WO2001028425A1 - Multislice x-ray ct apparatus - Google Patents

Multislice x-ray ct apparatus Download PDF

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Publication number
WO2001028425A1
WO2001028425A1 PCT/JP2000/007271 JP0007271W WO0128425A1 WO 2001028425 A1 WO2001028425 A1 WO 2001028425A1 JP 0007271 W JP0007271 W JP 0007271W WO 0128425 A1 WO0128425 A1 WO 0128425A1
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WIPO (PCT)
Prior art keywords
slice
projection data
detector
spiral
weight
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Ceased
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PCT/JP2000/007271
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English (en)
French (fr)
Japanese (ja)
Inventor
Osamu Miyazaki
Taiga Goto
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Hitachi Healthcare Manufacturing Ltd
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Hitachi Medical Corp
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Priority to US10/111,170 priority Critical patent/US6826251B1/en
Publication of WO2001028425A1 publication Critical patent/WO2001028425A1/ja
Anticipated expiration legal-status Critical
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    • AHUMAN NECESSITIES
    • A61MEDICAL OR VETERINARY SCIENCE; HYGIENE
    • A61BDIAGNOSIS; SURGERY; IDENTIFICATION
    • A61B6/00Apparatus or devices for radiation diagnosis; Apparatus or devices for radiation diagnosis combined with radiation therapy equipment
    • A61B6/02Arrangements for diagnosis sequentially in different planes; Stereoscopic radiation diagnosis
    • A61B6/03Computed tomography [CT]
    • A61B6/032Transmission computed tomography [CT]
    • AHUMAN NECESSITIES
    • A61MEDICAL OR VETERINARY SCIENCE; HYGIENE
    • A61BDIAGNOSIS; SURGERY; IDENTIFICATION
    • A61B6/00Apparatus or devices for radiation diagnosis; Apparatus or devices for radiation diagnosis combined with radiation therapy equipment
    • A61B6/02Arrangements for diagnosis sequentially in different planes; Stereoscopic radiation diagnosis
    • A61B6/027Arrangements for diagnosis sequentially in different planes; Stereoscopic radiation diagnosis characterised by the use of a particular data acquisition trajectory, e.g. helical or spiral

Definitions

  • the present invention relates to a multi-slice X-ray CT apparatus (hereinafter, referred to as a multi-slice CT) capable of performing a spiral scan, and more particularly to a multi-slice X-ray CT apparatus having a correction processing means for performing a correction process on measured spiral projection data.
  • a multi-slice CT capable of performing a spiral scan
  • a correction processing means for performing a correction process on measured spiral projection data.
  • the mainstream of X-ray CT systems is the RZR (3rd generation) CT system, where an X-ray source and an arc-shaped detector that directs the focal point of the X-ray source face each other across the subject. Are located.
  • the X-rays from the X-ray source are collimated to form a fan-shaped X-ray beam, which is radiated to the imaging section of the subject.
  • the imaging operation is performed by measuring the transmitted X-ray attenuated by the subject while rotating it. Measurement operations during rotation are performed at angular intervals of about 0.1 to 0.5 degrees, and for example, projection data of a total of about 600 to 1200 angles is obtained.
  • the detector is composed of a large number of detection elements, and the output of each element is collected as digital data by the measurement circuit, and data (views) for the number of elements is formed for each measurement angle.
  • This view data is sequentially transferred from the rotating system to the stationary system via the transmission path.
  • the transferred measurement data is subjected to pre-processing such as characteristic correction of the detection element, radiation quality correction and log conversion by an image processing device in a stationary system, and then filter correction by a well-known algorithm such as back projection. Reconstructed as
  • a spiral CT helical CT
  • CT helical CT
  • An interpolation processing method is disclosed in, for example, US Pat. No. 4,789,929 (1988). It is shown. By performing interpolation processing, artifacts due to motion can be reduced.
  • multi-slice CT in which a detector is divided into a plurality of rows so that projection data of a plurality of cross sections can be measured simultaneously.
  • views are collected simultaneously for the number of rows, so that in the case of a normal table fixed scan, tomographic images of several cross sections can be taken at the same time.
  • an object of the present invention is to provide a multi-slice CT capable of always obtaining a high-quality tomographic image in response to a change in a helical pitch when a helical scan is performed in the multi-slice CT.
