CN103390265A - Texture image denoising filter based on fractional order evolution equation - Google Patents

Abstract

The invention provides a texture image denoising filter based on a fractional order evaluation equation belongs to the technical field of applied mathematics, digital image processing and digital circuit interdisciplines.. The texture image denoisting filter achieves fractional order, nonlinear, multi-scale and high-precision denoising on texture images on the basis of a special fractional order evaluation equation. The texture image denoising filter is formed by a buffer, a first difference value squarer, a v3-power power unit, a first multiplying unit, a lambda <n> generator, a generator, a second multiplying unit, a third multiplying unit and a first summator in a cascade mode. The filter has the remarkably advantages of being capable of maintaining low-frequency outlines in a smooth area while removing noises of texture images and maintaining high-frequency edges with relatively high gray value saltus amplitude and texture details with relatively low gray value saltus amplitude changing. The texture image denoising filter is particularly suitable for occasions of denoising images with complex texture detail characteristics.

Description

Texture image denoising filter based on fractional order development equation

Field of the invention

The invention provides a texture image denoising filter based on a fractional order development equation, which is based on a special fractional order development equation denoising algorithm to realize fractional order, nonlinear, multi-scale and high-precision denoising of a texture image. Order v of fractional calculus₁、v₂And v₃Not the conventional integer order but a non-integer order, and generally takes a fraction or rational fraction in engineering applications, and v₃Not equal to 1, 2, 3. Referring to FIG. 1, the filter uses a buffer 2, a difference squarer-3, -v₃Power squarer 4, multiplier 5, lambdaⁿA generator 11,

The generator 10, the multiplier two 8, the multiplier three 9 and the adder one 6 are constructed in a cascade manner. The filter has the obvious advantages that while texture image noise is removed, low-frequency contours in a smooth area can be reserved as much as possible, high-frequency edges with relatively large gray value jump amplitude can be non-linearly reserved, and texture details with relatively small gray value jump amplitude change can be non-linearly reserved. The filter is particularly suitable for the application occasion of denoising the image rich in complex texture detail characteristics. The invention belongs to the technical field of applied mathematics, digital image processing and digital circuit interdisciplinary disciplines.

Background

Digital image processing theory mainly includes three major methods: random modeling, wavelet theory, and partial differential equation methods. Among them, image processing based on partial differential equation is an important branch in the field of image processing, and is an important part in mathematical analysis. Partial differential equation method it is closely related to the physical world. The well-known wave equation and heat conduction equation belong to partial differential equations of integer order, and Euler equation, Poisson equation, Laplace equation, and the like. Integer-order partial differential equations in physics are often applied to other fields, such as biology, finance, etc., and have been applied to the field of digital image processing. Regarding digital image processing technology based on integral order partial differential equation, on one hand, the image processing method belongs to the category of low-level image processing, and the processing result is generally used as an intermediate result and provided for other image processing methods to further use; on the other hand, with the intensive research of the image processing method, people increasingly and deeply dig the essence of the image and the image processing and try to modify the existing traditional image processing method by strict mathematical theory, which is a challenge for the traditional image processing method mainly used for practical use.

At present, although partial differential equations have been applied to the field of digital image processing, most of the related research is limited to the application of integer-order partial differential equations, but few partial differential equations are studied at home and abroad in the field of digital image processing. The integer order partial differential equation is itself from a continuous domain, so it can be described essentially as an analog image, and once its solution has proven to exist and unique, we can use discrete numerical methods to solve the integer order partial differential equation for digital images. Because digital image processing based on integral order partial differential equations can be linked to some physical process, they are usually described in the continuous domain. In general, the integer-order partial differential equation method is relatively large in calculation amount compared to the general filtering method: the method needs iterative solution or solution of an equation set constructed by finite difference, and the method of the integer order partial differential equation has the main advantages that: first, integer order partial differential equations and corresponding curved (surface) flows give a continuous model of the analysis image, and discrete filtering appears as a continuous differential operator, thus making the partitioning of the mesh, local nonlinear analysis easy to implement. On the other hand, when the image is represented as a joinContinuous signal, integral order partial differential equations can be considered as iterations of local filtering in the tiny sub-neighborhood, this property allows existing filtering methods to be synthesized and classified, and new filtering methods can be formed. Second, processing digital images using integer-order partial differential equations facilitates direct understanding and manipulation of visually important geometric features such as gradients, tangents, curvatures, and level sets, and also effectively simulates visually meaningful dynamics such as linear and non-linear diffusion and information transfer mechanisms. Thirdly, the unique analysis theory in the field of integral order partial differential equations provides possibility for researching better digital image processing algorithm and meaningful theoretical results, such as existence, uniqueness and the like of solutions. Particularly, the most notable advantage is that the integer-order partial differential equation method can obtain better image quality and has certain stability. The flexible and diversified numerical scheme provides great help for numerical calculation of the image processing equation. The introduction of integral order partial differential equations, which are more effective in the field of digital image processing, dates back to the end of the 80 s of the last century, and has been a very long development in the 90 s. The study can be traced back to Nagao, Rudin, etc. for image smoothing and image enhancement and konderink's exploration of image structure. A number of classical, integer order partial differential equations are applied to digital image processing, such as the thermal conduction equation, schrodinger equation, convection conduction equation, and the like. At present, image processing technology based on integral order partial differential equation has been applied well, for example, the french space agency has adopted the AMSS method as a standard method for enhancing images of aerial images. The integer order partial differential equation method is itself a matter of physics, and the earliest applications in digital image processing might be to calculate the heat conduction equation in isotropic media. If the gray-scale image is considered as a temperature field in an isotropic medium, the heat conduction process of the temperature field exactly corresponds to the gaussian smoothing process of the image, and the variance parameter of the gaussian filter is related to the conduction time. However, since gaussian smoothing is isotropic, the destructive effect on the edges is large, and the application thereof is limited. In 1987, Kass et al utilized the internal and external constraints required by image edgesAn energy function for representing the goodness of the contour curve is defined, wherein the internal constraint mainly considers the smoothness and the curvature of the contour, and the external energy represents the attraction of the contour of the image edge. By optimizing (minimizing) this energy function, the initially given contour can be converged on the adjacent image edges. The physical meaning of this method is clear, but since the object (contour) to be considered is [ alpha ]]²Is detected (assuming a contour in a two-dimensional image, if a three-dimensional image, such as a three-dimensional medical image, is considered, this contour corresponds to [ [ solution ] ]]³A two-dimensional surface) is described, the description mode and the discretization are limited, and the description mode directly limits the topological change of the contour curve, such as splitting, merging and the like. In 1989, Mumford and Shah proposed a variational model for image segmentation. In 1992, Chan and Vese constructed an active contour model without edges using the idea of minimum intra-class distance in pattern recognition, and then Yezzi and the like constructed a new active contour model using the idea of maximum inter-class distance. In 1995, Osher et al proposed the use of level sets to describe a curvature-dependent wavefront propagation process. The essence of this method is that the image contour is regarded as a zero level set of a two-dimensional function, and then the variation mode of the contour can be known by studying the variation behavior of the two-dimensional function. Also, since the object under study is a two-dimensional function, it is in]²Is easy to describe and solve (relative to a one-dimensional object), and the one-dimensional contour is not the object of direct solution, so that the processing of topological changes such as splitting, merging and the like of the contour is relatively easy through the change of a two-dimensional function. Since the late 90 s of the last century, integer-order partial differential equations have begun to be applied to digital image restoration, i.e., filling missing portions in digital images, or removing obstacles and the like in digital images, so that the resulting images look like real, which is a difficult task in the field of image editing. Image restoration based on integral order partial differential equation has two major directions, and Chan, Shen and the like of UCLA utilize energy optimization to deal with the problem, which is mainly to make assumptions on some properties (such as simplicity, small curvature and the like) of the structure image edge and then construct corresponding energy functions to traceConverting the partial differential equation into an integer order partial differential equation by an integer order variational method for solving; and the other genre represented by Bertalmio directly considers the diffusion process of certain properties in the image and directly provides an integral order partial differential equation to evolve and solve the partial differential equation. Both of these approaches have met with great success. In addition, in the field of image editing, the Poisson equation also plays a leading role in seamless pasting of images.

In image processing based on integral-order partial differential equation, image denoising is one of the most important research contents. Image denoising based on integer order partial differential equation is divided into two categories: a nonlinear diffusion-based method and a variational method based on minimization of an energy norm. The two basic models corresponding to this are: the anisotropic diffusion (PM) model proposed by Perona and Malik and the Total variation (ROF) model proposed by Rudin, Osher and Fatemi. The PM model simulates the denoising process of an image by using the diffusion process of thermal energy, and the denoising result of the image is the state when the thermal energy diffusion reaches the balance. The heat energy is described by total variation, namely an ROF model. On the basis, the researchers respectively popularize the PM model and the ROF model into color image processing. The learner has studied the choice of parameters in the model and how to compute the optimal stopping point for the iterative solution process. Rudin et al propose a variable time step method to solve the Euler-Lagrange equation. C.r.vogel and m.e.oman use a fixed point iteration method to improve the stability of the ROF model. D.c.dobson and c.r.vogel modify the fully variant form to ensure convergence of the ROF model numerical calculation. Chambole proposes a fast algorithm based on dual formula. Darbon and m.sigelle decompose the original problem into mutually independent optimization problems of the markov random field by using a level set method, and obtain a global optimal solution through reconstruction. The scholars propose an iterative weighted norm to solve the total variation to improve the calculation efficiency. Cotte et al first pass the original image through gaussian smoothing once to make the PM model have fitness. Both the PM model and the ROF model have significant disadvantages of easily causing contrast information loss, texture information loss, step effect, and the like. In response to these shortcomings, many improved models have been proposed. In order to maintain contrast information and texture informationThe students use L¹Norm substituted for L²And (4) norm. Osher et al propose an iterative regularization method. G.gilboa, y.y.zeevi and n.sochen propose a method of adaptive numerical fidelity term that varies with space. S.esedoglu and s.osher propose direction information that keeps a specific edge; to eliminate the step effect, p.blomgren proposes a model in which the total variation component varies with the gradient. The learners also introduce the high-order derivative into the energy model function, or combine the high-order derivative with the original ROF model, or propose an improved method such as two-stage denoising and the like. The image denoising improving method based on the integer order partial differential equation obtains certain effects on keeping contrast information and edge information of an image and eliminating a step effect.

