^{v_1}s\right|}^{v_2-2k-2}{D}_x^{v_1}s\right)\\ +D_y^1\left({\left|D^{v_1}s\right|}^{v_2-2k-2}{D}_y^{v_1}s\right)\end{array}\right]\frac{{\mathrm{\&lambda;s}}_0}{\mathrm{\&Gamma;}\left(1-v_3\right)\mathrm{\&Gamma;}\left(2-v_3\right)}.$

Wherein, be realize according to fractional order difference method, only converge to unique v ₃rank fractional order extreme point.So the present invention can derive

$\mathrm{\&lambda;}\left(t\right)=\frac{-\mathrm{\&Gamma;}\left(1-v_1\right)\mathrm{\&Gamma;}\left(1-v_3\right)\mathrm{\&Gamma;}\left(2-v_3\right)}{{\mathrm{\&sigma;}}^2\mathrm{\&Gamma;}\left(-v_1\right)\mathrm{\&Gamma;}\left(-v_3\right)\mathrm{\&Gamma;}\left(3-v_3\right)s_0}\underset{\mathrm{\&Omega;}}{\&Integral;\&Integral;}\underset{k=0}{\overset{\mathrm{\&infin;}}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}\left(v_2-\mathrm{\&tau;}+1\right){\&lsqb;\sqrt{\left|\frac{4\mathrm{\&Gamma;}\left(2-v_3\right)-\mathrm{\&Gamma;}\left(3-v_3\right)}{\mathrm{\&Gamma;}\left(3-v_3\right)}\right|}\&rsqb;}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left({\left|D^{v_1}s\right|}^{v_2-2k-2}{D}_x^{v_1}s\right)\\ +D_y^1\left({\left|D^{v_1}s\right|}^{v_2-2k-2}{D}_y^{v_1}s\right)\end{array}\right]{\left(s-s_0\right)}^{2}dxdy.$

Formula $\frac{{\&part;}^{v_3}s}{\&part;t^{v_3}}=\frac{-\mathrm{\&Gamma;}\left(1-v_1\right)}{\mathrm{\&Gamma;}\left(-v_1\right)\mathrm{\&Gamma;}\left(-v_3\right)\mathrm{\&Gamma;}\left(3-v_3\right)}\underset{k=0}{\overset{\mathrm{\&infin;}}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}\left(v_2-\mathrm{\&tau;}+1\right){\&lsqb;\sqrt{\left|\frac{4\mathrm{\&Gamma;}\left(2-v_3\right)-\mathrm{\&Gamma;}\left(3-v_3\right)}{\mathrm{\&Gamma;}\left(3-v_3\right)}\right|}\&rsqb;}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left({\left|D^{v_1}s\right|}^{v_2-2k-2}{D}_x^{v_1}s\right)\\ +D_y^1\left({\left|D^{v_1}s\right|}^{v_2-2k-2}{D}_y^{v_1}s\right)\end{array}\right]\frac{{\mathrm{\&lambda;s}}_0}{\mathrm{\&Gamma;}\left(1-v_3\right)\mathrm{\&Gamma;}\left(2-v_3\right)}$ With $\mathrm{\&lambda;}\left(t\right)=\frac{-\mathrm{\&Gamma;}\left(1-v_1\right)\mathrm{\&Gamma;}\left(1-v_3\right)\mathrm{\&Gamma;}\left(2-v_3\right)}{{\mathrm{\&sigma;}}^2\mathrm{\&Gamma;}\left(-v_1\right)\mathrm{\&Gamma;}\left(-v_3\right)\mathrm{\&Gamma;}\left(3-v_3\right)s_0}\underset{\mathrm{\&Omega;}}{\&Integral;\&Integral;}\underset{k=0}{\overset{\mathrm{\&infin;}}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}\left(v_2-\mathrm{\&tau;}+1\right){\&lsqb;\sqrt{\left|\frac{4\mathrm{\&Gamma;}\left(2-v_3\right)-\mathrm{\&Gamma;}\left(3-v_3\right)}{\mathrm{\&Gamma;}\left(3-v_3\right)}\right|}\&rsqb;}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left({\left|D^{v_1}s\right|}^{v_2-2k-2}{D}_x^{v_1}s\right)\\ +D_y^1\left({\left|D^{v_1}s\right|}^{v_2-2k-2}{D}_y^{v_1}s\right)\end{array}\right]{\left(s-s_0\right)}^{2}dxdy$

Represented fractional order evolution equation denoising model is the mathematical model of a kind of Denoising of Texture Images wave filter based on fractional order evolution equation of the present invention.

In order to enable above-mentioned fractional order evolution equation denoising model, completely filtering is at the very low frequency (VLF) of signal and the residual faint noise of direct current component, and the present invention, when iterative numerical realizes, also needs to carry out low-pass filtering to the very low frequency (VLF) of signal and direct current component simultaneously.In addition, above-mentioned fractional order evolution equation denoising model is by increasing in the molecule and increase Г (3-v in the denominator ₃) form enhance more greatly fractional order differential order v ₃to the Nonlinear Adjustment effect of denoising.In addition, v is worked as ₃when=0, above-mentioned fractional order evolution equation denoising model is traditional potential equation (elliptic equation); Work as v ₃when=1, above-mentioned fractional order evolution equation denoising model is traditional heat-conduction equation (parabolic equation); Work as v ₃when=2, above-mentioned fractional order evolution equation denoising model is traditional wave equation (hyperbolic equation); As 0 < v ₃during < 1, above-mentioned fractional order evolution equation denoising model is the continuous interpolation between traditional potential equation and traditional heat-conduction equation; As 1 < v ₃during < 2, above-mentioned fractional order evolution equation denoising model is the continuous interpolation between traditional heat-conduction equation and wave equation.Visible, on mathematics and physical significance, traditional image procossing based on partial differential equation has been generalized to more wide field from the general base conditioning method of the anisotropy parameter based on traditional heat-conduction equation by above-mentioned fractional order evolution equation denoising model.