  • Another object of the present invention is to reduce the exposure dose in a multi-slice CT. Disclosure of the invention
  • the multi-slice CT apparatus of the present invention that achieves the above object has a plurality of rows of multi-element detectors in the body axis direction, while moving the patient table on which the subject is placed in the body axis direction,
  • a multi-slice CT that measures transmitted X-rays of the subject by rotating an X-ray source and the detector and obtains a plurality of spiral projection data, a correction processing unit that performs a correction process on the measured spiral projection data; Image reconstruction means for reconstructing the projection data after correction to obtain a tomographic image, wherein the correction processing means corresponds to a helical pitch which is a table movement amount per one rotation with respect to a row interval of the detector.
  • the method is characterized in that one of the slice spiral weights is selected and applied to the spiral projection data of each column, and the spiral projection data of each column after weight application is synthesized.
  • the correction processing means changes a weighted region of spiral projection data to be processed according to a spiral pitch at the time of measurement.
  • data close to the actual measurement position can be used as data used for interpolation, and high image quality can be achieved.
  • the correction processing means includes means for generating a multi-slice spiral weight to be applied to the spiral projection data of each column, and a spiral projection data of each column after applying the weight.
  • the method is characterized in that a multi-slice spiral weight is set for.
  • the total number of columns including the real detector and the virtual detector can be set to ⁇ .
  • the virtual detector can be arranged between adjacent real detectors, and the weight applied to the projection data of the virtual detector is distributed to the weight of the projection data of the adjacent real detector. Can be.
  • the virtual detector is arranged outside the measurement range of the real detector row in the body axis direction, uses facing data as projection data of the virtual detector, and assigns a weight for the facing data to the projection of the neighboring real detector. Can be distributed to data.
  • the concept of a virtual detector even when the spiral pitch is larger than the number of detector rows, interpolation using an appropriate weight can be performed, and high image quality can be achieved.
  • the X-ray source controls the number of rows of the detectors in accordance with a helical pitch, which is a table movement amount per rotation with respect to a row spacing of the detectors. Means.
  • the optimal number of rows can be set according to the helical pitch. If the number of rows is large, the number of rows can be limited to reduce the exposure dose.
  • FIG. 1 is a view showing an entire X-ray CT apparatus to which the present invention is applied
  • FIG. 2 is a view showing an example of a main part of the X-ray CT apparatus according to the present invention
  • FIG. 3 is an image of the X-ray CT apparatus according to the present invention
  • Fig. 4 shows an example of the treatment device
  • Fig. 4 illustrates spiral scanning in multi-slice CT
  • Fig. 5 shows an example of multi-slice spiral weight (in-phase interpolation)
  • Fig. 6 shows various spiral pitches and columns.
  • Fig. 7 (a) shows the relationship between the fan beam and the opposing beam
  • Fig. 7 (b) shows the relationship between the parallel beam
  • Fig. 8 shows the multi-slice spiral weight (inverse).
  • FIG. 9 is a diagram illustrating the concept between in-phase interpolation and inverse complementation
  • FIG. 10 is a sinogram showing a fan beam, a counter beam, and a parallel beam
  • FIG. 11 is a variable view according to the present invention.
  • FIG. 12 shows weights according to the present invention.
  • FIG. 13 is a diagram showing an example of weights including a virtual detector according to the present invention;
  • FIG. 14 is a diagram showing another example of weights including a virtual detector according to the present invention;
  • 15 is a diagram showing another example of the weight including the virtual detector according to the present invention,
  • FIG. 16 is a diagram showing another example of the weight including the virtual detector according to the present invention, and FIG.
  • FIG. 17 is a diagram of the real detector and the virtual detector.
  • Fig. 18 shows the relationship with the opposing data
  • Fig. 18 shows the weight (in the case of anti-phase interpolation) including the virtual detector
  • Fig. 19 shows the setting of the virtual detector for eliminating discontinuity
  • FIG. 20 is a diagram showing a flow of weight generation when a virtual detector is set
  • FIG. 21 is a diagram showing an example of a mechanism for controlling the number of detector rows.
  • FIG. 1 is a diagram showing an outline of an X-ray CT apparatus to which the present invention is applied.
  • the X-ray CT apparatus includes a scanner 11 having an X-ray generation unit (X-ray source), a detector, a measurement circuit unit, a rotary scanning mechanism control unit, and a patient as a subject.
