CN100589126C - Utilize the CT image voxel formation method of transfer function of opacity - Google Patents

Abstract

The invention provides a kind of building method that can realize the transfer function of opacity of the multiple material classification in the three-dimensional data.At first test specimen is carried out CT scan, obtain the two-dimensional ct image sequence, by obtaining volume data in the two-dimensional ct image sequence, set up the volume data histogram and histogram is carried out segmentation, in every section, ask for the waypoint of a threshold value, determine transfer function of opacity, determine part to be shown in the object and each voxel is carried out the opacity assignment according to transfer function of opacity as transfer function of opacity, utilize mistake cut-deformation algorithm carries out voxel view, shows subject image at last.

Description

Utilize the CT image voxel formation method of transfer function of opacity

(1) technical field

The present invention relates to a kind of CT image voxel formation method, be specifically related to a kind of CT image voxel formation method that utilizes transfer function of opacity.

(2) background technology

In CT image voxel imaging process, in order in final visual image, correctly to express the distribution of the multiple material of coexistence, just need classify to volume data with different colors, find out the corresponding relation between data and the different material.Classification of substances is realized by transfer function of opacity, is embodied in each voxel that need treat in the data presented set and specifies an opacity value, and the transparency of its expression object, it is worth between 0.0 to 1.0.Specifying opacity for voxel is very complex calculations, realizes by transfer function of opacity.Thereby give high opacity according to the classification results of volume data with the point in certain structure and make this part structure as seen, on the contrary, it is transparent to allow uninterested part become.

Traditional KSW entropy (Kapur J N, Sahoo P K, Wong AKC.A new method for gray-level picturethresholding using the entropy of the histogram.Computer Vision, Graphics, and Image Processing, 1985,29 (3): 273～285P) and overall entropy algorithm (N.R.Pal.IEEE.SMC-21, No.5,1991.) be two dimensional image to be carried out single threshold cut apart, the CT three-dimensional data can't be handled, more multiple classification of substances can not be used for.

(3) summary of the invention

The object of the present invention is to provide a kind of CT image voxel formation method that utilizes transfer function of opacity that can realize the multiple material classification in the CT three-dimensional data.

The object of the present invention is achieved like this:

At first test specimen is carried out CT scan, obtain the two-dimensional ct image sequence, by obtaining volume data in the two-dimensional ct image sequence, set up the volume data histogram and histogram is carried out segmentation, in every section, ask for the waypoint of a threshold value, determine transfer function of opacity, determine part to be shown in the object and each voxel is carried out the opacity assignment according to transfer function of opacity as transfer function of opacity, utilize mistake cut-deformation algorithm carries out voxel view, shows subject image at last;

The described volume data histogram of setting up is to set up CT three-dimensional data histogram, and its method is: establish three-dimensional data and be of a size of C * N * G, f (x _{L, j, k}) expression voxel x _{L, j, k}Gray-scale value, the histogram h (u) of three-dimensional data is defined as:

$h\left(u\right)=\frac{\underset{i=1}{\overset{C}{\mathrm{\Σ}}}\underset{j=1}{\overset{N}{\mathrm{\Σ}}}\underset{k=1}{\overset{G}{\mathrm{\Σ}}}g\left(u\right)}{C\×N\×G};$

It is that its segmentation method is to CT three-dimensional data histogram h (u) segmentation that described histogram carries out segmentation:

At first calculate the non-zero end points f of h (u) _MinAnd f _Max, again with f _Min～f _MaxBe equally divided into the n section, promptly

Calculate the accumulation histogram of this n section then respectively

If Then this section is expanded, promptly made

Concrete steps are:

Step (1):

By gray scale with histogram h (u), u=0 wherein, 1 ..., 255, be divided into the n section, end points is f ₁, f ₂..., f _N-1

Step (2):

Calculate every section accumulative histogram h _m,

$h_m=\underset{u=f_m}{\overset{f_{m+1}}{\mathrm{\Σ}}}h\left(u\right)$ m＝1，2，...，n-2

Step (3):

Compare h _mWith the size of 1/n+2, if satisfy following formula, then turn to step (5), otherwise turn to step (4);

h _m≥1/n+2 m＝1，2，...，n-2

Step (4):