Further, the present invention needs the mathematical model of the above-mentioned a kind of Denoising of Texture Images wave filter based on fractional order evolution equation of Numerical Implementation.The first, the present invention needs the fractional order differential of Numerical Implementation two-dimensional digital image in x-axis and y-axis direction.For the Gr ü mwald-Letnikov definition of fractional calculus, when enough large of N, limit symbol can be removed.In order to improve speed of convergence and convergence precision, the present invention introduces the signal value of signal s (x) at not a node place in Gr ü mwald-Letnikov definition, namely $\frac{d^v}{{\mathrm{dx}}^v}s\left(x\right){|}_{G-L}\&cong;\frac{x^{-v}{N}^v}{\mathrm{\&Gamma;}(-v)}\underset{k=0}{\overset{N-1}{\&Sigma;}}\frac{\mathrm{\&Gamma;}(k-v)}{\mathrm{\&Gamma;}(k+1)}s(x+\frac{vx}{2N}-\frac{kx}{N}).$ So when v ≠ 1, apply Lagrangian three point interpolation formula and mark interpolation is carried out to signal s (x), the fractional order differential operator of digital picture on the direction of x-axis and y-axis can be constructed respectively.For gray level image, the numerical operation rule of fractional order differential operator adopts the airspace filter scheme of operator convolution.The present invention is chosen at the fractional order differential value of the maximum partial fractional differential value of modulus value on x-axis and y-axis direction as this pixel.The second, the present invention needs the 1 rank differential of Numerical Implementation two-dimensional digital image in x-axis and y-axis direction.In order to keep the stability of numerical evaluation, the present invention adopts $D_x^{1}s(x,y)=\frac{2\&lsqb;s(x+1,y)-s(x-1,y)\&rsqb;+s(x+1,y+1)-s(x-1,y+1)+s(x+1,y-1)-s(x-1,y-1)}{4}$ With $D_y^{1}s(x,y)=\frac{2\&lsqb;s(x,y+1)-s(x,y-1)\&rsqb;+s(x+1,y+1)-s(x+1,y-1)+s(x-1,y+1)-s(x-1,y-1)}{4}$ Carry out approximate first differential.3rd, the present invention needs Numerical Implementation two-dimensional digital image for the fractional order differential of time t.If time decile is spaced apart Δ t, i.e. unit iteration time interval, the n moment is t _n=n Δ t, n=0,1 ... (t ₀=0 represents initial time).The present invention get unit iteration time interval of delta t (0,0.1] in get arbitrarily less arithmetic number.So the digital picture in n moment is treat that the original image of denoising is s ₀for desirable noise-free picture, it is a steady state value, therefore s ₀(x, y, t ₀)=s ₀(x, y, t _n).So the fractional order differential of two-dimensional digital image for time t can be approximately by the present invention $\frac{{\&part;}^{v}s}{\&part;t^v}={\mathrm{\&Delta;t}}^{-v}\&lsqb;s_{x,y}^{n+1}-s_{x,y}^n+\frac{2\mathrm{\&mu;}\mathrm{\&eta;}}{\mathrm{\&Gamma;}\left(3-v\right)}{\left(s_{x,y}^n-s_{x,y}^{v*}\right)}^2{\left(s_{x,y}^n\right)}^{-v}\&rsqb;,when,v\&NotEqual;1.$ Wherein, it is best denoising image.In addition, due to desirable noise-free picture s ₀(x, y, t ₀) do not know in advance, but the denoising intermediate result of each iterative numerical all to desirable noise-free picture s ₀(x, y, t ₀) once approach, namely therefore in order to approach s-s when iterative numerical as far as possible ₀, the present invention makes due to best denoising image do not know in advance, but the denoising intermediate result of each iterative numerical all right once approach, namely therefore in order to approach when iterative numerical as far as possible the present invention makes simultaneously in order to simplify calculating, the present invention gets μ=0.005 and η=1, only gets k=0, and 1 carries out approximate treatment.So, formula of can deriving to obtain $\frac{{\&part;}^{v_3}s}{\&part;t^{v_3}}=\frac{-\mathrm{\&Gamma;}\left(1-v_1\right)}{\mathrm{\&Gamma;}\left(-v_1\right)\mathrm{\&Gamma;}\left(-v_3\right)\mathrm{\&Gamma;}\left(3-v_3\right)}\underset{k=0}{\overset{\mathrm{\&infin;}}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}\left(v_2-\mathrm{\&tau;}+1\right){\&lsqb;\sqrt{\left|\frac{4\mathrm{\&Gamma;}\left(2-v_3\right)-\mathrm{\&Gamma;}\left(3-v_3\right)}{\mathrm{\&Gamma;}\left(3-v_3\right)}\right|}\&rsqb;}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left({\left|D^{v_1}s\right|}^{v_2-2k-2}{D}_x^{v_1}s\right)\\ +D_y^1\left({\left|D^{v_1}s\right|}^{v_2-2k-2}{D}_y^{v_1}s\right)\end{array}\right]\frac{{\mathrm{\&lambda;s}}_0}{\mathrm{\&Gamma;}\left(1-v_3\right)\mathrm{\&Gamma;}\left(2-v_3\right)}$ With $\mathrm{\&lambda;}\left(t\right)=\frac{-\mathrm{\&Gamma;}\left(1-v_1\right)\mathrm{\&Gamma;}\left(1-v_3\right)\mathrm{\&Gamma;}\left(2-v_3\right)}{{\mathrm{\&sigma;}}^2\mathrm{\&Gamma;}\left(-v_1\right)\mathrm{\&Gamma;}\left(-v_3\right)\mathrm{\&Gamma;}\left(3-v_3\right)s_0}\underset{\mathrm{\&Omega;}}{\&Integral;\&Integral;}\underset{k=0}{\overset{\mathrm{\&infin;}}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}\left(v_2-\mathrm{\&tau;}+1\right){\&lsqb;\sqrt{\left|\frac{4\mathrm{\&Gamma;}\left(2-v_3\right)-\mathrm{\&Gamma;}\left(3-v_3\right)}{\mathrm{\&Gamma;}\left(3-v_3\right)}\right|}\&rsqb;}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left({\left|D^{v_1}s\right|}^{v_2-2k-2}{D}_x^{v_1}s\right)\\ +D_y^1\left({\left|D^{v_1}s\right|}^{v_2-2k-2}{D}_y^{v_1}s\right)\end{array}\right]{\left(s-s_0\right)}^{2}dxdy$ Numerical Implementation equation be respectively $s_{x,y}^{n+1}=Q\left(s_{x,y}^n{\mathrm{\&Delta;t}}^{v_3}-\frac{{\mathrm{\&lambda;}}^n{\mathrm{\&Delta;t}}^{v_3}}{\mathrm{\&Gamma;}\left(1-v_3\right)\mathrm{\&Gamma;}\left(2-v_3\right)}\right)s_{x,y}^n+s_{x,y}^n-\frac{2\mathrm{\&mu;}}{\mathrm{\&Gamma;}\left(3-v_3\right)}{\left(s_{x,y}^{n-1}-s_{x,y}^n\right)}^2{\left(s_{x,y}^n\right)}^{-v_3},v_3\&NotEqual;1,2,3$ With ${\mathrm{\&lambda;}}^n=\frac{\mathrm{\&Gamma;}(1-v_3)\mathrm{\&Gamma;}(2-v_3)}{{\mathrm{\&sigma;}}^{n^2}{s}_{x,y}^n}\underset{x,y}{\&Sigma;}Q\left(s_{x,y}^n\right){(s_{x,y}^0-s_{x,y}^n)}^2.$ Wherein, $\underset{\mathrm{\&tau;}=1}{\overset{n}{\mathrm{\&Pi;}}}\left(v_2-\mathrm{\&tau;}+1\right)\stackrel{n=0}{=}1,$ $Q\left(s_{x,y}^n\right)=\frac{-\mathrm{\&Gamma;}\left(1-v_1\right)}{\mathrm{\&Gamma;}\left(-v_1\right)\mathrm{\&Gamma;}\left(-v_3\right)\mathrm{\&Gamma;}\left(3-v_3\right)}\underset{k=0}{\overset{1}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}\left(v_2-\mathrm{\&tau;}+1\right){\&lsqb;\sqrt{\left|\frac{4\mathrm{\&Gamma;}\left(2-v_3\right)-\mathrm{\&Gamma;}\left(3-v_3\right)}{\mathrm{\&Gamma;}\left(3-v_3\right)}\right|}\&rsqb;}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n\right)\\ +D_y^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_y^{v_1}{s}_{x,y}^n\right)\end{array}\right].$

