Patent Number: 
Section: description

The theory and formation of gridline artifacts is well established (D. R. Bednareck, Radiology 147:255-258, 1983; M. A. King and G. T. Barnes, Med. Phys. 10:4-9, 1983, D. M. Gauntt and G. T. Barnes, submitted to Med Phys., 2002). As discussed above, the use of an anti-scatter grid in an X-ray imaging system can create a gridline artifact, which is a visible X-ray shadow of the septa of the grid. For example, suppose that a stationary anti-scatter grid is placed over the image receptor. The intensity of the primary X-ray radiation falling on the image receptor would then be proportional to the grid transmission function, which describes the transmission of primary X-rays through the grid as a function of position. A typical grid transmission function is shown in FIG. 3. Under the grid septa (indicated in FIG. 3 as xe2x80x9cdxe2x80x9d), the transmission function is zero because the septa absorb essentially all of the primary X-rays. Under the interspace material (indicated in FIG. 3 as xe2x80x9cDxe2x80x9d), the function is a uniform value (Tmax) as close to 100% as possible since the interspace materially allows a substantial portion of the X-rays to pass through. The grid pitch (which equals xe2x80x9cd+Dxe2x80x9d) is defined as xe2x80x9cP.xe2x80x9d Depending on the precise application, the grid is periodic either in linear distance or in angular distance. The angular distance refers to the angle between the center of the grid and a position on the grid, referenced back to the focal axis. A grid designed for linear motion is built to be periodic in linear distance; one designed for pendulous motion is built to be periodic in angular distance. Conventional high frequency X-ray generators produce pseudo-rect tube voltage (kV) and tube current (mA) functions (FIG. 4). A rect tube current function is generated if the X-ray tube current is instantly turned on at the start of an exposure, is constant during the entire period of the exposure and is instantly turned of at the end of the exposure. In a practical X-ray generator, the time required to turn the tube current and tube voltage on and off is on the order of one millisecond, and both the tube current and tube voltage vary slightly during the exposure, generally with less than 10% variation; therefore, we describe these functions as pseudo-rect. For true rect functions, gridline artifacts are eliminated when the product of X-ray exposure time and grid velocity is an exact integral multiple of the grid pitch (xe2x80x9cPxe2x80x9d). However, when the product of X-ray exposure time and grid velocity is not an exact integral multiple of the grid pitch, not all points on the image receptor are exposed to primary radiation for the same duration. This situation creates gridline artifacts. The conventional approach to solve this problem is either to move the grid through a large number ( greater than 20) of grid pitches, or to use such a small septum spacing that the grid line shadows are imperceptible. It is possible to calculate the intensity profile for a given form of the intensity function. The intensity function is the intensity of the primary x-ray radiation incident on a given point of the grid as a function of time. The intensity profile is the total primary radiation incident on the image receptor as a function of position integrated over the length of the exposure. Any variations in the intensity profile due to the grid can manifest as a gridline artifact. FIG. 5 shows the intensity profile calculated using a rect intensity function for different amounts of grid travel ranging from 0.5 to 3.5 grid pitches. As can be seen, the amplitude of the artifact falls off approximately as the reciprocal of the number of grid pitches traversed. It has long been established (H. E. Potter, Am. J. Roent., Vol XXV, May 1931, pp. 677-683) that a grid motion of 20 grid pitches provides an adequate amount of gridline artifact suppression. However, as discussed above, such a large grid motion puts certain constraints on grid design. To maintain a compact grid housing and allow the grid to move 20 times the grid pitch requires that the grid pitch xe2x80x9cPxe2x80x9d be fairly small (a high grid density). Certain kinds of grids, such as articulated grids, are impractical to build with a fine pitch, so there is a need to design an X-ray imaging system that does not require a large grid motion. The gridline artifact can be characterized as an amplitude; that is, the difference between the maximum and minimum primary radiation intensity across the receptor, divided by the maximum intensity present with a stationary grid. FIG. 6 shows the amplitude of the intensity profile calculated using a rect tube current function for different amounts of travel by the anti-scatter grid (calculated by dividing the distance the grid travels, or grid motion, by the grid pitch). The amplitude of the intensity profile generally decreases with increasing distance traveled, but is zero for specific values of the grid travel distance. As discussed above, it has long been known shown that the gridline artifacts are completely suppressed if the grid travels a distance equal to a positive integral number of grid pitches during the X-ray exposure. However, even a small error in the grid velocity can cause the grid to travel a distance not equal to a positive integral number of grid pitches, resulting in a visible gridline artifact. In addition, small errors in the tube current function and/or timing can also cause a visible gridline artifact. One goal of the present disclosure is to suppress gridline artifacts when the anti-scatter grid moves through a short but arbitrary distance. This is accomplished by using a dynamic tube current function. We distinguish between dynamic functions, in which the parameter of interest (tube current or tube voltage) is varied or modulated by design, as opposed to stabilized or pseudo-rect functions, in which the design intention is to hold the parameter of interest at a constant value during the exposure. In practical X-ray generators, stabilized parameters often vary by several percent during an exposure. In one embodiment, the dynamic tube current function is a symmetric trapezoidal function. However, other tube current functions can be used; in particular, any arbitrary