CN101825433A - Measuring method of offset of rotating center of rotating table of fan beam 2D-CT scanning system - Google Patents

Abstract

The invention discloses a measuring method of the offset of a rotating center of a rotating table of a fan beam 2D-CT scanning system, which comprises the following steps: rearranging fan beam projections into parallel beam projections; according to the principle that the parallel beam projections of which the projection angles have a difference pi are similar, extracting the projection information corresponding to [0-180 degrees] and [180-360 degrees] in a sinusoidal chart of the parallel beams which are rearranged, thereby acquiring a first sinusoidal chart and a second sinusoidal chart; and carrying out cross-correlation operation on the first sinusoidal chart and a third sinusoidal chart formed by horizontally turning the second sinusoidal chart, thereby computing the offset tau 0 of the rotating center of the rotating table. The method of the invention does not need to manufacture special correcting specimens and directly utilizes the sinusoidal chart of the scanned fault for carrying out measurement, thereby effectively ensuring the measuring accuracy of the offset of the rotating center of the rotating table of the fan beam 2D-CT scanning system and improving the quality of reconstructed images.

Description

The measuring method of fan beam 2 D-CT scanning system turntable rotation center side-play amount

Technical field

The present invention relates to a kind of measurement that is applicable to fan beam 2 D-CT (Two Dimensional ComputedTomography) scanning system turntable rotation center side-play amount, can be used for the measurement of correlation in medical science and industrial circle ray digital imaging (DR-Digital Radiography), two dimension, three-dimensional computed tomography (2D-CT/3D-CT) imaging process.

Background technology

For the fladellum ray scanning 2D-CT based on linear array detector, its scanning theory as shown in Figure 1.Turntable 4 is between ray source focus 1 and linear array detector 6; Turntable 4 is in the irradiation field of fan-beam ray 3; Central ray 2 is perpendicular to linear array detector 6, and its intersection point is designated as S ₀, S ₀It also is the coordinate of the center of linear array detector 6; According to the requirement of 2D-CT reconstruction algorithm, central ray 2 will be by the rotary middle point O of turntable 4 and perpendicular to linear array detector 6.Yet,, because the error installed of machinery, can make rotary middle point O off center ray 2, this side-play amount τ for the fan beam 2 D-CT system of reality ₀Existence can influence the reconstruction quality of CT image, the appearance that causes rebuilding pseudo-shadow.In order to obtain accurate reconstructed results, must accurately measure the side-play amount τ of rotary middle point O with respect to central ray 2 ₀, then with the side-play amount τ that obtains ₀In the cent(e)ring parameter input picture reconstruction algorithm as turntable 4.

Summary of the invention

The objective of the invention is to propose a kind of measuring method that is applicable to fan beam 2 D-CT scanning system turntable rotation center side-play amount, be fan-beam projection to be reset be the collimated beam projection, the principle similar according to the collimated beam projection of projected angle phase difference of pi, with [0 ° 180 °] in the collimated beam sinogram after resetting and Corresponding projection information takes out, and obtains first sinogram and second sinogram; The 3rd sinogram and first sinogram that form behind the second sinogram flip horizontal are carried out computing cross-correlation, thereby obtain turntable rotation center side-play amount τ ₀

The present invention is a kind of measuring method that is applicable to fan beam 2 D-CT scanning system turntable rotation center side-play amount, and this measuring method includes following implementation step:

Step 1: in the 2D-CT scanning system, linear array detector 6 and ray source focus 1 are motionless, allow turntable 4 make circumference around rotary middle point O and rotate;

In the present invention, turntable 4 makes linear array detector 6 can collect the data for projection information that is scanned tomography 5 under the different projected angles around the rotation of rotary middle point O, and the data for projection under all projected angles is synthesized a width of cloth fan-beam ray scanning sinogram p ^Fan(θ, s).Setting about rotational angle (also being projected angle) can be as accurate as 0.1 °～1 ° collection of carrying out a data for projection of revolution.