Wherein, when iterative numerical calculates, on the one hand, the present invention does not need to know in advance or the variance of estimating noise, and only needs when making first time iterative numerical it is a less positive number.The present invention gets will bring into ${\mathrm{\&lambda;}}^n=\frac{\mathrm{\&Gamma;}(1-v_3)\mathrm{\&Gamma;}(2-v_3)}{{\mathrm{\&sigma;}}^{n^2}{s}_{x,y}^n}\underset{x,y}{\&Sigma;}Q\left(s_{x,y}^n\right){(s_{x,y}^0-s_{x,y}^n)}^2$ To start the process that iterative numerical calculates, so iteration gained each time all different, but each all once approaching the real variance of noise; On the other hand, in the process that iterative numerical calculates, may occur situation, in order to make meaningful, when time, the present invention gets in order to make meaningful, when time, the present invention gets

Based on above-mentioned to a kind of deduce mathematical of the Denoising of Texture Images wave filter based on fractional order evolution equation proposed by the invention and the brief description of numerical operation rule, the circuit illustrating this wave filter is below formed:

See Fig. 1, a kind of Denoising of Texture Images wave filter based on fractional order evolution equation proposed by the invention realizes the fractional order to texture image, non-linear, multiple dimensioned, high precision denoising based on a kind of special fractional order evolution equation denoise algorithm.The order v of the fractional calculus that the present invention relates to ₁, v ₂and v ₃not traditional integer rank, but non-integral order, generally get mark or rational fractional in engineer applied, and v ₃≠ 1,2,3.See Fig. 1, this wave filter adopts buffer 2, squared difference device 1 ,-v ₃power power side device 4, multiplier 1, λ ⁿgenerator 11, generator 10, multiplier 28, multiplier 39 and totalizer 1 are formed in cascaded fashion.This wave filter has while removal texture image noise, the low frequency profile in smooth region can be retained as far as possible, simultaneously again can the relatively large high frequency edge of non-linear reservation gray-scale value transition amplitude, and can also the remarkable advantage of the relatively little grain details of non-linear reservation gray-scale value transition changes in amplitude.This wave filter is specially adapted to the application scenario of the image being rich in complex texture minutia being carried out to denoising.

