CN101815165A - Real-time high-efficiency digital image fractional order integration filter - Google Patents

Abstract

The invention provides a real-time high-efficiency digital image fractional order integration filter, which is a circuit device for denoising a noise polluted digital image. The filter comprises a digital video stream line memory group 9, a phase lock/shift circuit group 10, a fractional order integration mask convolution circuit 11 and an average calculator 12 in a cascading way; and the operational rule of 8 specific direction algorithm unit circuits in the fractional order integration mask convolution circuit adopts the fractional order integration mask convolution scheme based on Riemann-Liouville definition so as to realize the spatial filter of digital image fractional order integration. The real-time high-efficiency digital image fractional order integration filter has the advantages of simple circuit structure, simple operational rule, good enhancing effect and high real-time for the noise polluted digital image, and is particularly suitable for denoising processing on the digital image which is unavoidably polluted by noise in an acquiring or transmitting process.

Description

Real-time and high-efficiency fractional-order integral filter for digital image

Field

The real-time and high-efficiency digital image fractional integral filter proposed by the present invention is a signal processing circuit device for real-time enhancement of noise-contaminated digital images. The coefficients of the integral filter involved in the present invention are derived according to the Riemann-Liouville definition of fractional integration. In engineering applications, the order of this fractional integration generally takes negative fractions or negative rational decimals. See Fig. 1, this real-time high-efficiency digital image fractional order integral filter is made up of digital video popular memory group 9, phase lock/shift circuit group 10, fractional order integral mask convolution circuit 11 and average value calculator 12 in stages formed in a linked manner. The operating rules of the 8 specific direction algorithm unit circuits in its fractional integral mask convolution circuit 11 are realized by using the fractional integral mask convolution scheme defined by Riemann-Liouville. filter. The real-time and high-efficiency digital image fractional-order integral filter proposed by the present invention has simple circuit structure, concise operation rules, good enhancement effect on digital images polluted by noise, and high real-time performance, and is especially suitable for the acquisition or transmission process. Applications that inevitably suffer from noise pollution in digital camera images, scanner images, satellite remote sensing images, etc. for real-time enhancement processing. The invention belongs to the technical field of interdisciplinary subjects of applied mathematics, digital image processing and digital circuits.

Background technique

Image is an important carrier for people to record and transmit information. In the process of collecting, acquiring, encoding, storing and transmitting images, images will inevitably be polluted by different types and degrees of noise, which not only reduces the quality of images , It affects people's acquisition and application of information, hinders the process of informatization construction, also restricts the social and economic level and the development of science and technology, and delays the improvement of people's living standards and quality of life. In recent years, with the continuous improvement of people's requirements for the image effect of digital equipment such as digital cameras; with the continuous improvement of ultra-high-definition scanning equipment for scanning images in terms of contrast, clarity and real-time performance; with satellite remote sensing images The increasing requirements for the clarity of the detailed features of complex geographic textures urgently require the construction of a new filter scheme for real-time and efficient enhancement of digital images contaminated by noise.

The process of enhancing the noise image is also the process of smoothing and filtering the image, that is, performing an integral summation operation on the pixels of the digital image participating in the operation, how to achieve edge preservation and denoising (that is, retaining the image while removing noise) Details such as edges and textures) are the key to this process. The existing noise image enhancement algorithms based on spatial domain can be divided into five categories: the first category, Gaussian smoothing algorithm (Gaussians smoothing filter, GSF). This type of algorithm takes the variance of the noise as the variance of the normal distribution and the mean of the noise as the mean of the normal distribution, and derives the coefficient value of the filter according to this normal distribution (that is, the Gaussian function). This type of algorithm has good real-time performance, and is suitable for the enhancement processing of smooth images polluted by Gaussian noise and little detail information. Mean filtering algorithm, Gaussian blur algorithm, and isotropic linear filtering algorithm (ILF) all belong to this type of algorithm. The second category, anisotropic diffusion algorithm (anisotropic filter, AF). This type of algorithm assumes that the digital image is a piecewise smooth geometric figure, and regards the digital image as composed of transitional boundary parts and smooth regions. The denoising process is similar to the heat conduction and diffusion process. The boundary is regarded as adiabatic, and the heat conduction and diffusion are only carried out in the smooth area. When implementing the algorithm, first set the gradient threshold T. When the gradient value of a pixel is greater than the gradient threshold T, the pixel is considered to be a boundary point, and the algorithm diffuses along the tangent direction of the gradient to avoid blurring the edges and details of the image. ; When the gradient value of the pixel is less than or equal to the gradient threshold T, the pixel is considered to be in a smooth area, and the algorithm uses isotropic diffusion to enhance the image contaminated by noise. This type of algorithm selectively smooths the image according to the set gradient threshold T, but it will still destroy the details and texture information of the image while reducing noise. The third category is a denoising algorithm based on total variation (TV). This type of algorithm is based on the assumption that the most important information in the image is the discontinuous boundary features, and adopts the TV norm, which is essentially the ^L1 norm of the derivative. Unlike the L ² norm based on the least squares estimation, the L ¹ norm is non-linear, so the calculation of this type of algorithm is complex and time-consuming, and it is only suitable for cartoon images with smooth slices. The fourth category, neighborhood filter algorithm (neighborhood filter, NF). This type of algorithm first defines the pixel neighborhood according to the proximity of the image pixel position and the similarity of the pixel gray value, and then calculates the similarity between the current pixel and its neighbor pixels in the defined neighborhood, and finally puts the similarity The degree is used as the weight and the pixel points in its neighborhood are weighted and summed, and the result is used as the pixel value of the point. Bilateral linear interpolation filter algorithm (Bilateral filter, BF), Yaroslavsky neighborhood filter algorithm (Yaroslavsky neighborhood filter, YNF), non-local mean filter algorithm (non-local mean filter, NLMF) all belong to this type of algorithm. This kind of algorithm preserves the texture and detail information of the image well while denoising, but the computational complexity is high, which greatly consumes computing resources, reduces processing efficiency, and cannot meet the real-time requirements of computing.

Fractional Calculus was born 300 years ago, and has almost the same long history as integer-order calculus operations. It was proposed relative to integer-order calculus in the traditional sense. How to apply fractional calculus to modern signal analysis and processing (especially image signal processing) is a new topic worth studying both at home and abroad. The applicant of the present invention has done comprehensive and systematic research on this subject, among which, one of the applicants of the present invention applied for the invention patent on August 30, 2006 Fractional order differential filter of digital image (patent number: ZL200610021702.3) Authorized on September 2, 2009, the fractional differential filter has the advantages of enhancing image texture details in real time, multi-scale, and non-linearly, but at the same time, it also has the following shortcomings that need to be further improved and improved : First, the patent only realizes the fractional-order differential operation, but does not involve the fractional-order integral operation; secondly, the fractional-order differential filter of the patent is derived according to the Grümwald-Letnikov definition of fractional-order calculus, but does not The Riemann-Liouville definition, which has been widely studied in the field of fractional calculus, is used for derivation; the third is that the patent performs nonlinear enhancement operations on the R, G, and B components respectively when performing fractional differential filtering on color images, so it will destroy The correlation between the R, G, and B components causes color distortion or distortion of the color image. In response to the above three shortcomings, the applicant of the present invention has done a detailed and in-depth study on the digital image denoising processing based on the fractional integral operation defined by Riemann-Liouville on the basis of the existing fractional calculus in modern signal analysis and processing results. The research, the theoretical and experimental results of the research show that: the fractional order integral operation can not only remove the noise in the polluted image, but also will not destroy the details and texture information of the image while denoising. It has a good enhancement effect and has the characteristics of real-time and high efficiency. In addition, when processing color images, the color space conversion is first performed: from the RGB color space to the HSI space, and then only the fractional integral operation is performed on the I component. After the operation is completed, the HSI color space is converted to RGB Space, avoiding the color distortion or distortion caused by processing the R, G, and B components separately.

