CN101625763A - Method for measuring similarity of spatial color histogram - Google Patents

Abstract

The invention discloses a method for measuring similarity of a spatial color histogram, belonging to a digital image processing and analysis method, which aims at the problem of insufficient accuracy in the existing measurement method so as to further raise the accuracy of image similarity calculations. The method comprises the following steps: computing proportional similarity of pixels; computing the similarity of positional information of the pixels; measuring the similarity of the spatial color histograms, which means traversing all the three-dimensional squares in two spatial color histograms S and S' according to the step 1 and 2 to obtain the similarity of the two spatial color histograms. The invention approximates pixel distribution in each three-dimensional square in the spatial color histogram to Gaussian distribution, the function spaces of the probability density functions thereof constitute Lie group spaces, on such a basis, similarity of the positional information of the pixels is provided, with which the measurement method of the invention is induced in combination with the proportional similarity of the pixels. Accordingly, compared with the present method, the method has better tracking effect in tracking algorithms.

Description

Method for measuring similarity of spatial color histogram

Technical field

The invention belongs to Digital Image Processing and analytical approach, be specifically related to a kind of method for measuring similarity of spatial color histogram.

Background technology

Histogram is a kind of important means of describing image, and spatial color histogram is histogrammic a kind of expansion, has write down the colouring information of image, has also write down location of pixels information in the two field picture to a certain extent.Spatial color histogram comprises color histogram and location of pixels information two parts, and wherein location of pixels information is made of the coordinate mean vector and the covariance matrix of all pixels in a color square again.Spatial color histogram is formed s={s by m 3 D stereo square _u} _U=1...mU 3 D stereo square s _uBe expressed as tlv triple a: s _u=＜h _u, μ _u, ∑ _u; This tlv triple is accounted for the amount of pixels ratio h of the total number of pixels of image by the number of pixels in this square _u, the pixel coordinate average μ in this square _uAnd the pixel coordinate covariance matrix ∑ in this square _uForm:

$h_u=\frac{1}{N}\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{\mathrm{\δ}}_{\mathrm{iu}};$

${\mathrm{\μ}}_u=\frac{1}{\underset{j=1}{\overset{N}{\mathrm{\Σ}}}{\mathrm{\δ}}_{\mathrm{ju}}}\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{X}_i{\mathrm{\δ}}_{\mathrm{iu}};$

${\mathrm{\Σ}}_u=\frac{1}{\underset{j=1}{\overset{N}{\mathrm{\Σ}}}{\mathrm{\δ}}_{\mathrm{ju}}-1}\underset{i=1}{\overset{N}{\mathrm{\Σ}}}(X_i-{\mathrm{\μ}}_u){(X_i-{\mathrm{\μ}}_u)}^T{\mathrm{\δ}}_{\mathrm{iu}};$

In the following formula, X _iBe vector [x _i, y _i] ^T, the two-dimensional space coordinate of the i pixel that expression is lined by line scan to image; N is the total number of pixels of image; δ _IuBe a δ function, in the time of in the i pixel drops on u 3 D stereo square, value is 1, otherwise is 0.

The location of pixels information description in each square (bin) the coordinate average and the coordinate covariance of all pixels.Because average and covariance are described the unique effect in the positional information, people more and more are applied to them to image tracing and image retrieval.At computer vision field, determine the similarity degree of original image and target image by calculating corresponding spatial color histogram similarity.

Adopt the distance of two vectorial μ of mahalanobis distance (Mahalanobis distance) expression and μ ', its computing formula is $M(\mathrm{\μ},{\mathrm{\μ}}^{\′})=\sqrt{{(\mathrm{\μ}-{\mathrm{\μ}}^{\′})}^T{A}^{-1}(\mathrm{\μ}-{\mathrm{\μ}}^{\′})},$ Here A is a matrix, be used for given vector in calculating each the dimension weights.Based on mahalanobis distance, the formula that proposes computer memory color histogram similarity is Birchfiled in " Spatiograms versus histograms forregion-based tracking " (Conference on Computer Vision and PatternRecognition (CVPR 2005)) literary composition:

$\mathrm{\ρ}(s,s^{\′})=\underset{u-1}{\overset{m}{\mathrm{\Σ}}}\frac{\sqrt{h_u{h}_u^{\′}}}{2\mathrm{\π}{\left|{\widehat{\mathrm{\Σ}}}_u\right|}^{\frac{1}{2}}}{e}^{-\frac{1}{2}{({\mathrm{\μ}}_u-{{\mathrm{\μ}}^{\′}}_u)}^T{\widehat{\mathrm{\Σ}}}_u^{-1}({\mathrm{\μ}}_u-{{\mathrm{\μ}}^{\′}}_u)}$

In the formula, ${\widehat{\mathrm{\Σ}}}_u^{-1}={\mathrm{\Σ}}_u^{-1}+{\left({\mathrm{\Σ}}_u^{\′}\right)}^{-1}.$ Conaire is at " An improved spatiogram similaritymeasure for robust object localization " (In International Conference onAcoustics, Speech, and Signal Processing (ICASSP 2007)) in the literary composition based on mahalanobis distance, the formula that proposes computer memory color histogram similarity is:

$\mathrm{\ρ}(s,s^{\′})=\underset{u=1}{\overset{m}{\mathrm{\Σ}}}\frac{2\sqrt{h_u{h}_u^{\′}}{\left|{\mathrm{\Σ}}_u{\mathrm{\Σ}}_u^{\′}\right|}^{\frac{1}{4}}}{{|{\mathrm{\Σ}}_u+{\mathrm{\Σ}}_u^{\′}|}^{\frac{1}{2}}}{e}^{-\frac{1}{2}{({\mathrm{\μ}}_u-{\mathrm{\μ}}_u^{\′})}^T{\left(2({\mathrm{\Σ}}_u+{\mathrm{\Σ}}_u^{\′})\right)}^{-1}({\mathrm{\μ}}_u-{\mathrm{\μ}}_u^{\′})}$

But the location of pixels information in the spatial color histogram is not to define in vector space, and in the scalar space, mahalanobis distance is inapplicable, and is not accurate enough based on mahalanobis distance computer memory color histogram similarity.

Summary of the invention

The invention provides a kind of method for measuring similarity of spatial color histogram,, further improve the accuracy that image similarity calculates at the existing not accurate enough problem of measure.

A kind of method for measuring similarity of spatial color histogram of the present invention comprises the steps:

One. calculating pixel amount degree of being in similar proportion step: the amount of pixels ratio h of u 3 D stereo square of computer memory color histogram s and the middle correspondence of s ' _uWith h ' _uSimilarity ψ _u:

${\mathrm{\ψ}}_u=\sqrt{h_u{h}_u^{\′}}$

Two. calculating pixel positional information similarity step: the location of pixels information similarity of u 3 D stereo square of computer memory color histogram s and the middle correspondence of s ' comprises following substep:

2-1. with square interior pixel positional information approximate representation is Gaussian distribution;

With Gaussian distribution N (μ _u, ∑ _u) the location of pixels information of u corresponding 3 D stereo square among the representation space color histogram s, the average of this Gaussian distribution is square interior pixel coordinate mean vector μ _u, the covariance matrix of this Gaussian distribution is the pixel coordinate covariance matrix ∑ in this square _u, the function space of the Gaussian probability-density function of this Gaussian distribution correspondence constitutes a Lie group space;

Equally, with Gaussian distribution N (μ ' _u, ∑ ' _u) the location of pixels information of u 3 D stereo square of the middle correspondence of representation space color histogram s ';

2-2. Gaussian distribution is converted to the positive definite affine transformation matrix;

With described Gaussian distribution N (μ _u, ∑ _u) use affine transformation matrix M _uExpression:

$M_u=\left[\begin{array}{cc}{P}_u& {\mathrm{\&mu;}}_{\mathrm{<figure-callout\; id="0"\; label="u"\; filenames="A2009100617010014H1.png"\; state="\{\{state\}\}">u</figure-callout>}}\\ 0& 1\end{array}\right],$ Be positive definite matrix, wherein a μ _uBe the average of Gaussian distribution, P _uIt is covariance matrix ∑ with Gaussian distribution _uDecompose the matrix that obtains by Cholesky, decomposition formula is: ∑ _u=P _uP _u ^TLimiting the matrix ∑ _uWhen being the matrix of symmetric positive definite, decomposing could unique P that obtains _u, and P _uMatrix is the positive definite lower triangular matrix; So with the affine transformation matrix M that ties up _uAlso constitute a Lie group space;

