CN107818572A - Improvement particle image velocimetry robustness optical flow approach based on physics - Google Patents
Improvement particle image velocimetry robustness optical flow approach based on physics Download PDFInfo
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Abstract
The problem of present invention influences to cause optical flow computation robustness poor for visualization flow image edge-diffusion and noise and abnormity point, a kind of sane optical flow computation method based on physics is proposed to improve optical flow computation robustness.Carried algorithm introduces anisotropic filter to strengthen edge light stream robustness in based on physics optical flow approach, and increase penalty factor to reduce the influence of noise and abnormity point to optical flow computation, velocity field is then tried to achieve finally by alternative manner to solve Eular-Lagrange equations based on variational method minimization light stream energy function.Simulation result shows, compared with traditional Lucas Kanade, Horn Schunck, pyramid Lucas Kanade and optical flow computation method based on physics, carried algorithm can substantially reduce edge and corner regions light stream diffusion, improve the robustness for noise and abnormity point, so as to obtain that there is the velocity field of preferable robustness.
Description
Technical field
The invention belongs to UAV Video detection field, and in particular to a kind of improvement particle image velocimetry based on physics
Robustness optical flow approach.
Background technology
Fluid velocity field measurement is significant for understanding complex fluid in hydrodynamics and aerodynamic studies.
Fluid motion is a kind of typical non-rigid motion, and its calculating need to be based on image processing techniques.Can be by motion image sequence
Analysis obtains partial fluid motion vector size, direction and distribution situation, and then can obtain viscosity and vortex field distribution etc.
Physical characteristic.Under normal circumstances, due to moving object is transparent or not easily pass through optical device observation, need to will be seen that particle insert by
Thing is surveyed, to obtain fluid motion feature indirectly by estimation particle motion vector, this calls particle image velocimetry (PIV：
particle image velocimetry)。
Common optical flow computation method is (such as：Variation optical flow approach) resolution ratio can reach sub-pixel, meanwhile, fluid is continuous
Change in time and space characteristic micro- can assume that essence is identical with image sequence local space time in optical flow equation, thus, the PIV based on light stream is obtained
Extensive concern is arrived.
The it is proposeds such as Ma Pengfei are based on the PIV methods of estimation of Lucsa-kanade (LK) local light stream, and this algorithm calculates complicated
Spend low, robustness is preferable, but light stream field border is more fuzzy, and edge pixel point light stream estimation is poor, so as to gained optical flow field
It is more sparse.It is proposed to propose calculation based on pyramid LK multiple dimensioned optical flow algorithm to calculate PIV for this problem, Sun Lizhi etc.
Method can solve that velocity to moving target is excessive, and optical flow computation problem when consecutive frame continuity is not strong, gained light stream precision is higher than
LK algorithms, so obtained from optical flow field it is still more sparse.Therefore, the PIV that the proposition such as yellow Zhan is based on Horn-Schunck (HS) is counted
Calculation method, this algorithm can obtain dense optical flow field, but border easily obscures, and computation complexity is high, and robustness is poor.To enter one
Step improves light stream estimated accuracy, remaining to learn quick grade based on method that is global and locally combining to obtain higher light stream estimated accuracy, so
And light stream obtained by the method is poor in border district robustness.Tianshu L. etc. are based on the optical flow approach of physics from visualization
Image obtains high-resolution velocity field, yet with the influence of fringe region light stream diffusion and noise and abnormity point, light stream robustness
It is poor.
The content of the invention
Technical problem solved by the invention is to propose a kind of sane optical flow computation method based on physics to improve light
Stream calculation robustness.Anisotropic filter is introduced in the optical flow approach based on physics to strengthen the sane of edge light stream
Property, increase penalty factor is to reduce the influence of noise and abnormity point to optical flow computation, then based on variational method minimization light stream
Energy function tries to achieve velocity field to solve Euler-Lagrange equation, finally by alternative manner.
The present invention is that technical scheme is to provide a kind of improvement particle based on physics used by solving its technical problem
Image speed measurement robustness optical flow approach, this method comprise the following steps：
Step 1：Light stream based on physics solves
1. the optical flow constraint equation based on physics
Wherein, optical flow velocity in u=(u, v) representative image plane, g represent normalized image intensity, are included in f and ignore one
Rank is approximately spread and boundary condition, Calculated for Laplce
Son, D1It is diffusion coefficient, c represents KPT Scatter or absorption coefficient,Represent
Border item,Zero flux condition is represented, ψ represents scalar density, it is generally the case that when ψ is constant,Xiang Ke
It is ignored,Expression is projected in plane (x1,x2) on gradient operator, Γ1,Γ2Value and x1,x2Correlation,
λ1Represent Lagrange multiplier.
2. light stream smoothness constraint equation
Minimization smoothness constraint term EsIt is expressed as：
3. the light stream based on physics solves
With reference to optical flow equation (1) and light stream smoothness constraint equation (2) based on physics, following light stream energy letter can be obtained
Number：
Wherein, λ is Lagrange multiplier, a zonule in Ω representative pictures.
Step 2：Sane light stream based on physics solves
4. anisotropic filter
To try to achieve the higher light stream of robustness, the present invention uses the optical flow computation method of anisotropy parameter, anisotropy
The diffusion coefficient equation of the optical flow computation method of diffusion is：
Wherein, constant K controls the susceptibility at edge, generally determines its numerical value by experiment.
Based on formula (4), light stream energy function formula (3) is further rewritten as:
5. penalty term
Quadratic term is added in formula (5)As penalty term, after adding penalty term, formula (5) is rewritable to be：
Wherein, β2It is a small normal number.