See Fig. 1,1 is the input point of a kind of Denoising of Texture Images wave filter based on fractional order evolution equation of the present invention, i.e. the digital picture of n-th iteration input point.Buffer 2 role is by the output image of a kind of Denoising of Texture Images wave filter based on fractional order evolution equation of the present invention carry out the buffer memory of Liang Ge unit iteration time, thus obtain the digital picture of (n-1)th iteration the E input point of squared difference device 1 is E points is weights input point.The calculating that squared difference device 1 completes is -v ₃the calculating that power power side device 4 completes is the F input point of multiplier 1 is weights input point.The calculating that multiplier 1 completes is the calculating that generator 10 completes is $Q\left(s_{x,y}^n\right)=\frac{-\mathrm{\&Gamma;}(1-v_1)}{\mathrm{\&Gamma;}(-v_1)\mathrm{\&Gamma;}(-v_3)\mathrm{\&Gamma;}(3-v_3)}\underset{k=0}{\overset{1}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}(v_2-\mathrm{\&tau;}+1){\left[\sqrt{\left|\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}\right|}\right]}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n\right)\\ +D_y^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_y^{v_1}{s}_{x,y}^n\right)\end{array}\right].$ λ ⁿthe calculating that generator 11 completes is ${\mathrm{\&lambda;}}^n=\frac{\mathrm{\&Gamma;}(1-v_3)\mathrm{\&Gamma;}(2-v_3)}{{\mathrm{\&sigma;}}^{n^2}{s}_{x,y}^n}\underset{x,y}{\mathrm{\&Sigma;}}P\left(s_{x,y}^n\right){(s_{x,y}^0-s_{x,y}^n)}^2.$ The H input point of multiplier 39 is weights Δ t ^v3input point.The calculating that multiplier 39 completes is the G input point of multiplier 28 is weights input point.The calculating that multiplier 28 completes is the calculating that totalizer 1 completes is $s_{x,y}^{n+1}=Q\left(s_{x,y}^n\right){\mathrm{\&Delta;t}}^{v_3}-\frac{{\mathrm{\&lambda;}}^n{\mathrm{\&Delta;t}}^{v_3}}{\mathrm{\&Gamma;}\left(1-v_3\right)\mathrm{\&Gamma;}\left(2-v_3\right)}{s}_{x,y}^n+s_{x,y}^n-\frac{2\mathrm{\&mu;}}{\mathrm{\&Gamma;}\left(3-v_3\right)}{\left(s_{x,y}^{n-1}-s_{x,y}^n\right)}^2{\left(s_{x,y}^n\right)}^{-v_3},v_3\&NotEqual;1,2,3.$

Wherein, application claims order v ₃meet v ₃≠ 1,2,3.The function that the output point 7 of a kind of Denoising of Texture Images wave filter based on fractional order evolution equation of the present invention completes is the digital picture of output (n+1)th iteration

See Fig. 4, the calculating that generator 10 completes is $Q\left(s_{x,y}^n\right)=\frac{-\mathrm{\&Gamma;}\left(1-v_1\right)}{\mathrm{\&Gamma;}\left(-v_1\right)\mathrm{\&Gamma;}\left(-v_3\right)\mathrm{\&Gamma;}\left(3-v_3\right)}\underset{k=0}{\overset{1}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}\left(v_2-\mathrm{\&tau;}+1\right){\&lsqb;\sqrt{\left|\frac{4\mathrm{\&Gamma;}\left(2-v_3\right)-\mathrm{\&Gamma;}\left(3-v_3\right)}{\mathrm{\&Gamma;}\left(3-v_3\right)}\right|}\&rsqb;}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n\right)\\ +D_y^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_y^{v_1}{s}_{x,y}^n\right)\end{array}\right].$

The I input point of squared difference device 2 12 is weights of squared difference device two input point, i.e. the original digital image of the 0th iteration input point.The calculating that squared difference device 2 12 completes is the calculating that generator 13 completes is the calculating that multiplier 4 14 completes is the calculating that multiplier 5 15 completes is the calculating that totalizer 1 completes is the J input point of divider 17 is weights Г (1-v ₃) Г (2-v ₃) input point.The calculating that divider 17 completes is the calculating that multiplier 6 18 completes is ${\mathrm{\&lambda;}}^n=\frac{\mathrm{\&Gamma;}(1-v_3)\mathrm{\&Gamma;}(2-v_3)}{{\mathrm{\&sigma;}}^{n^2}{s}_{x,y}^n}\underset{x,y}{\&Sigma;}Q\left(s_{x,y}^n\right){(s_{x,y}^0-s_{x,y}^n)}^2.$ λ ⁿthe function that the output point 19 of generator completes exports λ ⁿvalue.