function convolved with a rect function of the appropriate width will substantially eliminate gridline artifacts. The present disclosure thereby provides a means of suppressing gridline artifacts without requiring that the grid move through a large number of grid pitches, thus allowing the construction of relatively coarse grids. These grids can be made with a high grid ratio and high primary transmission at relatively low cost. If the sharp edges of the rect function are changed to more gentle slopes, then the amplitude of the gridline artifact reduces. For example, FIG. 7 illustrates a symmetric trapezoidal current function. FIG. 8 shows a series of intensity profiles calculated using a symmetric trapezoidal current function for a total grid distance traveled of 3.4 grid pitches with various widths of the symmetric trapezoidal current ramp time (tramp). The width of tramp is expressed in grid repeat time (GRT), which is calculated by dividing the grid pitch (xe2x80x9cPxe2x80x9d) by the grid velocity. A GRT equal to 1 corresponds to the time for the grid to travel a distance of 1 grid pitch. Therefore, if tramp is equal to 1, then the width of tramp correspond to the time it takes the grid to move a distance equal to 1 grid pitch. FIG. 9A shows the amplitude of the gridline artifact as a function of the ramp time (tramp) when the total grid motion is 3.4 grid pitches. When the current ramp time (tramp) is exactly equal to the GRT, the gridline artifact disappears. In addition, the gridline artifact will disappear whenever either the current ramp time (tramp), or the difference between the current ramp time (tramp) and the total exposure time, is a positive integral multiple of the GRT, regardless of the total exposure time. When both conditions are true, an additional degree of gridline artifact suppression is obtained; this additional suppression can help minimize the effect of small errors in the grid velocity or tube current. FIG. 9B shows the amplitude of the intensity profile as a function of the amount of grid motion when the ramp time is set equal to the GRT. The intensity profile is zero for all grid motion of more than 2 grid pitches. Minimum length of this tube current function is 2 times the GRT, so motion of less than 2 grid pitches is not possible. A comparison of this figure to FIG. 6 shows the advantage of radiographic exposures taken using the process of the present disclosure. The present discussion describes a dynamic tube current function with a stabilized tube voltage function. It is also possible to modulate the tube output by modulating both the tube current function and the tube voltage function during the exposure. Varying only the tube current function is preferred, since the intensity of radiation reaching the image receptor is proportional to the tube current regardless of the breast thickness or composition. In contrast, the radiation intensity at the image receptor is a non-linear function of the tube voltage, and the shape of the non-linear function changes with changes in breast thickness and composition. Thus, the task of suppressing the gridline artifacts by modulating the tube output is simplified when the tube voltage is stabilized during the exposure. The use of dynamic tube output functions is not by itself novel. In 1931, Potter suggested the use of a trapezoidal xe2x80x9cintensityxe2x80x9d function to suppress gridline artifacts, a technique that he described as xe2x80x9cfeatheringxe2x80x9d (H. E. Potter, Am. J. Roent., Vol XXV (May 1931), pp. 677-683). The current disclosure builds on Potter""s suggestion in several ways. The first is the constraint on timing the intensity function and the grid repeat time. While Potter suggested that a minimum ramp time may be necessary to suppress gridline artifacts, he does not mention the possibility of completely suppressing gridline artifacts by matching the ramp time to the grid repeat time. The second improvement lies in the use of more practical functions. Designing a clinical system to follow a trapezoidal function is not trivial; in his paper, Potter admits to be unaware how to do so. In the present disclosure, we describe a broader class of functions that can be used to completely suppress the gridline artifacts; this class includes functions that unlike the symmetrical trapezoidal function can be implemented using a modern x-ray generator by modulating the filament current. Finally, Potter refers to varying the x-ray xe2x80x9cintensityxe2x80x9d, without reference to whether the tube current or the tube voltage is varied. As is described in the current disclosure, the intensity function seen by the image receptor varies with breast thickness, unless the tube voltage is held fixed. Other functions with soft edges are more suited for use as tube current functions than symmetric trapezoidal functions. One such function (the truncated dual exponential function) is shown in phantom in FIG. 10. This function has an exponential rising edge, a flat plateau, and an exponential falling edge. The rising edge is characterized by a time constant, and the falling edge is characterized by a different time constant, both of which may differ between X-ray tubes of different manufacture and model number. With the use of the proper time constants, this current function can be achieved on a typical X-ray tube by modulating the filament current. The time constants of the rising and falling edges can be set to match the characteristics of a given tube type. Typical intensity profiles at different amounts of grid motion calculated with the truncated dual exponential function are shown in FIG. 11. The amplitude of the gridline artifact intensity applying a truncated dual exponential function for different amounts of travel by the anti-scatter grid (calculated by dividing the distance the grid travels, or grid motion, by the grid pitch) is shown in FIG. 12. The truncated dual exponential function does not reduce the gridline artifact as was seen with the symmetric trapezoidal function (FIG. 7), and even at the minima of the function the amplitude of the intensity profile remains non-zero. It is possible to smooth the tube current function by convolving it with a rect function. Convolution is a standard mathematical procedure (F. B. Hilderbrand, Advanced Calculus for Applications, Second