Step 2: with fan-beam ray scanning sinogram p ^Fan(θ s) resets that (θ, s), (θ s) be two-dimensional matrix to p, and the data for projection when the capable data for projection of representing under each projected angle of each of this matrix, θ=β is designated as p for collimated beam scanning sinogram p _β(s), p _β(s) be p (θ, certain delegation s);

Step 3: (θ, s) the corresponding data for projection in middle θ ∈ [the 0 ° 180 °] scope takes out the synthetic first sinogram p with sinogram p _β ¹(s); (θ, s) the corresponding data for projection in middle θ ∈ [the 180 ° 360 °] scope takes out the synthetic second sinogram p with sinogram p _β ²(s); As can be seen, p _β ¹(s) and p _β ²(s) be two-dimensional matrix, its highly be sinogram p (θ, s) 1/2, p _β ¹(s) be p (θ, first half s), p _β ²(s) be p (θ, latter half s);

Step 4: with p _β ²(s) carry out flip horizontal, obtain the 3rd sinogram p _β ³(s), p _β ³(s) and p _β ²(s) pass is:

N is the total number of linear array detector probe unit.

Step 5: with the first sinogram p _β ¹(s) with the 3rd sinogram p _β ³(s) corresponding row is carried out computing cross-correlation, and the coordinate of the cross correlation function maximal value correspondence of corresponding row is exactly p so _β ¹(s) and p _β ³(s) the displacement difference ordered series of numbers between the corresponding row is designated as d (i), i=1～M, and M represents p _β ¹(s) or p _β ³(s) height.

Step 6: ask the average τ ' of ordered series of numbers d (i), 1/2 of this average τ ' equals the side-play amount τ of turntable rotary middle point ₀, promptly ${\mathrm{\τ}}_0=\frac{1}{2M}\underset{i=1}{\overset{M}{\mathrm{\Σ}}}d\left(i\right).$

Adopt measuring method of the present invention to carry out fan beam 2 D-CT scanning system turntable rotation center side-play amount and measure, have following advantage: (1) does not need to make special correction test specimen, directly utilizes the sinogram that is scanned tomography to measure.Effectively guarantee the measuring accuracy of fan beam 2 D-CT system turntable rotation center side-play amount, improved quality of reconstructed images.(2) cross-correlation method is applied to scanning the resolving in the process of sinogram, has improved the anti-noise ability of measuring method of the present invention.(3) Measurement Algorithm is simple, can be used as the auxiliary examination module of fan beam 2 D-CT system, monitors the variation of turntable rotation center automatically.

Description of drawings

Fig. 1 is based on the fladellum ray 2D-CT scanning theory figure of linear array detector.

Figure 1A resets the fladellum projection among Fig. 1 to be the scanning theory figure of collimated beam projection.

Collimated beam ray scanning schematic diagram when Fig. 2 is θ=β.

The former figure of collimated beam ray scanning when Fig. 2 A is θ=β+π.

Fig. 3 is p _β(s), p _β+π(s) with p ' _β+π(s) grey scale curve figure.

Fig. 4 resets to be the sinogram of collimated beam ray.

Fig. 4 A is first sinogram.

Fig. 4 B is second sinogram.

Fig. 4 C is the 3rd sinogram.

Fig. 4 D is the cross correlation image of the 3rd sinogram and first sinogram.

Fig. 5 A and Fig. 5 B are center offset τ ₀Reconstructed image before and after proofreading and correct.

Embodiment

The present invention is described in further detail below in conjunction with drawings and Examples.

Referring to shown in Figure 1, the turntable 4 in the fan beam 2 D-CT system is between ray source focus 1 and linear array detector 6; Turntable 4 is in the irradiation field of fan-beam ray 3; Central ray 2 is perpendicular to linear array detector 6, and its intersection point is designated as S ₀, S ₀It also is the coordinate of the center of linear array detector 6.The coordinate of turntable 4 is xoy.

Linear array detector 6 and ray source focus 1 are motionless, turntable 4 rotates (being that turntable 4 is made circular motion around rotary middle point O) around rotary middle point O in 360 ° of scopes, be scanned tomography 5 rotation synchronously under the dragging of turntable 4, intensity can change after fan-beam ray 3 passed and is scanned tomography 5, and linear array detector 6 is crossed the station acquisition transmitted intensity information of a fixed angle (being generally 0.1 °～1 °) at turntable 4 revolutions.