See Fig. 5, the calculating that differentiator 20 completes is v in the direction of the x axis ₁rank fractional order differential. simultaneously the calculating that differentiator 21 completes carry out v in the direction of the x axis with on y-axis direction ₁rank fractional order differential. the calculating that differentiator 22 completes is v in the y-axis direction ₁rank fractional order differential.V ₂the calculating that-2k-2 power side device 23 completes is the calculating that multiplier 7 24 completes is the calculating that multiplier 8 25 completes is the calculating that differentiator 26 completes is 1 rank differential in the direction of the x axis. the calculating that differentiator 27 completes is 1 rank differential in the y-axis direction.The calculating that totalizer 2 28 completes is $\&lsqb;D_x^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n\right)+D_y^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_y^{v_1}{s}_{x,y}^n\right)\&rsqb;.$ The K input point of multiplier 9 32 is weights input point.The calculating that multiplier 9 32 completes is $\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}\left(v_2-\mathrm{\&tau;}+1\right){\&lsqb;\sqrt{\left|\frac{4\mathrm{\&Gamma;}\left(2-v_3\right)-\mathrm{\&Gamma;}\left(3-v_3\right)}{\mathrm{\&Gamma;}\left(3-v_3\right)}\right|}\&rsqb;}^{v_2-2k-2}}{\left(2k\right)!}\&lsqb;D_x^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n\right)+D_y^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_y^{v_1}{s}_{x,y}^n\right)\&rsqb;.$ the calculating that totalizer 31 completes is $\underset{k=0}{\overset{1}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}\left(v_2-\mathrm{\&tau;}+1\right){\&lsqb;\sqrt{\left|\frac{4\mathrm{\&Gamma;}\left(2-v_3\right)-\mathrm{\&Gamma;}\left(3-v_3\right)}{\mathrm{\&Gamma;}\left(3-v_3\right)}\right|}\&rsqb;}^{v_2-2k-2}}{\left(2k\right)!}\&lsqb;D_x^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n\right)+D_y^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_y^{v_1}{s}_{x,y}^n\right)\&rsqb;.$ The L input point of multiplier 10 is weights input point.The calculating that multiplier 10 completes is $\frac{-\mathrm{\&Gamma;}\left(1-v_1\right)}{\mathrm{\&Gamma;}\left(-v_1\right)\mathrm{\&Gamma;}\left(-v_3\right)\mathrm{\&Gamma;}\left(3-v_3\right)}\underset{k=0}{\overset{1}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}\left(v_2-\mathrm{\&tau;}+1\right){\&lsqb;\sqrt{\left|\frac{4\mathrm{\&Gamma;}\left(2-v_3\right)-\mathrm{\&Gamma;}\left(3-v_3\right)}{\mathrm{\&Gamma;}\left(3-v_3\right)}\right|}\&rsqb;}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n\right)\\ +D_y^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_y^{v_1}{s}_{x,y}^n\right)\end{array}\right].$ the function that the output point 29 of generator completes exports value.

See Fig. 6, the M input point of squared difference device 3 33 is weights input point, i.e. the original digital image of the 0th iteration input point.The calculating that squared difference device 3 33 completes is the calculating that totalizer 2 34 completes is the function that the output point 35 of generator completes exports value.

New departure below in conjunction with accompanying drawing and example in detail a kind of Denoising of Texture Images wave filter based on fractional order evolution equation of the present invention:

Accompanying drawing explanation

Fig. 1 is the schematic diagram of a kind of Denoising of Texture Images wave filter based on fractional order evolution equation of the present invention.

Fig. 2 is the schematic diagram of simply connected region Ω and smooth boundary curve C thereof.

Fig. 3 is the schematic diagram of two-dimentional simply connected image-region Ω and sectionally smooth boundary curve C thereof.

Fig. 4 is λ ⁿthe schematic diagram of generator.

Fig. 5 is the schematic diagram of generator.

Fig. 6 is the schematic diagram of generator.

Wherein, 1 is the input point of a kind of Denoising of Texture Images wave filter based on fractional order evolution equation of the present invention, i.e. the digital picture of n-th iteration input point; 2 is buffers; 3 is squared difference devices one; 4 is-v ₃power power side device; 5 is multipliers one; 6 is totalizers one; 7 is output points of a kind of Denoising of Texture Images wave filter based on fractional order evolution equation of the present invention, i.e. the digital picture of (n+1)th iteration output point; 8 is multipliers two; 9 is multipliers three; 10 are generator; 11 is λ ⁿgenerator; 12 is squared difference devices two; 13 are generator; 14 is multipliers four; 15 is multipliers five; 16 are totalizer one; 17 is dividers; 18 is multipliers six; 19 is λ ⁿthe output point of generator, i.e. λ ⁿoutput point; 20 are differentiator; 21 are differentiator; 22 are differentiator; 23 is v ₂-2k-2 power side device; 24 is multipliers seven; 25 is multipliers eight; 26 are differentiator; 27 are differentiator; 28 is totalizers two; 29 are the output point of generator, namely output point; 30 is multipliers ten; 31 are totalizer; 32 is multipliers nine; 33 is squared difference devices three; 34 are totalizer two; 35 are the output point of generator, namely output point.In addition, 3,12 and 33 is squared difference devices that function is identical with parameter; 5,8,9,14,15,18,24,25,30 and 32 is multipliers that function is identical with parameter; 6 and 28 is totalizers that function is identical with parameter; 16 and 34 is that function is identical with parameter totalizer.

Wherein, A point is the upper arbitrfary point of piecewise smooth curve C; B point is the upper arbitrfary point of piecewise smooth curve C; C is piecewise smooth curve; C ₁point is the upper arbitrfary point of piecewise smooth curve C; C ₂point is the upper arbitrfary point of piecewise smooth curve C; A point is the subpoint of A point in x-axis; B point is the subpoint of B point in x-axis; C point is C ₁the subpoint of point in y-axis; D point is C ₂the subpoint of point in y-axis; The plane simply connected region that Ω is is border with piecewise smooth curve C; E point is weights input point; F point is weights input point; G point is weights input point; H point is weights input point; I point is the weights of squared difference device two input point; J point is weights Г (1-v ₃) Г (2-v ₃) input point; K point is weights input point; L point is weights input point; M point is the weights of squared difference device three input point.