Equally, with described Gaussian distribution N (μ ' _u, ∑ ' _u) usefulness affine transformation matrix M ' _uExpression;

2-3. the geodesic line of calculating two affine transformation matrixs is apart from d (M _u, M ' _u):

d(M _u，M′ _u)＝||log(M _u ^-1M′ _u)||，

Here || || mould formula, M are asked in expression _u, M ' _uBe affine transformation matrix, log () asks the logarithm operation;

2-4. with geodesic distance d (M _u, M ' _u) be converted to location of pixels information similarity

In the formula, mapping parameters θ is 0～1, is estimated to obtain by test;

Three. metric space color histogram similarity step: by above-mentioned steps one～two, travel through all 3 D stereo squares among two spatial color histogram s and the s ', the amount of pixels degree of being in similar proportion of each 3 D stereo square and location of pixels information similarity are multiplied each other and superpose, obtain the similarity ρ (s, s ') of two spatial color histograms:

Described method for measuring similarity of spatial color histogram is characterized in that:

Among the substep 2-4 of described calculating pixel positional information similarity step, mapping parameters θ obtains by analyzing experimental data: location of pixels information similarity

For geodesic line apart from d (M _u, M ' _u) monotonic decreasing function, the value of mapping parameters θ is estimated by test, makes at d (M _u, M ' _u) in the span [0, D], location of pixels information similarity

All greater than 0.05; Wherein D is d (M in the real image _u, M ' _u) maximum occurrences, be arithmetic number.

In the step 2 of the present invention, location of pixels information similarity

Can not measure with Pasteur's distance simply, satisfy Gaussian distribution because the space distribution of pixel is approximate, and the function space of Gaussian probability-density function is not a linear space, but this similarity must be asked with the method for Lie group in a Lie group space.

Among the substep 2-2 of step 2, the affine transformation matrix M of gained _uCan represent a Gaussian distribution N (μ _u, ∑ _u), its reason is as follows: make z ₀Be a n dimension random vector, this vectorial element is independent identically distributed variable, and its average and variance are respectively 0 and 1.Utilize M _uThe matrixing expression formula of carrying out is $\left[\begin{array}{c}\mathrm{<figure-callout\; id="1"\; label="Z"\; filenames="A2009100617010013H1.png"\; state="\{\{state\}\}">Z</figure-callout>}\\ 1\end{array}\right]=M_u*\left[\begin{array}{c}{Z}_0\\ 1\end{array}\right]=\left[\begin{array}{cc}{P}_u& {\mathrm{\&mu;}}_u\\ 0& 1\end{array}\right]\left[\begin{array}{c}{\mathrm{<figure-callout\; id="0"\; label="Z"\; filenames="A2009100617010014H1.png"\; state="\{\{state\}\}">Z</figure-callout>}}_0\\ 1\end{array}\right],$ The Z=P that this conversion of process obtains _uZ ₀+ μ _uBe a random vector that satisfies Gaussian distribution, its average and covariance are respectively μ _uAnd P _uP _u ^T

Because affine transformation matrix all seals for the matrix multiplication and the operation of inverting, and constitutes a Lie group, the similarity that need measure two elements on the Lie group space with the affine transformation matrix of tieing up.

Among the substep 2-3 of step 2, the Lie group space is a stream shape, calculates in the general computer memory of distance of two elements in Lie group space 2 curve distance, and curve distance is the shortest between 2 is called as the geodesic line distance.Calculate this curve distance, must earlier element be projected on the tangent space, again at the mould of asking vector between element on the tangent space.Above-mentioned M _uBe an element in the Lie group, M _u ^-1M ' _uIt also is an element in the Lie group.So with above-mentioned M _u ^-1M ' _uBe mapped to the tangent space, the mapping formula is m _u=log (M _u ^-1M ' _u), m wherein _u∈ g, M _u ^-1M ' _u∈ G, g represents the tangent space, G represents the Lie group space.Again with m _uAsk mould, ask the mould formula to be || m _u||.