Edition, 1976, Prentice-Hall, Inc., Edgewood Cliffs, N.J.). FIG. 10 shows the truncated dual exponential function before (in phantom) and after convolution with a rect function of width xe2x80x9cWxe2x80x9d. FIG. 13 shows the intensity profile calculated for such a convolved tube current function for different widths of the convolving rect function (expressed in GRT). In FIG. 13, the total distance traveled by the grid is 2 grid pitches. FIG. 14 illustrates the amplitude of the intensity profile as a function of the convolving rect function width xe2x80x9cWxe2x80x9d. When the width of the convolving rect function is equal to the grid repeat time, the grid artifact disappears entirely for any grid motion of more than the grid pitch. This effect is similar to artifact suppression seen with the symmetric trapezoidal function (FIG. 9B). It should be possible to improve the robustness of the gridline artifact suppression by careful optimization of the tube current function and the grid velocity. FIG. 14 shows calculations for a total grid travel of both 3.4 and of 5.5 grid pitches. Clearly, the latter case is more robust against errors in the convolution width, and is likely more robust against errors in other parameters. Gridline artifacts should be entirely suppressed whenever the tube current function is equal to any function with a finite time integral convolved with a rect function whose width is equal to a positive integral of the grid repeat time, for any exposure time larger than the GRT. For example, a symmetric trapezoidal function may be considered a rect function convolved with a rect function; when either one of the rect functions has the appropriate width, then the gridline artifacts are completely suppressed. In this case, the ramp time tramp is given by the width xe2x80x9cWxe2x80x9d of the narrower of the two rect functions. This phenomenon is one foundation of the current disclosure. By modulating the tube current function in an appropriate way it is possible to completely suppress the gridline artifact for any exposure time greater than the GRT. When the grid velocity is constant, one class of tube current functions that does this is generated by convolving an arbitrary function by a rect function with a width equal to a positive integer multiple of the GRT. Equivalently, any tube current function whose Fourier transform has zero amplitude at all frequencies equal to all positive integer multiples of the reciprocal of the GRT will completely suppress the gridline artifact if the grid velocity is constant during the exposure. There is a more general rule for gridline artifact suppression that includes situations where the velocity changes during the exposure; this rule is described in Example 1 of these specifications, and in a paper published by the inventors (D. M. Gauntt and G. T. Barnes, submitted to Med. Phys., 2002). Typical specifications of a compact, high ratio, high primary transmission anti-scatter grid for use in mammography according to the present disclosure are listed in Table 1. While it is appreciated that a grid according to the present disclosure is readily constructed having values that differ from those given in Table 1, it is preferred that the width of the radiolucent interspace material (xe2x80x9cDxe2x80x9d) is more than 8 times greater than the width of the radiopaque adjoining septa (xe2x80x9cdxe2x80x9d) in order to achieve superior primary X-ray transmission, and therefore, superior dose efficiency. The thickness and material of the septa should also be sufficient to effectively eliminate X-ray transmission through the septa. In one embodiment, a grid according to the present disclosure is flat. In an alternate embodiment, the grid is curved. Whether the grid is flat or curved, the grid will preferably have septa focused to the appropriate X-ray focus-grid distance. In one embodiment, the flat grid is mounted in a Bucky housing on a curved track, with the curved track having a radius of curvature corresponding to the X-ray focus-grid distance. This allows the grid to remain aligned on the X-ray focus while moving transversely. In an alternate embodiment that is substantially the same each end of the grid is mounted on a straight track, where the two straight tracks are angled relative to each other, and tangent to the curved track just described. In a further embodiment, the grid may be mounted on a linear track, but be built in such a way that the individual septa articulate within the grid to remain focused on the X-ray focus. Generally, the distance between the septa of the grid is fixed while the grid is moving and the image receptor is exposed to X-rays. In one embodiment, the top surface of the septa will be bonded to a thin radiolucent sheeting material such as Mylar (DuPont de Nemours Co.) or a thin carbon fiber composite in order to maintain septal spacing and damp vibrations within the grid. In the case of non-articulating grid septa, the damping may be accomplished alternatively by using an interspace material with X-ray attenuation approaching that of air, such as aerogels or polymer foams. In the case of grids with articulating septa, small tabs on the top of the grid slats may be designed to extend through corresponding holes in the radiolucent sheet. One embodiment of the present invention is shown in FIGS. 15, 16 and 17. As shown in FIG. 15, a fixed focal-length high-ratio anti-scatter grid 14A is equipped with bearings 15 that ride along a curved guide track 16. The starting position of the grid 14A is indicated, with intermediate and finishing positions shown in phantom. The center of curvature of the track 16 is coincident with grid focal axis, which passes through the X-ray focus 4. A drive means (not shown) moves the grid 14A at a constant angular velocity along the track 16. As the grid 14A is driven along the track 16, the grid remains aligned on the X-ray focus 4. The amount of motion is exaggerated for clarity. In practical mammography applications, a grid 14A according to the present disclosure will rotate through an arc of approximately 0.5xc2x0 to 2xc2x0. As shown in FIG. 17, the high ratio anti-scatter grid 14A is equipped with bearings 15 which are configured to engage the curved guide tracks 16. In an alternate embodiment, the tracks 16 may be straight, but angled so