Unfortunately, when we directly apply the conventional image denoising method based on integer-order partial differential equation to texture image denoising, it is generally difficult to achieve better processing effect. On the one hand, the conventional image denoising method based on the integral order partial differential equation is essentially based on the integral order calculus operation. It is difficult to handle some non-linear, non-causal, non-minimum phase systems, non-gaussian, non-stationary, non-integer dimensional (fractal) signals, non-white additive noise, etc. If we directly apply the image denoising method based on the integer order partial differential equation to the texture image denoising, there are the following limitations: firstly, an integer order partial differential equation processes an image only according to local information, so that periodic texture features cannot be maintained, and global features of the image cannot be recovered; secondly, an integer order partial differential equation model based on an integer order variation principle realizes image processing by optimizing an energy functional. The energy functional realizes the optimization in a local neighborhood, so that a blocky effect exists in the processed image; thirdly, the integral order partial differential equation only containing forward or backward diffusion has limited processing capability, and the equation of the bidirectional diffusion can generate a phenomenon of canceling diffusion information in two directions in the diffusion process, so that the final processing result is influenced. On the other hand, for texture images rich in complex texture detail information, the texture detail information of the images has great value to the accuracy of interpretation. The texture image denoising method has special requirements of unchanged contrast, unchanged texture characteristics and the like. Since the integer order differential value of the constant or the direct current component is zero, and the integer order differential value of the slightly fluctuating alternating current component is also zero after the current limitation of the specific threshold, the integer order differential operation causes great loss to the image complex texture detail information. When the traditional image denoising method based on the integral order partial differential equation is directly applied to texture image denoising, under the condition of low resolution, the processing result can be barely accepted because the texture detail information in the original image is originally not rich and clear; however, when the resolution is gradually increased, the complex texture detail information in the processing result image will be greatly lost, so that it is difficult to accurately interpret the texture detail of the processing result image. Therefore, for denoising a texture image rich in complex texture detail information, in order to more effectively maintain and utilize the complex texture detail information while denoising, a class of texture image denoising methods based on fractional order partial differential equations, which can process image complex texture detail features in a fractional order, non-linear and multi-scale manner, is urgently required.

Disclosure of Invention

The invention provides a texture image denoising filter based on a fractional order development equation, which is based on a special fractional order development equation denoising algorithm to realize fractional order, nonlinear, multi-scale and high-precision denoising of a texture image. Order v of fractional calculus₁、v₂And v₃Not the conventional integer order but a non-integer order, and generally takes a fraction or rational fraction in engineering applications, and v₃Not equal to 1, 2, 3. Referring to FIG. 1, the filter uses a buffer 2, a difference squarer-3, -v₃Power squarer 4, multiplier 5, lambdaⁿA generator 11, Generator 10, multiplier two 8, multiplier three 9 and adder one 6 are cascadedIs formed by the formula (I). The filter has the obvious advantages that while texture image noise is removed, low-frequency contours in a smooth area can be reserved as much as possible, high-frequency edges with relatively large gray value jump amplitude can be non-linearly reserved, and texture details with relatively small gray value jump amplitude change can be non-linearly reserved. The filter is particularly suitable for the application occasion of denoising the image rich in complex texture detail characteristics.

Referring to fig. 1, in order to clearly illustrate the circuit configuration of the texture image denoising filter based on the fractional order partial differential equation of the present invention, it is necessary to first briefly describe the mathematical formula derivation and the numerical operation rule of the high-precision filter suitable for texture image denoising based on the fractional order partial differential equation as follows:

it is well known that fractal mathematical theory produces a transition in measured magnitude and fractal geometry negates the presence of newton-leibraz derivatives. Fractal theory based on the Hausdorff measure, although over 90 years of research, is still a very incomplete mathematical theory to date. The construction of the mathematical theory of calculus under the Hausdorff measure has not been completed to date. The current development is well-established by fractional calculus defined under the euclidean measure, which mathematically requires that the euclidean measure must be used. Fractional calculus is most commonly used under the Euclidean measure, with both the Grumwald-Letnikov definition and the Riemann-Liouville definition.

The v-order calculus of the Grumwald-Letnikov definition signal s (x) is $D_{G-L}^{v}s\left(x\right)=\frac{d^v}{{\left[d(x-a)\right]}^v}s\left(x\right){|}_{G-L}=\underset{N\&RightArrow;\&infin;}{\lim }\left\{\frac{{\left(\frac{x-a}{N}\right)}^{-v}}{\mathrm{\&Gamma;}(-v)}\underset{k=0}{\overset{N-1}{\mathrm{\&Sigma;}}}\frac{\mathrm{\&Gamma;}(k-v)}{\mathrm{\&Gamma;}(k+1)}s(x-k\left(\frac{x-a}{N}\right))\right\}.$ Wherein the duration of the signal s (x) is [ a, x ]]V is any real number (including fractional),

denotes a fractional order differential operator defined based on Grumwald-Letnikov, where Γ is the Gamma function. According to the Grimwald-Letnikov definition formula of fractional calculus, Grumwald-Letnikov definition generalizes the integer step length of the integral calculus to the fractional step length under Euclidean measurement, so that the integer step length of the calculus is generalized to the fractional step length. The calculation of the Griirnwald-Letnikov definition of fractional calculus is simple and easy to implement, which only requires the correlation with the signal s (x) itself

Without the need for derivatives and integrals of the signal s (x). The Riemann-Liouville definition signal s (x) has a v-order integral (v π 0) of $D_{R-L}^{v}s\left(x\right)=\frac{d^v}{{\left[d(x-a)\right]}^v}s\left(x\right){|}_{R-L}=\frac{1}{\mathrm{\&Gamma;}(-v)}\underset{a}{\overset{x}{\&Integral;}}{(x-\mathrm{\&eta;})}^{-v-1}s\left(\mathrm{\&eta;}\right)\mathrm{d\&eta;}=\frac{-1}{\mathrm{\&Gamma;}(-v)}\underset{a}{\overset{x}{\&Integral;}}s\left(\mathrm{\&eta;}\right)d{(x-\mathrm{\&eta;})}^{-v},\mathrm{v\&pi;}0.$ Wherein,

representing a fractional order differential operator defined based on Riemann-Liouville. For the v-order differential (v ≧ 0) of the signal s (x), n satisfies n-l < v ≦ n. Thus, the present invention derives a v-order differential of signal s (x) defined by the Riemann-Liouville integral definition as $D_{R-L}^{y}s\left(x\right)=\frac{d^v}{{\left[d(x-a)\right]}^v}s\left(x\right){|}_{R-L}=\frac{d^n}{{\mathrm{dx}}^n}\frac{d^{v-n}}{{\left[d(x-a)\right]}^{v-n}}s\left(x\right){|}_{R-L}=\underset{k=0}{\overset{n-1}{\mathrm{\&Sigma;}}}\frac{{(x-a)}^{k-v}{s}^{\left(k\right)}\left(a\right)}{\mathrm{\&Gamma;}(k-v+1)}+\frac{1}{\mathrm{\&Gamma;}(n-v)}\underset{a}{\overset{x}{\&Integral;}}\frac{s^{\left(n\right)}\left(\mathrm{\&eta;}\right)}{{(x-\mathrm{\&eta;})}^{v-n+1}}\mathrm{d\&eta;},0\&le;\mathrm{v\&pi;n}.$ The present invention can derive the Fourier transform of the signal s (x) from the fractional differential Riemann-Liouville definition $\mathrm{FT}\left[D^{v}s\left(x\right)\right]={\left(\mathrm{i\&omega;}\right)}^v\mathrm{FT}\left[s\left(x\right)\right]-\underset{k=0}{\overset{n-1}{\mathrm{\&Sigma;}}}{\left(\mathrm{i\&omega;}\right)}^k\frac{d^{v-1-k}}{d{x}^{v-1-k}}s\left(0\right).$ Where i is an imaginary unit and ω is a digital frequency. When signal s (x) is a causal signal, the above equation can be simplified to FT [ D ]^vs(x)]＝(iω)^vFT[s(x)]。

In the invention, to construct a fractional partial differential equation suitable for denoising texture images, a fractional Euler-Lagrange equation must be constructed, and a fractional Green formula is a necessary premise for constructing the fractional Euler-Lagrange equation. Therefore, the method has to firstly popularize the traditional integer-order Green formula to the fractional order, and deduce and construct the fractional-order Green formula.

Referring to fig. 2, in the present invention, Ω is a planar single-connected region bounded by a piecewise smooth curve c, the integrable functions P (x, y) and Q (x, y) are continuous over Ω and c, and there are fractional continuous partial derivatives of x and y; let D¹Denotes the differential operator of order 1, D^vRepresenting a fractional order differential operator of order v, I¹＝D^-1Denotes the 1 st order integration operator, I^v＝D^-vRepresenting a v > 0 order fractional order integration operator,

represents the v-order fractional surface integrator on the plane omega (generalizing the Riemann-Liouville definition of fractional integration from one dimension to two),

AC shown in curve c₁The upper edge of the section B

A directional v-order fractional order curve integration operator,

a v-order fractional order closed curve integral operator representing a counterclockwise direction on the closed curve c; let the boundary c of the region Ω be formed by two curves y ═ φ₁(x)，y＝φ₂(x) X is not less than a and not more than b or x is psi₁(y)，x＝ψ₂And (y) c is less than or equal to y and less than or equal to d.

For the micro-product function P (x, y), when $P-D^{{-v}_1}{D}^{v_1}P\&NotEqual;0,$ Then $D^{v_1}{D}^{v_2}P={\mathrm{<figure-callout\; id="1"\; label="D\; v"\; filenames="HSA0000092593440000011.png"\; state="\{\{state\}\}">D</figure-callout>}}^{v_{}}P-{\mathrm{<figure-callout\; id="1"\; label="D\; v"\; filenames="HSA0000092593440000011.png"\; state="\{\{state\}\}">D</figure-callout>}}^{v_{}}(P-D^{{-v}_1}{D}^{v_1}P).$ As a result of this, the number of the, $I_x^{v_2}{I}_y^{v_2}{D}_y^{v_1}P(x,y)=I_x^{v_2}\{D_y^{v_1-v_2}P(x,y)-D_y^{v_1-v_2}[P(x,y)-D_y^{-v_1}{D}_y^{v_1}P(x,y)\left]\right\},$ the invention can be derived $\underset{\mathrm{\&Omega;}}{I_x^{v_2}{I}_y^{v_2}}{D}_y^{v_1}P(x,y)=I_{x{\mathrm{\&phi;}}_1\left(x\right)}^{v_2{\mathrm{\&phi;}}_2\left(x\right)}{}_a{}^b{I}_y^{v_2}{D}_y^{v_1}P(x,y)=I_x^{v_2}{}_a{}^b\{D_y^{v_1-v_2}P(x,y)-D_y^{v_1-v_2}[P(x,y)-D_y^{-v_1}{D}_y^{v_1}P(x,y)\left]\right\}{|}_{{\mathrm{\&phi;}}_1\left(x\right)}^{{\mathrm{\&phi;}}_2\left(x\right)}$ $=-\underset{C\left({\mathrm{BC}}_{2}A\right)}{I_x^{v_2}}\{D_y^{v_1-v_2}P(x,y)-D_y^{v_1-v_2}[P(x,y)-D_y^{-v_1}{D}_y^{v_1}P(x,y)\left]\right\}-\underset{C\left(A{C}_{1}B\right)}{I_x^{v_2}}\{D_y^{v_1-v_2}P(x,y)-D_y^{v_1-v_2}[P(x,y)-D_y^{{-v}_1}{D}_y^{v_1}P(x,y)\left]\right\}$ $=-\underset{C-}{I_x^{v_2}}\{D_y^{v_1-v_2}P(x,y)-D_y^{v_1-v_2}[P(x,y)-D_y^{-v_1}{D}_y^{v_1}P(x,y)\left]\right\}$