Contents of the invention

The purpose of the present invention is to construct a digital image filter, which can be completed at one time, and has the characteristics of real-time, high efficiency, and simplicity. To enhance the texture detail features with relatively small gray value jumps and frequency changes in digital images, and to preserve the low-frequency contour features in the smooth areas of digital images, the applicant of the present invention has thoroughly studied the use of fractional integrals to enhance noise-affected The basic principles and operation rules of polluting digital images, on this basis, aiming at the core content of how to construct the signal processing circuit device of digital image fractional integral filter, according to the nature of digital image fractional integral and digital image processing, digital circuit, According to the input characteristics of the serial digital video code stream, a new scheme of signal processing circuit device for enhancing the digital image polluted by noise is proposed, that is, a real-time and efficient digital image fractional-order integral filter. See Fig. 1, this real-time high-efficiency digital image fractional order integral filter is made up of digital video popular memory group 9, phase lock/shift circuit group 10, fractional order integral mask convolution circuit 11 and average value calculator 12 in stages formed in a linked manner. The operation rules of the 8 direction algorithm unit circuits in the fractional integral mask convolution circuit 11 of the real-time and efficient digital image fractional integral filter are realized by using the fractional integral mask convolution scheme defined by Riemann-Liouville Spatial filters for fractional integration of digital images.

Before explaining the content of the present invention in detail, it is necessary to explain the meaning of the symbols used in this manual and their value ranges in three points: The first point is to follow the customary x and y coordinates used in traditional image processing to represent the vertical axis and horizontal axis of image pixels respectively Coordinates (different from the general mathematical representation of Euclidean space, it is customary to use x and y coordinates to represent the horizontal axis and vertical axis coordinates respectively), and use s(x, y) to represent the gray of the pixel on the coordinates (x, y) degree value or RGB value; when x and y take continuous analog values, S(x, y) represents an analog image; when x and y take discrete digital values, S(x, y) represents a digital image (x and y y represents the row coordinates and column coordinates), which is a pixel matrix; the second point, in order to make the fractional integral mask (it is an n×n square matrix) have a clear axisymmetric center, the fractional integral mask The size number n is an odd number; the minimum value of n is 3, and the maximum value of n is smaller than the size number of the digital image to be processed by the fractional-order integral filter (if the digital image S(x, y) to be processed by the fractional-order integral filter is a pixel matrix of L×H, when L=H, its size is L; when L≠H, its size is the minimum value of L and H); the third point, in actual engineering application, to be The processed digital image S(x, y) (it is a L×H pixel matrix, L represents the number of rows of S(x, y), H represents the number of columns of S(x, y), that is, each row has H pixels, x takes an integer between 0 and (L-1), and y takes an integer between 0 and (H-1)), the gray value or RGB value of the L row of pixels is generally not parallel input (L row The gray value or RGB value of the pixel is input at the same time), but serial input (the gray value or RGB value of the L row of pixels is input one row of pixels one by one, and the gray value or RGB value of H pixels is input for each row, form a serial digital video code stream) image processing device; according to the input characteristics of the serial digital video code stream, use S _x (k) to represent the pixels in the serial digital video code stream (the subscript x represents each frame of digital image S (x, y) forms a serial digital video stream by inputting one row of pixels one by one, S(x, y) starts from the bottom row (L row) and inputs from bottom to top, and k represents the pixel The pixel sequence number of S _x (k) in the serial digital video code stream, k starts counting from L×H-1, and the value of k is input pixel by pixel minus one until it is zero); if S _x (k) corresponds to the serial input The pixel S(x, y) on the previous coordinates (x, y), then S _x (k±mH±b) corresponds to the pixel S(x±mH±b) on the serial input front coordinates ((x±m, y±b) m, y ± b).

See Fig. 1, the real-time efficient digital image fractional order integral filter of the present invention is made of digital video popular memory group 9, phase lock/shift circuit group 10, fractional order integral mask convolution circuit 11 and average value calculator 12 It is formed by cascading; the serial digital video code stream S _x (k) is input into a real-time and efficient digital image fractional-order integral filter and then divided into three paths: the first path sequentially passes through line memory group 9 and phase-lock/shift circuit group 10 After processing by the fractional-order integral mask convolution circuit 11, the approximate values of 8 v-order fractional integrals of the pixel S _x (k+(n-1)(H+1)) in its 8-neighborhood direction are respectively output, After being processed by the average calculator 12, the average value of the above-mentioned 8 approximate values is output as the v-order fractional integral value S _x ^(v) (k+ ₍ n-1) (H+1)); the second trigger sequence control circuit produces a corresponding sequence control signal; the third path and the output of the line memory group 9 are fed into the phase lock/shift circuit group 10 to generate (2n- 1) × (2n-1) pixel array. Wherein, the order v of the fractional-order integral mask convolution circuit 11 in the real-time and efficient digital image fractional-order integral filter can take negative fractions or negative rational decimals. The real-time efficient digital image fractional-order integral filter that the present invention proposes comprises following circuit parts, and its specific structure is as follows:

See Fig. 1, line memory group 9 is made up of sequence control circuit, read-write address generator and dual-port RAM group; Sequence control circuit produces corresponding control read-write address generator, Timing control signals required for the operation of dual-port RAM group, phase-lock/shift circuit group 10, fractional-order integral mask convolution circuit 11 and average value calculator 12; read and write address generators are generated under the action of timing control signals The read-write address of the dual-port RAM is responsible for processing the read-write address initialization and the problem of rotation; the line memory group 9 utilizes the current input pixel according to the input characteristics of the serial digital video code stream, and the line The memory group 9 is divided into two structures: the first structure, when processing digital grayscale images, the line memory group 9 uses 2n-2 line memories to complete the acquisition of 2n-1 line video image data; the second structure: when When processing color images, the row memory group 9 is composed of one RGB-to-HSI color space converter and three identical and parallel-processed row memory group sub-circuits. Wherein each line memory group sub-circuit is completely identical with the circuit structure and parameters of the above-mentioned line memory group 9 when processing digital gray-scale images; these 3 line memory group sub-circuits respectively store digital color images in parallel through RGB to HSI The H, S, and I component values processed by the color space converter; the line memory group 9 uses 6n-6 line memories in total, wherein each line memory group sub-circuit uses 2n-2 line memories to complete 2n-1 line digital video color Storage of H, S, I component values of the image.

See Fig. 1, phase lock/shift circuit group 10 adopts 3n ² -3n D flip-flops in total, generates and calculates digital grayscale image (or color image) by point delay to digital grayscale image (or the I component of color image) The (2n-1)×(2n-1) pixel array required for the fractional integration of the I component of the image; the first row of the (2n-1)×(2n-1) pixel array uses 2n-2 D triggers The 2nd row adopts 2n-3 D flip-flops, and the number of D flip-flops used in each row until the n-1th row is reduced by one row by row, and the n-1th row adopts n D flip-flops;( The nth row of 2n-1)×(2n-1) pixel array uses 2n-2 D flip-flops; the n+1th row of (2n-1)×(2n-1) pixel array uses n D flip-flops , row n+2 uses n+1 D flip-flops, and the number of D flip-flops used in each row up to row 2n-1 is increased by one row by row, and row 2n-1 uses 2n-2 D flip-flops device.