6. the sane light stream based on physics solves
Formula (6) is minimized according to variation principle, following Euler-Lagrange (Euler-Lagrange) equation can be obtained：
Wherein：
By formula (8) and (9) bring into formula (7) can abbreviation be：
Wherein,Certain point and the difference approximate representation of its circumferential velocity average value can be used, i.e.,：
Convolution (11), formula (10) can be expressed as again：
Formula (12), which is arranged, to be obtained：
Solution formula (13) obtains un+1And vn+1：
Wherein, u, v mean value computation formula are as follows：
The derivative in x direction, y direction is sought filtered image using sobel operators；Subtracted each other using respective pixel in two frames
Mode seek the derivative in t directions, sobel templates used are as follows：
Step 3：Iterative
According to formula (14), based on Gauss-Seidel method iteratives, given if the difference of the adjacent light stream of iteration twice is less than
Determine threshold value, or iterations exceedes set-point, then terminates iteration.
The present invention is based on physics optical flow approach, introduces anisotropic filter and penalty factor, can substantially reduce edge
Spread with corner regions light stream, improve the robustness for noise and abnormity point, so as to obtain the speed with high-resolution
.Influence for visualization flow image edge-diffusion and noise spot and abnormity point causes optical flow computation robustness is poor to ask
Topic, the present invention propose a kind of sane optical flow approach based on physics to improve optical flow computation robustness.Carried algorithm based on
Anisotropic filter is introduced in the optical flow approach of physics to strengthen the robustness of edge light stream, increases penalty factor to reduce
The influence of noise and abnormity point to optical flow computation.Euler-glug is then solved based on variational method minimization light stream energy function
Bright day equation, velocity field is tried to achieve finally by alternative manner.
Brief description of the drawings
Fig. 1 is the flow chart that the present invention realizes；
Fig. 2 is original image；
Fig. 3 is based on light stream comparison diagram obtained by physics optical flow algorithm and classical optical flow algorithm；
Fig. 4 is penalty factor to the influence based on physics optical flow method；
Fig. 5 is anisotropy parameter to the influence based on physics optical flow method；
Fig. 6 is the robust algorithm comparison diagram based on physical method and based on physics.
Embodiment
Improvement particle image velocimetry robustness optical flow approach of the present invention based on physics is done into one below in conjunction with the accompanying drawings
Step is described in detail.
Optical flow constraint equation：
Under the hypothesis of adjacent image time interval and variation of image grayscale very little, Horn et al. proposes following gray scale light stream
Computation schema：When object of which movement can cause therewith, so as to form consecutive variations optical flow field, it is counted respective pixel brightness consecutive variations
It is as follows to learn model：
If some brightness of (x, y) t is I (x, y, t) in image, time Δt brightness is I (x+ Δs x, y+ Δ y, t+
Δ t), it is believed that brightness is constant when Δ t infinitesimals, then can obtain following equation：
I (x, y, t)=I (x+ Δs x, y+ Δ y, t+ Δ t) (1)
To above formula Taylor series expansion, when Δ t → 0 can obtain：
WillIt is rewritten as Ix,Iy,It, and makeThen above formula can be deformed into：
Ixu+Iyv+It=0 (3)
Wherein, u, v represent the two dimensional component of velocity.It can thus be concluded that the basic equation of optical flow computation.It should be noted that
There are two unknown parameters in formula (3), thus this problem can not solve.
Based on this, the present invention proposes a kind of improvement particle image velocimetry robustness optical flow approach based on physics to solve
Certainly above-mentioned technical problem.
As described in Figure 1, the flow chart realized for the present invention.
The first step：Optical flow algorithm based on physics
1. the optical flow constraint equation based on physics
For above-mentioned algorithm resolution ratio it is relatively low the problem of, Tianshu L. etc. propose a kind of high-resolution based on physics
Optical flow computation method.This algorithm gives the physical significance of light stream in detail, i.e., light stream and fluid in visual image or
The speed of grain is directly proportional.In addition, in image coordinate system, object space coordinate system can be stated by perspective projection transformation, institute
There is the project motion equation in the case of these to be stated with based on the optical flow equation of physics：
Wherein, optical flow velocity in u=(u, v) representative image plane, g represent normalized image intensity, are included in f and ignore one
Rank is approximately spread and boundary condition, Calculated for Laplce
Son, D1It is diffusion coefficient, c represents KPT Scatter or absorption coefficient,Represent
Border item,Zero flux condition is represented, ψ represents scalar density, it is generally the case that when ψ is constant,Xiang Ke
It is ignored.Expression is projected in plane (x1,x2) on gradient operator, Γ1,Γ2Value and x1,x2Correlation,
λ1Represent Lagrange multiplier.WhenAnd during f=0, the optical flow equation based on physics can be reduced to be based on HS light streams
The brightness constraint equation of algorithm.
2. light stream smoothness constraint equation
The constraints of the propositions such as Horn is then minimization smoothness constraint term Es
3. the light stream based on physics solves
With reference to the optical flow equation based on physics and light stream smoothness constraint equation, following light stream energy function can be obtained：
Wherein, λ is Lagrange multiplier, a zonule in Ω representative pictures.Minimization formula (6) can obtain following Euler-
Lagrange's equation：
Formula (7) can carry out light stream solution by standard finite difference method.Optical flow computation method based on physics, can
To obtain the light stream compared with HS method higher resolutions.However, due to easily being spread in border district based on the light stream of physics optical flow approach,
And easily influenceed by noise and abnormity point, light stream entirety robustness is poor.
Second step：Sane optical flow algorithm based on physics
1. anisotropic filter
It is proposed that anisotropy expands to reduce the influence, Marius D. etc. in corner and edge light stream diffusion couple optical flow computation
Scattered optical flow computation method, and give diffusion coefficient equation.The method can reduce propagation of the light stream at edge, strengthen edge-light
Flow robustness.Carrying two diffusion coefficient equations can represent as follows：
Wherein, sane light stream solves in the case of formula (8) is more suitable for high-contrast, and it is steady that formula (9) is more suitable for wide region
Strong light stream solves.