  • Scanner 11 Patient table to be transported to the measurement space inside 12, Image processing unit 13 that performs image processing such as pre-processing and reconstruction on data measured by the measurement circuit unit 13, Display unit 14 that displays reconstructed images, X-ray A high voltage generator 15 for supplying a high voltage to the generator and a host computer 16 for controlling the whole are provided.
  • the X-ray detector 17 has a plurality of rows of multi-element detectors in the body axis direction (slice direction).
  • the detector is divided into 16 columns, and the output of the divided 16 columns is configured to be output as four arbitrary systems by the output selector 171.
  • the output of the four systems may be an output of any four of the 16 columns, or an output obtained by adding a plurality of columns in an analog or digital manner. Images having different slice thickness characteristics and different noise characteristics can be obtained depending on how the four systems are set.
  • the description will be made assuming that the number of detector columns is four, but the number of columns referred to in the present invention includes the number of columns after addition as described above, and is limited to four columns. not.
  • the X-ray generator includes an X-ray tube 181 and a collimator 182 that collimates the X-rays emitted from the X-ray tube and controls the fan beam to have a predetermined width and a predetermined opening angle.
  • the collimator 182 has a mechanism for adjusting the width of the X-ray in the slice direction. By adjusting the width of the fan beam in the slice direction, the number of detector rows described above can be arbitrarily set. The mechanism for adjusting the width can be controlled by the host computer 16.
  • the X-ray CT apparatus configured as described above measures the transmitted X-rays of the subject by rotating the scanner 11 (X-ray source and detector) while moving the patient table 12 in the body axis direction.
  • Spiral projection data can be acquired for the number of rows of instruments (here, four systems).
  • the acquisition of such spiral projection data is known as a spiral scan, and the spiral pitch! 5 is defined as a table feed amount t (t / LI) per rotation with respect to the detector row interval ⁇ Z.
  • FIG. 3 is a diagram conceptually showing the configuration of the image processing unit.
  • the projection data memory 131 stores four columns of projection data output from the output selector 171 of the detector 17, and the projection data
  • the projection distribution device 132 that distributes the data to the elements, and the weighting factors used in the arithmetic element
  • a selector 134 an operation unit 135 composed of operation elements PE0 to PE3 for multiplying the weighting coefficient in the weighting coefficient table and the corresponding projection data, a projection data synthesis device 136 for synthesizing the operation result of each operation element, and projection data synthesis
  • An image processing device 137 is provided for subjecting the projection data for a predetermined slice obtained by the device 136 to known image processing such as filter-corrected backprojection and performing image reconstruction.
  • the weight generation device 133 creates a plurality of types of weighting factors according to differences in the helical pitch, mode, and the like, and stores them as a weighting factor table.
  • One feature of the multi-slice CT apparatus of the present invention is that when a predetermined helical pitch is set, an optimum weighting factor according to the set helical pitch is applied.
  • the weight selector 134 selects one weight coefficient table from a plurality of weight coefficient tables, and loads the selected weight coefficient table into each operation element.
  • Fig. 4 shows the above-mentioned four sets of output data (hereinafter referred to as column data) R1 to R4, where the horizontal axis represents the slice direction Z and the vertical axis represents the projection angle, that is, the angular position of the focal point of the X-ray tube. is there.
  • column data the horizontal axis represents the slice direction Z
  • the vertical axis represents the projection angle, that is, the angular position of the focal point of the X-ray tube. is there.
  • each element row shifts linearly in the slice direction, so that four straight lines are obtained on the Z ⁇ / 3 plane as shown in the figure.
  • each column data is multiplied by a weight based on Zs (generally a weight based on the concept of interpolation). By cumulatively adding the weighted data for each projection angle, projection data for one rotation can be obtained for the slice.
  • the region indicated by a thick line on each column data is a portion where a weight coefficient other than 0 is multiplied. That is, the weighted addition portion (view length: Weight Length).
  • view length Weight Length
  • the difference V0, VI, V2, V3 between the predetermined view angle (for example, the view angle when the focal point of the X-ray tube passes through) / 3s and the starting point of the view length of each row for slice Zs is set to the offset view. Is a number.
  • Figure 5 shows an example of the weights applied here expressed on a sinogram.