Embodiment

Now citing is described below:

See Fig. 1, Fig. 4, Fig. 5 and Fig. 6, in practical implementation, the order v of a kind of fractional calculus related to based on the Denoising of Texture Images wave filter of fractional order evolution equation of the present invention ₁, v ₂and v ₃not traditional integer rank, but non-integral order, generally get mark or rational fractional.In addition, as shown in the above description if the present invention gets order v ₁=2.25, v ₂=2.5, v ₃=0.25, get unit iteration time interval of delta t=0.01, variance initial value the present invention gets parameter μ=0.005, so can obtain the physical circuit parameter of a kind of Denoising of Texture Images wave filter based on fractional order evolution equation of the present invention.See Fig. 1, the input weights of the F input point of multiplier 1 are the G input point of multiplier 28 is input weights $\frac{-{\mathrm{\&Delta;t}}^{v_3}}{\mathrm{\&Gamma;}\left(1-v_3\right)\mathrm{\&Gamma;}\left(2-v_3\right)}=\frac{-{\left(0.01\right)}^{0.25}}{\mathrm{\&Gamma;}\left(0.75\right)\mathrm{\&Gamma;}\left(1.75\right)}=-\mathrm{0.2808.}$ The input weights of the H input point of multiplier 39 are see Fig. 4, the input weights of the J input point of divider 17 are Г (1-v ₃) Г (2-v ₃)=Г (0.75) Г (1.75)=1.1262.See Fig. 5, the input weights of the K input point of multiplier 9 32 are $\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}\left(v_2-\mathrm{\&tau;}+1\right){\&lsqb;\sqrt{\left|\frac{4\mathrm{\&Gamma;}\left(2-v_3\right)-\mathrm{\&Gamma;}\left(3-v_3\right)}{\mathrm{\&Gamma;}\left(3-v_3\right)}\right|}\&rsqb;}^{v_2-2k-2}}{\left(2k\right)!}=\frac{{1.1339}^{0.5-2k}\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}\left(3.5-\mathrm{\&tau;}\right)}{\left(2k\right)!}.$ The input weights of the L input point of multiplier 10 are $\frac{-\mathrm{\&Gamma;}\left(1-v_1\right)}{\mathrm{\&Gamma;}\left(-v_1\right)\mathrm{\&Gamma;}\left(-v_3\right)\mathrm{\&Gamma;}\left(3-v_3\right)}=\frac{-\mathrm{\&Gamma;}\left(-1.25\right)}{\mathrm{\&Gamma;}\left(-2.25\right)\mathrm{\&Gamma;}\left(-0.25\right)\mathrm{\&Gamma;}\left(2.75\right)}=-\mathrm{0.2854.}$ So, as shown in Fig. 1, Fig. 4, Fig. 5 and Fig. 6, according to cascade circuit structure and the physical circuit parameter thereof of the of the present invention a kind of Denoising of Texture Images wave filter based on fractional order evolution equation described in detail in the summary of the invention of this instructions, this physical circuit being suitable for the high-precision filter of Denoising of Texture Images based on partial fractional differential equation just can be constructed easily.Under the prerequisite not affecting precise expression, in order to more clear the physical circuit describing a kind of Denoising of Texture Images wave filter based on fractional order evolution equation of the present invention, Fig. 1, Fig. 4, Fig. 5 and Fig. 6 do not draw sequential control circuit wherein and the timing control signal produced that is triggered.

Claims (6)

1. based on a Denoising of Texture Images wave filter for fractional order evolution equation, it is characterized in that: it is by buffer (2), squared difference device one (3) ,-v ₃power power side device (4), multiplier one (5), λ ⁿthe calculating that generator (11) completes is ${\mathrm{\&lambda;}}^n=\frac{\mathrm{\&Gamma;}\left(1-v_3\right)\mathrm{\&Gamma;}\left(2-v_3\right)}{{\mathrm{\&sigma;}}^{n^2}{s}_{x,y}^n}\underset{x,y}{\&Sigma;}Q\left(s_{x,y}^n\right){(s_{x,y}^0-s_{x,y}^n)}^2,$ the calculating that generator (10) completes is $Q\left(s_{x,y}^n\right)=\frac{-\mathrm{\&Gamma;}(1-v_1)}{\mathrm{\&Gamma;}(-v_1)\mathrm{\&Gamma;}(-v_3)\mathrm{\&Gamma;}(3-v_3)}\underset{k=0}{\overset{1}{\&Sigma;}}\frac{{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\&Pi;}}\left(v_2-\mathrm{\&tau;}+1\right)\&lsqb;\sqrt{\left|\frac{4\mathrm{\&Gamma;}\left(2-v_3\right)-\mathrm{\&Gamma;}\left(3-v_3\right)}{\mathrm{\&Gamma;}\left(3-v_3\right)}\right|}\&rsqb;}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left(\right|D^{v_1}{s}_{x,y}^n{|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n)\\ +D_y^1\left(\right|D^{v_1}{s}_{x,y}^n{|}^{v_2-2k-2}{D}_y^{v_1}{s}_{x,y}^n)\end{array}\right],$ Multiplier two (8), multiplier three (9) and totalizer one (6) are formed in cascaded fashion, wherein, and order v ₁, v ₂and v ₃get mark or rational fractional, and v ₃≠ 1,2,3, n gets any positive integer, and k gets 0 and 1, and the original digital image of the 0th iteration is the digital picture of n-th iteration is gamma function is Γ, variance variance initial value the calculating completed is v in the direction of the x axis ₁rank fractional order differential, simultaneously the calculating completed carry out v in the direction of the x axis with on y-axis direction ₁rank fractional order differential, the calculating completed is v in the y-axis direction ₁rank fractional order differential, the calculating completed is 1 rank differential in the direction of the x axis, the calculating completed is 1 rank differential in the y-axis direction, behalf digital picture, and x represents the variable in x-axis, and y represents the variable in y-axis, representative is sued for peace for variable x and y in digital picture s, and λ is regularization parameter, and when k gets 0, parameter τ gets 0 and 1, and when k gets 1, parameter τ gets 1 and 2.