See Fig. 1, the fractional-order integral mask convolution circuit 11 is the most critical circuit component among all circuit components of the real-time and efficient digital image fractional-order integral filter of the present invention, and is also the core content of the new solution proposed by the present invention. In order to clearly illustrate the circuit configuration of the fractional-order integral mask convolution circuit 11, it is necessary to briefly describe the operation rules of the fractional-order integral mask convolution circuit as follows:

将等分份数N代入分数阶积分的Riemann-Liouville定义式可推导出一维信号s(x)的分数阶积分的Riemann-Liouville定义的后向差分近似表达式：Since digital circuits or digital filters deal with digital quantities, their values are limited; the maximum change in grayscale of an image signal is limited; the shortest distance for a change in grayscale in a digital image can only be between two adjacent pixels, so The duration of a two-dimensional digital image s(x, y) in the direction of the x or y coordinate axis (the size of the image matrix) can only be measured in pixels, and s(x, y) in the direction of the x or y coordinate axis The smallest bisection interval on is only possible for h=1. If the duration of the one-dimensional signal s(x) is t∈[a, t], the signal duration [a, t] is equally divided by the unit interval h=1, and the number of equal parts is

Substituting the number of equal parts N into the Riemann-Liouville definition of fractional integration can derive the Riemann-Liouville-defined backward difference approximate expression of the fractional integration of one-dimensional signal s(x):

$\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span>}\underset{\mathrm{<span\; class="notranslate">\; a</span>}}{\overset{\mathrm{<span\; class="notranslate">\; t</span>}}{<span\; class="notranslate">\; \&Integral;</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&xi;</span>}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&xi;</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; d\&xi;</span>}{<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; R</span>}<span\; class="notranslate">\; \_</span>\mathrm{<span\; class="notranslate">\; L</span>}}<span\; class="notranslate">\; \&ap;</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span>}\underset{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}<span\; class="notranslate">\; (</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}}<span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>$

$<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; no</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}}<span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; <</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ;</span>$

表示Gamma函数。本发明定义s(x，y)在x和y坐标轴负方向上分数阶积分的后向差分近似表达式分别为：where v is the order of fractional integration. In the present invention, the order v can take a negative fraction or a negative rational decimal value (due to the limited calculation length of the digital circuit, when v is an irrational decimal, it can be approximately equal to an approximate rational decimal);

Represents the Gamma function. The present invention defines s (x, y) in the negative direction of x and y coordinate axis, and the backward difference approximate expression of fractional order integral is respectively:

$\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span>}\underset{\mathrm{<span\; class="notranslate">\; a</span>}}{\overset{\mathrm{<span\; class="notranslate">\; t</span>}}{<span\; class="notranslate">\; \&Integral;</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&xi;</span>}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&xi;</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; d\&xi;</span>}{<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; R</span>}<span\; class="notranslate">\; \_</span>\mathrm{<span\; class="notranslate">\; L</span>}}<span\; class="notranslate">\; \&ap;</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span>}\underset{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}<span\; class="notranslate">\; (</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}}<span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span>$

$<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; no</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}}<span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; <</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ;</span>$

$\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span>}\underset{\mathrm{<span\; class="notranslate">\; a</span>}}{\overset{\mathrm{<span\; class="notranslate">\; t</span>}}{<span\; class="notranslate">\; \&Integral;</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&xi;</span>}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; \&xi;</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; d\&xi;</span>}{<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; R</span>}<span\; class="notranslate">\; \_</span>\mathrm{<span\; class="notranslate">\; L</span>}}<span\; class="notranslate">\; \&ap;</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span>}\underset{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}<span\; class="notranslate">\; (</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}}<span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>$

$<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; no</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}}<span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; <</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ;</span>$

The present invention selects the preceding n items in above-mentioned two difference approximation expressions and respectively as s (x, y) the approximation value of fractional order integral on x and y coordinate axis negative direction:

$\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span>}\underset{\mathrm{<span\; class="notranslate">\; a</span>}}{\overset{\mathrm{<span\; class="notranslate">\; t</span>}}{<span\; class="notranslate">\; \&Integral;</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&xi;</span>}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&xi;</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; d\&xi;</span>}{<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; R</span>}<span\; class="notranslate">\; \_</span>\mathrm{<span\; class="notranslate">\; L</span>}}<span\; class="notranslate">\; \&ap;</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\frac{{\mathrm{<span\; class="notranslate">\; 2</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}}}{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\frac{{\mathrm{<span\; class="notranslate">\; 3</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}}}{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span>$

$<span\; class="notranslate">\; +</span>\frac{{\mathrm{<span\; class="notranslate">\; 4</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; 2</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}}}{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; +</span>\frac{{\mathrm{<span\; class="notranslate">\; no</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}}<span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}}}{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; <</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ;</span>$

$\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span>}\underset{\mathrm{<span\; class="notranslate">\; a</span>}}{\overset{\mathrm{<span\; class="notranslate">\; t</span>}}{<span\; class="notranslate">\; \&Integral;</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&xi;</span>}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; \&xi;</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; d\&xi;</span>}{<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; R</span>}<span\; class="notranslate">\; \_</span>\mathrm{<span\; class="notranslate">\; L</span>}}<span\; class="notranslate">\; \&ap;</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\frac{{\mathrm{<span\; class="notranslate">\; 2</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}}}{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\frac{{\mathrm{<span\; class="notranslate">\; 3</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}}}{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$

$<span\; class="notranslate">\; +</span>\frac{{\mathrm{<span\; class="notranslate">\; 4</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; 2</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}}}{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; +</span>\frac{{\mathrm{<span\; class="notranslate">\; no</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}}<span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}}}{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; <</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ;</span>$

这n个非零系数值都是严格按照分数阶积分的Riemann-Liouville定义推导而来，都是阶次v的函数。在数字图像中，邻域内像素与像素之间的灰度值具有很大的相关性，为了加强分数阶积分掩模卷积电路11的抗图像旋转性，见图2，有必要分别计算出像素s(x，y)在其8-邻域方向上的v阶分数阶积分的近似值；本发明将s(x，y)在上述8个方向上的v阶分数阶积分的近似值的平均值作为s(x，y)的v阶分数阶积分值。见图3，在n×n全零方阵沿方阵左上对角线方向上，用

这n个非零系数值按顺序置换掉n×n全零方阵中左上对角线相应位置上的零值，从而构造出135°方向上的积分掩模(用W_135°表示)。见图4，在n×n全零方阵沿x坐标轴负方向的中心对称轴上，用

这n个非零系数值按顺序置换掉n×n全零方阵中相应位置上的零值，从而构造出90°方向上的积分掩模(用W_90°表示)。另外，45°方向上的积分掩模(用W_45°表示，见图5)、0°方向上的积分掩模(用W_0°表示，见图6)、180°方向上的积分掩模(用W_180°表示，见图7)、315°方向上的积分掩模(用W_315°表示，见图8)、270°方向上的积分掩模(用W_270°表示，见图9)、225°方向上的积分掩模(用W_225°表示，见图10)与W_135°和W_90°的构造原理和方法类似，这里不再赘述。It can be seen that the coefficient value of each corresponding summation term in the approximation (n term sum) of the fractional integration of s(x, y) in the negative direction of the x and y coordinate axes is the same. The n non-zero coefficient values are, in order, respectively:

These n non-zero coefficient values are derived strictly according to the Riemann-Liouville definition of fractional integration, and are functions of order v. In a digital image, the grayscale values between pixels in the neighborhood have a great correlation. In order to strengthen the anti-image rotation of the fractional-order integral mask convolution circuit 11, as shown in Fig. 2, it is necessary to calculate the pixel s (x, y) in its 8-neighborhood directions v order fractional integral approximation; The present invention uses the average value of the v order fractional integral s (x, y) in above-mentioned 8 directions as the approximate value The v-order fractional integral value of s(x, y). As shown in Figure 3, in the direction of the n×n all-zero square matrix along the upper left diagonal of the square matrix, use

These n non-zero coefficient values replace the zero values at the corresponding positions on the upper left diagonal in the n×n all-zero square matrix in order, thereby constructing the integral mask in the 135° direction (denoted by W _135° ). As shown in Figure 4, on the central symmetry axis of the n×n all-zero square matrix along the negative direction of the x-coordinate axis, use

These n non-zero coefficient values replace the zero values at the corresponding positions in the n×n all-zero square matrix in sequence, thereby constructing an integral mask in the 90° direction (denoted by W _90° ). In addition, the integral mask in the 45° direction (indicated by W _45° , see Figure 5), the integral mask in the 0° direction (indicated by W _0° , see Figure 6), the integral mask in the 180° direction (Denoted by W _180° , see Figure 7), integral mask on 315° direction (expressed by W _315° , see Figure 8), integral mask on 270° direction (expressed by W _270° , see Figure 9 ), the integral mask in the 225° direction (represented by W _225° , see Figure 10) is similar to the construction principles and methods of W _135° and W _90° , and will not be repeated here.