that a normal through the center of each track passes through the focal spot. Throughout the motion of the grid, including the extreme end of motion (as shown in FIG. 17), the anti-scatter grid 14A completely covers the image receptor 17 and the AEC sensor 18. An alternate embodiment of the high ratio, anti-scatter grid of the present disclosure is shown in FIGS. 18 through 21. The articulated grid 14B is equipped with a drive mechanism (not shown) that moves the grid 14B at a constant linear velocity along a track (not shown). The drive mechanism moves the upper and lower bars of the articulated grid at different velocities, in such a way that the septa of the grid 14B remain oriented on the X-ray focal spot 4. FIGS. 18, 20 and 21, show the articulation of the high ratio grid 14B. FIG. 18 shows how the articulating grid stays aligned on the focal spot while moving. As illustrated in FIG. 20, the high ratio grid 14B is composed of a series of anti-scatter septa 20 which are mounted by a first hinge means 22 to an upper support 21. The septa 20 engage a lower support 23 by a second hinge means 24. The first hinge means 22 are fixedly mounted to the septa 20. The second hinge means 24 may be sliding pins or rolling pins. The septa 20 remain aligned on the focus by moving the lower support 23 and the upper support 21 transversely by different amounts as shown in FIG. 21. Again, the grid completely covers the image receptor 17 and the AEC sensor 18 throughout the range of its motion. The motion of the grid is shown schematically in FIG. 21, which shows the grid in its leftmost position 26, its center position 27, and its rightmost position 28. One feature of the present disclosure is the use of a dynamic tube output function in combination with the high ratio, anti-scatter grid described above during exposure of the image receptor. Representative tube output functions are presented in FIGS. 7 and 10. Other functions may be used as discussed below, with the functions disclosed being exemplary and not intended to limit the scope of the disclosure. As discussed above, if the time W is a positive integral multiple of the GRT, then the gridline artifact is completely suppressed for any exposure time greater than GRT. The GRT has been defined above as the grid pitch divided by the grid velocity. The GRT may also be thought of as the time taken for the shadows of the grid lines to move the same distance as the separation between shadows. Even with other values of W, the gridline artifact would be considerably smaller than with a conventional rect function, so long as W is not much less than the grid repeat time. There are multiple ways to achieve the dynamic functions described. One technique would be to hold the tube voltage fixed while varying the tube current, either by varying the voltage on a control grid that shields the cathode from the anode or by varying the filament current. The former technique has the advantage of a rapid response to the control signal, so the symmetric trapezoidal function described above could be generated very accurately. However, gridded tubes are more expensive than conventional tubes to design and produce. It is possible to vary the filament current with conventional tubes, but the response of the tube current to changes in filament current is slower, and may require the use of a dynamic function other than the symmetric trapezoidal function (FIG. 7), such as, but not limited to, the convolved truncated dual exponential function (FIG. 10). However, any function that has zero frequency components at positive integer multiples of the reciprocal of the grid repeat time will completely suppress gridline artifacts. Any function that is equal to the convolution of an arbitrary function with a rect function whose width is a positive multiple of the grid repeat time will fit this criterion, and its use will completely suppress gridline artifacts. The use of tube output modulation has been described here for use with linear grids, in which the septa are all parallel to each other. In principle it can be used with any grid that has a septum pattern that is periodic with linear or angular distance in the direction of the grid motion. This includes cellular grids, in which multiple sets of septa intersect each other to produce a pattern of radiolucent cells. Examples of patents teaching cellular grids are U.S. Pat. No. 1,164,987 and U.S. Pat. No. 5,606,589. The following is a mathematical description of the theory behind gridline artifact suppression as described in the present disclosure. Suppose that a system incorporates a stationary anti-scatter grid located immediately over the image receptor, and that the x-ray intensity falling on the anti-scatter grid is uniform. In such a system, the x-ray flux density (i.e. the number of x-rays per area) falling on the image receptor would be "psgr"(x,y)="PHgr"0I0texpTgrid(x)xe2x80x83xe2x80x83(1) where the grid transmission Tgrid(x) is shown in FIG. 2, texp is the exposure time, I0 is the tube current, and "PHgr"0 is the x-ray intensity (x-rays per area per second per mA) measured without a grid. It is important to note that the grid transmission function Tgrid is periodic, with period P called the grid pitch. While the incident x-ray intensity is uniform, the flux density after the grid is not and so a series of gridline shadows falls on the image receptor. In some x-ray systems, such as fluoroscopic systems, the spatial resolution of the image receptor is relatively poor. In these systems, it is possible to use a stationary anti-scatter grid with septa so closely spaced that they are not visible to the image receptor. In other systems, the gridline shadows would appear in the final image as gridline artifacts. In these systems, the grid is moved during the exposure in order to blur out the grid lines. Since the x-ray flux rate at each point changes during the exposure, we need to integrate the flux rate over the exposure time:   "AutoLeftMatch"                                                                                          Ψ                  ⁡                                      (                                          x                      ,                      y                                        )                                                  =                                  