And then carry out segmentation expansion by following formula, at first judge whether be at the accumulative histogram of first section or left side section whether less than the accumulative histogram of right section, if, then expansion to the right; Otherwise, judge whether be at the accumulative histogram of latter end or right section whether smaller or equal to the accumulative histogram of left side section, if, then expansion left;

f _M+1=f _M+1+ 1 m=1 or h _M-1＜h _M+1

f _M-1=f _M-1-1 m=n-2 or h _M-1〉=h1 _M+1

Turn to step (2) then;

Step (5):

Determine that end points is f ₁, f ₂..., f _N-1

After with above-mentioned segmentation method CT three-dimensional data histogram being divided into the n section, in every section, ask for the waypoint of a threshold value, be respectively t as transfer function of opacity ₁, t ₂... t _n

If f is (x _{I, j, k}),

α (x _{I, j, k}) be respectively voxel x _{I, j, k}Gray-scale value, Grad and opacity value; Suppose the material that has the n kind to be shown in volume data, the tonal range of the material that the v kind is to be shown is t _v～t _V+1, t wherein _v＜t _V+1, v=1,2 ..., n; With t _vCorresponding opacity value is α _v

Described definite transfer function of opacity is selected piecewise function, and model is:

$\▿f\left(x_{i,j,k}\right)=\left[\begin{array}{c}\frac{1}{2}(f\left(x_{i+1,j,k}\right)-f\left(x_{i-1,j,k}\right)),\frac{1}{2}(f\left(x_{i,j+1,k}\right)-f\left(x_{i,j-1,k}\right))\\ ,\frac{1}{2}(f\left(x_{i,j,k+1}\right)-f\left(x_{i,j,k-1}\right))\end{array}\right]$

α (x _{I, j, k}) be f (x _{I, j, k}) and

Function, n=4 wherein.

The present invention also has some technical characterictics like this:

1, describedly in every section, asks for a threshold value and adopt many threshold values sorting technique based on CT three-dimensional data entropy of histogram as the waypoint of transfer function of opacity.

2, described method based on measurement CT three-dimensional data entropy of histogram in many threshold values sorting technique of CT three-dimensional data entropy of histogram is:

If the histogram of each segmentation of volume data is h _m(i),

(1) KSW entropy method

The hypothesis test result is made up of S elementary event, and the probability of r incident appearance is p _r, then the quantity of information H that obtains of whole test result is:

$H=E\left(\mathrm{\ΔI}\right)=-\underset{r=1}{\overset{S}{\mathrm{\Σ}}}{p}_r\ln p_r$

The total entropy of three-dimensional data is shown below, and asks to make the maximum t value of H (t) be the threshold value of three-dimensional data;

$H\left(t\right)=\ln P_t(1-P_t)+\frac{H_t}{P_t}+\frac{H_{\max }-H_t}{1-P_t}$

In the formula:

$P_t=\underset{i=f_{\min }}{\overset{t}{\mathrm{\Σ}}}{h}_m\left(i\right)$

$H_{\max }=-\underset{i=f_{\min }}{\overset{f_{\max }}{\mathrm{\Σ}}}{h}_m\left(i\right)\ln h\left(i\right)$

$H_t=-\underset{i=f_{\min }}{\overset{t}{\mathrm{\Σ}}}{h}_m\left(i\right)\ln h\left(i\right)$

(2) overall entropy method

Overall situation entropy is defined as:

$H=\underset{i}{\mathrm{\Σ}}{h}_m\left(i\right)e^{1-h\left(i\right)}$

The total entropy of three-dimensional data wherein is shown below, and asks to make the minimum t value of H (t) be the threshold value of volume data;

$H\left(t\right)=\underset{i=t+1}{\overset{f_{\max }}{\mathrm{\Σ}}}[h_m\left(i\right)/(P-P_A)]\×e^{1-[h_m\left(i\right)/(P-P_A)]}+\underset{i=f_{\min }}{\overset{t}{\mathrm{\Σ}}}(h_m\left(i\right)/P_A)\×e^{1-h_m\left(i\right)/P_A}$

In the formula:

$P_A=\underset{i=f_{\min }}{\overset{t}{\mathrm{\Σ}}}{h}_m\left(i\right)$

$P=\underset{i=f_{\min }}{\overset{f_{\max }}{\mathrm{\Σ}}}{h}_m\left(i\right).$

Principle of work of the present invention is: at first obtain volume data from the two-dimensional ct image sequence, draw the histogram of volume data then, the volume data histogram is carried out segmentation, in each section, construct transfer function of opacity according to histogrammic KSW entropy of CT three-dimensional data and overall entropy, voxel is carried out the opacity assignment, utilize mistake to cut-the deformable body rendering algorithm, show the 3-d modelling of any part interested in the object.