The operation rule of the fractional integral mask convolution circuit 11 is to adopt the scheme based on the fractional integral mask convolution defined by Riemann-Liouville to realize the spatial domain filter of the digital image fractional integral, which is suitable for realizing the digital image signal with a hardware circuit. processing. The operation rules of the fractional-order integral mask convolution circuit 11 for digital images are as follows: Step 1, input serially input digital video signals into W _135° , W _90° , W _45° , W _0° , W Coefficient values in the 8 integral masks of _180° , W _315° , W _270° and W _225° The coordinates (x, y) where it is located and the coordinate position (x, y) of the pixel s(x, y) to be integrated in fractional order must keep coincident; in the second step, the coefficient values of the above 8 integration masks are respectively compared with Multiply the gray value of the corresponding input pixel (or the I component value of the color image), and then add the respective product terms (ie weighted sum) to obtain the weighted sum corresponding to the above 8 integral masks value; in the 3rd step, these 8 values are respectively used as the processing results of the fractional order integral mask convolution circuit 11 in the directions corresponding to the 8 integral masks (that is, the pixel s (x, y) is in its 8-neighborhood Approximation of v-order fractional integration in the direction); step 4, pixel by pixel translation of W _135° , W _90° , W _45° , W _0° , W _180° , W The 8 integral masks of _315° , W _270° and W _225° repeat the operation rules of the above steps 1 to 3 respectively, and traverse the entire digital image to be fractionally integrated to calculate the entire digital image The approximation of the v-order fractional integral of the pixel in its 8-neighborhood direction; in addition, when translating pixel by pixel, in order not to make the row or column of the integral mask outside the digital image plane to be subjected to fractional integration, it is necessary Make the distance between the central point of the integral mask and the edge pixels of the digital image to be fractionally integrated not less than (n-1)/2 pixels, that is, not to n-1 rows or columns from the edge of the digital image to be fractionally integrated The pixels of the fractional order integration operation.

见图1，分数阶积分掩模卷积电路11由如下8个特定的方向算法单元电路构成：The circuit structure of the fractional integral mask convolution circuit 11 will be specifically described below. As shown in Figure 1, the fractional order integral mask convolution circuit 11 is composed of 8 specific 8 direction algorithm unit circuits for parallel calculation: 135° direction algorithm unit circuit 1, 90° direction algorithm unit circuit 2 and 45° direction algorithm unit Circuit 3, 0 ° direction algorithm unit circuit 4 and 180 ° direction algorithm unit circuit 5, 315 ° direction algorithm unit circuit 6, 270 ° direction algorithm unit circuit 7 and 225 ° direction algorithm unit circuit 8, respectively calculate the digital image pixel in its The fractional integral approximation in the 8-neighborhood direction, see Figure 2, the 8-neighborhood direction of the pixel includes 135° direction, 90° direction, 45° direction, 0° direction, 180° direction, 315° direction, 270° direction These 8 directions of direction and 225 ° direction; See Fig. 1 and Fig. 11, each arithmetic unit circuit is by n (fractional order integral mask size number) first multiplier to the third multiplier 13～15 and an adder 16; the weights of the n multipliers are in order:

As shown in Figure 1, the fractional-order integral mask convolution circuit 11 is composed of the following eight specific direction algorithm unit circuits:

分别馈入第一乘法器13，相乘后馈入加法器16；像素S_x(k+(n-1)(H+1)-1-H)的灰度值或I分量值与权值

分别馈入第二乘法器14，相乘后馈入加法器16；依此类推，若1≤m≤n，像素S_x(k+(n-1)(H+1)-(m-1)-(m-1)H)的灰度值或I分量值与权值

相乘后馈入加法器16，从而得到像素S_x(k+(n-1)(H+1))在其8-邻域方向中的135°方向上的v阶分数阶积分的近似值。135 ° direction algorithm unit circuit 1 calculates the approximate value of the v-order fractional integral of pixel S _x (k+(n-1)(H+1)) under the action of integral mask W _{135 °} ; pixel S _x (k+(n- 1) Gray value or I component value and weight of (H+1))

Feed into the first multiplier 13 respectively, and feed into the adder 16 after being multiplied; the gray value or the I component value and the weight of the pixel S _x (k+(n-1)(H+1)-1-H)

Feed into the second multiplier 14 respectively, feed into the adder 16 after being multiplied; By analogy, if 1≤m≤n, the pixel S _x (k+(n-1)(H+1)-(m-1) -(m-1)H) gray value or I component value and weight

Feed into the adder 16 after multiplication, so as to obtain the approximate value of v order fractional integral of pixel S _x (k+(n-1)(H+1)) in the direction of 135° in its 8-neighborhood direction.

分别馈入第一乘法器13，相乘后馈入加法器16；像素S_x(k+(n-1)(H+1)-H)的灰度值或I分量值与权值

分别馈入第二乘法器14，相乘后馈入加法器16；依此类推，若1≤m≤n，像素S_x(k+(n-1)(H+1)-(m-1)H)的灰度值或I分量值与权值相乘后馈入加法器16，从而得到像素S_x(k+(n-1)(H+1))在其8-邻域方向中的90°方向上的v阶分数阶积分的近似值。The 90° direction algorithm unit circuit 2 calculates the approximate value of the v-order fractional integral of the pixel S _x (k+(n-1)(H+1)) under the integral mask W _90° ; the pixel S _x (k+(n-1) 1) Gray value or I component value and weight of (H+1))

Feed into the first multiplier 13 respectively, and feed into the adder 16 after being multiplied; the gray value or the I component value and the weight of the pixel S _x (k+(n-1)(H+1)-H)

Feed into the second multiplier 14 respectively, feed into the adder 16 after being multiplied; By analogy, if 1≤m≤n, the pixel S _x (k+(n-1)(H+1)-(m-1) H) gray value or I component value and weight Feed into the adder 16 after multiplication, so as to obtain the approximate value of v order fractional integral of pixel S _x (k+(n-1)(H+1)) in the 90° direction in its 8-neighborhood directions.

分别馈入第一乘法器13，相乘后馈入加法器16；像素S_x(k+(n-1)(H+1)+1-H)的灰度值或I分量值与权值

分别馈入第二乘法器14，相乘后馈入加法器16；依此类推，若1≤m≤n，像素S_x(k+(n-1)(H+1)+(m-1)-(m-1)H)的灰度值或I分量值与权值相乘后馈入加法器16，从而得到像素S_x(k+(n-1)(H+1))在其8-邻域方向中的45°方向上的v阶分数阶积分的近似值。The 45 ° direction algorithm unit circuit 3 calculates the approximate value of the v-order fractional integral of the pixel S _x (k+(n-1)(H+1)) under the integral mask W _{45 °} ; the pixel S _x (k+(n- 1) Gray value or I component value and weight of (H+1))

Feed into the first multiplier 13 respectively, and feed into the adder 16 after being multiplied; the gray value or the I component value and the weight of the pixel S _x (k+(n-1)(H+1)+1-H)

Feed into the second multiplier 14 respectively, and feed into the adder 16 after being multiplied; and so on, if 1≤m≤n, the pixel S _x (k+(n-1)(H+1)+(m-1) -(m-1)H) gray value or I component value and weight Feed into the adder 16 after multiplication, so as to obtain the approximate value of v order fractional integral of pixel S _x (k+(n-1)(H+1)) in the direction of 45° in its 8-neighborhood directions.