2. a kind of Denoising of Texture Images wave filter based on fractional order evolution equation according to claim 1, is characterized in that: the digital picture of input point (1) input n-th iteration of this wave filter buffer (2) is by a kind of output image of the Denoising of Texture Images wave filter based on fractional order evolution equation carry out the buffer memory of Liang Ge unit iteration time, thus obtain the digital picture of (n-1)th iteration the input weights of squared difference device one (3) are the calculating that squared difference device one (3) completes is -v ₃the calculating that power power side device (4) completes is the input weights of multiplier one (5) are the calculating that multiplier one (5) completes is $-\frac{2\mathrm{\&mu;}}{\mathrm{\&Gamma;}\left(3-v_3\right)}{(s_{x,y}^{n-1}-s_{x,y}^n)}^2{\left(s_{x,y}^n\right)}^{-v_3},$ the calculating that generator (10) completes is $Q\left(s_{x,y}^n\right)=\frac{-\mathrm{\&Gamma;}(1-v_1)}{\mathrm{\&Gamma;}(-v_1)\mathrm{\&Gamma;}(-v_3)\mathrm{\&Gamma;}(3-v_3)}\underset{k=0}{\overset{1}{\&Sigma;}}\frac{{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\&Pi;}}\left(v_2-\mathrm{\&tau;}+1\right)\&lsqb;\sqrt{\left|\frac{4\mathrm{\&Gamma;}\left(2-v_3\right)-\mathrm{\&Gamma;}\left(3-v_3\right)}{\mathrm{\&Gamma;}\left(3-v_3\right)}\right|}\&rsqb;}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left(\right|D^{v_1}{s}_{x,y}^n{|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n)\\ +D_y^1\left(\right|D^{v_1}{s}_{x,y}^n{|}^{v_2-2k-2}{D}_y^{v_1}{s}_{x,y}^n)\end{array}\right],$ λ ⁿthe calculating that generator (11) completes is ${\mathrm{\&lambda;}}^n=\frac{\mathrm{\&Gamma;}\left(1-v_3\right)\mathrm{\&Gamma;}\left(2-v_3\right)}{{\mathrm{\&sigma;}}^{n^2}{s}_{x,y}^n}\underset{x,y}{\&Sigma;}Q\left(s_{x,y}^n\right){(s_{x,y}^0-s_{x,y}^n)}^2,$ The input weights of multiplier three (9) are the calculating that multiplier three (9) completes is the input weights of multiplier two (8) are the calculating that multiplier two (8) completes is the calculating that totalizer one (6) completes is $s_{x,y}^{n+1}=Q\left(s_{x,y}^n\right){\mathrm{\&Delta;t}}^{v_3}-\frac{{\mathrm{\&lambda;}}^n{\mathrm{\&Delta;t}}^{v_3}}{\mathrm{\&Gamma;}\left(1-v_3\right)\mathrm{\&Gamma;}\left(2-v_3\right)}{s}_{x,y}^n+s_{x,y}^n-\frac{2\mathrm{\&mu;}}{\mathrm{\&Gamma;}\left(3-v_3\right)}{(s_{x,y}^{n-1}-s_{x,y}^n)}^2{\left(s_{x,y}^n\right)}^{-v_3},$ V ₃≠ 1,2,3, wherein, parameter μ=0.005, unit iteration time interval of delta t is (0,0.1] get arbitrary arithmetic number in, the digital picture of (n+1)th iteration should be exported based on the output point (7) of the Denoising of Texture Images wave filter of fractional order evolution equation

3. a kind of Denoising of Texture Images wave filter based on fractional order evolution equation according to claim 1, is characterized in that: λ ⁿgenerator (11) comprises generator (10), squared difference device two (12), generator (13), multiplier four (14), multiplier five (15), totalizer one (16), divider (17), multiplier six (18), wherein the calculating that generator (10) completes is $Q\left(s_{x,y}^n\right)=\frac{-\mathrm{\&Gamma;}(1-v_1)}{\mathrm{\&Gamma;}(-v_1)\mathrm{\&Gamma;}(-v_3)\mathrm{\&Gamma;}(3-v_3)}\underset{k=0}{\overset{1}{\&Sigma;}}\frac{{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\&Pi;}}\left(v_2-\mathrm{\&tau;}+1\right)\&lsqb;\sqrt{\left|\frac{4\mathrm{\&Gamma;}\left(2-v_3\right)-\mathrm{\&Gamma;}\left(3-v_3\right)}{\mathrm{\&Gamma;}\left(3-v_3\right)}\right|}\&rsqb;}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left(\right|D^{v_1}{s}_{x,y}^n{|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n)\\ +D_y^1\left(\right|D^{v_1}{s}_{x,y}^n{|}^{v_2-2k-2}{D}_y^{v_1}{s}_{x,y}^n)\end{array}\right],$ The input weights of squared difference device two (12) are the original digital image of the 0th iteration the calculating that squared difference device two (12) completes is the calculating that generator (13) completes is the calculating that multiplier four (14) completes is the calculating that multiplier five (15) completes is the calculating that totalizer one (16) completes is the input weights of divider (17) are Γ (1-v ₃) Γ (2-v ₃), the calculating that divider (17) completes is the calculating that multiplier six (18) completes is ${\mathrm{\&lambda;}}^n=\frac{\mathrm{\&Gamma;}\left(1-v_3\right)\mathrm{\&Gamma;}\left(2-v_3\right)}{{\mathrm{\&sigma;}}^{n^2}{s}_{x,y}^n}\underset{x,y}{\&Sigma;}Q\left(s_{x,y}^n\right){(s_{x,y}^0-s_{x,y}^n)}^2,$ λ ⁿthe output point (19) of generator exports λ ⁿvalue.