Design at present transfer function of opacity is still comparatively difficult.Transfer function of opacity of the present invention is a piecewise function, wherein waypoint determine be very crucial, and imaging results is had very big influence.The present invention mainly solves definite problem of transfer function of opacity waypoint.The present invention improves traditional KSW entropy and overall entropy algorithm, has proposed the construction algorithm of two kinds of transfer function of opacity, has realized the multiple material classification in the CT three-dimensional data.

(4) description of drawings

Fig. 1 is the opacity function synoptic diagram of protuberate intensity and object entity sense;

Fig. 2 is a three-dimensional data histogram segmentation method flow process;

Fig. 3 is the histogram based on the three-dimensional data of the transfer function of opacity structure of KSW entropy;

Fig. 4 is the opacity function synoptic diagram based on the transfer function of opacity structure of KSW entropy;

Fig. 5 is the histogram based on the volume data of the transfer function of opacity structure of overall entropy;

Fig. 6 is the opacity function synoptic diagram based on the transfer function of opacity structure of overall entropy.

(5) embodiment

The present invention is described in further detail below in conjunction with the drawings and specific embodiments:

Method of the present invention is: test specimen is carried out CT scan, obtain the two-dimensional ct image sequence, obtain volume data from the two-dimensional ct image sequence.Determine transfer function of opacity, set up the volume data histogram and histogram is carried out segmentation, in every section, ask for the waypoint of a threshold value as transfer function of opacity.Determine part to be shown in the object and each voxel carried out the opacity assignment according to transfer function of opacity, utilize mistake cut-deformation algorithm shows interested part in the object.

1, the structure of transfer function of opacity

In order to show any interested part in the object neatly, transfer function of opacity is generally piecewise function.The transfer function of opacity model that the present invention adopts is as follows, and this function can protuberate intensity and object entity sense.

If f is (x _{I, j, k}),

α (x _{I, j, k}) be respectively voxel x _{I, j, k}Gray-scale value, Grad and opacity value; Suppose the material that has the n kind to be shown in volume data, the tonal range of the material that the v kind is to be shown is t _v～t _V+1, t wherein _v＜t _V+1, v=1,2 ..., n; With t _vCorresponding opacity value is α _v

$\▿f\left(x_{i,j,k}\right)=\left[\begin{array}{c}\frac{1}{2}(f\left(x_{i+1,j,k}\right)-f\left(x_{i-1,j,k}\right)),\frac{1}{2}(f\left(x_{i,j+1,k}\right)-f\left(x_{i,j-1,k}\right))\\ ,\frac{1}{2}(f\left(x_{i,j,k+1}\right)-f\left(x_{i,j,k-1}\right))\end{array}\right]---\left(2\right)$

α (x _{I, j, k}) be f (z _{I, j, k}) and

Function, three's relation can represent with Fig. 1, wherein n=4.

2, the transport function waypoint determines

Many threshold values sorting technique based on the three-dimensional data entropy of histogram is proposed in the present invention, the histogram of three-dimensional data is divided into the n section by its gray scale size, determine a threshold value in each section, can obtain n threshold value, be n waypoint in the transfer function of opacity, thereby volume data is divided into n+1 class.

2.1, the histogrammic calculating of volume data

If volume data is of a size of C * N * G, f (x _{L, j, k}) expression voxel x _{L, j, k}Gray-scale value, the present invention is defined as the histogram h (u) of three-dimensional data:

$h\left(u\right)=\frac{\underset{l=1}{\overset{C}{\mathrm{\Σ}}}\underset{j=1}{\overset{N}{\mathrm{\Σ}}}\underset{k=1}{\overset{G}{\mathrm{\Σ}}}g\left(u\right)}{C\×N\×G}---\left(4\right)$

2.2, the histogrammic segmentation of three-dimensional data

The h that proposes among the present invention (u) segmentation method is as follows.