Constant K controls the susceptibility at edge, generally determines its numerical value by experiment.
In order to take into account the influence of contrast and area size to optical flow computation, the higher light stream of robustness is tried to achieve, this
Invention uses following diffusion coefficient equation：
Based on formula (10), light stream energy function formula (6) is rewritable to be:
Robustness preferable light stream in edge can be tried to achieve by minimization formula (11), yet with depositing for noise and abnormity point
The overall robustness of optical flow computation is not very high.In order to reduce the influence of noise and abnormity point to optical flow computation, strengthen light stream
Overall robustness.
2. penalty term
Quadratic term is added in formula (11)As penalty term, when noise and big abnormity point, light stream can be reduced
Calculation error, strengthen overall optical flow computation robustness.
3. the sane light stream based on physics solves
Penalty term is added, formula (11) is rewritable to be：
Wherein, β2It is a small normal number.
E (u, v) is minimized according to variation principle, following Euler-Lagrange (Euler-Lagrange) equation can be obtained：
Wherein：
By formula (14) and (15) bring into formula (13) can abbreviation be：
Wherein,Certain point and the difference approximate representation of its circumferential velocity average value can be used, i.e.,：
Convolution (17), formula (16) can be expressed as again：
Formula (18), which is arranged, to be obtained：
Solution formula (19) obtains un+1And vn+1：
So far, the sane optical flow computation method based on physics can be obtained.The method can be described in detail below：
(1) two continuous frames particle picture is read, it is filtered to reduce noise, and parameter needed for initialization.
(2) x direction, y directional derivative are asked filtered image using sobel operators；Subtracted each other using respective pixel in two frames
Mode seeks t directional derivatives.
(3) calculating u, v average is carried out using nine-point scheme.
(4) solution point shade of gray is calculated, setting speed smoothing weights coefficient (being set to 1), initial velocity is set to 0.
(5) according to formula (20), based on Gauss-Seidel method iteratives.
(6) if the difference of the adjacent light stream of iteration twice is less than given threshold value, or iterations exceedes set-point, then terminates and change
Generation.
The effect of the present invention can be further illustrated by following emulation：
(1) simulated conditions：
The present invention is by with classical HS, LK optical flow algorithm and pyramid LK optical flow algorithms and the optical flow algorithm based on physics
Contrasted, checking is carried based on the robust algorithm of physics in gained light stream density and the validity of edge light stream diffusion effect.
Simulation parameter sets as follows：Anisotropic filter relevant parameter α=5, β1=0.8, penalty relevant parameter β2=0.02,
Iteration error ε=1 × 10-3, iterations n=200.
Emulation uses sobel templates as follows：
U, v mean value computation formula are as follows：
(2) emulation content：
Emulation 1：
Fig. 2 is the continuous particle picture of two frames under uniform flow state, and following algorithm is all based on this calculating light stream.Fig. 3 (a),
(b) it in density is LK algorithms in the case of 5 that, (c), (d), which is, HS algorithms, pyramid LK algorithms and is obtained based on physical algorithms
Light stream image.From the figure 3, it may be seen that compared with HS optical flow methods, the light stream that LK optical flow algorithms obtain is sparse, and robustness is preferable.HS light streams
Though method can obtain dense optical flow, robustness is poor.Compared with HS algorithms and LK algorithms, light stream precision obtained by pyramid LK optical flow methods is higher,
Robustness is stronger, but light stream is sparse.This three kinds of optical flow algorithms, gained light stream resolution ratio are relatively low.Optical flow method based on physics can
Secure satisfactory grades and distinguish light stream, but edge easily spreads, and easily influenceed by noise and abnormity point, light stream is sparse, and robustness is poor.
Emulation 2：
Fig. 4 (a), (b) are respectively the light stream based on physics after optical flow computation method and addition penalty factor based on physics
Light stream image obtained by computational methods, β in penalty factor2=0.02.As shown in Figure 4, on the basis of the optical flow algorithm based on physics
Add preferable to the robustness of noise and abnormity point after penalty factor, but because light stream is spread in border district, cause gained light
Stream is more sparse.
Emulation 3：
Fig. 5 (a), after (b) is successively the optical flow computation method based on physics and the addition anisotropy parameter based on physics
The obtained light stream image of optical flow computation method, wherein diffusion coefficient equation parameter α=5, β1=0.8.It can be obtained by Fig. 5, the latter
Diffusion of the light stream in border district can be significantly reduced, light stream is more dense, but because noise and abnormity point influence, light stream robustness
It is poor.
Emulation 4：
Fig. 6 (a), the light stream image that (b) is the optical flow method based on physics and the sane optical flow algorithm based on physics obtains.
Wherein α=5, β1=0.8, β2=0.02, it can be obtained by Fig. 6, the sane optical flow algorithm based on physics can obviously reduce edge light stream and expand
Dissipate, and significantly reduce the influence of noise and abnormity point to optical flow computation, closeer light stream is obtained, so as to improve the steady of optical flow computation
Strong property.
In summary, compared with LK, HS, based on pyramid LK and based on the optical flow approach of physics, what the present invention was carried
Optical flow computation can be strengthened to noise and different by introducing the sane optical flow algorithm based on physics of anisotropic filter and penalty factor
The robustness often put, while significantly reduce border district light stream and spread and then obtain dense light stream, so as to significantly improve light
The robustness of stream calculation.