  • the sinogram is a two-dimensional measurement space in which the horizontal axis is the opening angle ⁇ of the detector channel and the vertical axis is the X-ray tube angle (projection angle).
  • the area indicated by hatching is an area to which weights are set, and the area indicated by the same hatching is a pair for interpolation.
  • the area between Rowl's) 31 and) 31- and the area between Row2's / 32 + and 32 are an interpolation pair, and Row2's / 32)
  • the area between 3 2 ⁇ and the area between 3+ and / 3 3 of Row 3 are interpolation pairs.
  • the reference view angle 3 n is obtained by the following equation (1) using the view angle 3 s when the focal point passes through the slice Zs.
  • V takes +1 or 1 1 depending on the table feed direction.
  • ⁇ 3 is ⁇ / ⁇ .
  • the weight is suitable for the area where the upper limit is / 3 ⁇ + and the lower limit is ⁇ - centering on the view angle ⁇ . Used. 3n + and n-, which represent the upper and lower limits, respectively
  • Wn (ct,) 3) does not need to be linear as in the above equation, and may satisfy the following equation with respect to the interpolation position ⁇ (the ratio of the interpolation position to the sample interval).
  • Typical examples of f (6) + f (1- ⁇ ) 1.0 (8) f (S) , there are 3 ⁇ 2 +2 ⁇ 3.
  • ⁇ - ⁇ ⁇ / 3 ⁇ + corresponds to the Wn ( ⁇ ,) force ⁇ in the above equation.
  • Figure 5 shows the force in the case of multi-slice CT, which illustrates the weight in the case of interpolation between data with the same phase (in-phase interpolation) using fan beam projection data parallel to the ⁇ axis.
  • the opposing beam is a beam in the range of S1 to S2 (0 + ⁇ ⁇ 2 ⁇ ) for the fan beam whose focal point is at position SO () 30), as shown in Fig. It is represented as an area with
  • FIG. 8 shows an example of weights in the case of anti-phase interpolation using such opposed beams.
  • Rowl to Row3 correspond to Rowl to Row3 in FIG.
  • Rl ', R2, R3 correspond to the opposing beams of Rowl, Row2, and Row3, respectively.
  • Such anti-phase interpolation can be realized as weighting for data in the range of ⁇ ⁇ / 2.
  • Figures 9 (a) and 9 (b) show the differences between in-phase and anti-phase interpolation in multi-slice CT. This will be described with reference to FIG. Fig. 9 (a) shows the case of in-phase interpolation, where the projection data of a given cross-section SP is the interpolation of the in-phase data of the closest column, and the weight is set to maximize SP as shown in the lower row. It becomes a ramp function. The width of this weight in the Z direction is soil ⁇ Z.
  • the width of the weight in the ⁇ direction is soil ⁇ / 2
  • the width of the weight in the ⁇ direction can be reduced to half of that in the case of inphase interpolation, so that the effective slice width can be halved.
  • a function other than the linear function shown in the figure can be used as the weighting function, provided that the above-mentioned equation (8) is satisfied.
  • the view to be handled is made variable according to the helical pitch (the inclination of the helical). .
  • the variable view interpolation will be described.
  • Fig. 7 (a) shows the relationship between the fan beam parallel to the ⁇ -axis and the opposing fan beam
  • Fig. 7 (b) shows the fan beam parallel to the ⁇ -axis and the parallel beam with the same projection angle of each channel. This shows the relationship.
  • Figure 10 shows these different view types in sinogram ( ⁇ , ⁇ ) space. As can be seen from Fig. 10, the projection angle of each line is
  • the reference line of each column can be selected as in the following equation.
  • the reference line is a parallel beam
  • each region is a parallel beam data set having a spread of ⁇ . Therefore, weighting corresponding to in-phase interpolation between parallel beam projection data is performed.
  • the weighting corresponding to the same phase interpolation between the fan beam and the opposite-beam projection data is performed.
  • Such a view type (M) is generally changed according to the helical pitch (table feed amount).
  • the selection of M may be linked with the inclination of the spiral, or may be arbitrarily selected by providing a dedicated mode.
  • Figure 12 shows the flow of generating the weight for one column. Note that the weight generation is realized by the weight generation device 133 of the image processing device 13 shown in FIG.
  • a reference angle 3 s corresponding to a slice position to be reconstructed is determined (step 121).