At first calculate the non-zero end points f of h (u) _MinAnd f _Max, again with f _Min～f _MaxBe equally divided into the n section, promptly

Calculate the accumulation histogram of this n section then respectively

If

Then, this section should be expanded, promptly make in order to guarantee the quantity of information in each segmentation And the tonal range that finally guarantees every section is not overlapping.The idiographic flow of this method as shown in Figure 2.

Step 1:

Press gray scale with histogram h (u), (u=0,1 ..., 255) be divided into the n section, end points is f ₁, f ₂..., f _N-1

Step 2:

Calculate every section accumulative histogram h _m,

$h_m=\underset{u=f_m}{\overset{f_{m+1}}{\mathrm{\Σ}}}h\left(u\right)$ m＝1，2，...，n-2 (5)

Step 3:

Compare h _mWith the size of 1/n+2, if satisfy formula (6), then turn to step 5, otherwise turn to step 4.

h _m≥1/n+2 m＝1，2，...，n-2 (6)

Step 4:

And then carry out segmentation by (7) formula and expand.Promptly judge whether be at the accumulative histogram of first section or left side section whether less than the accumulative histogram of right section, if, then expansion to the right; Otherwise, whether be at the accumulative histogram of latter end or right section whether smaller or equal to the accumulative histogram of left side section, if, then expansion left.

f _M+1=f _M+1+ 1 m=1 or h _M-1＜h _M+1

(7)

f _M-1=f _M-1-1 m=n-2 or h _M-1〉=h _M+1

Turn to step 2 then.

Step 5:

Determine that end points is f ₁, f ₂..., f _N-1

2.3, threshold value asks in each segmentation

Make probabilistic tolerance with entropy in the information theory, in other words random test result's ignorant is measured.If a test findings is made up of the several separate elementary event, the probability that each elementary event occurs is known, and the probability of occurrence of testing each known elementary event of resulting quantity of information in back and people so is inversely proportional to.For determining to want event, do not obtain any information; And, then obtain maximum information for least determining whether event.

Also have no talent at present and the notion of entropy in the information theory is applied to the classification of volume data.In our algorithm, need the entropy of measuring body data grey level histogram, and find out optimal threshold thus automatically volume data is classified.If the histogram of each segmentation of volume data is h _m(i), maximum gradation value is f _Max, minimum gradation value is f _Min, h then _m(i) show that gray-scale value is i (f in the three-dimensional data _Min＜i＜f _Max) the probability that occurs of voxel, two kinds of methods of measuring the three-dimensional data entropy of histogram have been proposed among the present invention.

(1) KSW entropy method

The hypothesis test result is made up of S elementary event, and the probability of r incident appearance is p _r, then the quantity of information H that obtains of whole test result is:

$H=E\left(\mathrm{\ΔI}\right)=-\underset{r=1}{\overset{n}{\mathrm{\Σ}}}{p}_r\ln p_r---\left(8\right)$

Entropy in the information theory that Here it is.

The total entropy of the three-dimensional data among the present invention is asked to make the maximum t value of H (t) be the threshold value of three-dimensional data as the formula (9).

$H\left(t\right)=\ln P_t(1-P_t)+\frac{H_t}{P_t}+\frac{H_{\max }-H_t}{1-P_t}---\left(9\right)$

In the formula:

$P_t=\underset{i=f_{\min }}{\overset{t}{\mathrm{\Σ}}}{h}_m\left(i\right)---\left(10\right)$

$H_{\max }=-\underset{i=f_{\min }}{\overset{f_{\max }}{\mathrm{\Σ}}}{h}_m\left(i\right)\ln h\left(i\right)---\left(11\right)$

$H_t=-\underset{i=f_{\min }}{\overset{t}{\mathrm{\Σ}}}{h}_m\left(i\right)\ln h\left(i\right)---\left(12\right)$

Fig. 3-4 is the histogram of three-dimensional data and based on the transfer function of opacity of KSW entropy.With size is that 512 * 512 * 15 volume data is an example, wherein Fig. 3 is the histogram of volume data, because the ratio of the pairing pixel of different gray-scale values differs bigger, in order to make histogram more clear, different gray scale sections are adopted different displaying ratios, and will mark among the engineer's scale figure, Fig. 4 is the transfer function of opacity based on the KSW entropy.

(2) overall entropy method

Overall situation entropy is defined as

$H=\underset{i}{\mathrm{\Σ}}{h}_m\left(i\right)e^{1-h\left(i\right)}---\left(13\right)$

The total entropy of the three-dimensional data among the present invention is asked to make the minimum t value of H (t) be the threshold value of volume data as the formula (14).