分别馈入第一乘法器13，相乘后馈入加法器16；像素S_x(k+(n-1)(H+1)+1)的灰度值或I分量值与权值分别馈入第二乘法器14，相乘后馈入加法器16；依此类推，若1≤m≤n，像素S_x(k+(n-1)(H+1)+(m-1))的灰度值或I分量值与权值相乘后馈入加法器16，从而得到像素S_x(k+(n-1)(H+1))在其8-邻域方向中的0°方向上的v阶分数阶积分的近似值。0° direction algorithm unit circuit 4 calculates the approximate value of the v order fractional integral of pixel S _x (k+(n-1)(H+1)) under the action of integral mask W _0° ; pixel S _x (k+(n- 1) Gray value or I component value and weight of (H+1))

Feed into the first multiplier 13 respectively, and feed into the adder 16 after being multiplied; the gray value or the I component value and the weight of the pixel S _x (k+(n-1)(H+1)+1) Feed into the second multiplier 14 respectively, and feed into the adder 16 after being multiplied; and so on, if 1≤m≤n, the pixel S _x (k+(n-1)(H+1)+(m-1) ) gray value or I component value and weight Feed into the adder 16 after being multiplied, so as to obtain the approximate value of the fractional integral of order v of the pixel S _x (k+(n-1)(H+1)) in the direction of 0° in its 8-neighborhood directions.

分别馈入第一乘法器13，相乘后馈入加法器16；像素S_x(k+(n-1)(H+1)-1)的灰度值或I分量值与权值分别馈入第二乘法器14，相乘后馈入加法器16；依此类推，若1≤m≤n，像素S_x(k+(n-1)(H+1)-(m-1))的灰度值或I分量值与权值

相乘后馈入加法器16，从而得到像素S_x(k+(n-1)(H+1))在其8-邻域方向中的180°方向上的v阶分数阶积分的近似值。The 180 ° direction algorithm unit circuit 5 calculates the approximate value of the v-order fractional integral of the pixel S _x (k+(n-1)(H+1)) under the integral mask W _{180 °} ; the pixel S _x (k+(n- 1) Gray value or I component value and weight of (H+1))

Feed into the first multiplier 13 respectively, and feed into the adder 16 after being multiplied; the gray value or the I component value and the weight of the pixel S _x (k+(n-1)(H+1)-1) Feed into the second multiplier 14 respectively, feed into the adder 16 after being multiplied; By analogy, if 1≤m≤n, the pixel S _x (k+(n-1)(H+1)-(m-1) ) gray value or I component value and weight

Feed into the adder 16 after multiplication, so as to obtain the approximate value of the fractional integral of order v of the pixel S _x (k+(n-1)(H+1)) in the direction of 180° in its 8-neighborhood direction.

分别馈入第一乘法器13，相乘后馈入加法器16；像素S_x(k+(n-1)(H+1)+1+H)的灰度值或I分量值与权值

分别馈入第二乘法器14，相乘后馈入加法器16；依此类推，若1≤m≤n，像素S_x(k+(n-1)(H+1)+(m-1)+(m-1)H)的灰度值或I分量值与权值

相乘后馈入加法器16，从而得到像素S_x(k+(n-1)(H+1))在其8-邻域方向中的315°方向上的v阶分数阶积分的近似值。315 ° direction algorithm unit circuit 6 calculates the approximate value of the v-order fractional integral of pixel S _x (k+(n-1)(H+1)) under the action of integral mask W _{315 °} ; pixel S _x (k+(n- 1) Gray value or I component value and weight of (H+1))

Feed into the first multiplier 13 respectively, and feed into the adder 16 after being multiplied; the gray value or the I component value and the weight of the pixel S _x (k+(n-1)(H+1)+1+H)

Feed into the second multiplier 14 respectively, and feed into the adder 16 after being multiplied; and so on, if 1≤m≤n, the pixel S _x (k+(n-1)(H+1)+(m-1) +(m-1)H) gray value or I component value and weight

Feed into the adder 16 after being multiplied, so as to obtain the approximate value of the fractional integral of order v of the pixel S _x (k+(n-1)(H+1)) in the direction of 315° in its 8-neighborhood direction.

分别馈入第一乘法器13，相乘后馈入加法器16；像素S_x(k+(n-1)(H+1)+H)的灰度值或I分量值与权值

分别馈入第二乘法器14，相乘后馈入加法器16；依此类推，若1≤m≤n，像素S_x(k+(n-1)(H+1)+(m-1)H)的灰度值或I分量值与权值

相乘后馈入加法器16，从而得到像素S_x(k+(n-1)(H+1))在其8-邻域方向中的270°方向上的v阶分数阶积分的近似值。The 270 ° direction algorithm unit circuit 7 calculates the approximate value of the v-order fractional integral of the pixel S _x (k+(n-1)(H+1)) under the integral mask W _{270 °} ; the pixel S _x (k+(n- 1) Gray value or I component value and weight of (H+1))

Feed into the first multiplier 13 respectively, and feed into the adder 16 after being multiplied; the gray value or the I component value and the weight of the pixel S _x (k+(n-1)(H+1)+H)

Feed into the second multiplier 14 respectively, and feed into the adder 16 after being multiplied; and so on, if 1≤m≤n, the pixel S _x (k+(n-1)(H+1)+(m-1) H) gray value or I component value and weight

Feed into the adder 16 after multiplication, so as to obtain the approximate value of v order fractional integral of pixel S _x (k+(n-1)(H+1)) in the direction of 270° in its 8-neighborhood directions.

分别馈入第一乘法器13，相乘后馈入加法器16；像素S_x(k+(n-1)(H+1)-1+H)的灰度值或I分量值与权值

分别馈入第二乘法器14，相乘后馈入加法器16；依此类推，若1≤m≤n，像素S_x(k+(n-1)(H+1)-(m-1)+(m-1)H)的灰度值或I分量值与权值

相乘后馈入加法器16，从而得到像素S_x(k+(n-1)(H+1))在其8-邻域方向中的225°方向上的v阶分数阶积分的近似值。225 ° direction algorithm unit circuit 8 calculates the approximate value of the v order fractional integral of pixel S _x (k+(n-1)(H+1)) under the action of integral mask W _{225 °} ; Pixel S _x (k+(n- 1) Gray value or I component value and weight of (H+1))

Feed into the first multiplier 13 respectively, and feed into the adder 16 after being multiplied; the gray value or the I component value and the weight of the pixel S _x (k+(n-1)(H+1)-1+H)

Feed into the second multiplier 14 respectively, feed into the adder 16 after being multiplied; By analogy, if 1≤m≤n, the pixel S _x (k+(n-1)(H+1)-(m-1) +(m-1)H) gray value or I component value and weight

Feed into the adder 16 after multiplication, so as to obtain the approximate value of v order fractional integral of pixel S _x (k+(n-1)(H+1)) in the direction of 225° in its 8-neighborhood direction.

See Fig. 1, the average value calculator 12 calculates the average value of the output values of the 8 direction algorithm unit circuits of the fractional order integral mask convolution circuit 11, and the average value calculator 12 is divided into two types according to the different properties of the processed digital image Structure: in the first structure, when processing digital gray scale images, the average value calculator 12 has 8 inputs and 1 output, which are respectively fed into the gray levels of the 135° direction algorithm unit circuit 1 to the 225° direction algorithm unit circuit 8 Value modulus value, output the average value of the above 8 feed-in gray value modulus values (that is, the v-order fractional integral value of the pixel gray value of the digital gray image image); the second structure, when processing a digital color image, the average The value calculator 12 is composed of an average value calculator subcircuit and a color space converter from HSI to RGB, wherein the average value calculator subcircuit is the same as the average value used in the first structure for processing digital grayscale images The circuit structure and parameters of the value calculator 12 are exactly the same, and are used to calculate the average value of the v-order fractional integral approximation modulus of the 8 I component values fed in; the color space converter will feed in the I component and H, S The components are converted into the corresponding R, G, and B components in the RGB color space, and the v-order fractional integral value of the color image pixel s(x, y) is output.