For visualization flow image edge-diffusion and noise spot and abnormity point influence cause optical flow computation robustness compared with
The problem of poor, the sane optical flow approach of the present invention is to improve optical flow computation robustness.Carried algorithm is in the light stream side based on physics
Method proposes a kind of middle introducing anisotropic filter based on physics to strengthen the robustness of edge light stream, increases penalty factor
To reduce the influence of noise and abnormity point to optical flow computation.Europe is then solved based on variational method minimization light stream energy function
Drawing-Lagrange's equation, velocity field is tried to achieve finally by alternative manner.Simulation result shows, with traditional Lucas-Kanade,
Horn-Schunck, pyramid Lucas-Kanade and optical flow computation method based on physics are compared, and carrying algorithm can be notable
Edge and corner regions light stream diffusion are reduced, improves the robustness for noise and abnormity point, it is preferably sane so as to obtain having
The velocity field of property.
Claims (1)
1. the improvement particle image velocimetry robustness optical flow approach based on physics, it is characterised in that：This method includes following step
Suddenly：
Step 1：Light stream based on physics solves
1. the optical flow constraint equation based on physics
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Show Lagrange multiplier；
2. light stream smoothness constraint equation
Minimization smoothness constraint term EsIt is expressed as：
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With reference to optical flow equation (1) and light stream smoothness constraint equation (2) based on physics, following light stream energy function can be obtained：
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Step 2：Sane light stream based on physics solves
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To try to achieve the higher light stream of robustness, the present invention uses the optical flow computation method of anisotropy parameter, anisotropy parameter
The diffusion coefficient equation of optical flow computation method be：
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Based on formula (4), light stream energy function formula (3) is further rewritten as:
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</msup>
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<mo>|</mo>
<mo>|</mo>
<mi>D</mi>
<mo>&dtri;</mo>
<mi>v</mi>
<mo>|</mo>
<msup>
<mo>|</mo>
<mn>2</mn>
</msup>
<mo>)</mo>
</mrow>
<msub>
<mi>dx</mi>
<mn>1</mn>
</msub>
<msub>
<mi>dx</mi>
<mn>2</mn>
</msub>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>5</mn>
<mo>)</mo>
</mrow>
</mrow>
5. penalty term
Quadratic term is added in formula (5)As penalty term, after adding penalty term, formula (5) is rewritable to be：
<mrow>
<mi>E</mi>
<mrow>
<mo>(</mo>
<mi>u</mi>
<mo>,</mo>
<mi>v</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msub>
<mo>&Integral;</mo>
<mi>&Omega;</mi>
</msub>
<mrow>
<msup>
<mrow>
<mo>&lsqb;</mo>
<mfrac>
<mrow>
<mo>&part;</mo>
<mi>g</mi>
</mrow>
<mrow>
<mo>&part;</mo>
<mi>t</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mo>&dtri;</mo>
<mo>.</mo>
<mrow>
<mo>(</mo>
<mi>g</mi>
<mi>u</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mi>f</mi>
<mo>&rsqb;</mo>
</mrow>
<mn>2</mn>
</msup>
<msub>
<mi>dx</mi>
<mn>1</mn>
</msub>
<msub>
<mi>dx</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>+</mo>
<msup>
<msub>
<mi>&beta;</mi>
<mn>2</mn>
</msub>
<mn>2</mn>
</msup>
<mrow>
<mo>(</mo>
<msup>
<mi>u</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<mi>v</mi>
<mn>2</mn>
</msup>
<mo>)</mo>
</mrow>
<msub>
<mi>dx</mi>
<mn>1</mn>
</msub>
<msub>
<mi>dx</mi>
<mn>2</mn>
</msub>
<mo>+</mo>
<mi>&lambda;</mi>
<msub>
<mo>&Integral;</mo>
<mi>&Omega;</mi>
</msub>
<mrow>
<mo>(</mo>
<mo>|</mo>
<mo>|</mo>
<mi>D</mi>
<mo>&dtri;</mo>
<mi>u</mi>
<mo>|</mo>
<msup>
<mo>|</mo>
<mn>2</mn>
</msup>
<mo>+</mo>
<mo>|</mo>
<mo>|</mo>
<mi>D</mi>
<mo>&dtri;</mo>
<mi>v</mi>
<mo>|</mo>
<msup>
<mo>|</mo>
<mn>2</mn>
</msup>
<mo>)</mo>
</mrow>
<msub>
<mi>dx</mi>
<mn>1</mn>
</msub>
<msub>
<mi>dx</mi>
<mn>2</mn>
</msub>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>6</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, β2It is a small normal number；
6. the sane light stream based on physics solves
Formula (6) is minimized according to variation principle, following Euler-Lagrange (Euler-Lagrange) equation can be obtained：
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mi>L</mi>
<mi>u</mi>
</msub>
<mo>-</mo>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>L</mi>
<msub>
<mi>u</mi>
<mi>x</mi>
</msub>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<mi>x</mi>
</mrow>
</mfrac>