4. a kind of Denoising of Texture Images wave filter based on fractional order evolution equation according to claim 1, is characterized in that: generator (10) comprises differentiator (20), differentiator (21), differentiator (22), v ₂-2k-2 power side device (23), multiplier seven (24), multiplier eight (25), differentiator (26), differentiator (27), totalizer two (28), multiplier nine (32), totalizer (31), multiplier ten (30) ", wherein the calculating that differentiator (20) completes is v in the direction of the x axis ₁rank fractional order differential, simultaneously the calculating that differentiator (21) completes carry out v in the direction of the x axis with on y-axis direction ₁rank fractional order differential, the calculating that differentiator (22) completes is v in the y-axis direction ₁rank fractional order differential, v ₂the calculating that-2k-2 power side device (23) completes is the calculating that multiplier seven (24) completes is the calculating that multiplier eight (25) completes is the calculating that differentiator (26) completes is 1 rank differential in the direction of the x axis, the calculating that differentiator (27) completes is 1 rank differential in the y-axis direction, and the calculating that totalizer two (28) completes is $\&lsqb;D_x^1\left(\right|D^{v_1}{s}_{x,y}^n{|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n)+D_y^1\left(\right|D^{v_1}{s}_{x,y}^n{|}^{v_2-2k-2}{D}_y^{v_1}{s}_{x,y}^n)\&rsqb;,$ The input weights of multiplier nine (32) are $\frac{{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\&Pi;}}\left(v_2-\mathrm{\&tau;}+1\right)\&lsqb;\sqrt{\left|\frac{4\mathrm{\&Gamma;}\left(2-v_3\right)-\mathrm{\&Gamma;}\left(3-v_3\right)}{\mathrm{\&Gamma;}\left(3-v_3\right)}\right|}\&rsqb;}^{v_2-2k-2}}{\left(2k\right)!},$ The calculating that multiplier nine (32) completes is $\frac{{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\&Pi;}}\left(v_2-\mathrm{\&tau;}+1\right)\&lsqb;\sqrt{\left|\frac{4\mathrm{\&Gamma;}\left(2-v_3\right)-\mathrm{\&Gamma;}\left(3-v_3\right)}{\mathrm{\&Gamma;}\left(3-v_3\right)}\right|}\&rsqb;}^{v_2-2k-2}}{\left(2k\right)!}\&lsqb;D_x^1\left(\right|D^{v_1}{s}_{x,y}^n{|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n)+D_y^1\left(\right|D^{v_1}{s}_{x,y}^n{|}^{v_2-2k-2}{D}_y^{v_1}{s}_{x,y}^n)\&rsqb;,$ the calculating that totalizer (31) completes is $\underset{k=0}{\overset{1}{\&Sigma;}}\frac{{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\&Pi;}}\left(v_2-\mathrm{\&tau;}+1\right)\&lsqb;\sqrt{\left|\frac{4\mathrm{\&Gamma;}\left(2-v_3\right)-\mathrm{\&Gamma;}\left(3-v_3\right)}{\mathrm{\&Gamma;}\left(3-v_3\right)}\right|}\&rsqb;}^{v_2-2k-2}}{\left(2k\right)!}\&lsqb;D_x^1\left(\right|D^{v_1}{s}_{x,y}^n{|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n)+D_y^1\left(\right|D^{v_1}{s}_{x,y}^n{|}^{v_2-2k-2}{D}_y^{v_1}{s}_{x,y}^n)\&rsqb;,$ The input weights of multiplier ten (30) are the calculating that multiplier ten (30) completes is $\frac{-\mathrm{\&Gamma;}(1-v_1)}{\mathrm{\&Gamma;}(-v_1)\mathrm{\&Gamma;}(-v_3)\mathrm{\&Gamma;}(3-v_3)}\underset{k=0}{\overset{1}{\&Sigma;}}\frac{{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\&Pi;}}\left(v_2-\mathrm{\&tau;}+1\right)\&lsqb;\sqrt{\left|\frac{4\mathrm{\&Gamma;}\left(2-v_3\right)-\mathrm{\&Gamma;}\left(3-v_3\right)}{\mathrm{\&Gamma;}\left(3-v_3\right)}\right|}\&rsqb;}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left(\right|D^{v_1}{s}_{x,y}^n{|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n)\\ +D_y^1\left(\right|D^{v_1}{s}_{x,y}^n{|}^{v_2-2k-2}{D}_y^{v_1}{s}_{x,y}^n)\end{array}\right],$ the output point (29) of generator exports value.

5. a kind of Denoising of Texture Images wave filter based on fractional order evolution equation according to claim 3, is characterized in that: generator (13) comprise squared difference device three (33), totalizer two (34), wherein the input weights of squared difference device three (33) are the original digital image of the 0th iteration the calculating that squared difference device three (33) completes is the calculating that totalizer two (34) completes is the original digital image of the 0th iteration is the digital picture of n-th iteration is variance variance initial value the output point (35) of generator exports value.

6. a kind of Denoising of Texture Images wave filter based on fractional order evolution equation according to claim 1, is characterized in that: do not know in advance or the variance of estimating noise, and only get when first time iterative computation will bring λ into ⁿgenerator is to start the process of iterative computation; In the process of iterative computation, when $\left|D^{v_1}{s}_{x,y}^n\right|\&le;0.0689$ Time, get $\left|D^{v_1}{s}_{x,y}^n\right|=0.0689;$ When $s_{x,y}^n=0$ Time, get $s_{x,y}^n=\mathrm{0.0000.1.}$