$H\left(t\right)=\underset{i=t+1}{\overset{f_{\max }}{\mathrm{\Σ}}}[h_m\left(i\right)/(P-P_A)]\×e^{1-[h_m\left(i\right)/(P-P_A)]}+\underset{i=f_{\min }}{\overset{t}{\mathrm{\Σ}}}(h_m\left(i\right)/P_A)\×e^{1-h_m\left(i\right)/P_A}---\left(14\right)$

In the formula:

$P_A=\underset{i=f_{\min }}{\overset{t}{\mathrm{\Σ}}}{h}_m\left(i\right)---\left(15\right)$

$P=\underset{i=f_{\min }}{\overset{f_{\max }}{\mathrm{\Σ}}}{h}_m\left(i\right)---\left(16\right)$

Fig. 5-6 is the histogram of three-dimensional data and based on the transfer function of opacity of overall entropy.With size is that 512 * 512 * 253 volume data is an example, wherein Fig. 5 is the histogram of volume data, because the ratio of the pairing pixel of different gray-scale values differs bigger, in order to make histogram more clear, different gray scale sections are adopted different displaying ratios, and will mark among the engineer's scale figure, Fig. 6 is the transfer function of opacity based on overall entropy.

Claims (3)

1, a kind of CT image voxel formation method that utilizes transfer function of opacity, at first test specimen is carried out CT scan, obtain the two-dimensional ct image sequence, by obtaining the CT volume data in the two-dimensional ct image sequence, set up CT volume data histogram and histogram is carried out segmentation, in every section, ask for the waypoint of a threshold value as transfer function of opacity, determine transfer function of opacity, determine part to be shown in the object and each voxel is carried out the opacity assignment according to transfer function of opacity, utilize mistake to cut a deformation algorithm and carry out voxel view, show subject image at last; It is characterized in that:

The described CT of foundation volume data histogram is to set up the three-dimensional data histogram, and its method is: establish the CT three-dimensional data and be of a size of C * N * G, f (x _{I, j, k}) expression voxel x _{I, j, k}Gray-scale value, the histogram h (u) of CT three-dimensional data is defined as:

$h\left(u\right)=\frac{\underset{i=1}{\overset{C}{\mathrm{\Σ}}}\underset{j=1}{\overset{N}{\mathrm{\Σ}}}\underset{k=1}{\overset{G}{\mathrm{\Σ}}}g\left(u\right)}{C\×N\×G};$

It is that its segmentation method is to CT three-dimensional data histogram h (u) segmentation that described histogram carries out segmentation:

At first calculate the non-zero end points f of h (u) _MinAnd f _Max, again with f _Min～f _MaxBe equally divided into the n section, promptly $f_{\min }~\frac{f_{\max }+(n-1)f_{\min }}{n},\frac{f_{\max }+(n-1)f_{\min }}{n}~\frac{2\×f_{\max }+(n-2)f_{\min }}{n}...\frac{(n-1)f_{\max }+f_{\min }}{n}~f_{\max };$ Calculate the accumulation histogram of this n section then respectively

If $\underset{u}{\mathrm{\&Sigma;}}h\left(u\right)<\frac{1}{n+2},$ Then this section is expanded, promptly made $\underset{u}{\mathrm{\&Sigma;}}h\left(u\right)\&GreaterEqual;\frac{1}{n+2};$ Concrete steps are:

Step (1):

By gray scale with histogram h (u), u=0 wherein, 1 .., 255, be divided into the n section, end points is f ₁, f ₂.., f _N-1

Step (2):

Calculate every section accumulative histogram h _m,

$h_m=\underset{u=f_m}{\overset{f_{m+1}}{\mathrm{\&Sigma;}}}h\left(u\right)$ m＝1，2，..，n-2

Step (3):

Compare h _mWith the size of 1/n+2, if satisfy following formula, then turn to step (5), otherwise turn to step (4);

$h_m\&GreaterEqual;\frac{1}{n+2}$ m＝1，2，..，n-2

Step (4):