<mo>-</mo>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>L</mi>
<msub>
<mi>u</mi>
<mi>y</mi>
</msub>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<mi>y</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mi>L</mi>
<mi>v</mi>
</msub>
<mo>-</mo>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>L</mi>
<msub>
<mi>v</mi>
<mi>x</mi>
</msub>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<mi>x</mi>
</mrow>
</mfrac>
<mo>-</mo>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>L</mi>
<msub>
<mi>v</mi>
<mi>y</mi>
</msub>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<mi>y</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>7</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein：
<mrow>
<mi>L</mi>
<mo>=</mo>
<mi>L</mi>
<mrow>
<mo>(</mo>
<mi>u</mi>
<mo>,</mo>
<mi>v</mi>
<mo>,</mo>
<msub>
<mi>u</mi>
<mi>x</mi>
</msub>
<mo>,</mo>
<msub>
<mi>u</mi>
<mi>y</mi>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mi>x</mi>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mi>y</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msup>
<mrow>
<mo>&lsqb;</mo>
<mfrac>
<mrow>
<mo>&part;</mo>
<mi>g</mi>
</mrow>
<mrow>
<mo>&part;</mo>
<mi>t</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mo>&dtri;</mo>
<mo>&CenterDot;</mo>
<mrow>
<mo>(</mo>
<mi>g</mi>
<mi>u</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mi>f</mi>
<mo>&rsqb;</mo>
</mrow>
<mn>2</mn>
</msup>
<mo>+</mo>
<msubsup>
<mi>&beta;</mi>
<mn>2</mn>
<mn>2</mn>
</msubsup>
<mrow>
<mo>(</mo>
<msup>
<mi>u</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<mi>v</mi>
<mn>2</mn>
</msup>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>&lambda;</mi>
<mrow>
<mo>(</mo>
<mo>|</mo>
<mo>|</mo>
<mi>D</mi>
<mo>&dtri;</mo>
<mi>u</mi>
<mo>|</mo>
<msup>
<mo>|</mo>
<mn>2</mn>
</msup>
<mo>+</mo>
<mo>|</mo>
<mo>|</mo>
<mi>D</mi>
<mo>&dtri;</mo>
<mi>v</mi>
<mo>|</mo>
<msup>
<mo>|</mo>
<mn>2</mn>
</msup>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>8</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mi>L</mi>
<mi>u</mi>
</msub>
<mo>=</mo>
<mn>2</mn>
<mo>&CenterDot;</mo>
<mo>&lsqb;</mo>
<msub>
<mi>g</mi>
<mi>t</mi>
</msub>
<mo>+</mo>
<mo>&dtri;</mo>
<mo>&CenterDot;</mo>
<mrow>
<mo>(</mo>
<mi>g</mi>
<mi>u</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mi>f</mi>
<mo>&rsqb;</mo>
<mo>*</mo>
<mfrac>
<mrow>
<mo>&part;</mo>
<mo>&dtri;</mo>
<mo>&CenterDot;</mo>
<mrow>
<mo>(</mo>
<mi>g</mi>
<mi>u</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mo>&part;</mo>
<mi>u</mi>
</mrow>
</mfrac>
<mo>-</mo>
<mn>2</mn>
<msup>
<msub>
<mi>&beta;</mi>
<mn>2</mn>
</msub>
<mn>2</mn>
</msup>
<mi>u</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>L</mi>
<msub>
<mi>u</mi>
<mi>x</mi>
</msub>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<mi>x</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>L</mi>
<msub>
<mi>u</mi>
<mi>y</mi>
</msub>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<mi>y</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>2</mn>
<msup>
<mi>&lambda;D</mi>
<mn>2</mn>
</msup>
<msup>
<mo>&dtri;</mo>
<mn>2</mn>
</msup>
<mi>u</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>9</mn>
<mo>)</mo>
</mrow>
</mrow>
By formula (8) and (9) bring into formula (7) can abbreviation be：
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mo>&lsqb;</mo>
<msub>
<mi>g</mi>
<mi>t</mi>
</msub>
<mo>+</mo>
<mo>&dtri;</mo>
<mo>&CenterDot;</mo>
<mrow>
<mo>(</mo>
<mi>g</mi>
<mi>u</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mi>f</mi>
<mo>&rsqb;</mo>
<mi>*</mi>
<mfrac>
<mrow>
<mo>&part;</mo>
<mo>&dtri;</mo>
<mo>&CenterDot;</mo>
<mrow>
<mo>(</mo>
<mi>g</mi>
<mi>u</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mo>&part;</mo>
<mi>u</mi>
</mrow>
</mfrac>
<mo>-</mo>
<msup>
<msub>
<mi>&beta;</mi>
<mn>2</mn>
</msub>
<mn>2</mn>
</msup>
<mi>u</mi>
<mo>-</mo>
<msup>
<mi>&lambda;D</mi>
<mn>2</mn>
</msup>
<msup>
<mo>&dtri;</mo>
<mn>2</mn>
</msup>
<mi>u</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>&lsqb;</mo>
<msub>
<mi>g</mi>
<mi>t</mi>
</msub>
<mo>+</mo>
<mo>&dtri;</mo>
<mo>&CenterDot;</mo>
<mrow>
<mo>(</mo>
<mi>g</mi>
<mi>u</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mi>f</mi>
<mo>&rsqb;</mo>
<mi>*</mi>
<mfrac>
<mrow>
<mo>&part;</mo>
<mo>&dtri;</mo>
<mo>&CenterDot;</mo>
<mrow>
<mo>(</mo>
<mi>g</mi>
<mi>u</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mo>&part;</mo>
<mi>v</mi>
</mrow>
</mfrac>
<mo>-</mo>
<msup>
<msub>
<mi>&beta;</mi>
<mn>2</mn>
</msub>
<mn>2</mn>
</msup>
<mi>v</mi>
<mo>-</mo>
<msup>
<mi>&lambda;D</mi>
<mn>2</mn>
</msup>
<msup>
<mo>&dtri;</mo>
<mn>2</mn>
</msup>
<mi>v</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>10</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, ▽ (gu)=g ▽ (u)+u ▽ g, certain point and the difference approximate representation of its circumferential velocity average value can be used, i.e.,：
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<msup>