The inventor of the present invention has deeply studied the basic principles and operation rules of using fractional integrals to enhance noise-contaminated digital images, and based on this, aims at the core content of how to construct signal processing circuit devices for digital image fractional integral filters , according to the nature of digital image fractional integral and the input characteristics of digital image processing, digital circuit, and serial digital video stream, a new scheme of signal processing circuit device for enhancing noise-contaminated digital image is proposed, that is, real-time high-efficiency The digital image fractional order integral filter. Its promotion will have a profound impact on the application of fractional calculus in the analysis and processing of modern signals, especially in the analysis and processing of digital image signals.

Below in conjunction with accompanying drawing and the example of the digital image fractional integral filter of real-time high efficiency describe in detail the new scheme of the present invention strengthens the signal processing circuit device of the digital image polluted by noise:

Description of drawings

FIG. 1 is a schematic diagram of the circuit structure of the real-time and high-efficiency digital image fractional-order integral filter of the present invention.

Fig. 2 is a schematic diagram of 8-neighborhood directions of a digital image pixel s(x, y).

FIG. 3 is a schematic diagram of an n×n square matrix of the fractional integral mask W _135° of the 135° direction arithmetic unit circuit 1 in the 135° direction.

FIG. 4 is a schematic diagram of an n×n square matrix of the fractional order integral mask W _90° in the 90° direction algorithm unit circuit 2 in the 90° direction.

FIG. 5 is a schematic diagram of an n×n square matrix of the fractional integral mask W _45° in the 45° direction algorithm unit circuit 3 in the 45° direction.

FIG. 6 is a schematic diagram of an n×n square matrix of the fractional order integral mask W _0° of the 0° direction arithmetic unit circuit 4 in the 0° direction.

FIG. 7 is a schematic diagram of an n×n square matrix of the fractional-order integral mask W 180° in the 180° direction of the 180 _° direction algorithm unit circuit 5 .

FIG. 8 is a schematic diagram of an n×n square matrix of the fractional integral mask W _135° of the 315° direction algorithm unit circuit 6 in the 315° direction.

FIG. 9 is a schematic diagram of an n×n square matrix of the fractional-order integral mask W 270° in the 270° direction of the 270 _° direction arithmetic unit circuit 7 .

FIG. 10 is a schematic diagram of an n×n square matrix of the fractional integral mask W 225 _° of the 225° direction arithmetic unit circuit 8 in the 225° direction.

FIG. 11 is a schematic diagram of the common circuit structure of the 135° direction algorithm unit circuit 1 to the 225° direction algorithm unit circuit 8 .

Figure 12 is a real-time and efficient digital image fractional-order integral filter in the fractional-order integral mask W _135° , W _90° , W _45° , W _0° , W _180° , W _315° in a 3×3 square matrix , W _270° and W _225° are circuit schematic diagrams for fractional-order integral filter processing of digital images.

的输入点；E点是权值

的输入点；F点是像素S_x(k+2H+2)的灰度值的输出点。上述的8个方向算法单元电路分别输出像素S_x(k+(n-1)(H-1))在W1_35°、W_90°、W_45°、W_0°、W_180°、W_315°、W_270°和W_225°这8个积分掩模作用后的v阶分数阶积分的近似值。Among them, 1 is the 135° direction algorithm unit circuit; 2 is the 90° direction algorithm unit circuit; 3 is the 45° direction algorithm unit circuit; 4 is the 0° direction algorithm unit circuit; 5 is the 180° direction algorithm unit circuit; 6 is the 315 ° direction algorithm unit circuit; 7 is 270 ° direction algorithm unit circuit; 8 is 225 ° direction algorithm unit circuit; 9 is row memory group; 10 is phase-lock/shift circuit group; 11 is fractional order integral mask convolution circuit 12 is an average value calculator; 13 is a first multiplier; 14 is a second multiplier; 15 is a third multiplier; 16 is an adder; The average value calculator with the same function and parameters; A point is the input point of the serial digital video code stream S _x (k) of the fractional-order integral filter of the digital image; B point is the weight The input point; C point is the weight

The input point; E point is the weight

The input point of F; the output point of the gray value of the pixel S _x (k+2H+2). The above-mentioned 8 direction algorithm unit circuits respectively output the pixel S _x (k+(n-1)(H-1)) at W1 _35° , W _90° , W _45° , W _0° , W _180° , W _315° , W _270° and W _225° are the approximate values of the v-th order fractional integral after the 8 integral masks.

Detailed ways

这n个非零系数值都是严格按照分数阶积分的Riemann-Liouville定义推导而来，都是阶次v的函数。②为了使分数阶积分掩模(它是一个n×n的方阵)有明确的轴对称中心，分数阶积分掩模的尺寸数n是奇数；n的最小取值是3，n的最大取值小于待进行分数阶积分的数字图像的尺寸数(若待进行分数阶积分的数字图像S(x，y)是L×H的像素矩阵，当L＝H时，其尺寸数为L；当L≠H时，其尺寸数为L和H中的最小值)；③根据待处理的数字图像的性质不同，对于数字灰度图像和彩色图像的处理应该区别对待。本发明的行存储器9针对彩色图像的结构不同于其针对灰度图像的结构，不同之处在于它是由1个RGB到HSI的色彩空间转换器和3个完全相同且并行处理的行存储器组子电路构成；另外，本发明的平均值计算器12针对彩色图像的结构也不同于其针对灰色图像的结构，不同之处在于它是由1个最大值比较器子电路和1个HSI到RGB的色彩空间转换器构成。这避免了对彩色图像像素的R、G、B各分量分别进行分数阶积分运算而引起的色彩失真或畸变。④本发明的实时高效的数字图像分数阶积分滤波器中的分数阶积分掩模卷积电路11不限于图11所采用的方案，一般说来它可以采用图1中的分数阶积分掩模卷积电路11所示结构，任何一种用硬件电路实现本发明的分数阶积分掩模卷积电路11的运算规则的具体措施均可导出实时高效的数字图像分数阶积分滤波器的具体方案，这需要针对具体使用背景来加以选取。See Fig. 1 and Fig. 12, according to the cascaded circuit structure of the real-time efficient digital image fractional-order integral filter of the present invention described in detail in the content of the invention of this specification sheet and its line memory group 9, phase lock/shift circuit group 10. The specific circuit structure and circuit parameters of the fractional-order integral mask convolution circuit 11 and the average value calculator 12 can construct the specific circuit of the real-time and efficient digital image fractional-order integral filter. In addition, in the process of specific implementation, it should also be noted that: ① The fractional order integral masks W _135° , W _90° , W _45° , W _0° , W _180° , W _315° , W _270° proposed by the present invention and the n nonzero coefficient values in W _225° are, in order, respectively:

These n non-zero coefficient values are derived strictly according to the Riemann-Liouville definition of fractional integration, and are functions of order v. ②In order to make the fractional integral mask (it is an n×n square matrix) have a clear axisymmetric center, the size n of the fractional integral mask is an odd number; the minimum value of n is 3, and the maximum value of n is The value is less than the size number of the digital image to be carried out fractional order integration (if the digital image S(x, y) to be carried out fractional order integration is a pixel matrix of L×H, when L=H, its size number is L; when When L≠H, its size is the minimum value of L and H); ③ According to the different properties of the digital images to be processed, the processing of digital grayscale images and color images should be treated differently. The structure of the line memory 9 of the present invention for color images is different from its structure for grayscale images, the difference is that it consists of 1 color space converter from RGB to HSI and 3 identical and parallel-processed line memory groups In addition, the average value calculator 12 of the present invention is also different from its structure for gray images for the structure of the color image, and the difference is that it is composed of 1 maximum value comparator subcircuit and 1 HSI to RGB The color space converter constitutes. This avoids color distortion or distortion caused by fractional integral operations on R, G, and B components of color image pixels. 4. the fractional order integral mask convolution circuit 11 in the real-time efficient digital image fractional integral filter of the present invention is not limited to the scheme adopted in Fig. 11, generally speaking it can adopt the fractional order integral mask volume in Fig. 1 The structure shown in the product circuit 11, any concrete measure that realizes the operating rules of the fractional-order integral mask convolution circuit 11 of the present invention with a hardware circuit can derive the concrete scheme of the real-time efficient digital image fractional-order integral filter, which It needs to be selected according to the specific use background.