Linear array detector 6 is arranged in delegation by a plurality of independently probe units, transmitted intensity information is gathered by each probe unit and is transferred to computing machine through A/D conversion (computing machine is a kind ofly can carry out the modernized intelligent electronic device of massive values computation and various information processings automatically, at high speed according to prior program stored.Minimalist configuration is CPU 2GHz, internal memory 2GB, hard disk 180GB; Operating system is windows2000/2003/XP.), the data of each detector cells collection are called projection value.Linear array detector 6 collects data line at every turn, and its all line data that collect in 360 ° of scopes are arranged in order, and just constitutes two-dimensional matrix, is called sinogram.Fan-beam ray scanning sinogram is designated as p ^Fan(θ, S), θ represents the anglec of rotation of turntable, also represents projected angle, S to represent the detector cells coordinate.

According to sky, village dagger-axe disclosed " CT principle and algorithm " as can be known, the fan-beam ray projection data can be reset into the collimated beam ray projection data, shown in Figure 1A.Measuring method of the present invention is based on the collimated beam data for projection that the fan-beam projection data rearrangement is become and carries out turntable rotation center side-play amount τ ₀Accurate measurement.At document (Zhuan Tiange, CT principle and algorithm, publishing house of Shanghai Communications University, 1992) in, disclose based on the fan-beam projection data rearrangement is the image reconstruction algorithm of collimated beam data for projection, its basic thought is: according to the geometric relationship of fan-ray beam scanning, utilize the mathematics interpolation method, with the fan-beam projection data rearrangement is the collimated beam data for projection, again the collimated beam data for projection after resetting is carried out filtering, utilize the perspective geometry relation of collimated beam to calculate the projection address, carry out backprojection operation at last and obtain reconstructed image.

If the coordinate of turntable 4 is xoy, the central point O ' (also claiming to be scanned tomography central point O ') that is scanned tomography 5 is designated as projected angle θ with the line O ' O of rotary middle point O and the angle of x axle, and θ ∈ [0 ° 360 °], θ also represents the corner of turntable with respect to x.

When the angle value of projected angle θ is β, (during θ=β), the data for projection that is scanned tomography 5 that linear array detector 6 collects is designated as p to β ∈ [0 ° 180 °] _βOr claim the first projection information p (s), _β(s), the grey scale curve of these data is shown in curve among Fig. 3 (a).

(θ=β+π), the data for projection that is scanned tomography 5 that linear array detector 6 collects is designated as p during at π+β when the angle value of projected angle θ _β+πOr claim the second projection information p (s), _β+π(s), the grey scale curve of these data is shown in curve among Fig. 3 (b).

Contrast different projection angles (the projection gray level curve map when θ=β or θ=β+π), the first projection information p as can be known _β(s) and the second projection information p _β+π(s) existing inherent contact is:

Fig. 3 (a) represents the first projection information p _β(s) grey scale curve, coordinate axis O _dS represents the coordinate axis of the length direction of linear array detector 6, and the length of linear array detector 6 is designated as N, and this has also represented the sum of linear array detector probe unit.Q ₁Be grey scale curve p _β(s) peak, it is at coordinate axis O _dThe last respective coordinates of S is S ₂, the center of linear array detector 6 is at coordinate axis O _dThe last respective coordinates of S is S ₀, S ₀Also be the intersection point of central ray 2 perpendicular to linear array detector 6, the rotary middle point O of turntable 4 is at coordinate axis O _dThe last respective coordinates value of S is designated as R ₀, as seen from the figure,

S ₀-R ₀=τ ₀, τ ₀The side-play amount of the rotary middle point O of expression central ray 2 skew turntables 4.

Fig. 3 (b) represents the second projection information p _β+π(s) grey scale curve, Q ₂Be grey scale curve p _β+π(s) peak, it is at coordinate axis O _dThe last respective coordinates of S is S ₁

By the projection theory of collimated beam scanning as can be known, the grey scale curve of Fig. 3 (a) and Fig. 3 (b) is at coordinate axis O _dS is last with R ₀Point is left-right symmetric relation, i.e. Q for the center ₁Point and Q ₂Point is about R ₀Point symmetry, satisfied pass is:

$|S_1-R_0|=|S_2-R_0|=\frac{1}{2}|S_2-{\mathrm{<figure-callout\; id="1"\; label="S"\; filenames="HSA00000073718400031.png"\; state="\{\{state\}\}">S</figure-callout>}}_1|$