<mo>&dtri;</mo>
<mn>2</mn>
</msup>
<mi>u</mi>
<mo>=</mo>
<msup>
<mi>u</mi>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>-</mo>
<msup>
<mover>
<mi>u</mi>
<mo>&OverBar;</mo>
</mover>
<mi>n</mi>
</msup>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msup>
<mo>&dtri;</mo>
<mn>2</mn>
</msup>
<mi>v</mi>
<mo>=</mo>
<msup>
<mi>v</mi>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>-</mo>
<msup>
<mover>
<mi>v</mi>
<mo>&OverBar;</mo>
</mover>
<mi>n</mi>
</msup>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>11</mn>
<mo>)</mo>
</mrow>
</mrow>
Convolution (11), formula (10) can be expressed as again：
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mo>(</mo>
<msub>
<mi>g</mi>
<mi>t</mi>
</msub>
<mo>+</mo>
<msub>
<mi>ug</mi>
<mi>x</mi>
</msub>
<mo>+</mo>
<msub>
<mi>vg</mi>
<mi>y</mi>
</msub>
<mo>+</mo>
<msub>
<mi>gu</mi>
<mi>x</mi>
</msub>
<mo>+</mo>
<msub>
<mi>gv</mi>
<mi>y</mi>
</msub>
<mo>-</mo>
<mi>f</mi>
<mo>)</mo>
<mo>&CenterDot;</mo>
<msub>
<mi>g</mi>
<mi>x</mi>
</msub>
<mo>-</mo>
<msub>
<mi>&beta;</mi>
<mn>2</mn>
</msub>
<mi>u</mi>
<mo>-</mo>
<msup>
<mi>&lambda;D</mi>
<mn>2</mn>
</msup>
<mo>(</mo>
<msup>
<mi>u</mi>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>-</mo>
<msup>
<mover>
<mi>u</mi>
<mo>&OverBar;</mo>
</mover>
<mi>n</mi>
</msup>
<mo>)</mo>
<mo>=</mo>
<mn>0</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>(</mo>
<msub>
<mi>g</mi>
<mi>t</mi>
</msub>
<mo>+</mo>
<msub>
<mi>ug</mi>
<mi>x</mi>
</msub>
<mo>+</mo>
<msub>
<mi>vg</mi>
<mi>y</mi>
</msub>
<mo>+</mo>
<msub>
<mi>gu</mi>
<mi>x</mi>
</msub>
<mo>+</mo>
<msub>
<mi>gv</mi>
<mi>y</mi>
</msub>
<mo>-</mo>
<mi>f</mi>
<mo>)</mo>
<mo>&CenterDot;</mo>
<msub>
<mi>g</mi>
<mi>y</mi>
</msub>
<mo>-</mo>
<msub>
<mi>&beta;</mi>
<mn>2</mn>
</msub>
<mi>v</mi>
<mo>-</mo>
<msup>
<mi>&lambda;D</mi>
<mn>2</mn>
</msup>
<mo>(</mo>
<msup>
<mi>v</mi>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>-</mo>
<msup>
<mover>
<mi>v</mi>
<mo>&OverBar;</mo>
</mover>
<mi>n</mi>
</msup>
<mo>)</mo>
<mo>=</mo>
<mn>0</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>12</mn>
<mo>)</mo>
</mrow>
</mrow>
Formula (12), which is arranged, to be obtained：
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mo>(</mo>
<msup>
<mi>&lambda;D</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<msubsup>
<mi>g</mi>
<mi>x</mi>
<mn>2</mn>
</msubsup>
<mo>-</mo>
<msubsup>
<mi>&beta;</mi>
<mn>2</mn>
<mn>2</mn>
</msubsup>
<mo>)</mo>
<msup>
<mi>u</mi>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>+</mo>
<msup>
<mi>v</mi>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>.</mo>
<msub>
<mi>g</mi>
<mi>x</mi>
</msub>
<msub>
<mi>g</mi>
<mi>y</mi>
</msub>
<mo>+</mo>
<msub>
<mi>g</mi>
<mi>t</mi>
</msub>
<msub>
<mi>g</mi>
<mi>x</mi>
</msub>
<mo>+</mo>
<msub>
<mi>gg</mi>
<mi>x</mi>
</msub>
<msub>
<mi>u</mi>
<mi>x</mi>
</msub>
<mo>+</mo>
<msub>
<mi>gg</mi>
<mi>x</mi>
</msub>
<msub>
<mi>v</mi>
<mi>y</mi>
</msub>
<mo>-</mo>
<mi>f</mi>
<mo>&CenterDot;</mo>
<msub>
<mi>g</mi>
<mi>x</mi>
</msub>
<mo>-</mo>
<msup>
<mi>&lambda;D</mi>
<mn>2</mn>
</msup>
<msup>
<mover>
<mi>u</mi>
<mo>&OverBar;</mo>
</mover>
<mi>n</mi>
</msup>
<mo>=</mo>
<mn>0</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>(</mo>
<msup>
<mi>&lambda;D</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<msubsup>
<mi>g</mi>
<mi>y</mi>
<mn>2</mn>
</msubsup>
<mo>-</mo>
<msubsup>
<mi>&beta;</mi>
<mn>2</mn>
<mn>2</mn>
</msubsup>
<mo>)</mo>
<msup>
<mi>v</mi>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>+</mo>
<msup>
<mi>u</mi>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>.</mo>
<msub>
<mi>g</mi>
<mi>x</mi>
</msub>
<msub>
<mi>g</mi>
<mi>y</mi>
</msub>
<mo>+</mo>
<msub>
<mi>g</mi>
<mi>t</mi>
</msub>
<msub>
<mi>g</mi>
<mi>y</mi>
</msub>
<mo>+</mo>
<msub>
<mi>gg</mi>
<mi>y</mi>
</msub>
<msub>
<mi>u</mi>
<mi>x</mi>
</msub>
<mo>+</mo>
<msub>
<mi>gg</mi>
<mi>y</mi>
</msub>
<msub>
<mi>v</mi>
<mi>y</mi>
</msub>
<mo>-</mo>
<mi>f</mi>
<mo>&CenterDot;</mo>
<msub>
<mi>g</mi>
<mi>y</mi>
</msub>
<mo>-</mo>
<msup>
<mi>&lambda;D</mi>
<mn>2</mn>
</msup>
<msup>
<mover>
<mi>v</mi>
<mo>&OverBar;</mo>
</mover>
<mi>n</mi>
</msup>
<mo>=</mo>
<mn>0</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>13</mn>
<mo>)</mo>
</mrow>
</mrow>
Solution formula (13) obtains un+1And vn+1：
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<msup>
<mi>u</mi>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>=</mo>
<mfrac>
<mrow>
<mo>(</mo>
<msup>
<mi>&lambda;D</mi>
<mn>2</mn>
</msup>
<mo>-</mo>
<msup>
<msub>
<mi>&beta;</mi>
<mn>2</mn>
</msub>
<mn>2</mn>
</msup>
<mo>)</mo>
<mo>&lsqb;</mo>
<msub>
<mi>g</mi>
<mi>x</mi>
</msub>
<mo>(</mo>
<msub>
<mi>g</mi>
<mi>t</mi>
</msub>
<mo>+</mo>
<mi>g</mi>
<mo>&CenterDot;</mo>
<msubsup>
<mover>
<mi>u</mi>
<mo>&OverBar;</mo>
</mover>
<mi>x</mi>
<mi>n</mi>
</msubsup>
<mo>+</mo>
<mi>g</mi>
<mo>&CenterDot;</mo>