The same can be obtained $\underset{\mathrm{\&Omega;}}{I_x^{v_2}{I}_y^{v_2}}{D}_x^{v_1}Q(x,y)=\underset{C-}{I_y^{v_2}}\{D_x^{v_1-v_2}Q(x,y)-D_x^{v_1-v_2}[Q(x,y)-D_x^{-v_1}{D}_x^{v_1}Q(x,y)\left]\right\}.$ From the above two formulas, the present invention can derive a fractional order Green formula as $\underset{\mathrm{\&Omega;}}{I_x^{v_2}{I}_y^{v_2}}(D_x^{v_1}Q(x,y)-D_y^{v_1}P(x,y))=\underset{C-}{I_x^{v_2}}\{D_y^{v_1-v_2}P(x,y)-D_y^{v_1-v_2}[P(x,y)-D_y^{-v_1}{D}_y^{v_1}P(x,y)\left]\right\}+\underset{C-}{I_y^{v_2}}\{D_x^{v_1-v_2}Q(x,y)-D_x^{v_1-v_2}[Q(x,y)-D_x^{-v_1}{D}_x^{v_1}Q(x,y)\left]\right\}.$ In particular when

And

when the conditions are mutually inverse (the conditions are high and generally difficult to meet), i.e.

Then

The simplified table of the fractional order Green formula obtained by the above formula under specific conditions is shown as $\underset{\mathrm{\&Omega;}}{I_x^{v_2}{I}_y^{v_2}}(D_x^{v_1}Q(x,y)-D_y^{v_1}P(x,y))=\underset{C-}{I_x^{v_2}}{D}_y^{v_1-v_2}P(x,y)+\underset{C-}{I_y^{v_2}}{D}_x^{v_1-v_2}Q(x,y).$ As can be seen from the above formula, when v₁＝v₂When 1, it can be derived $\underset{\mathrm{\&Omega;}}{I_x^1{\mathrm{<figure-callout\; id="1"\; label="I\; y"\; filenames="HSA0000092593440000011.png"\; state="\{\{state\}\}">I</figure-callout>}}_y^{}}(D_x^{1}Q(x,y)-D_y^{1}P(x,y))=\underset{C-}{I_x^1}P(x,y)+\underset{C-}{I_y^1}Q(x,y),$ The traditional integer order Green formula is only a special case of a fractional order Green formula; when v is₁＝v₂When v, it can be derived

This is a fractional order Green formula that is widely applicable.

Based on the derived fractional order Green formula, the invention can further derive a fractional order Euler-Lagrange equation for two-dimensional image processing.

Referring to FIG. 3, the present invention makes the two-dimensional space have a differentiable product quantity function of u (x, y) and a differentiable product vector function ofFractional order differential operator of order v $D^v=i\frac{{\&PartialD;}^v}{{\&PartialD;x}^v}+j\frac{{\&PartialD;}^v}{{\&PartialD;y}^v}=i{D}_x^v+j{D}_y^v=(D_x^v,D_y^v),$ D^vIs a linear operator (when v is 0, D⁰Representing neither differentiation nor integration, is an identity operator), where i and j represent unit vectors in the x and y directions, respectively. In general, the two-dimensional image region Ω is a rectangular single-connected region, and thus the piecewise smooth boundary c of Ω is a closed rectangular curve.

Is obtained from the fractional order integral Riemann-Liouville definition

$I_y^{v}s(x,y)=\frac{1}{\mathrm{\&Gamma;}\left(v\right)}\underset{a_y}{\overset{y}{\&Integral;}}{(y-\mathrm{\&xi;})}^{v-1}s(x,\mathrm{\&xi;})\mathrm{d\&xi;}$ And is $I_x^v{I}_y^{v}s(x,y)=\frac{1}{{\mathrm{\&Gamma;}}^2\left(v\right)}\underset{a_x}{\overset{x}{\&Integral;}}\underset{a_y}{\overset{y}{\&Integral;}}{(x-\mathrm{\&eta;})}^{v-1}{(y-\mathrm{\&xi;})}^{v-1}s(\mathrm{\&eta;},\mathrm{\&xi;})\mathrm{d\&eta;d\&xi;}.$ So that the above fractional order Green formula can be deduced

This is always true. Due to the fact that $\underset{m=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\underset{n=0}{\overset{m}{\mathrm{\&Sigma;}}}\&equiv;\underset{n=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\underset{m=n}{\overset{\&infin;}{\mathrm{\&Sigma;}}}$ And is $\left(\begin{array}{c}v\\ r+n\end{array}\right)\left(\begin{array}{c}r+n\\ n\end{array}\right)\&equiv;\left(\begin{array}{c}v\\ n\end{array}\right)\left(\begin{array}{c}v-n\\ r\end{array}\right),$ The invention can then be derivedFrom the homogeneous nature of fractional calculus

Thus, the present invention can be derived

In the above, the symbol · represents an inner product. And fractional divergence operator

The invention makes the v-order fractional order differential operator $P^v=\underset{n=1}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\left(\begin{array}{c}v\\ n\end{array}\right)[i\frac{\left(D_x^{v-n}u\right)}{u}{D}_x^n+j\frac{\left(D_y^{v-n}u\right)}{u}{D}_y^n],$ v-order fractional order divergence operator

divP^vAnd P^vAre all linear operators, so the invention can be derived according to Hilbert-adjoint operator theory

Wherein,denotes v₂Integral form of fractional order inner product of order, (D)^v)^*Represents D^vThe v fractional order hilbert companion operator of (a). The present invention can provide (D) from the above formula^v)^*＝-divP^v. As can be seen from the above formula, the fractional order Hilbert syndrome (D)^v)^*Is a linear operator. When v is₁＝v₂When 1, the invention can be derived

Wherein, the¹An integral form of the first order inner product is represented,

represents the first-order divergence operator, (D)¹)^*Represents D¹First order hilbert adjoint operator. For digital images, the invention can derive (D)¹)^*＝-div¹. Thus, it is understood that the first order hilbert operator is only a special case of the fractional order hilbert operator. When in use

Then, then obtain

In the above formula, due to the orthogonality in the x-direction and the y-direction,

for any two-dimensional quantity function u (test function), corresponding

And

having randomness, the principle of variation is that the above formula is only needed to be established

Since n is a positive integer of 1 → ∞ to be used

Only need to makeThe equivalence of the above formula is

Then, if and only if the above formula holds

This is true.

That is to

The corresponding fractional order Euler-Lagrange equation.

Further, if

Is a function of a vector

A quantity function of;

is a micro-product vector functionAs a function of the quantity of (c). In the same way, when

Then, the present invention can be obtained

Then, if and only if the above formula holds

This is true.

That is to

The corresponding fractional order Euler-Lagrange equation.

Since fractional calculus exists for all v

The reason for (1) is that the fractional Euler-Lagrange equation and the fractional surface integral

Integral order v of₂Irrelevant, therefore, the invention does not adopt fractional order area when constructing the energy functional of the fractional order partial differential equation model suitable for denoising the texture imageIs divided intoBut only one-order integrationIn the form of (1).

As described in the following, the invention adopts a first-order extreme value to construct an energy functional based on fractional order total variation, thereby constructing a texture image denoising filter based on a fractional order development equation. The invention makes the curved family s + (beta-1) xi containing small parameter beta, when beta is 1, it is v₃An order extremum surface s. First, the invention order ${\mathrm{\&Psi;}}_1\left(\mathrm{\&beta;}\right)=\underset{\mathrm{\&Omega;}}{I_x^1{\mathrm{<figure-callout\; id="1"\; label="I\; y"\; filenames="HSA0000092593440000011.png"\; state="\{\{state\}\}">I</figure-callout>}}_y^{}}\left[f\left({\left|\right|\stackrel{\mathrm{\&rho;}}{\mathrm{\&phi;}}\left|\right|}^{v_2}\right)\right]=\underset{\mathrm{\&Omega;}}{\&Integral;\&Integral;}f\left({\left|\right|\stackrel{\mathrm{\&rho;}}{\mathrm{\&phi;}}\left|\right|}^{v_2}\right)\mathrm{dxdy}.$ Meanwhile, the present invention orders the vector $\stackrel{\&RightArrow;}{\mathrm{\&phi;}}[D^{v_1}s,(\mathrm{\&beta;}-1){\mathrm{<figure-callout\; id="1"\; label="D\; v"\; filenames="HSA0000092593440000011.png"\; state="\{\{state\}\}">D</figure-callout>}}^{v_{}}\mathrm{\&xi;}]=(\mathrm{\&beta;}-1){\mathrm{<figure-callout\; id="1"\; label="D\; v"\; filenames="HSA0000092593440000011.png"\; state="\{\{state\}\}">D</figure-callout>}}^{v_{}}\mathrm{\&xi;}-\frac{2\mathrm{\&Gamma;}(2-v_3)}{\mathrm{\&Gamma;}(3-v_3)}{D}^{v_1}s,$ And defining a vector

Norm of $\left|\right|\stackrel{\&RightArrow;}{\mathrm{\&phi;}}\left|\right|=\sqrt{\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}{\left(D^{v_1}s\right)}^2-\frac{4(\mathrm{\&beta;}-1)\mathrm{\&Gamma;}(2-v_3)}{\mathrm{\&Gamma;}(3-v_3)}{D}^{v_1}s\&CenterDot;{\mathrm{<figure-callout\; id="1"\; label="D\; v"\; filenames="HSA0000092593440000011.png"\; state="\{\{state\}\}">D</figure-callout>}}^{v_{}}\mathrm{\&xi;}+{(\mathrm{\&beta;}-1)}^2{\left({\mathrm{<figure-callout\; id="1"\; label="D\; v"\; filenames="HSA0000092593440000011.png"\; state="\{\{state\}\}">D</figure-callout>}}^{v_{}}\mathrm{\&xi;}\right)}^2}=\sqrt{{\left(\stackrel{\&RightArrow;}{\mathrm{\&phi;}}\right)}^2+C},$ Wherein $C=\frac{4\mathrm{\&Gamma;}(2-v_3)\mathrm{\&Gamma;}(3-v_3)-{\mathrm{\&Gamma;}}^2(3-v_3)-4{\mathrm{\&Gamma;}}^2(2-v_3)}{{\mathrm{\&Gamma;}}^2(3-v_3)}{\left(D^{v_1}s\right)}^2.$ In the above, the symbol · represents an inner product. For vectors