Be S ₁=S ₂-(S ₂-S ₁)=S ₂-2 (S ₂-R ₀)=2R ₀-S ₂

With the second projection information p _β+π(s) at coordinate axis O _dUpset just obtains the projection information p ' that overturns about carrying out on the S _β+π(s), p ' _β+π(s) and p _β+π(s) pass is:

p′ _β+π(s)＝p _β+π(N-s) s∈[0 N] (2)

Fig. 3 (c) is upset projection information p ' _β+π(s) grey scale curve, Q ₃Be p ' _β+π(s) peak, it is at coordinate axis O _dThe last respective coordinates of S is S ₃, simultaneous formula (1), formula (2) get p ' _β+π(s) projection coordinate's relation:

S ₃＝N-S ₁＝N-(2R ₀-S ₂) (3)

Comparison diagram 3 (a) and Fig. 3 (c) as can be known, p _β(s) with p ' _β+π(s) grey scale curve shape is identical, just at coordinate axis O _dCertain displacement difference is arranged on the S direction, and this displacement difference promptly is a Q ₁With Q ₃The coordinate difference be designated as τ ', its expression formula is:

τ′＝S ₃-S ₂ (4)

Formula (3) substitution formula (4) is got:

${\mathrm{\&tau;}}^{\&prime;}=S_3-{\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="HSA00000073718400011.png,HSA00000073718400021.png"\; state="\{\{state\}\}">S</figure-callout>}}_2=N-(2R_0-S_2)-{\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="HSA00000073718400011.png,HSA00000073718400021.png"\; state="\{\{state\}\}">S</figure-callout>}}_2=N-2R_0$

$=2\&times;(\frac{N}{2}-R_0)=2\&times;(S_0-R_0)=2{\mathrm{\&tau;}}_0$

That is: ${\mathrm{\&tau;}}_0=\frac{{\mathrm{\&tau;}}^{\&prime;}}{2}.$

Can be reached a conclusion by formula (4): the data for projection that utilizes two projected angles to differ 180 ° can be obtained turntable rotation center side-play amount τ ₀, its method is for to be projected in coordinate axis O with one of them _dUpset about carrying out on the S, the data for projection after asking for another data for projection and overturning is at coordinate axis O _dDisplacement difference τ ' on the S, the side-play amount τ of turntable rotation center so ₀Equal half of τ '.

The present invention is a kind of measuring method that is applicable to fan beam 2 D-CT scanning system turntable rotation center side-play amount, and this measuring method includes following implementation step:

Step 1: in the 2D-CT scanning system, linear array detector 6 and ray source focus 1 are motionless, allow turntable 4 make circumference around rotary middle point O and rotate;

In the present invention, turntable 4 makes linear array detector 6 can collect the data for projection information that is scanned tomography 5 under the different projected angles around the rotation of rotary middle point O, and the data for projection under all projected angles is synthesized a width of cloth fan-beam ray scanning sinogram p ^Fan(θ, s).Setting about rotational angle (also being projected angle) can be as accurate as 0.1 °～1 ° collection of carrying out a data for projection of revolution.

Step 2: with fan-beam ray scanning sinogram p ^Fan(θ s) resets that (θ, s), (θ s) be two-dimensional matrix to p, and the data for projection when the capable data for projection of representing under each projected angle of each of this matrix, θ=β is designated as p for collimated beam scanning sinogram p _β(s), p _β(s) be p (θ, certain delegation s);

Step 3: (θ, s) the corresponding data for projection in middle θ ∈ [the 0 ° 180 °] scope takes out the synthetic first sinogram p with sinogram p _β ¹(s); (θ, s) the corresponding data for projection in middle θ ∈ [the 180 ° 360 °] scope takes out the synthetic second sinogram p with sinogram p _β ²(s); As can be seen, p _β ¹(s) and p _β ²(s) be two-dimensional matrix, its highly be sinogram p (θ, s) 1/2, p _β ¹(s) be p (θ, first half s), p _β ²(s) be p (θ, latter half s);

Step 4: with p _β ²(s) carry out flip horizontal, obtain the 3rd sinogram p _β ³(s), p _β ³(s) and p _β ²(s) pass is:

N is the total number of linear array detector probe unit.

Step 5: with the first sinogram p _β ¹(s) with the 3rd sinogram p _β ³(s) corresponding row is carried out computing cross-correlation, and the coordinate of the cross correlation function maximal value correspondence of corresponding row is exactly p so _β ¹(s) and p _β ³(s) the displacement difference ordered series of numbers between the corresponding row is designated as d (i), i=1～M, and M represents p _β ¹(s) or p _β ³(s) height.