An example is introduced as follows:

所以，其中行存储器组9采用2n-2|_n＝3＝4个行存储器完成2n-1|_n＝3＝5行视频图像数据的获取；其中锁相/移位电路组10共采用3n²-3n|_n＝3＝18个D触发器，通过对数字图像进行点延时产生计算数字图像分数阶积分所需的(2n-1)×(2n-1)|_n＝3＝5×5像素阵列；其中8个方向算法单元电路共有8×n|_n＝3＝24个乘法器，每个算法单元电路中n＝3个乘法器的非零权值按顺序分别是：

于是，如图12所示，按照本说明书发明内容中所述的本发明的实时高效的数字图像分数阶积分滤波器的级联电路结构及其行存储器组9、锁相/移位电路组10、分数阶积分掩模卷积电路11和平均值计算器12的具体电路结构和电路参数，就可以方便地构造出该实时高效的数字图像分数阶积分滤波器的具体电路°在不影响准确表述该滤波器的具体电路的前提下，为了更加清晰明了地描述其中的8个方向算法单元电路的具体电路，图12未画出其中的时序控制电路及其被触发产生的时序控制信号。See Fig. 1 and Fig. 12, if want to construct the concrete circuit of a real-time high-efficiency fractional-order integral filter of digital gray-scale image, in engineering practical application, the operation rule of the fractional-order integral mask convolution circuit 11 in this filter The 3×3 fractional integral mask convolution scheme is often used to realize the v-order fractional integration of the pixel s(x, y) of the digital grayscale image. From the above description, it can be known that the v-order fraction in 8 directions Order integral masks W _135° , W _90° , W _45° , W _0° , W _180° , W _315° , W _270° and W _225° the number of dimensions n=3, 3 non-zero in the mask The coefficient values are, in order, respectively:

Therefore, wherein row memory group 9 adopts 2n-2| _n=3 =4 row memories to complete the acquisition of 2n-1| _n=3 =5 rows of video image data; wherein phase lock/shift circuit group 10 adopts 3n ² altogether -3n| _n=3 =18 D flip-flops, which generate (2n-1)×(2n-1)| _n=3 =5×5 needed to calculate the fractional integral of the digital image by performing point delay on the digital image Pixel array; wherein 8 direction arithmetic unit circuits have 8×n| _n=3 =24 multipliers in total, and the non-zero weights of n=3 multipliers in each arithmetic unit circuit are respectively in order:

Then, as shown in FIG. 12 , according to the cascaded circuit structure of the real-time and efficient digital image fractional-order integral filter of the present invention described in the summary of the invention of this specification and its line memory group 9 and phase-lock/shift circuit group 10 , the specific circuit structure and circuit parameters of the fractional-order integral mask convolution circuit 11 and the average value calculator 12, the concrete circuit of this real-time efficient digital image fractional-order integral filter can be constructed easily. Without affecting the accurate expression On the premise of the specific circuit of the filter, in order to more clearly describe the specific circuits of the 8 direction algorithm unit circuits, FIG. 12 does not show the timing control circuit and the timing control signals generated by triggering.

Claims (4)

1. a real-time efficient digital image fractional-order integral filter is characterized in that: it is composed of digital video popular memory group (9), phase-lock/shift circuit group (10), fractional-order integral mask convolution circuit (11) is cascaded with the average value calculator (12); the serial digital video stream S _x (k) is divided into three paths after being input into the real-time and efficient digital image fractional order integral filter: the first path passes through the row memory in sequence Group (9), phase lock/shift circuit group (10), fractional order integral mask convolution circuit (11) after processing, respectively output pixel S _x (k+(n-1)(H+1)) in its The approximate value of 8 v-order fractional integrals on the 8-neighborhood direction, after being processed by the average calculator (12), the average value in the above-mentioned 8 approximate values is output as the pixel S _x (k+(n-1)( H+1)) v-order fractional integral value S _x ^(v) (k+(n-1)(H+1)); the second trigger timing control circuit generates a corresponding timing control signal; the third and row The outputs of the memory group (9) are fed into the phase lock/shift circuit group (10) together to generate a (2n-1)×(2n-1) pixel array. Among them, the value of k is reduced one by one from L×H-1 until it is zero; the value of L is equal to the positive integer of the number of digital image lines to be subjected to fractional integration; the value of H is equal to the A positive integer of the number of columns in a digital image; n is any odd number between [3, min(L, H)], where min(L, H) means finding the minimum value of L and H; v is a negative fraction or Negative rational decimals. 2. the real-time efficient digital image fractional order integral filter according to claim 1, is characterized in that: wherein line memory group (9) is made up of sequential control circuit, read-write address generator and dual-port RAM; Generate corresponding control read-write address generator, dual-port RAM group, phase-lock/shift circuit group (10), fractional order integral mask convolution circuit (11) and The timing control signal required for the operation of the average value calculator (12); the read-write address generator generates the read-write address of the dual-port RAM under the action of the timing control signal, and is responsible for processing the read-write address initialization and rotation problems; the row memory Group (9) is divided into two structures according to the input characteristics of the serial digital video code stream, using the current input pixels, and according to the different properties of the processed digital images: the first structure, when processing digital gray When processing a color image, the line memory group (9) uses 2n-2 line memory to complete the acquisition of 2n-1 row video image data; the second structure: when processing a color image, the line memory group (9) consists of 1 RGB to The color space converter of HSI is composed of three identical and parallel-processed line memory group sub-circuits. Wherein each line memory group sub-circuit is completely identical with the circuit structure and parameters of the above-mentioned line memory group (9) when processing digital grayscale images; these 3 line memory group sub-circuits store digital color images in parallel respectively through RGB to The H, S, and I component values processed by the color space converter of HSI; the line memory group (9) adopts 6n-6 line memory in total, wherein each line memory group subcircuit adopts 2n-2 line memory to complete 2n-1 Storage of H, S, I component values of digital video color images. 3. the real-time efficient digital image fractional order integral filter according to claim 1, is characterized in that: the fractional order integral mask convolution circuit (11) wherein is made up of 8 specific arithmetic unit circuits of parallel calculation: 135 ° direction algorithm unit circuit (1), 90° direction algorithm unit circuit (2), 45° direction algorithm unit circuit (3), 0° direction algorithm unit circuit (4), 180° direction algorithm unit circuit (5), 315 The ° direction algorithm unit circuit (6), the 270° direction algorithm unit circuit (7) and the 225° direction algorithm unit circuit (8), respectively calculate the digital image pixel in the integral mask W _{135 °} , W _{90 °} , W _{45 °} , Approximate values of fractional integrals after W _0° , W1 _80° , W _315° , W _270° and W _225° . The 135° direction arithmetic unit circuit (1) to the 225° direction arithmetic unit circuit (8) consists of n (fractional order integral mask size numbers) first to third multipliers (13) to (15) and an addition Device (16) constitutes; The weights of these n multipliers are respectively in order:

The fractional order integral mask convolution circuit (11) is composed of the following 8 specific arithmetic unit circuits: 135 ° direction algorithm unit circuit (1) calculates the approximate value of the v-order fractional integral of pixel S _x (k+(n-1) (H+1)) under the action of integral mask W _{135 °} ; pixel S _x (k+( n-1)(H+1)) gray value or I component value and weight