And

in the case of a non-woven fabric, ${\left(\stackrel{\&RightArrow;}{\mathrm{\&phi;}}\right)}^2={\left|\stackrel{\&RightArrow;}{\mathrm{\&phi;}}\right|}^2={\left(\sqrt{{\left(\stackrel{\&RightArrow;}{\mathrm{\&phi;}}\right)}^2}\right)}^2=\stackrel{\&RightArrow;}{\mathrm{\&phi;}}\&CenterDot;\stackrel{\&RightArrow;}{\mathrm{\&phi;}},$ ${\left(D^{v_1}s\right)}^2={\left|D^{v_1}s\right|}^2={\left(\sqrt{{\left(D^{v_1}s\right)}^2}\right)}^2=D^{v_1}s\&CenterDot;D^{v_1}s$ and is ${\left({\mathrm{<figure-callout\; id="1"\; label="D\; v"\; filenames="HSA0000092593440000011.png"\; state="\{\{state\}\}">D</figure-callout>}}^{v_{}}\mathrm{\&xi;}\right)}^2={\left|{\mathrm{<figure-callout\; id="1"\; label="D\; v"\; filenames="HSA0000092593440000011.png"\; state="\{\{state\}\}">D</figure-callout>}}^{v_{}}\mathrm{\&xi;}\right|}^2={\left(\sqrt{{\left(D^{{\mathrm{<figure-callout\; id="1"\; label="v"\; filenames="HSA0000092593440000011.png"\; state="\{\{state\}\}">v</figure-callout>}}_1}\mathrm{\&xi;}\right)}^2}\right)}^2={\mathrm{<figure-callout\; id="1"\; label="D\; v"\; filenames="HSA0000092593440000011.png"\; state="\{\{state\}\}">D</figure-callout>}}^{v_{}}\mathrm{\&xi;}\&CenterDot;{\mathrm{<figure-callout\; id="1"\; label="D\; v"\; filenames="HSA0000092593440000011.png"\; state="\{\{state\}\}">D</figure-callout>}}^{v_{}}\mathrm{\&xi;}.$ Further, the invention makes

Solving for Ψ₂The process of minimizing (β) is the process of solving for the minimum similarity between noisy and noise-free signals. Ψ₂And (beta) plays a role in nonlinear fidelity in the denoising process, and lambda is a regularization parameter. A fractional order energy functional based on fractional order total variation on the family of surfaces s + (beta-1) xi is then defined as $\mathrm{\&Psi;}\left(\mathrm{\&beta;}\right)={\mathrm{\&Psi;}}_1\left(\mathrm{\&beta;}\right)+{\mathrm{\&Psi;}}_2\left(\mathrm{\&beta;}\right)=I_x^1\underset{\mathrm{\&Omega;}}{{\mathrm{<figure-callout\; id="1"\; label="I\; y"\; filenames="HSA0000092593440000011.png"\; state="\{\{state\}\}">I</figure-callout>}}_y^{}}[f\left({\left|\right|\stackrel{\mathrm{\&rho;}}{\mathrm{\&phi;}}\left|\right|}^{v_2}\right)+\mathrm{\&lambda;}[(s+(\mathrm{\&beta;}-1)\mathrm{\&xi;}-s_0\left](s-s_0)\right]=\underset{\mathrm{\&Omega;}}{\&Integral;\&Integral;}[f\left({\left|\right|\stackrel{\mathrm{\&rho;}}{\mathrm{\&phi;}}\left|\right|}^{v_2}\right)+\mathrm{\&lambda;}[s+(\mathrm{\&beta;}-1)\mathrm{\&xi;}-s_0\left](s-s_0)\right]\mathrm{dxdy}.$ Thus, can be derived $D_{\mathrm{\&beta;}}^{{\mathrm{<figure-callout\; id="3"\; label="v"\; filenames="HSA0000092593440000011.png,HSA0000092593440000041.png"\; state="\{\{state\}\}">v</figure-callout>}}_3}{\mathrm{\&Psi;}}_1\left(\mathrm{\&beta;}\right){|}_{\mathrm{\&beta;}=1}=\frac{{\&PartialD;}^{{\mathrm{<figure-callout\; id="3"\; label="v"\; filenames="HSA0000092593440000011.png,HSA0000092593440000041.png"\; state="\{\{state\}\}">v</figure-callout>}}_3}}{\&PartialD;{\mathrm{\&beta;}}^{{\mathrm{<figure-callout\; id="3"\; label="v"\; filenames="HSA0000092593440000011.png,HSA0000092593440000041.png"\; state="\{\{state\}\}">v</figure-callout>}}_3}}\underset{\mathrm{\&Omega;}}{\&Integral;\&Integral;}f\left({\left|\right|\stackrel{\&RightArrow;}{\mathrm{\&phi;}}\left|\right|}^{v_2}\right)\mathrm{dxdy}{|}_{\mathrm{\&beta;}=1}=\underset{\mathrm{\&Omega;}}{\&Integral;\&Integral;}\frac{{\&PartialD;}^{{\mathrm{<figure-callout\; id="3"\; label="v"\; filenames="HSA0000092593440000011.png,HSA0000092593440000041.png"\; state="\{\{state\}\}">v</figure-callout>}}_3}}{\&PartialD;{\mathrm{\&beta;}}^{v_3}}f\left({\left|\right|\stackrel{\&RightArrow;}{\mathrm{\&phi;}}\left|\right|}^{v_2}\right)\mathrm{dxdy}{|}_{\mathrm{\&beta;}=1}=0.$ Unlike the conventional first order variational method based on first order extrema, the above formula is psi₁(β) v with respect to β₃The order fractional order extremum aims to utilize the special properties of the image fractional order calculus (not only can reserve low-frequency contour characteristics in a smooth region of an image as much as possible, but also can reserve high-frequency edge characteristics with relatively large gray value jump amplitude in a fractional order and multi-scale enhanced image, and can reserve complex texture detail characteristics of the texture image in a fractional order and multi-scale enhanced image in a non-linear way as much as possible while denoising the texture image. Further, can be derived $\underset{\mathrm{\&Omega;}}{\&Integral;\&Integral;}\frac{{\left[\sqrt{\left|\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}\right|}\right]}^{{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="HSA0000092593440000011.png,HSA0000092593440000021.png"\; state="\{\{state\}\}">v</figure-callout>}}_2}{\left|D^{v_1}s\right|}^{{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="HSA0000092593440000011.png,HSA0000092593440000021.png"\; state="\{\{state\}\}">v</figure-callout>}}_2}}{\mathrm{\&Gamma;}(1-v_3)}+\underset{n=1}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\frac{{(-1)}^n\mathrm{\&Gamma;}(n-v_3)}{\mathrm{\&Gamma;}(-v_3)\mathrm{\&Gamma;}(n-{\mathrm{<figure-callout\; id="3"\; label="v"\; filenames="HSA0000092593440000011.png,HSA0000092593440000041.png"\; state="\{\{state\}\}">v</figure-callout>}}_3+1)}\frac{D_{\mathrm{\&beta;}}^n\left({\left|\right|\stackrel{\&RightArrow;}{\mathrm{\&phi;}}\left|\right|}^{v_2}\right){|}_{\mathrm{\&beta;}=1}}{n!}\mathrm{dxdy}=0.$ Therein, in order to ensureAlways is positive so that

It is significant that the invention takes the absolute value of the value

When n is an odd number (n ═ 2k +1, k ═ 0, 1, 2, 3 Λ) and when n is an even number (n ═ 2k, k ═ 1, 2, 3 Λ),

with different expressions. Thus, the invention can be derived separately $D_{\mathrm{\&beta;}}^n\left({\left|\right|\stackrel{\&RightArrow;}{\mathrm{\&phi;}}\left|\right|}^{v_2}\right){|}_{\mathrm{\&beta;}=1}^{n=2k+1}=\frac{-2\mathrm{\&Gamma;}(2-v_3)\underset{\mathrm{\&tau;}=1}{\overset{n}{\mathrm{\&Pi;}}}(v_2-\mathrm{\&tau;}+1)}{\mathrm{\&Gamma;}(3-v_3)}{\left[\sqrt{\left|\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}\right|}\right]}^{v_2-n-1}{\left|D^{v_1}s\right|}^{v_2-n-1}{\left|{\mathrm{<figure-callout\; id="1"\; label="D\; v"\; filenames="HSA0000092593440000011.png"\; state="\{\{state\}\}">D</figure-callout>}}^{v_{}}\mathrm{\&xi;}\right|}^{n-1}\left({\mathrm{<figure-callout\; id="1"\; label="D\; v"\; filenames="HSA0000092593440000011.png"\; state="\{\{state\}\}">D</figure-callout>}}^{v_{}}\mathrm{\&xi;}\right)\&CenterDot;\left(D^{v_1}s\right)$ And $D_{\mathrm{\&beta;}}^n\left({\left|\right|\stackrel{\&RightArrow;}{\mathrm{\&phi;}}\left|\right|}^{v_2}\right){|}_{\mathrm{\&beta;}=1}^{n=2k}=\underset{\mathrm{\&tau;}=1}{\overset{n}{\mathrm{\&Pi;}}}(v_2-\mathrm{\&tau;}+1){\left[\sqrt{\left|\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}\right|}\right]}^{v_2-n}{\left|D^{v_1}s\right|}^{v_2-n}{\left|{\mathrm{<figure-callout\; id="1"\; label="D\; v"\; filenames="HSA0000092593440000011.png"\; state="\{\{state\}\}">D</figure-callout>}}^{v_{}}\mathrm{\&xi;}\right|}^n.$ further, the present invention can be derived $\underset{\mathrm{\&Omega;}}{\&Integral;\&Integral;}\underset{k=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}(v_2-\mathrm{\&tau;}+1){\left[\sqrt{\left|\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}\right|}\right|D^{v_1}s\left|\right]}^{v_2-2k-2}{\left|{\mathrm{<figure-callout\; id="1"\; label="D\; v"\; filenames="HSA0000092593440000011.png"\; state="\{\{state\}\}">D</figure-callout>}}^{v_{}}\mathrm{\&xi;}\right|}^{2k}}{\mathrm{\&Gamma;}(-v_3)\mathrm{\&Gamma;}(3-v_3)\left(2k\right)!}\left(D^{v_1}s\right)\&CenterDot;{\mathrm{<figure-callout\; id="1"\; label="D\; v"\; filenames="HSA0000092593440000011.png"\; state="\{\{state\}\}">D</figure-callout>}}^{v_{}}\left[\begin{array}{c}\frac{\mathrm{\&Gamma;}(2k-v_3)[4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)]}{\mathrm{\&Gamma;}(2k-{\mathrm{<figure-callout\; id="3"\; label="v"\; filenames="HSA0000092593440000011.png,HSA0000092593440000041.png"\; state="\{\{state\}\}">v</figure-callout>}}_3+1)}\text{s}\\ +\frac{2(v_2-2k)\mathrm{\&Gamma;}(2k-{\mathrm{<figure-callout\; id="3"\; label="v"\; filenames="HSA0000092593440000011.png,HSA0000092593440000041.png"\; state="\{\{state\}\}">v</figure-callout>}}_3+1)\mathrm{\&Gamma;}(2-v_3)}{(2k+1)\mathrm{\&Gamma;}(2k-{\mathrm{<figure-callout\; id="3"\; label="v"\; filenames="HSA0000092593440000011.png,HSA0000092593440000041.png"\; state="\{\{state\}\}">v</figure-callout>}}_3+2)}\mathrm{\&xi;}\end{array}\right]\mathrm{dxdy}=0.$ Wherein, the invention is as followsFor renFor two-dimensional quantity function xi (test function), corresponding