And then carry out segmentation expansion by following formula, at first judge whether be at the accumulative histogram of first section or left side section whether less than the accumulative histogram of right section, if, then expansion to the right; Otherwise, judge whether be at the accumulative histogram of latter end or right section whether smaller or equal to the accumulative histogram of left side section, if, then expansion left;

f _M+1=f _M+1+ 1 m=1 or f _M-1＜h _M+1

f _M-1=f _M-1-1 m=n-2 or h _M-1〉=h _M+1

Turn to step (2) then;

Step (5):

Determine that end points is f ₁, f ₂..., f _N-1

After with above-mentioned segmentation method CT three-dimensional data histogram being divided into the n section, in every section, ask for the waypoint of a threshold value, be respectively t as transfer function of opacity ₁, t ₂... t _n

If f is (x _{I, j, k}),

α (x _{I, j, k}) be respectively voxel x _{I, j, k}Gray-scale value, Grad and opacity value; Suppose the material that has the n kind to be shown in volume data, the tonal range of the material that the v kind is to be shown is t _v～t _V+1, t wherein _v＜t _V+1, v=1,2 ..., n; With t _vCorresponding opacity value is α _v

Described definite transfer function of opacity is selected piecewise function, and model is:

$\&dtri;f\left(x_{i,j,k}\right)=\left[\begin{array}{c}\frac{1}{2}(f\left(x_{i+1,j,k}\right)-f\left(x_{i-1,j,k}\right)),\frac{1}{2}(f\left(x_{i,j+1,k}\right)-f\left(x_{i,j-1,k}\right))\\ ,\frac{1}{2}(f\left(x_{i,j,k+1}\right)-f\left(x_{i,j,k-1}\right))\end{array}\right]$

α (x _{I, j, k}) be f (x _{I, j, k}) and

Function, n=4 wherein.

2, the CT image voxel formation method that utilizes transfer function of opacity according to claim 1 is characterized in that describedly asking for a threshold value adopt many threshold values sorting technique based on CT three-dimensional data entropy of histogram as the waypoint of transfer function of opacity in every section.

3, the CT image voxel formation method that utilizes transfer function of opacity according to claim 2 is characterized in that described method based on measurement CT three-dimensional data entropy of histogram in many threshold values sorting technique of CT three-dimensional data entropy of histogram is:

If the histogram of each segmentation of volume data is h _m(i),

(1) KSW entropy method

The hypothesis test result is made up of S elementary event, and the probability of r incident appearance is p _r, then the quantity of information H that obtains of whole test result is:

$H=E\left(\mathrm{\&Delta;I}\right)=-\underset{r=1}{\overset{S}{\mathrm{\&Sigma;}}}{p}_r\ln p_r$

The total entropy of three-dimensional data is shown below, and asks to make the maximum t value of H (t) be the threshold value of three-dimensional data;

$H\left(t\right)=\ln P_t(1-P_t)+\frac{H_t}{P_t}+\frac{H_{\max }-H_t}{1-P_t}$

In the formula:

$P_t=\underset{i=f_{\min }}{\overset{t}{\mathrm{\&Sigma;}}}{h}_m\left(i\right)$

$H_{\max }=-\underset{i=f_{\min }}{\overset{f_{\max }}{\mathrm{\&Sigma;}}}{h}_m\left(i\right)\ln h\left(i\right)$

$H_t=-\underset{i=f_{\min }}{\overset{t}{\mathrm{\&Sigma;}}}{h}_m\left(i\right)\ln h\left(i\right)$

(2) overall entropy method

Overall situation entropy is defined as:

$H=\underset{i}{\mathrm{\&Sigma;}}{h}_m\left(i\right)e^{1-h\left(i\right)}$

The total entropy of three-dimensional data wherein is shown below, and asks to make the minimum t value of H (t) be the threshold value of volume data;

$H\left(t\right)=\underset{i=t+1}{\overset{f_{\max }}{\mathrm{\&Sigma;}}}[h_m\left(i\right)/(P-P_A)]\&times;e^{1-[h_m\left(i\right)/(P-P_A)]}+\underset{i=f_{\min }}{\overset{t}{\mathrm{\&Sigma;}}}(h_m\left(i\right)/P_A)\&times;e^{1-h_m\left(i\right)/P_A}$

In the formula:

$P_A=\underset{i=f_{\min }}{\overset{t}{\mathrm{\&Sigma;}}}{h}_m\left(i\right)$

$P=\underset{i=f_{\min }}{\overset{f_{\max }}{\mathrm{\&Sigma;}}}{h}_m\left(i\right).$

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