Step 6: ask the average τ ' of ordered series of numbers d (i), 1/2 of this average τ ' equals the side-play amount τ of turntable rotary middle point ₀, promptly ${\mathrm{\&tau;}}_0=\frac{1}{2M}\underset{i=1}{\overset{M}{\mathrm{\&Sigma;}}}d\left(i\right).$

Below to the measuring method of the present invention checking that experimentizes:

Image-forming condition: ray source focus 1 is of a size of 2.5mm, tube voltage 420kV, tube current 2.5mA.The spy elemental size of linear array detector 6 is 0.083mm, and the probe unit number is 2048, and focal length is 2600mm, and the projected angle number is 1800.

Under this image-forming condition, turbo blade is scanned, obtain the fan-beam sinogram, as shown in Figure 4.The projection information taking-up of preceding 1/2 in fan-beam sinogram shown in Figure 4 and back 1/2 correspondence is obtained first sinogram (as Fig. 4 A) and second sinogram (as Fig. 4 B), be designated as p _β ¹(s) and p _β ²(s).With p _β ²(s) carry out flip horizontal and obtain the 3rd sinogram p _β ³(s) (as Fig. 4 C).With the first sinogram p _β ¹(s) with the 3rd sinogram p _β ³(s) corresponding row is carried out computing cross-correlation and is obtained cross correlation image (as Fig. 4 D), and the coordinate of the every capable gray scale maximal value correspondence of cross correlation image is designated as d (i), asks the average of ordered series of numbers d (i) to be designated as τ ', and 1/2 of this value equals the side-play amount τ of turntable rotation center ₀, ${\mathrm{\&tau;}}_0=\frac{1}{2M}\underset{i=1}{\overset{M}{\mathrm{\&Sigma;}}}d\left(i\right).$

The turntable rotation center side-play amount of the original demarcation of fan beam 2 D-CT system that experiment is adopted is 0, but after being to use 3 years, the rotation center position of turntable changes to some extent, if utilize original calibration value can cause reconstructed image pseudo-shadow to occur this moment, Fig. 5 A is a reconstructed results of utilizing original value; (unit: pixel), utilize this value that reconstruction parameter is revised, the reconstructed image that obtains is shown in Fig. 5 B for-2.9 to utilize side-play amount that measuring method of the present invention records this system's turntable rotation center.As seen, utilize measuring method of the present invention can effectively improve the image reconstruction precision of fan beam 2 D-CT system, improve the details resolving power of CT image.

Claims (3)

1. the measuring method of a fan beam 2 D-CT scanning system turntable rotation center side-play amount is characterized in that including following measuring process:

Step 1: in the 2D-CT scanning system, linear array detector (6) and ray source focus (1) are motionless, allow turntable (4) make circumference around rotary middle point O and rotate; Described turntable (4) makes linear array detector (6) can collect the data for projection information that is scanned tomography (5) under the different projected angles around the rotation of rotary middle point O, and the data for projection under all projected angles is synthesized a width of cloth fan-beam ray scanning sinogram p ^Fan(θ, s);

Step 2: with fan-beam ray scanning sinogram p ^Fan(θ s) resets that (θ, s), (θ s) be two-dimensional matrix to p, and the data for projection when the capable data for projection of representing under each projected angle of each of this matrix, θ=β is designated as p for collimated beam scanning sinogram p _β(s), p _β(s) be p (θ, certain delegation s);

Step 3: (θ, s) the corresponding data for projection in middle θ ∈ [the 0 ° 180 °] scope takes out the synthetic first sinogram p with sinogram p _β ¹(s); (θ, s) the corresponding data for projection in middle θ ∈ [the 180 ° 360 °] scope takes out the synthetic second sinogram p with sinogram p _β ²(s); p _β ¹(s) and p _β ²(s) be two-dimensional matrix, p _β ¹(s) be p (θ, first half s), p _β ²(s) be p (θ, latter half s);

Step 4: with p _β ²(s) carry out flip horizontal, obtain the 3rd sinogram p _β ³(s), p _β ³(s) and p _β ²(s) pass is p _β ³(s)=p _β ²(N-s), N is the total number of probe unit in the linear array detector (6);