Has randomness, thus $D^{v_1}[\frac{\mathrm{\&Gamma;}(2k-v_3)[4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)]}{\mathrm{\&Gamma;}(2k-{\mathrm{<figure-callout\; id="3"\; label="v"\; filenames="HSA0000092593440000011.png,HSA0000092593440000041.png"\; state="\{\{state\}\}">v</figure-callout>}}_3+1)}s+\frac{2(v_2-2k)\mathrm{\&Gamma;}(2k-{\mathrm{<figure-callout\; id="3"\; label="v"\; filenames="HSA0000092593440000011.png,HSA0000092593440000041.png"\; state="\{\{state\}\}">v</figure-callout>}}_3+1)\mathrm{\&Gamma;}(2-v_3)}{(2k+1)\mathrm{\&Gamma;}(2k-{\mathrm{<figure-callout\; id="3"\; label="v"\; filenames="HSA0000092593440000011.png,HSA0000092593440000041.png"\; state="\{\{state\}\}">v</figure-callout>}}_3+2)}\mathrm{\&xi;}]$ It also has randomness. Thus, the present invention can derive the corresponding fractional order pan Euler-Lagrange equation as $\frac{\mathrm{\&Gamma;}(1-v_1)}{\mathrm{\&Gamma;}(-v_1)\mathrm{\&Gamma;}(-v_3)\mathrm{\&Gamma;}(3-v_3)}\underset{k=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}(v_2-\mathrm{\&tau;}+1){\left[\sqrt{\left|\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}\right|}\right]}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left({\left|D^{v_1}s\right|}^{v_2-2k-2}{D}_x^{v_1}s\right)\\ +D_y^1\left({\left|D^{v_1}s\right|}^{v_2-2k-2}{D}_y^{v_1}s\right)\end{array}\right]=0.$ Wherein,

thus, the present invention takes v ═ v₃Not equal to 1, 2, 3, derivable $\frac{{\&PartialD;}^{v_3}s}{{\&PartialD;\mathrm{<figure-callout\; id="3"\; label="t\; v"\; filenames="HSA0000092593440000011.png,HSA0000092593440000041.png"\; state="\{\{state\}\}">t</figure-callout>}}^{v_{}}}=\frac{-\mathrm{\&Gamma;}(1-v_1)}{\mathrm{\&Gamma;}(-v_1)\mathrm{\&Gamma;}(-v_3)\mathrm{\&Gamma;}(3-v_3)}\underset{k=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}(v_2-\mathrm{\&tau;}+1){\left[\sqrt{\left|\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}\right|}\right]}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left({\left|D^{v_1}s\right|}^{v_2-2k-2}{D}_x^{v_1}s\right)\\ +D_y^1\left({\left|D^{v_1}s\right|}^{v_2-2k-2}{D}_y^{v_1}s\right)\end{array}\right]\frac{\mathrm{\&lambda;}{s}_0}{\mathrm{\&Gamma;}(1-v_3)\mathrm{\&Gamma;}(2-v_3)}.$ Wherein,

is realized according to a fractional order difference method and only converges to a unique v₃Order fraction order extreme points. Thus, the present invention can be derived $\mathrm{\&lambda;}\left(t\right)=\frac{-\mathrm{\&Gamma;}(1-v_1)\mathrm{\&Gamma;}(1-v_3)\mathrm{\&Gamma;}(2-v_3)}{{\mathrm{\&sigma;}}^2\mathrm{\&Gamma;}(-v_1)\mathrm{\&Gamma;}(-v_3)\mathrm{\&Gamma;}(3-v_3){\mathrm{<figure-callout\; id="0"\; label="s"\; filenames="HSA0000092593440000021.png,HSA0000092593440000022.png"\; state="\{\{state\}\}">s</figure-callout>}}_0}\underset{\mathrm{\&Omega;}}{\&Integral;\&Integral;}\underset{k=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}(v_2-\mathrm{\&tau;}+1){\left[\sqrt{\left|\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}\right|}\right]}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left({\left|D^{v_1}s\right|}^{v_2-2k-2}{D}_x^{v_1}s\right)\\ +D_y^1\left({\left|D^{v_1}s\right|}^{v_2-2k-2}{D}_y^{v_1}s\right)\end{array}\right]{(s-s_0)}^2\mathrm{dxdy}.$ Formula (II) $\frac{{\&PartialD;}^{v_3}s}{\&PartialD;{\mathrm{<figure-callout\; id="3"\; label="t\; v"\; filenames="HSA0000092593440000011.png,HSA0000092593440000041.png"\; state="\{\{state\}\}">t</figure-callout>}}^{v_{}}}=\frac{-\mathrm{\&Gamma;}(1-v_1)}{\mathrm{\&Gamma;}(-v_1)\mathrm{\&Gamma;}(-v_3)\mathrm{\&Gamma;}(3-v_3)}\underset{k=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}(v_2-\mathrm{\&tau;}+1){\left[\sqrt{\left|\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}\right|}\right]}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left({\left|D^{v_1}s\right|}^{v_2-2k-2}{D}_x^{v_1}s\right)\\ +D_y^1\left({{\left|D^{v_1}s\right|}^{v_2-2k-2}}^{}{D}_y^{v_1}s\right)\end{array}\right]\frac{\mathrm{\&lambda;}{s}_0}{\mathrm{\&Gamma;}(1-v_3)\mathrm{\&Gamma;}(2-v_3)}$ And $\mathrm{\&lambda;}\left(t\right)=\frac{-\mathrm{\&Gamma;}(1-v_1)\mathrm{\&Gamma;}(1-v_3)\mathrm{\&Gamma;}(2-v_3)}{{\mathrm{\&sigma;}}^2\mathrm{\&Gamma;}(-v_1)\mathrm{\&Gamma;}(-v_3)\mathrm{\&Gamma;}(3-v_3){\mathrm{<figure-callout\; id="0"\; label="s"\; filenames="HSA0000092593440000021.png,HSA0000092593440000022.png"\; state="\{\{state\}\}">s</figure-callout>}}_0}\underset{\mathrm{\&Omega;}}{\&Integral;\&Integral;}\underset{k=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}(v_2-\mathrm{\&tau;}+1){\left[\sqrt{\left|\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}\right|}\right]}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left({\left|D^{v_1}s\right|}^{v_2-2k-2}{D}_x^{v_1}s\right)\\ +D_y^1\left({\left|D^{v_1}s\right|}^{v_2-2k-2}{D}_y^{v_1}s\right)\end{array}\right]{(s-s_0)}^2\mathrm{dxdy}$ the represented fractional order development equation denoising model is the mathematical model of the texture image denoising filter based on the fractional order development equation.

In order to enable the fractional order development equation denoising model to completely filter residual weak noise in the very low frequency and direct current parts of the signal, the invention also needs to perform low-pass filtering on the very low frequency and direct current parts of the signal simultaneously when the numerical iteration is realized. In addition, the fractional order equation denoising model is added into molecules

And increasing gamma (3-v) in the denominator₃) Is greatly enhanced in the form of fractional order differential order v₃And (3) nonlinear adjustment effect on denoising. In addition, when v is₃When the integral equation is equal to 0, the denoising model of the fractional order development equation is a traditional potential equation (elliptic equation); when v is₃When the integral equation is 1, the fractional order development equation denoising model is a traditional heat conduction equation (parabolic equation); when v is₃When the integral equation is 2, the denoising model of the fractional order development equation is a traditional wave equation (hyperbolic equation); when 0 < v₃When the frequency is less than 1, the fractional order development equation denoising model is continuous interpolation between the traditional potential equation and the traditional heat conduction equation; when 1 < v₃When the frequency is less than 2, the fractional order development equation denoising model is the continuous interpolation between the traditional heat conduction equation and the wave equation. Therefore, in mathematical and physical meanings, the fractional order development equation denoising model can be used for popularizing the traditional partial differential equation-based image processing from the traditional anisotropic diffusion-based general basic processing method of the heat conduction equation to a wider field.