  • the reference angle 0 s is, for example, the view angle when the focal point passes through the slice Zs.
  • the reference angle n of each column is obtained by the above-mentioned equation (1) (Step 122), and based on the M value selected or set in advance, the upper limit of the weighting area of the column data) 3 n + and Lower limit) 3 n- is defined (step 123). Further, the spread of the weights (weighting range) is set (step 124).
  • G is a parameter for controlling the weight range.
  • the spread of the weight can be arbitrarily varied according to the image slice thickness.
  • the weight Wn is determined by applying, for example, a ramp function represented by equations (4) to (7). Yes (step 125).
  • Figure 13 shows a case where the pitch is 5 with a multi-slice of 4 rows. Since the fifth column set here does not actually exist, it can be obtained by extrapolating from, for example, Row3 and Row4, or by substituting the counter beam in the fifth column and applying a weight to this.
  • half-period phase difference
  • the area indicated by the same hatching in the figure is a pair for interpolation (for example, Q1 of Rowl and Q5 of Row5), but the data corresponding to the fifth column does not need to be actually obtained, but simply
  • the weights of the virtual detectors may be distributed to the real detectors according to the interpolation weights for obtaining the data of the virtual detectors. For example, when interpolation is performed from Row2 and Row3, 0.5 is distributed to Row2 and Row3.
  • W2 (a, / 3) W2 (a, / 3) +0.5 X W5 (a, ⁇ )
  • the column data of the virtual detector may be obtained by calculation, and weights based on equations (4) to (7) may be applied thereto.
  • the weight setting for the opposing beam is the same as in the case of FIG.
  • FIG. 14 shows the case where the difference between the helical pitch and the number of rows is 1, but the same can be applied when the difference is 2 or more.
  • This embodiment can be applied not only to the case where the spiral pitch is an integer, but also to the case where the spiral pitch has a fraction.
  • the detector array is further subdivided. For example, in the case of 4 rows and a pitch of 4.25, the weight is configured as 16 rows and a pitch of 17. By subdividing the detector array in this way, the problem can be simplified, and the concept of the virtual detector array can be easily applied.
  • antiphase interpolation is effective for odd pitches. That is, as can be seen from a comparison between FIGS. 15 and 16, when the spiral pitch is odd, the opposing data of the virtual detector row is located in the middle of the row data of the actual detector row, whereas the spiral pitch is If is an even number, the opposite data of the virtual detector row matches the real detector row, The effect of resolution cannot be expected. This is shown in FIG.
  • FIG. 18 shows an example of antiphase interpolation when a virtual detector is set in Row5.
  • antiphase interpolation can be realized by setting the width of the weight created by the in-phase interpolation to ⁇ ) 3/2, and a high resolution of 20% or more can be achieved.
  • the width of the weight has been exemplified in the case of the basic ⁇ and in the case of ⁇ / 2 in the case of an odd-number pitch capable of achieving the highest resolution, but the width of the weight is arbitrarily set to ⁇ / 3/2 as a minimum. It is possible to change. By changing the width, the effective slice thickness of the image can be arbitrarily changed.
  • a pair of interpolation is different for each ⁇ 3.
  • This change causes discontinuity in the corrected projection data.
  • This discontinuity '14 becomes noticeable when the width of the weight is small. Therefore, in the present embodiment, as shown in FIG. 19, a virtual detector VR is arranged between each column, and weights are set for all columns including this virtual detector.
  • the spiral projection data obtained with, for example, 4 rows and a spiral pitch of 3 is equivalent to 8 rows or 7 rows of the spiral pitch 8.
  • the weight for the virtual detector VR the weight when three columns and three pitches are applied considering only the virtual detector is applied.
  • the weight for the virtual detector is distributed to the peripheral weights, and is added to the weight of each column.
  • 4 rows and helical pitch of 1.25 can be treated as 8 rows and helical pitch of 2.5.
  • efficient weight setting is possible.
  • Fig. 20 shows the flow of weight generation when a virtual detector is used.
  • a weight is generated for an actual detector array (step 201). If the number of columns is N and N is satisfied, N-P columns overlap. In this case, the overlapping columns are averaged to avoid redundancy (step 202). On the other hand, if N ⁇ P, the number of columns is insufficient, so a virtual detector is assumed, its weight is generated (step 203), and this weight is distributed to the weight of the real detector (step 204).