Feed into the first multiplier (13) respectively, feed into the adder (16) after being multiplied; The gray value or I component value of pixel S _x (k+(n-1)(H+1)-1-H) and weight

Feed into the second multiplier (14) respectively, and feed into the adder (16) after being multiplied; By analogy, if 1≤m≤n, the pixel S _x (k+(n-1)(H+1)-( m-1)-(m-1)H) gray value or I component value and weight

Feed into the adder (16) after being multiplied, thereby obtain the approximation value of the v order fractional integral of pixel S _x (k+(n-1)(H+1)) in the direction of 135 ° in its 8-neighborhood direction ; 90 ° direction algorithm unit circuit (2) calculates the approximate value of the v-order fractional integral of pixel S _x (k+(n-1) (H+1)) under the action of integral mask W _{90 °} ; pixel S _x (k+( n-1)(H+1)) gray value or I component value and weight

Feed into the first multiplier (13) respectively, feed into the adder (16) after being multiplied; The gray value of pixel S _x (k+(n-1)(H+1)-H) or I component value and weight value

Feed into the second multiplier (14) respectively, and feed into the adder (16) after being multiplied; By analogy, if 1≤m≤n, the pixel S _x (k+(n-1)(H+1)-( m-1)H) gray value or I component value and weight

Feed into the adder (16) after being multiplied, thereby obtain the approximation of the v order fractional integral of pixel S _x (k+(n-1)(H+1)) in the 90° direction in its 8-neighborhood direction ; 45 ° direction algorithm unit circuit (3) calculates the approximate value of the v-order fractional integral of pixel S _x (k+(n-1) (H+1)) under the action of integral mask W _{45 °} ; pixel S _x (k+( n-1)(H+1)) gray value or I component value and weight

Feed into the first multiplier (13) respectively, and feed into the adder (16) after being multiplied; Gray value or I component value of pixel S _x (k+(n-1)(H+1)+1-H) and weight

Feed into the second multiplier (14) respectively, and feed into the adder (16) after being multiplied; By analogy, if 1≤m≤n, the pixel S _x (k+(n-1)(H+1)+( m-1)-(m-1)H) gray value or I component value and weight Feed into the adder (16) after being multiplied, thereby obtain the approximation of the v order fractional integral of pixel S _x (k+(n-1)(H+1)) in the direction of 45 ° in its 8-neighborhood direction ; 0° direction algorithm unit circuit (4) calculates the approximate value of the v-order fractional integral of pixel S _x (k+(n-1)(H+1)) under the action of integral mask W _0° ; pixel S _x (k+( n-1)(H+1)) gray value or I component value and weight

Feed into first multiplier (13) respectively, feed into _adder (16) after being multiplied; value

Feed into the second multiplier (14) respectively, and feed into the adder (16) after being multiplied; By analogy, if 1≤m≤n, the pixel S _x (k+(n-1)(H+1)+( Gray value or I component value and weight of m-1))

Feed into the adder (16) after being multiplied, thereby obtain the approximation of the v order fractional integral of the pixel S _x (k+(n-1)(H+1)) in the 0° direction in its 8-neighborhood direction ; The 180 ° direction algorithm unit circuit (5) calculates the approximate value of the v-order fractional integral of the pixel S _x (k+(n-1) (H+1)) under the action of the integral mask W _{180 °} ; the pixel S _x (k+( n-1)(H+1)) gray value or I component value and weight

Feed into first multiplier (13) respectively, feed into _adder (16) after being multiplied; value

Feed into the second multiplier (14) respectively, and feed into the adder (16) after being multiplied; By analogy, if 1≤m≤n, the pixel S _x (k+(n-1)(H+1)-( Gray value or I component value and weight of m-1))

Feed into the adder (16) after being multiplied, thereby obtain the approximate value of the v order fractional integral of pixel S _x (k+(n-1)(H+1)) in the direction of 180 ° in its 8-neighborhood direction ; The 315 ° direction algorithm unit circuit (6) calculates the approximate value of the v-order fractional integral of the pixel S _x (k+(n-1) (H+1)) under the integral mask W _{315 °} ; the pixel S _x (k+( n-1)(H+1)) gray value or I component value and weight Feed into the first multiplier (13) respectively, and feed into the adder (16) after being multiplied; The gray value or I component value of pixel S _x (k+(n-1)(H+1)+1+H) and weight

Feed into the second multiplier (14) respectively, and feed into the adder (16) after being multiplied; By analogy, if 1≤m≤n, the pixel S _x (k+(n-1)(H+1)+( Gray value or I component value and weight of m-1)+(m-1)H) Feed into the adder (16) after being multiplied, thereby obtain the approximation value of the v order fractional integral of pixel S _x (k+(n-1)(H+1)) in the direction of 315° in its 8-neighborhood direction ; The 270 ° direction algorithm unit circuit (7) calculates the approximate value of the v-order fractional integral of the pixel S _x (k+(n-1) (H+1)) under the integral mask W _{270 °} ; the pixel S _x (k+( n-1)(H+1)) gray value or I component value and weight

Feed into the first multiplier (13) respectively, feed into the adder (16) after being multiplied; The gray value of pixel S _x (k+(n-1)(H+1)+H) or I component value and weight value

Feed into the second multiplier (14) respectively, and feed into the adder (16) after being multiplied; By analogy, if 1≤m≤n, the pixel S _x (k+(n-1)(H+1)+( m-1)H) gray value or I component value and weight

Feed into the adder (16) after being multiplied, thereby obtain the approximation value of the v order fractional integral of pixel S _x (k+(n-1)(H+1)) in the direction of 270 ° in its 8-neighborhood direction ; 225 ° direction algorithm unit circuit (8) calculates the approximate value of the v-order fractional integral of pixel S _x (k+(n-1) (H+1)) under the action of integral mask W _{225 °} ; Pixel S _x (k+( n-1)(H+1)) gray value or I component value and weight

Feed into the first multiplier (13) respectively, and feed into the adder (16) after being multiplied; The gray value or I component value of pixel S _x (k+(n-1)(H+1)-1+H) and weight

Feed into the second multiplier (14) respectively, and feed into the adder (16) after being multiplied; By analogy, if 1≤m≤n, the pixel S _x (k+(n-1)(H+1)-( Gray value or I component value and weight of m-1)+(m-1)H)

Feed into the adder (16) after being multiplied, thereby obtain the approximation of the v order fractional integral of the pixel S _x (k+(n-1)(H+1)) in the direction of 225 ° in its 8-neighborhood direction . 4. the real-time efficient digital image fractional order integral filter according to claim 1, is characterized in that: the average value calculator (12) wherein calculates 135 ° of direction algorithm unit circuits (1) to 225 ° of direction algorithm unit circuits (8) The average value of these 8 direction algorithm unit circuit output values is different according to the character of the digital image of processing, and average value calculator (12) is divided into two kinds of structures: the first kind of structure, when processing digital gray scale image , the average value calculator (12) has 8 road inputs, 1 road output, feeds respectively the gray scale value modulus of 135 ° direction algorithm unit circuit (1) to 225 ° direction algorithm unit circuit (8), outputs above-mentioned 8 Feed in the average value of the gray value modulus (i.e. the v-order fractional integral value of the digital gray image pixel gray value); the second structure, when processing digital color images, the average value calculator (12) consists of 1 An average value calculator sub-circuit and a color space converter from HSI to RGB are formed, wherein the average value calculator sub-circuit and the average value calculator (12) used for processing digital grayscale images in the first structure The circuit structure and parameters are exactly the same, and it is used to calculate the average value of the v-order fractional integral approximation modulus of the 8 fed-in I component values; the color space converter converts the fed-in I component and H, S components into RGB The corresponding R, G, and B components in the color space output the v-order fractional integral value of the color image pixel s(x, y).

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