Further, the present invention requires numerical values to implement the above mathematical model of the texture image denoising filter based on the fractional order evolution equation. First, the present invention requires numerical values to achieve fractional order differentiation of two-dimensional digital images in the x-axis and y-axis directions. For the Grumwald-Letnikov definition of fractional calculus, the limit symbols can be removed when N is sufficiently large. In order to improve convergence speed and convergence accuracy, the invention introduces signal s (x) at non-node in Grumwald-Letnikov definition formula, namely signal value $\frac{d^v}{{\mathrm{dx}}^v}s\left(x\right){|}_{G-L}\&cong;\frac{x^{-v}{N}^v}{\mathrm{\&Gamma;}(-v)}\underset{k=0}{\overset{N-1}{\mathrm{\&Sigma;}}}\frac{\mathrm{\&Gamma;}(k-v)}{\mathrm{\&Gamma;}(k+1)}s(x+\frac{\mathrm{vx}}{2N}-\frac{\mathrm{kx}}{N}).$ And when v ≠ 1, performing fractional interpolation on the signal s (x) by applying a Lagrange three-point interpolation formula, and respectively constructing fractional differential operators of the digital image in the directions of the x axis and the y axis. For the digital gray level image, the numerical operation rule of the fractional order differential operator adopts a spatial filtering scheme of operator convolution. The fractional order partial differential value with the maximum modulus values in the x-axis direction and the y-axis direction is selected as the fractional order differential value of the pixel point. Second, the present invention requires numerical values to achieve 1 st order differentiation of two-dimensional digital images in the x-axis and y-axis directions. In order to maintain the stability of numerical calculation, the invention adopts $D_x^{1}s(x,y)=\frac{2[s(x+1,y)-s(x-1,y)]+s(x+1,y+1)-s(x-1,y+1)+s(x+1,y-1)-s(x-1,y-1)}{4}$ And $D_y^{1}s(x,y)=\frac{2[s(x,y+1)-s(x,y-1)]+s(x+1,y+1)-s(x+1,y-1)+s(x-1,y+1)-s(x-1,y-1)}{4}$ to approximate a first order differential. Third, the present invention requires numerical values to achieve fractional order differentiation of the two-dimensional digital image with respect to time t. If the time equal division interval is delta t, namely the unit iteration time interval, n is t_n＝nΔt，n＝0，1，Λ(t ₀0 denotes the initial time). The invention takes the unit iteration time interval delta t to be (0, 0.1)]Any small positive real number is interpolated. The digital image at time n is then

The original image to be denoised is

s₀Is an ideal noise-free image, which is a constant value, so s₀(x，y，t₀)＝s₀(x，y，t_n). Thus, the present invention can approximate the fractional order differential of a two-dimensional digital image over time t as $\frac{{\&PartialD;}^{v}s}{{\&PartialD;t}^v}={\mathrm{\&Delta;t}}^{-v}[s_{x,y}^{n+1}-s_{x,y}^n+\frac{2\mathrm{\&mu;\&eta;}}{\mathrm{\&Gamma;}(3-v)}{(s_{x,y}^n-s_{x,y}^{v*})}^2{\left(s_{x,y}^n\right)}^{-v}],$ while v ≠ 1. Wherein,

is the best denoised image. In addition, due to the ideal noiseless image s₀(x，y，t₀) Intermediate results of denoising with each numerical iteration, not known in advanceAre all for an ideal noiseless image s₀(x，y，t₀) By a first approximation, i.e.

So as to approach s-s as much as possible during numerical iteration₀The invention is to

Due to the best de-noising image

Intermediate results of denoising with each numerical iteration, not known in advance

Are all pairsA first approximation of, i.e.

So as to approach as much as possible during numerical iteration

The invention makes ${(s_{x,y}^n-s_{x,y}^{v*})}^2\&cong;{(s_{x,y}^{n-1}-s_{x,y}^n)}^2.$ Meanwhile, in order to simplify the calculation, the invention takes mu as 0.005 and eta as 1, and takes k as 0, 1 to perform approximate calculation. Thus, the formula can be derived $\frac{{\&PartialD;}^{v_3}s}{{\&PartialD;\mathrm{<figure-callout\; id="3"\; label="t\; v"\; filenames="HSA0000092593440000011.png,HSA0000092593440000041.png"\; state="\{\{state\}\}">t</figure-callout>}}^{v_{}}}=\frac{-\mathrm{\&Gamma;}(1-v_1)}{\mathrm{\&Gamma;}(-v_1)\mathrm{\&Gamma;}\left({-v}_3\right)\mathrm{\&Gamma;}(3-v_3)}\underset{k=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}(v_2-\mathrm{\&tau;}+1){\left[\sqrt{\left|\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}\right|}\right]}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left(\right|D^{y_1}s{|}^{v_2-2k-2}{D}_x^{y_1}s)\\ +D_y^1\left(\right|D^{y_1}s{|}^{v_2-2k-2}{D}_y^{y_1}s)\end{array}\right]\frac{{\mathrm{\&lambda;s}}_0}{\mathrm{\&Gamma;}(1-v_3)\mathrm{\&Gamma;}(2-v_3)}$ And $\mathrm{\&lambda;}\left(t\right)\frac{-\mathrm{\&Gamma;}(1-v_1)\mathrm{\&Gamma;}(1-v_3)\mathrm{\&Gamma;}(2-v_3)}{{\mathrm{\&sigma;}}^2\mathrm{\&Gamma;}\left({-v}_1\right)\mathrm{\&Gamma;}\left({-v}_3\right)\mathrm{\&Gamma;}(3-v_3){\mathrm{<figure-callout\; id="0"\; label="s"\; filenames="HSA0000092593440000021.png,HSA0000092593440000022.png"\; state="\{\{state\}\}">s</figure-callout>}}_0}\underset{\mathrm{\&Omega;}}{\&Integral;\&Integral;}\underset{k=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}(v_2-\mathrm{\&tau;}+1){\left[\sqrt{\left|\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}\right|}\right]}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left({\left|D^{y_1}s\right|}^{v_2-2k-2}{D}_x^{y_1}s\right)\\ +D_y^1\left({\left|D^{y_1}s\right|}^{v_2-2k-2}{D}_y^{y_1}s\right)\end{array}\right]{(s-s_0)}^2\mathrm{dxdy}$ are respectively as follows $s_{x,y}^{n+1}=Q\left(s_{x,y}^n\right){\mathrm{\&Delta;t}}^{v_3}-\frac{{\mathrm{\&lambda;}}^n{\mathrm{\&Delta;<figure-callout\; id="3"\; label="t\; v"\; filenames="HSA0000092593440000011.png,HSA0000092593440000041.png"\; state="\{\{state\}\}">t</figure-callout>}}^{v_{}}}{\mathrm{\&Gamma;}(1-v_3)\mathrm{\&Gamma;}(2-v_3)}{s}_{x,y}^n+s_{x,y}^n-\frac{2\mathrm{\&mu;}}{\mathrm{\&Gamma;}(3-v_3)}{(s_{x,y}^{n-1}-s_{x,y}^n)}^2{\left(s_{x,y}^n\right)}^{{-\mathrm{<figure-callout\; id="3"\; label="v"\; filenames="HSA0000092593440000011.png,HSA0000092593440000041.png"\; state="\{\{state\}\}">v</figure-callout>}}_3},$ v₃Not equal to 1, 2, 3 and ${\mathrm{\&lambda;}}^n=\frac{\mathrm{\&Gamma;}(1-v_3)\mathrm{\&Gamma;}(2-v_3)}{{\mathrm{\&sigma;}}^{n^2}{s}_{x,y}^n}\underset{x,y}{\mathrm{\&Sigma;}}Q\left(s_{x,y}^n\right){(s_{x,y}^0-s_{x,y}^n)}^2.$ wherein, $\underset{\mathrm{\&tau;}=1}{\overset{n}{\mathrm{\&Pi;}}}(v_2-\mathrm{\&tau;}+1)\stackrel{n=0}{=}1,$ $Q\left(s_{x,y}^n\right)=\frac{-\mathrm{\&Gamma;}(1-v_1)}{\mathrm{\&Gamma;}\left({-v}_1\right)\mathrm{\&Gamma;}\left({-v}_3\right)\mathrm{\&Gamma;}(3-v_3)}\underset{k=0}{\overset{1}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}(v_2-\mathrm{\&tau;}+1){\left[\sqrt{\left|\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}\right|}\right]}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n\right)\\ +D_y^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_y^{v_1}{s}_{x,y}^n\right)\end{array}\right].$ wherein,in the iterative computation of the values, on the one hand, the variance of the noise does not need to be known or estimated in advance, but only needs to be obtained when the values are iterated for the first time

A small positive number. The invention is to get

0.01. Will be provided with

Bringing in ${\mathrm{\&lambda;}}^n=\frac{\mathrm{\&Gamma;}(1-v_3)\mathrm{\&Gamma;}(2-v_3)}{{\mathrm{\&sigma;}}^{n^2}{s}_{x,y}^n}\underset{x,y}{\mathrm{\&Sigma;}}Q\left(s_{x,y}^n\right){(s_{x,y}^0-s_{x,y}^n)}^2$ To initiate a process of iterative computation of values, thus resulting from each iteration

Are all different, but eachAre all first-order approximations to the true variance of the noise; on the other hand, during the iterative calculation of the values, it may occurIn order to make

Is significant when

When the invention is used, the invention takesTo make it possible to

Is significant when

When the invention is used, the invention takes

Based on the above brief description of mathematical formula derivation and numerical operation rule of the texture image denoising filter based on the fractional order development equation provided by the present invention, the circuit structure of the filter is specifically described below:

referring to fig. 1, the texture image denoising filter based on the fractional order evolution equation provided by the invention is based on a special fractional order evolution equation denoising algorithm to realize fractional order, nonlinear, multi-scale and high-precision denoising of a texture image. Order v of fractional calculus₁、v₂And v₃Not the conventional integer order but a non-integer order, and generally takes a fraction or rational fraction in engineering applications, and v₃Not equal to 1, 2, 3. Referring to FIG. 1, the filter uses a buffer 2, a difference squarer-3, -v₃Power squarer 4, multiplier 5, lambdaⁿA generator 11,

The generator 10, the multiplier two 8, the multiplier three 9 and the adder one 6 are constructed in a cascade manner. The filter has the obvious advantages that while texture image noise is removed, low-frequency contours in a smooth area can be reserved as much as possible, high-frequency edges with relatively large gray value jump amplitude can be non-linearly reserved, and texture details with relatively small gray value jump amplitude change can be non-linearly reserved. The filter is particularly suitable for the application occasion of denoising the image rich in complex texture detail characteristics.

Referring to fig. 1, 1 is an input point of a texture image denoising filter based on a fractional order development equation of the present invention, i.e., a digital image of an nth iterationThe input point of (2). The buffer 2 is used for outputting an output image of the texture image denoising filter based on the fractional order development equation

Buffering two unit iteration time to obtain the digital image of the (n-1) th iteration

The E input point of the difference squarer-3 is the weight

The input point of (2). The calculation performed by the difference squarer-3 is

-v₃The power squarer 4 performs the calculation of

The F input point of the multiplier one 5 is the weight

The input point of (2). The multiplier one 5 performs the calculation of $-\frac{2\mathrm{\&mu;}}{\mathrm{\&Gamma;}(3-v_3)}{(s_{x,y}^{n-1}-s_{x,y}^n)}^2{\left(s_{x,y}^n\right)}^{{-\mathrm{<figure-callout\; id="3"\; label="v"\; filenames="HSA0000092593440000011.png,HSA0000092593440000041.png"\; state="\{\{state\}\}">v</figure-callout>}}_3}.$

The calculations performed by the generator 10 are $Q\left(s_{x,y}^n\right)=\frac{-\mathrm{\&Gamma;}(1-v_1)}{\mathrm{\&Gamma;}\left({-v}_1\right)\mathrm{\&Gamma;}\left({-v}_3\right)\mathrm{\&Gamma;}(3-v_3)}\underset{k=0}{\overset{1}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}(v_2-\mathrm{\&tau;}+1){\left[\sqrt{\left|\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}\right|}\right]}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n\right)\\ +D_y^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_y^{v_1}{s}_{x,y}^n\right)\end{array}\right].$ λⁿThe generator 11 performs the calculation that ${\mathrm{\&lambda;}}^n=\frac{\mathrm{\&Gamma;}(1-v_3)\mathrm{\&Gamma;}(2-v_3)}{{\mathrm{\&sigma;}}^{n^2}{s}_{x,y}^n}\underset{x,y}{\mathrm{\&Sigma;}}P\left(s_{x,y}^n\right){(s_{x,y}^0-s_{x,y}^n)}^2.$ The H input point of multiplier three 9 is the weight