  • the weight for setting the virtual detector between adjacent real detectors is distributed between the real detectors. After that, the weighting process is performed.
  • the collimator 182 for controlling the width and the opening angle of the X-ray is provided with a mechanism for adjusting the width of the X-ray in the slice direction.
  • this Figure 21 shows the mechanism.
  • the adjusting mechanism includes a control means 211 (FIG. 2) controlled by a command from the host computer 16 (FIG. 1), a motor 212 driven by the control means 211, a slice collimator 213 of the link mechanism, and a motor. Means for transmitting the rotation of 212 to the link mechanism of slice collimator 213.
  • the control means 211 drives the motor 212 to adjust the opening width of the slice collimator 213 in the slice direction. .
  • the link rotates clockwise, for example, by the rotation of the motor 212, the X-ray irradiation range in the Z direction is increased, and when the link rotates counterclockwise, the X-ray irradiation range in the Z direction decreases.
  • the irradiation range in the direction perpendicular to the Z direction is limited by the X-ray shielding container.
  • This adjustment mechanism is effective especially when the number of detector rows increases to 16 to 96 rows, and basically, a complete image can be reconstructed if the number of rows is about half the pitch P. Is possible.
  • the illustrated collimator employs a link mechanism
  • a known mechanism such as a slide mechanism can be employed as long as the mechanism can adjust the width in the Z direction.
  • the number of columns can be adjusted in relation to the weight algorithm described above, and the effectiveness of the interpolation processing according to the present invention is further improved. Can be done. In addition to eliminating measurement redundancy, it is possible to reduce the amount of exposure.
  • the present invention it is possible to set an optimum interpolation weight according to a helical pitch in a multi-slice CT, and to achieve high image quality. Further, according to the present invention, by using the concept of a virtual detector, it is possible to efficiently set an optimal interpolation weight in various cases where the relationship between the number of detector rows and the spiral pitch is various. It is possible to eliminate discontinuities that occur in weights. Further, according to the present invention, it is possible to reduce redundant measurement as much as possible and reduce exposure.

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PCT/JP2000/007271 1999-10-20 2000-10-19 Multislice x-ray ct apparatus Ceased WO2001028425A1 (en)

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EP1324271A3 (en) * 2001-11-09 2005-04-27 GE Medical Systems Global Technology Company LLC Row-wise full helical view weighting method and apparatus for CT scanners
US7221730B2 (en) 2001-10-24 2007-05-22 Hitachi Medical Corporation Multi-row detector x-ray CT apparatus and method for creating tomogram

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JP4056922B2 (ja) * 2003-04-21 2008-03-05 ジーイー・メディカル・システムズ・グローバル・テクノロジー・カンパニー・エルエルシー 放射線計算断層画像装置
JP2007505674A (ja) * 2003-09-18 2007-03-15 コニンクリユケ フィリップス エレクトロニクス エヌ.ブイ. イメージング処理方法及び装置
JP4091008B2 (ja) * 2004-03-09 2008-05-28 ジーイー・メディカル・システムズ・グローバル・テクノロジー・カンパニー・エルエルシー Ct画像生成方法およびx線ct装置
DE102004020861B4 (de) * 2004-04-28 2009-10-01 Siemens Ag Verfahren zur Rekonstruktion von Projektionsdatensätzen bei Dosis-reduzierter abschnittsweiser spiralförmiger Abtastung in der Computertomographie
KR100687846B1 (ko) * 2005-01-21 2007-02-27 경희대학교 산학협력단 국부 고해상도 엑스선 단층 영상 재구성 방법 및 국부고해상도 엑스선 단층 영상 재구성 장치
US7242749B2 (en) * 2005-11-15 2007-07-10 General Electric Company Methods and systems for dynamic pitch helical scanning
US7269244B2 (en) * 2006-01-25 2007-09-11 General Electric Company Methods and apparatus for generating thick images in cone beam volumetric CT
JP5872212B2 (ja) * 2011-09-08 2016-03-01 株式会社東芝 マルチスライスct装置及びデータ前処理方法
JP6280851B2 (ja) * 2014-09-30 2018-02-14 ジーイー・メディカル・システムズ・グローバル・テクノロジー・カンパニー・エルエルシー 放射線断層撮影装置及びプログラム

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