<msubsup>
<mover>
<mi>v</mi>
<mo>&OverBar;</mo>
</mover>
<mi>y</mi>
<mi>n</mi>
</msubsup>
<mo>-</mo>
<mi>f</mi>
<mo>)</mo>
<mo>-</mo>
<msup>
<mi>&lambda;D</mi>
<mn>2</mn>
</msup>
<msup>
<mover>
<mi>u</mi>
<mo>&OverBar;</mo>
</mover>
<mi>n</mi>
</msup>
<mo>&rsqb;</mo>
<mo>-</mo>
<msup>
<mi>&lambda;D</mi>
<mn>2</mn>
</msup>
<mo>(</mo>
<msup>
<mover>
<mi>u</mi>
<mo>&OverBar;</mo>
</mover>
<mi>n</mi>
</msup>
<msubsup>
<mi>g</mi>
<mi>y</mi>
<mi>n</mi>
</msubsup>
<mo>-</mo>
<msup>
<mover>
<mi>v</mi>
<mo>&OverBar;</mo>
</mover>
<mi>n</mi>
</msup>
<msub>
<mi>g</mi>
<mi>x</mi>
</msub>
<msub>
<mi>g</mi>
<mi>y</mi>
</msub>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<msup>
<mi>&lambda;D</mi>
<mn>2</mn>
</msup>
<mo>-</mo>
<msup>
<msub>
<mi>&beta;</mi>
<mn>2</mn>
</msub>
<mn>2</mn>
</msup>
<mo>)</mo>
<mo>&lsqb;</mo>
<msup>
<mrow>
<mo>(</mo>
<msup>
<mi>&lambda;D</mi>
<mn>2</mn>
</msup>
<mo>-</mo>
<msup>
<msub>
<mi>&beta;</mi>
<mn>2</mn>
</msub>
<mn>2</mn>
</msup>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
<mo>+</mo>
<msubsup>
<mi>g</mi>
<mi>y</mi>
<mn>2</mn>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>g</mi>
<mi>x</mi>
<mn>2</mn>
</msubsup>
<mo>&rsqb;</mo>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msup>
<mi>v</mi>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>=</mo>
<mfrac>
<mrow>
<mo>(</mo>
<msup>
<mi>&lambda;D</mi>
<mn>2</mn>
</msup>
<mo>-</mo>
<msup>
<msub>
<mi>&beta;</mi>
<mn>2</mn>
</msub>
<mn>2</mn>
</msup>
<mo>)</mo>
<mo>&lsqb;</mo>
<msub>
<mi>g</mi>
<mi>y</mi>
</msub>
<mo>(</mo>
<msub>
<mi>g</mi>
<mi>t</mi>
</msub>
<mo>+</mo>
<mi>g</mi>
<mo>&CenterDot;</mo>
<msup>
<msub>
<mover>
<mi>u</mi>
<mo>&OverBar;</mo>
</mover>
<mi>x</mi>
</msub>
<mi>n</mi>
</msup>
<mo>+</mo>
<mi>g</mi>
<mo>&CenterDot;</mo>
<msup>
<msub>
<mover>
<mi>v</mi>
<mo>&OverBar;</mo>
</mover>
<mi>y</mi>
</msub>
<mi>n</mi>
</msup>
<mo>-</mo>
<mi>f</mi>
<mo>)</mo>
<mo>-</mo>
<msup>
<mi>&lambda;D</mi>
<mn>2</mn>
</msup>
<msup>
<mover>
<mi>v</mi>
<mo>&OverBar;</mo>
</mover>
<mi>n</mi>
</msup>
<mo>&rsqb;</mo>
<mo>-</mo>
<msup>
<mi>&lambda;D</mi>
<mn>2</mn>
</msup>
<mo>(</mo>
<msup>
<mover>
<mi>v</mi>
<mo>&OverBar;</mo>
</mover>
<mi>n</mi>
</msup>
<msubsup>
<mi>g</mi>
<mi>x</mi>
<mn>2</mn>
</msubsup>
<mo>-</mo>
<msup>
<mover>
<mi>u</mi>
<mo>&OverBar;</mo>
</mover>
<mi>n</mi>
</msup>
<msub>
<mi>g</mi>
<mi>x</mi>
</msub>
<msub>
<mi>g</mi>
<mi>y</mi>
</msub>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<msup>
<mi>&lambda;D</mi>
<mn>2</mn>
</msup>
<mo>-</mo>
<msup>
<msub>
<mi>&beta;</mi>
<mn>2</mn>
</msub>
<mn>2</mn>
</msup>
<mo>)</mo>
<mo>&lsqb;</mo>
<msup>
<mrow>
<mo>(</mo>
<msup>
<mi>&lambda;D</mi>
<mn>2</mn>
</msup>
<mo>-</mo>
<msup>
<msub>
<mi>&beta;</mi>
<mn>2</mn>
</msub>
<mn>2</mn>
</msup>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
<mo>+</mo>
<msubsup>
<mi>g</mi>
<mi>y</mi>
<mn>2</mn>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>g</mi>
<mi>x</mi>
<mn>2</mn>
</msubsup>
<mo>&rsqb;</mo>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>14</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, u, v mean value computation formula are as follows：
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mover>
<mi>u</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
</msub>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>6</mn>
</mfrac>
<mrow>
<mo>(</mo>
<msub>
<mover>
<mi>u</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>-</mo>
<mn>1</mn>
<mo>,</mo>
<mi>k</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mover>
<mi>u</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mo>-</mo>
<mn>1</mn>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mover>
<mi>u</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
<mo>,</mo>
<mi>k</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mover>
<mi>u</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>i</mi>
<mo>+</mo>
<mn>1</mn>
<mo>,</mo>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>12</mn>
</mfrac>
<mrow>
<mo>(</mo>
<msub>
<mover>
<mi>u</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>i</mi>
<mo>-</mo>
<mn>1</mn>
<mo>,</mo>
<mi>j</mi>
<mo>-</mo>
<mn>1</mn>
<mo>,</mo>
<mi>k</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mover>
<mi>u</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>i</mi>
<mo>-</mo>
<mn>1</mn>
<mo>,</mo>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
<mo>,</mo>
<mi>k</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mover>
<mi>u</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>i</mi>
<mo>+</mo>
<mn>1</mn>
<mo>,</mo>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
<mo>,</mo>
<mi>k</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mover>
<mi>u</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>-</mo>
<mn>1</mn>
<mo>,</mo>
<mi>k</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mover>