The input point of (2). The multiplier three 9 performs the calculation of

The G input point of the second multiplier 8 is the weight

The input point of (2). The second multiplier 8 performs the calculation of

The adder-6 performs the calculation $s_{x,y}^{n+1}=Q\left(s_{x,y}^n\right){\mathrm{\&Delta;t}}^{v_3}-\frac{{\mathrm{\&lambda;}}^n{\mathrm{\&Delta;<figure-callout\; id="3"\; label="t\; v"\; filenames="HSA0000092593440000011.png,HSA0000092593440000041.png"\; state="\{\{state\}\}">t</figure-callout>}}^{v_{}}}{\mathrm{\&Gamma;}(1-v_3)\mathrm{\&Gamma;}(2-v_3)}{s}_{x,y}^n+s_{x,y}^n-\frac{2\mathrm{\&mu;}}{\mathrm{\&Gamma;}(3-v_3)}{(s_{x,y}^{n-1}-s_{x,y}^n)}^2{\left(s_{x,y}^n\right)}^{{-\mathrm{<figure-callout\; id="3"\; label="v"\; filenames="HSA0000092593440000011.png,HSA0000092593440000041.png"\; state="\{\{state\}\}">v</figure-callout>}}_3},$ v₃Not equal to 1, 2, 3. Wherein the invention requires an order v₃Satisfy v₃Not equal to 1, 2, 3. Texture based on fractional order development equationThe output point 7 of the image denoising filter completes the function of outputting a digital image of the (n + 1) th iteration

As can be seen in figure 4 of the drawings,

the calculations performed by the generator 10 are $Q\left(s_{x,y}^n\right)=\frac{-\mathrm{\&Gamma;}(1-v_1)}{\mathrm{\&Gamma;}\left({-v}_1\right)\mathrm{\&Gamma;}\left({-v}_3\right)\mathrm{\&Gamma;}(3-v_3)}\underset{k=0}{\overset{1}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}(v_2-\mathrm{\&tau;}+1){\left[\sqrt{\left|\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}\right|}\right]}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n\right)\\ +D_y^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_y^{v_1}{s}_{x,y}^n\right)\end{array}\right].$ The I input point of the second difference squarer 12 is the weight of the second difference squarer

The input point of (1), i.e. the original digital image of the 0 th iteration

The input point of (2). Calculation by the second difference squarer 12Is that

The generator 13 performs the calculation that

The multiplier four 14 performs the calculation of

The multiplier five 15 performs the calculation of

The adder-16 performs the calculation

The J input point of the divider 17 is the weight Γ (1-v)₃)Γ(2-v₃) The input point of (2). The divider 17 performs the calculation

The multiplier six 18 performs the calculation of ${\mathrm{\&lambda;}}^n=\frac{\mathrm{\&Gamma;}(1-v_3)\mathrm{\&Gamma;}(2-v_3)}{{\mathrm{\&sigma;}}^{n^2}{s}_{x,y}^n}\underset{x,y}{\mathrm{\&Sigma;}}Q\left(s_{x,y}^n\right){(s_{x,y}^0-s_{x,y}^n)}^2.$ λⁿThe output point 19 of the generator performs the function of outputting lambdaⁿThe value is obtained.

As can be seen in figure 5 of the drawings,the calculation performed by the differentiator 20 is v in the direction of the x-axis₁Fractional order differentiation of order.

The differentiator 21 performs the calculation by performing v in the x-axis direction and the y-axis direction simultaneously₁Fractional order differentiation of order.

The calculation performed by the differentiator 22 is v in the y-direction₁Fractional order differentiation of order. v. of₂Computation by the-2 k-2 power squarer 23Is that

The seventh multiplier 24 performs the calculation ofThe multiplier eight 25 performs the calculation of

The calculation performed by the differentiator 26 is a 1 st order differential in the x-axis direction.

The calculation performed by the differentiator 27 is a 1 st order differential in the y-axis direction. The second adder 28 performs the calculation $[D_x^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n\right)+D_y^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_y^{v_1}{s}_{x,y}^n\right)].$ The K input points of the multiplier nine 32 are weight values $\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}(v_2-\mathrm{\&tau;}+1){\left[\sqrt{\left|\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}\right|}\right]}^{v_2-2k-2}}{\left(2k\right)!}$ The input point of (2). The calculation performed by the multiplier nine 32 is $\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}(v_2-\mathrm{\&tau;}+1){\left[\sqrt{\left|\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}\right|}\right]}^{v_2-2k-2}}{\left(2k\right)!}[D_x^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n\right)+D_y^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_y^{v_1}{s}_{x,y}^n\right)].$ The adder 31 performs the calculation $\underset{k=0}{\overset{1}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}(v_2-\mathrm{\&tau;}+1){\left[\sqrt{\left|\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}\right|}\right]}^{v_2-2k-2}}{\left(2k\right)!}[D_x^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n\right)+D_y^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_y^{v_1}{s}_{x,y}^n\right)].$ The L input point of the multiplier ten 30 is the weightThe input point of (2). The multiplier ten 30 performs the calculation of $\frac{-\mathrm{\&Gamma;}(1-v_1)}{\mathrm{\&Gamma;}\left({-v}_1\right)\mathrm{\&Gamma;}\left({-v}_3\right)\mathrm{\&Gamma;}(3-v_3)}\underset{k=0}{\overset{1}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}(v_2-\mathrm{\&tau;}+1){\left[\sqrt{\left|\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}\right|}\right]}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n\right)\\ +D_y^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_y^{v_1}{s}_{x,y}^n\right)\end{array}\right].$

The output point 29 of the generator performs the function of an output

The value is obtained.

See FIG. 6, where the M input points of the difference squarer three 33 are weights

The input point of (1), i.e. the original digital image of the 0 th iterationThe input point of (2). The third 33 difference squarer performs the calculation

The second adder 34 performs the calculation

The output point 35 of the generator performs the function of an output

The value is obtained.

The following describes a new scheme of a texture image denoising filter based on a fractional order development equation in detail with reference to the accompanying drawings and examples:

drawings

FIG. 1 is a schematic diagram of a texture image denoising filter based on a fractional order evolution equation according to the present invention.

Fig. 2 is a schematic diagram of a single-connectivity region Ω and its smooth boundary curve C.

Fig. 3 is a schematic diagram of a two-dimensional single-connected image region Ω and its piecewise-smooth boundary curve C.

FIG. 4 is λⁿSchematic diagram of the generator.

FIG. 5 is

Schematic diagram of the generator.

FIG. 6 is

Schematic diagram of the generator.

Wherein, 1 is an input point of the texture image denoising filter based on the fractional order development equation of the invention, namely the digital image of the nth iteration

The input point of (a); 2 is a buffer; 3 is a difference squarer I; 4 is-v₃A power squarer; 5 is a multiplier one; 6 is adder one; 7 is the output point of the texture image denoising filter based on the fractional order development equation, namely the digital image of the (n + 1) th iterationAn output point of (a); 8 is a multiplier two; 9 is a multiplier three; 10 isA generator; 11 is λⁿA generator; 12 is a second difference squarer; 13 is

A generator; 14 is multiplier four; 15 is a multiplier five;16 is

A first adder; 17 is a divider; 18 is a multiplier six; 19 is λⁿOutput point of the generator, i.e. λⁿAn output point of (a); 20 is

A differentiator; 21 is

A differentiator; 22 is

A differentiator; 23 is v₂-a 2k-2 power squarer; 24 is a multiplier seven; 25 is a multiplier eight; 26 is

A differentiator; 27 is

A differentiator; 28 is adder two; 29 is

Output points of generators, i.e.An output point of (a); 30 is a multiplier ten; 31 is

An adder; 32 is a multiplier nine; 33 is a third difference squarer; 34 is

An adder II; 35 is

Output points of generators, i.e.

To the output point of (1). In addition, 3, 12 and 33 are difference squarers with the same function and parameters; 5. 8, 9, 14, 15, 18, 24, 25, 30 and 32 are multipliers with the same function and parameters; 6 and 28 are adders with the same function and parameters; 16 and 34 are functionally and parametrically identical

And an adder.

Wherein, the point A is an arbitrary point on the segmented smooth curve C; point B is an arbitrary point on the piecewise smooth curve C; c is a piecewise smooth curve; c₁The point is an arbitrary point on the segmented smooth curve C; c₂The point is an arbitrary point on the segmented smooth curve C; the point a is a projection point of the point A on the x axis; the point B is a projection point of the point B on the x axis; c is point C₁Projection points of the points on the y axis; d is C₂Projection points of the points on the y axis; omega is a planar single-connected region with a segmented smooth curve c as a boundary; e is the weight

The input point of (a); f is the weight

The input point of (a); g point is weight

The input point of (a); h point is weightThe input point of (a); the I point is the weight of the second difference squarer

The input point of (a); j points are weights Γ (1-v)₃)Γ(2-v₃) The input point of (a); k is the weight $\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}(v_2-\mathrm{\&tau;}+1){\left[\sqrt{\left|\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}\right|}\right]}^{v_2-2k-2}}{\left(2k\right)!}$ The input point of (a); l isThe point is the weight

The input point of (a); m point is weight of the third difference squarer

The input point of (2).