Among the substep 2-4 of step 2, because the geodesic line that calculates is apart from d (M _u, M ' _u) theoretical codomain be [0, ∞], d (M in real image _u, M ' _u) span is [0, D], and location of pixels information similarity

Codomain be [0,1], so will be with geodesic line apart from d (M _u, M ' _u) be mapped as location of pixels information similarity

Among the present invention, the pixel distribution in each square of spatial color histogram is approximately a Gaussian distribution, and uses mean vector μ _uWith the covariance matrix ∑ _uWith this Gaussian distribution parametrization, the function space of the probability density function of this Gaussian distribution constitutes a Lie group space, because the distance between the element in the Lie group space is represented with the geodesic line distance, the applicant has proposed a kind of location of pixels information method for measuring similarity on this basis, in conjunction with amount of pixels degree of being in similar proportion measure, reduce the measure of spatial color histogram.

Can find out obviously that from Fig. 3 the present invention has better tracking effect than existing two kinds of methods in track algorithm.Among the figure, transverse axis is represented frame number, and the longitudinal axis is represented error rate; Dot-and-dash line is represented the error rate of the present invention in application, and solid line is represented the error rate of method in application of Birchfiled, and dotted line is represented the error rate of method in application of Conaire.

Description of drawings

Fig. 1 is a FB(flow block) of the present invention;

Fig. 2 is a location of pixels information similarity

With geodesic line apart from d (M _u, M ' _u) the mapping function curve;

Fig. 3 is the effect comparison synoptic diagram of embodiments of the invention.

Embodiment

Below in conjunction with embodiment the present invention is specified.

In order to compare the performance of the present invention and existing measure, they are used in the vision track the inside simultaneously, in the comparison test, use and follow the tracks of the object of manual markings with the low but very sane track algorithm of a kind of efficient.

The general flow of track algorithm is tracing object initialization, object prediction, object location, template renewal etc.The present invention mainly is used in the object location, be exactly particularly select in the forecasting object with template object similarity maximum as the optimum target object, will use method and calculate similarity based on Lie group.

Flow process of the present invention as shown in Figure 1.The average that distributes for the coordinate that guarantees each 3 D stereo square interior pixel of spatial color histogram is 0, requirement is normalized to codomain [1 with the x and the y coordinate figure of correspondence image, 1] in, obtain the spatial color histogram s and the s ' of two images on this basis, the tlv triple of the 1st 3 D stereo square of two spatial color histograms is respectively s ₁=＜h ₁, μ ₁, ∑ ₁And s ' ₁=＜h ' ₁, μ ' ₁, ∑ ' ₁, wherein: h ₁=0.1476, h ₁'=0.1502; μ ₁=[0.1550 ,-0.0799] ^T, μ ' ₁=[0.1143 ,-0.3586] ^T ${\mathrm{\&Sigma;}}_1=\left[\begin{array}{cc}0.4007& 0.0074\\ 0.0074& 0.5196\end{array}\right],$ ${\mathrm{\&Sigma;}}_1^{\&prime;}=\left[\begin{array}{cc}0.4574& -0.0498\\ -0.0498& 0.3547\end{array}\right];$

One. calculating pixel amount degree of being in similar proportion step:

The amount of pixels ratio h of the 1st 3 D stereo square of computer memory color histogram s and the middle correspondence of s ' ₁With h ' ₁Similarity ψ ₁: ${\mathrm{\&psi;}}_1=\sqrt{h_1{\mathrm{<figure-callout\; id="1"\; label="h"\; filenames="A2009100617010013H1.png"\; state="\{\{state\}\}">h</figure-callout>}}_1^{\&prime;}}$

Obtain ψ ₁=0.1489;

Two. calculating pixel positional information similarity step: the location of pixels information similarity of u 3 D stereo square of computer memory color histogram s and the middle correspondence of s ' comprises following substep:

2-1. with square interior pixel positional information approximate representation is Gaussian distribution;

With Gaussian distribution N (μ ₁, ∑ ₁) the location of pixels information of the 1st corresponding 3 D stereo square among the representation space color histogram s, the average of this Gaussian distribution is square interior pixel coordinate mean vector μ ₁, the covariance matrix of this Gaussian distribution is the pixel coordinate covariance matrix ∑ in this square ₁, the function space of the Gaussian probability-density function of this Gaussian distribution correspondence constitutes a Lie group space;