Step 5: with the first sinogram p _β ¹(s) with the 3rd sinogram p _β ³(s) corresponding row is carried out computing cross-correlation, and the coordinate of the cross correlation function maximal value correspondence of corresponding row is exactly p so _β ¹(s) and p _β ³(s) the displacement difference ordered series of numbers between the corresponding row is designated as d (i), i=1～M, and M represents p _β ¹(s) or p _β ³(s) height;

Step 6: ask the average τ ' of displacement difference ordered series of numbers d (i), 1/2 of this average τ ' equals the side-play amount τ of turntable rotary middle point ₀, promptly

2. the measuring method of fan beam 2 D-CT scanning system turntable rotation center side-play amount according to claim 1 is characterized in that: turntable (4) is set to 0.1 °～1 ° around the each angle of rotating of rotary middle point O.

3. the measuring method of fan beam 2 D-CT scanning system turntable rotation center side-play amount according to claim 1 is characterized in that: the projection gray level curve map when contrasting different projection angle θ=β or θ=β+π, the first projection information p as can be known _β(s) and the second projection information p _β+π(s) existing inherent contact is:

What described β represented is the value of projected angle θ, β ∈ [0 ° 180 °];

At the first projection information p _β(s) in the grey scale curve, coordinate axis O _dS represents the coordinate axis of the length direction of linear array detector (6), and the length of linear array detector (6) is designated as N, and this has also represented the sum of linear array detector probe unit; Q ₁Be grey scale curve p _β(s) peak, it is at coordinate axis O _dThe last respective coordinates of S is S ₂, the center of linear array detector (6) is at coordinate axis O _dThe last respective coordinates of S is S ₀, S ₀Also be the intersection point of central ray (2) perpendicular to linear array detector (6), the rotary middle point O of turntable (4) is at coordinate axis O _dThe last respective coordinates value of S is designated as R ₀, then have

S ₀-R ₀=τ ₀, τ ₀The side-play amount of the rotary middle point O of expression central ray (2) skew turntable (4);

At the second projection information p _β+π(s) in the grey scale curve, Q ₂Be grey scale curve p _β+π(s) peak, it is at coordinate axis O _dThe last respective coordinates of S is S ₁By the projection theory of collimated beam scanning as can be known, the first projection information p _β(s) grey scale curve and and the second projection information p _β+π(s) grey scale curve is at coordinate axis O _dS is last with R ₀Point is left-right symmetric relation, i.e. Q for the center ₁Point and Q ₂Point is about R ₀Point symmetry, satisfied pass is:

$|S_1-R_0|=|S_2-R_0|=\frac{1}{2}|S_2-S_1|---\left(1\right)$

Be S ₁=S ₂-(S ₂-S ₁)=S ₂-2 (S ₂-R ₀)=2R ₀-S ₂

With the second projection information p _β+π(s) at coordinate axis O _dUpset just obtains the projection information p ' that overturns about carrying out on the S _β+π(s), p ' _β+π(s) and p _β+π(s) pass is:

p′ _β+π(s)＝p _β+π(N-s) s∈[0?N] (2)

At upset projection information p ' _β+π(s) grey scale curve, Q ₃Be p ' _β+π(s) peak, it is at coordinate axis O _dThe last respective coordinates of S is S ₃, simultaneous formula (1), formula (2) get p ' _β+π(s) projection coordinate's relation:

S ₃＝N-S ₁＝N-(2R ₀-S ₂) (3)

Contrast the first projection information p _β(s) grey scale curve and upset projection information p ' _β+π(s) grey scale curve then has p _β(s) with p ' _β+π(s) grey scale curve shape is identical, just at coordinate axis O _dCertain displacement difference is arranged on the S direction, and this displacement difference promptly is a Q ₁With Q ₃The coordinate difference be designated as τ ', its expression formula is:

τ′＝S ₃-S ₂ (4)

Formula (3) substitution formula (4) is got:

${\mathrm{\&tau;}}^{\&prime;}=S_3-S_2=N-(2R_0-S_2)-S_2=N-2R_0$

$=2\&times;(\frac{N}{2}-R_0)=2\&times;(S_0-R_0)=2{\mathrm{\&tau;}}_0$

That is: ${\mathrm{\&tau;}}_0=\frac{{\mathrm{\&tau;}}^{\&prime;}}{2}.$

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