Detailed Description

The following are presented by way of example:

referring to fig. 1, 4, 5 and 6, in practical engineering applications, the order v of the fractional calculus involved in the texture image denoising filter based on the fractional order evolution equation of the present invention₁、v₂And v₃The order is not a traditional integer order, but a non-integer order, typically taking a fraction or rational fraction. In addition, as is apparent from the above description

. If the invention takes the order v₁＝2.25，v₂＝2.5，v₃0.25, unit iteration time interval Δ t is 0.01, and variance is initialized

And taking the parameter mu to be 0.005, so that the invention can obtain the specific circuit parameter of the texture image denoising filter based on the fractional order development equation. Referring to FIG. 1, the F input point of the multiplier one 5 has an input weight of

The G input point of the second multiplier 8 is the input weight value $\frac{-{\mathrm{\&Delta;<figure-callout\; id="3"\; label="t\; v"\; filenames="HSA0000092593440000011.png,HSA0000092593440000041.png"\; state="\{\{state\}\}">t</figure-callout>}}^{v_{}}}{\mathrm{\&Gamma;}(1-v_3)\mathrm{\&Gamma;}(2-v_3)}=\frac{-{\left(0.01\right)}^{0.25}}{\mathrm{\&Gamma;}\left(0.75\right)\mathrm{\&Gamma;}\left(1.75\right)}=-0.2808.$ The input weight of the H input point of the multiplier three 9 is

. Referring to FIG. 4, the input weight of the J input point of the divider 17 is Γ (1-v)₃)Γ(2-v₃) Γ (0.75) Γ (1.75) ═ 1.1262. Referring to FIG. 5, the K input points of the multiplier nine 32 have input weights of $\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}(v_2-\mathrm{\&tau;}+1){\left[\sqrt{\left|\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}\right|}\right]}^{v_2-2k-2}}{\left(2k\right)!}=\frac{{1.1339}^{0.5-2k}\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}(3.5-\mathrm{\&tau;})}{\left(2k\right)!}.$ The input weight of the L input point of the multiplier ten 30 is $\frac{-\mathrm{\&Gamma;}(1-v_1)}{\mathrm{\&Gamma;}(-v_1)\mathrm{\&Gamma;}(-v_3)\mathrm{\&Gamma;}(3-v_3)}=\frac{-\mathrm{\&Gamma;}(-1.25)}{\mathrm{\&Gamma;}(-2.25)\mathrm{\&Gamma;}(-0.25)\mathrm{\&Gamma;}\left(2.75\right)}=-0.2854.$ Therefore, as shown in fig. 1, fig. 4, fig. 5 and fig. 6, according to the cascaded circuit structure of the texture image denoising filter based on the fractional order development equation and the specific circuit parameters thereof of the present invention, which are described in detail in the summary of the invention of the present specification, a specific circuit of the high-precision filter suitable for texture image denoising based on the fractional order partial differential equation can be conveniently constructed. In order to more clearly describe a specific circuit of the texture image denoising filter based on the fractional order development equation according to the present invention without affecting the accurate representation, the timing control circuit and the triggered timing control signal thereof are not shown in fig. 1, 4, 5, and 6.

Claims (6)

1. A texture image denoising filter based on a fractional order development equation is characterized in that: it is composed of buffer (2), difference squarer I (3), -v₃Power squarer (4), multiplier one (5), lambdaⁿA generator (11),The generator (10), the multiplier two (8), the multiplier three (9) and the adder one (6) are formed in a cascade mode. Wherein the order v₁、v₂And v₃Taking a fraction or rational decimalAnd v is₃Not equal to 1, 2, 3, n is any positive integer, k is 0 and 1, and the variance is initialized

The parameter μ is 0.005 and the unit iteration time interval Δ t is (0, 0.1)]Any small positive real number is interpolated.

2. The texture image denoising filter based on the fractional order evolution equation of claim 1, wherein: the input point (1) of the filter is input into the digital image of the nth iteration

The buffer (2) outputs an image of the texture image denoising filter based on the fractional order development equationBuffering two unit iteration time to obtain the digital image of the (n-1) th iterationThe input weight of the difference squarer I (3) is

The calculation performed by the difference squarer I (3) is

-v₃The power squarer (4) performs the calculation that

The input weight of the multiplier one (5) is

The calculation performed by the multiplier one (5) is

The generator (10) performs the calculation that $Q\left(s_{x,y}^n\right)=\frac{-\mathrm{\&Gamma;}(1-v_1)}{\mathrm{\&Gamma;}\left({-v}_1\right)\mathrm{\&Gamma;}\left({-v}_3\right)\mathrm{\&Gamma;}(3-v_3)}\underset{k=0}{\overset{1}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}(v_2-\mathrm{\&tau;}+1){\left[\sqrt{\left|\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}\right|}\right]}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n\right)\\ +D_y^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_y^{v_1}{s}_{x,y}^n\right)\end{array}\right].$ λⁿThe generator (11) performs the calculation that ${\mathrm{\&lambda;}}^n=\frac{\mathrm{\&Gamma;}(1-v_3)\mathrm{\&Gamma;}(2-v_3)}{{\mathrm{\&sigma;}}^{n^2}{s}_{x,y}^n}\underset{x,y}{\mathrm{\&Sigma;}}P\left(s_{x,y}^n\right){(s_{x,y}^0-s_{x,y}^n)}^2.$ The input weight of the multiplier three (9) is

The multiplier three (9) completes the calculation that

The input weight of the multiplier two (8) isThe second multiplier (8) performs the calculation that

The adder one (6) performs the calculation $s_{x,y}^{n+1}=Q\left(s_{x,y}^n\right){\mathrm{\&Delta;t}}^{v_3}-\frac{{\mathrm{\&lambda;}}^n{\mathrm{\&Delta;t}}^{v_3}}{\mathrm{\&Gamma;}(1-v_3)\mathrm{\&Gamma;}(2-v_3)}{s}_{x,y}^n+s_{x,y}^n-\frac{2\mathrm{\&mu;}}{\mathrm{\&Gamma;}(3-v_3)}{(s_{x,y}^{n-1}-s_{x,y}^n)}^2{\left(s_{x,y}^n\right)}^{{-v}_3},$ v₃Not equal to 1, 2, 3. The output point (7) of the texture image denoising filter based on the fractional order development equation outputs a digital image of the (n + 1) th iteration

3. The texture image denoising filter based on the fractional order evolution equation of claim 1, wherein: whereinThe generator (10) performs the calculation that $Q\left(s_{x,y}^n\right)=\frac{-\mathrm{\&Gamma;}(1-v_1)}{\mathrm{\&Gamma;}\left({-v}_1\right)\mathrm{\&Gamma;}\left({-v}_3\right)\mathrm{\&Gamma;}(3-v_3)}\underset{k=0}{\overset{1}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}(v_2-\mathrm{\&tau;}+1){\left[\sqrt{\left|\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}\right|}\right]}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n\right)\\ +D_y^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_y^{v_1}{s}_{x,y}^n\right)\end{array}\right].$ The I input weight of the difference squarer II (12) is the original digital image of the 0 th iteration

The second (12) difference squarer performs the calculation

The generator (13) performs the calculation that ${\mathrm{\&sigma;}}^{n^2}=\underset{x,y}{\mathrm{\&Sigma;}}{(s_{x,y}^0-s_{x,y}^n)}^2.$ The multiplier four (14) performs the calculation that $Q\left(s_{x,y}^n\right){(s_{x,y}^0-s_{x,y}^n)}^2.$ The multiplier five (15) completes the calculation that

The adder one (16) performs the calculation

The input weight value of the divider (17) is gamma (1-v)₃)Γ(2-v₃). The divider (17) performs the calculation thatThe calculation performed by the multiplier six (18) is ${\mathrm{\&lambda;}}^n=\frac{\mathrm{\&Gamma;}(1-v_3)\mathrm{\&Gamma;}(2-v_3)}{{\mathrm{\&sigma;}}^{n^2}{s}_{x,y}^n}\underset{x,y}{\mathrm{\&Sigma;}}Q\left(s_{x,y}^n\right){(s_{x,y}^0-s_{x,y}^n)}^2.$ λⁿThe output point (19) of the generator outputs lambdaⁿThe value is obtained.

4. The texture image denoising filter based on the fractional order evolution equation of claim 1, whereinIs characterized in that: wherein

The differentiator (20) performs the calculation as v in the direction of the x-axis₁Fractional order differentiation of order.

The differentiator (21) performs the calculation by performing v simultaneously in the x-axis direction and the y-axis direction₁Fractional order differentiation of order.The differentiator (22) performs the calculation as v in the y-direction₁Fractional order differentiation of order. v. of₂-the power of 2k-2 squarer (23) performs the calculation

The seventh multiplier (24) performs the calculation that ${\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n.$ The multiplier eight (25) performs the calculation that ${\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n.$

The differentiator (26) performs a 1 st order differentiation in the x-axis direction.

The differentiator (27) performs 1 st order differentiation in the y-axis direction. The adder two (28) performs the calculation $[D_x^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n\right)+D_y^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_y^{v_1}{s}_{x,y}^n\right)].$ The input weight of the multiplier nine (32) is $\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}(v_2-\mathrm{\&tau;}+1){\left[\sqrt{\left|\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}\right|}\right]}^{v_2-2k-2}}{\left(2k\right)!}.$ The multiplier nine (32) performs the calculation $\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}(v_2-\mathrm{\&tau;}+1){\left[\sqrt{\left|\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}\right|}\right]}^{v_2-2k-2}}{\left(2k\right)!}[D_x^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n\right)+D_y^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_y^{v_1}{s}_{x,y}^n\right)].$

The adder (31) performs the calculation that $\underset{k=0}{\overset{1}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}(v_2-\mathrm{\&tau;}+1){\left[\sqrt{\left|\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}\right|}\right]}^{v_2-2k-2}}{\left(2k\right)!}[D_x^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n\right)+D_y^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_y^{v_1}{s}_{x,y}^n\right)].$ The input weight of the multiplier ten (30) is

The multiplier ten (30) performs the calculation that $\frac{-\mathrm{\&Gamma;}(1-v_1)}{\mathrm{\&Gamma;}\left({-v}_1\right)\mathrm{\&Gamma;}\left({-v}_3\right)\mathrm{\&Gamma;}(3-v_3)}\underset{k=0}{\overset{1}{\mathrm{\&Sigma;}}}\frac{\underset{\mathrm{\&tau;}=1}{\overset{2k}{\mathrm{\&Pi;}}}(v_2-\mathrm{\&tau;}+1){\left[\sqrt{\left|\frac{4\mathrm{\&Gamma;}(2-v_3)-\mathrm{\&Gamma;}(3-v_3)}{\mathrm{\&Gamma;}(3-v_3)}\right|}\right]}^{v_2-2k-2}}{\left(2k\right)!}\left[\begin{array}{c}{D}_x^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_x^{v_1}{s}_{x,y}^n\right)\\ +D_y^1\left({\left|D^{v_1}{s}_{x,y}^n\right|}^{v_2-2k-2}{D}_y^{v_1}{s}_{x,y}^n\right)\end{array}\right].$

Output of the generator at output point (29)

The value is obtained.

5. The texture image denoising filter based on the fractional order evolution equation of claim 1, wherein: wherein the input weight of the third (33) difference squarer is the original digital image of the 0 th iteration

The third (33) of the difference value squarer performs the calculation that

The calculation performed by the adder two (34) is

Output of the generator at output point (35)

The value is obtained.

6. The texture image denoising filter based on the fractional order evolution equation of claim 1, wherein: the invention does not know or estimate the variance of the noise in advance, but only orders the first iteration of calculation

Is a small positive number, take

Will be provided with

Lambda is introduced intoⁿA generator to initiate a process of iterative computation; in the process of iterative computation, when

When the invention is used, the invention takesWhen in use

When the invention is used, the invention takes

Images