Equally, with Gaussian distribution N (μ ' ₁, ∑ ' ₁) the location of pixels information of the 1st 3 D stereo square of the middle correspondence of representation space color histogram s ';

2-2. Gaussian distribution is converted to the positive definite affine transformation matrix;

With described Gaussian distribution N (μ ₁, ∑ ₁) use affine transformation matrix M ₁Expression:

${\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="A2009100617010013H1.png"\; state="\{\{state\}\}">M</figure-callout>}}_1=\left[\begin{array}{cc}{\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="A2009100617010013H1.png"\; state="\{\{state\}\}">P</figure-callout>}}_1& {\mathrm{\&mu;}}_1\\ 0& 1\end{array}\right],$ Be positive definite matrix, wherein a μ ₁Be the average of Gaussian distribution, P ₁It is covariance matrix ∑ with Gaussian distribution ₁Decompose the matrix that obtains by Cholesky, decomposition formula is: ∑ ₁=P ₁P ₁ ^TLimiting the matrix ∑ ₁When being the matrix of symmetric positive definite, decomposing could unique P that obtains ₁, and P ₁Matrix is the positive definite lower triangular matrix; So with the affine transformation matrix M that ties up ₁Also constitute a Lie group space;

Equally, with described Gaussian distribution N (μ ' ₁, ∑ ' ₁) usefulness affine transformation matrix M ' ₁Expression;

Wherein ${\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="A2009100617010013H1.png"\; state="\{\{state\}\}">P</figure-callout>}}_1=\left[\begin{array}{cc}0.6330& 0\\ 0.0117& 0.7207\end{array}\right],$ ${\mathrm{<figure-callout\; id="1"\; label="P"\; filenames="A2009100617010013H1.png"\; state="\{\{state\}\}">P</figure-callout>}}_1^{\&prime;}=\left[\begin{array}{cc}0.6763& 0\\ -0.0736& 0.5910\end{array}\right],$ In conjunction with μ ₁And μ ' ₁Obtain ${\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="A2009100617010013H1.png"\; state="\{\{state\}\}">M</figure-callout>}}_1=\left[\begin{array}{ccc}0.6330& 0& -0.1550\\ 0.0117& 0.7207& -0.0799\\ 0& 0& 1\end{array}\right],$ ${\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="A2009100617010013H1.png"\; state="\{\{state\}\}">M</figure-callout>}}_1^{\&prime;}=\left[\begin{array}{ccc}0.6763& 0& -0.1143\\ -0.0736& 0.5910& -0.3586\\ 0& 0& 1\end{array}\right];$

2-3. the geodesic line of calculating two affine transformation matrixs is apart from d (M ₁, M ' ₁):

d(M ₁，M′ ₁)＝||log(M ₁ ^-1M′ ₁)||，

Obtain M ₁And M ' ₁Geodesic line apart from d (M ₁, M ' ₁)=0.4899;

2-4. with geodesic distance d (M ₁, M ' ₁) be converted to location of pixels information similarity

Mapping parameters θ is taken as at 0.5 o'clock, and effect is best, and geodesic line is apart from d (M ₁, M ' ₁) be mapped as similarity

Three. metric space color histogram similarity step: by above-mentioned steps one～two, travel through all 3 D stereo squares among two spatial color histogram s and the s ', the amount of pixels degree of being in similar proportion of each 3 D stereo square and location of pixels information similarity are multiplied each other and superpose, obtain the similarity ρ (s, s ') of two spatial color histograms:

, obtain ρ (s, s ')=1.5957.

Fig. 2 is a location of pixels information similarity

With geodesic line apart from d (M _u, M ' _u) the mapping function curve; Transverse axis is the geodesic line distance, and the longitudinal axis is a location of pixels information similarity, location of pixels information similarity For geodesic line apart from d (M _u, M ' _u) monotonic decreasing function,

Determine mapping parameters θ by analyzing experimental data: the value of mapping parameters θ makes at d (M _u, M ' _u) in the span [0, D], location of pixels information similarity All greater than 0.05;

Among Fig. 2, solid line is the mapping function curve that θ got 0.2 o'clock, and dotted line is the mapping function curve that θ got 0.5 o'clock, and dot-and-dash line is the mapping function curve that θ got 0.8 o'clock, and getting θ is 0.5, makes location of pixels information similarity

All, bigger discrimination is arranged greater than 0.05.