<mi>v</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
</msub>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>6</mn>
</mfrac>
<mrow>
<mo>(</mo>
<msub>
<mover>
<mi>v</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>-</mo>
<mn>1</mn>
<mo>,</mo>
<mi>k</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mover>
<mi>v</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mo>-</mo>
<mn>1</mn>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mover>
<mi>v</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
<mo>,</mo>
<mi>k</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mover>
<mi>v</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>i</mi>
<mo>+</mo>
<mn>1</mn>
<mo>,</mo>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>12</mn>
</mfrac>
<mrow>
<mo>(</mo>
<msub>
<mover>
<mi>v</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>i</mi>
<mo>-</mo>
<mn>1</mn>
<mo>,</mo>
<mi>j</mi>
<mo>-</mo>
<mn>1</mn>
<mo>,</mo>
<mi>k</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mover>
<mi>v</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>i</mi>
<mo>-</mo>
<mn>1</mn>
<mo>,</mo>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
<mo>,</mo>
<mi>k</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mover>
<mi>v</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>i</mi>
<mo>+</mo>
<mn>1</mn>
<mo>,</mo>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
<mo>,</mo>
<mi>k</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mover>
<mi>v</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>-</mo>
<mn>1</mn>
<mo>,</mo>
<mi>k</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
The derivative in x direction, y direction is sought filtered image using sobel operators；The side subtracted each other using respective pixel in two frames
Formula seeks the derivative in t directions, and sobel templates used are as follows：
<mfenced open = "" close = "">
<mtable>
<mtr>
<mtd>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mo>-</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mfenced>
</mtd>
<mtd>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>-</mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
</mtd>
</mtr>
</mtable>
</mfenced>
Step 3：Iterative
According to formula (14), based on Gauss-Seidel method iteratives, if the difference of the adjacent light stream of iteration twice is less than given threshold
Value, or iterations exceed set-point, then terminate iteration.
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Cited By (5)
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| CN108922627A (en) * | 2018-06-28 | 2018-11-30 | 福州大学 | Blood flow emulation mode based on data-driven |
| CN110223328A (en) * | 2019-05-14 | 2019-09-10 | 焦作大学 | A kind of improvement particle image velocimetry robustness optical flow approach based on physics |
| CN112446179A (en) * | 2020-12-10 | 2021-03-05 | 华中科技大学 | Fluid velocity measuring method based on variable split optical flow model |
| CN112884818A (en) * | 2019-11-29 | 2021-06-01 | 中移物联网有限公司 | Dense optical flow calculation method, dense optical flow calculation device, electronic device, and storage medium |
| CN117422735A (en) * | 2023-12-13 | 2024-01-19 | 南方科技大学 | Particle velocity measurement method, particle velocity measurement device, electronic apparatus, and storage medium |
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Cited By (9)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN108922627A (en) * | 2018-06-28 | 2018-11-30 | 福州大学 | Blood flow emulation mode based on data-driven |
| CN108922627B (en) * | 2018-06-28 | 2021-04-27 | 福州大学 | Blood flow simulation method based on data driving |
| CN110223328A (en) * | 2019-05-14 | 2019-09-10 | 焦作大学 | A kind of improvement particle image velocimetry robustness optical flow approach based on physics |
| CN112884818A (en) * | 2019-11-29 | 2021-06-01 | 中移物联网有限公司 | Dense optical flow calculation method, dense optical flow calculation device, electronic device, and storage medium |
| CN112884818B (en) * | 2019-11-29 | 2023-04-14 | 中移物联网有限公司 | Dense optical flow calculation method, dense optical flow calculation device, electronic device, and storage medium |
| CN112446179A (en) * | 2020-12-10 | 2021-03-05 | 华中科技大学 | Fluid velocity measuring method based on variable split optical flow model |
| CN112446179B (en) * | 2020-12-10 | 2024-05-14 | 华中科技大学 | Fluid velocity measurement method based on variational optical flow model |
| CN117422735A (en) * | 2023-12-13 | 2024-01-19 | 南方科技大学 | Particle velocity measurement method, particle velocity measurement device, electronic apparatus, and storage medium |
| CN117422735B (en) * | 2023-12-13 | 2024-03-26 | 南方科技大学 | Particle velocity measurement method, particle velocity measurement device, electronic apparatus, and storage medium |
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