In the concrete practice, follow the tracks of same object with existing other two kinds of measures, and compared the effect of following the tracks of, Fig. 3 has shown the effect comparison of following the tracks of, and transverse axis is represented frame number, and the longitudinal axis is represented error rate; Dot-and-dash line is represented the error rate of the present invention in application, and solid line is represented the error rate of method in application of Birchfiled, and dotted line is represented the error rate of method in application of Conaire; The present invention has better tracking effect than existing two kinds of methods in track algorithm.

Claims (2)

1. method for measuring similarity of spatial color histogram comprises:

One. calculating pixel amount degree of being in similar proportion step: the amount of pixels ratio h of u 3 D stereo square of computer memory color histogram s and the middle correspondence of s ' _uWith h ' _uSimilarity ψ _u:

${\mathrm{\&psi;}}_u=\sqrt{h_u{h}_u^{\text{'}}}$

Two. calculating pixel positional information similarity step: the location of pixels information similarity of u 3 D stereo square of computer memory color histogram s and the middle correspondence of s ' comprises following substep:

2-1. with square interior pixel positional information approximate representation is Gaussian distribution;

With Gaussian distribution N (μ _u, ∑ _u) the location of pixels information of u corresponding 3 D stereo square among the representation space color histogram s, the average of this Gaussian distribution is square interior pixel coordinate mean vector μ _u, the covariance matrix of this Gaussian distribution is the pixel coordinate covariance matrix ∑ in this square _u, the function space of the Gaussian probability-density function of this Gaussian distribution correspondence constitutes a Lie group space;

Equally, with Gaussian distribution N (μ ' _u, ∑ ' _u) the location of pixels information of u 3 D stereo square of the middle correspondence of representation space color histogram s ';

2-2. Gaussian distribution is converted to the positive definite affine transformation matrix;

With described Gaussian distribution N (μ _u, ∑ _u) use affine transformation matrix M _uExpression:

$M_u=\left[\begin{array}{cc}{P}_u& {\mathrm{\&mu;}}_u\\ 0& 1\end{array}\right],$ Be positive definite matrix, wherein a μ _uBe the average of Gaussian distribution, P _uIt is covariance matrix ∑ with Gaussian distribution _uDecompose the matrix that obtains by Cholesky, decomposition formula is: ∑ _u=P _uP _u ^TLimiting the matrix ∑ _uWhen being the matrix of symmetric positive definite, decomposing could unique P that obtains _u, and P _uMatrix is the positive definite lower triangular matrix; So with the affine transformation matrix M that ties up _uAlso constitute a Lie group space;

Equally, with described Gaussian distribution N (μ ' _u, ∑ ' _u) usefulness affine transformation matrix M ' _uExpression;

2-3. the geodesic line of calculating two affine transformation matrixs is apart from d (M _u, M ' _u):

D (M _u, M ' _u)=|| log (M _u ^-1M ' _u) ||, here || || mould formula, M are asked in expression _u, M ' _uBe affine transformation matrix, log () asks the logarithm operation;

2-4. with geodesic distance d (M _u, M ' _u) be converted to location of pixels information similarity

In the formula, mapping parameters θ is 0～1, is estimated to obtain by test;

Three. metric space color histogram similarity step: by above-mentioned steps one～two, travel through all 3 D stereo squares among two spatial color histogram s and the s ', the amount of pixels degree of being in similar proportion of each 3 D stereo square and location of pixels information similarity are multiplied each other and superpose, obtain the similarity ρ (s, s ') of two spatial color histograms:

2. method for measuring similarity of spatial color histogram as claimed in claim 1 is characterized in that:

Among the substep 2-4 of described calculating pixel positional information similarity step, mapping parameters θ obtains by analyzing experimental data: location of pixels information similarity For geodesic line apart from d (M _u, M ' _u) monotonic decreasing function, the value of mapping parameters θ is estimated by test, makes at d (M _u, M ' _u) in the span [0, D], location of pixels information similarity

All greater than 0.05; Wherein D is d (M in the real image _u, M ' _u) maximum occurrences, be arithmetic number.

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