xe2x80x83                                ⁢                                                      Φ                    0                                    ⁢                                                            QF                      grid                                        ⁡                                          (                      x                      )                                                                                                                                                                                    F                    grid                                    ⁡                                      (                    x                    )                                                  ≡                                  xe2x80x83                                ⁢                                                      ∫                    0                                          t                      exp                                                        ⁢                                                                                    T                        grid                                            ⁡                                              (                                                  x                          -                                                                                    x                              c                                                        ⁡                                                          (                              t                              )                                                                                                      )                                                              ⁢                                          i                      ⁡                                              (                        t                        )                                                              ⁢                                          xe2x80x83                                        ⁢                                          ⅆ                      t                                                                                                                                              Q                ≡                                  xe2x80x83                                ⁢                                                      ∫                    0                                          t                      exp                                                        ⁢                                                            I                      ⁡                                              (                        t                        )                                                              ⁢                                          xe2x80x83                                        ⁢                                          ⅆ                      t                                                                                                                                                                i                  ⁡                                      (                    t                    )                                                  ≡                                  xe2x80x83                                ⁢                                                      I                    ⁡                                          (                      t                      )                                                        Q                                                                                          (          2          )                     where xc(t) is the position of the center of the grid at time t. Note that we now allow the tube current I(t)=Q i(t) to change during the exposure. The function i(t) describes the tube current variation, and has a time integral of 1. The dimensionless function Fgrid(x) is analogous to the grid transmission. For the case where the tube current is constant and the grid is stationary, equation (2)reduces to equation (1). Most importantly, if the function Fgrid(x) is independent of x, then the grid is invisible to the image receptor and there is no grid line artifact. Otherwise, it adds to the image strips running in the y- direction. The function Fgrid(x) can be rewritten in terms of a mathematical entity called a convolution, indicated by a * symbol:   "AutoLeftMatch"                                                                                                              F                    grid                                    ⁡                                      (                    x                    )                                                  =                                  xe2x80x83                                ⁢                                                      ∫                    0                                          t                      exp                                                        ⁢                                                                                    T                        grid                                            ⁡                                              (                                                  x                          -                                                      x                            xe2x80x2                                                                          )                                                              ⁢                                          xe2x80x83                                        ⁢                                                                  i                        ⁡                                                  (                                                                                    t                              c                                                        ⁡                                                          (                                                              x                                xe2x80x2                                                            )                                                                                )                                                                                            v                        ⁡                                                  (                                                      x                            xe2x80x2                                                    )                                                                                      ⁢                                          xe2x80x83                                        ⁢                                          ⅆ                                              x                        xe2x80x2                                                                                                                                                                    =                                  xe2x80x83                                ⁢                                                                            T                      grid                                        ⁡                                          (                      x                      )                                                        *                                      (                                                                  i                        ⁡                                                  (                                                                                    t                              c                                                        ⁡                                                          (                              x                              )                                                                                )                                                                                            v                        ⁡                                                  (                          x                          )                                                                                      )                                                                                                                                            v                  ⁡                                      (                                          x                      xe2x80x2                                        )                                                  ≡                                  xe2x80x83                                ⁢                                                      ⅆ                                          x                      xe2x80x2                                                                            ⅆ                    t                                                                                                            (          3          )                     where v(xxe2x80x2) is the grid velocity when the center of the grid is at position xxe2x80x2, and tc(xxe2x80x2) is the time when the center of the grid is at position xxe2x80x2. It is possible to suppress the gridline artifacts entirely for any amount of grid motion more than 1 grid pitch. This is done by making use of the periodic nature of the grid shadow Fgrid. One characteristic of a periodic function with period P is that the average of the function over any domain of width P is the same, regardless of the locatoin of the center of the domain. To put it another way, the function Ggrid                                          G            grid                    ⁡                      (            x            )                          =                              1            P                    ⁢                                    ∫                              x                -                                  P                  /                  2                                                            x                +                                  P                  /                  2                                                      ⁢                                                            F                  grid                                ⁡                                  (                                      x                    xe2x80x2                                    )                                            ⁢                              xe2x80x83                            ⁢                              ⅆ                                  x                  xe2x80x2                                                                                        (        4        )             is independent of x. Equation (4) can be rewritten as a convolution over a rect function R:   "AutoLeftMatch"                                                                                                              G                    grid                                    ⁡                                      (                    x                    )                                                  =                                  xe2x80x83                                ⁢                                                      1                    P                                    ⁢                                      xe2x80x83                                    ⁢                                                            ∫                                              -                        ∞                                            ∞                                        ⁢                                                                                            F                          grid                                                ⁡                                                  (                                                      x                            xe2x80x2                                                    )                                                                    ⁢                                              R                        ⁡                                                  (                                                                                    x                              -                                                              x                                xe2x80x2                                                                                      P                                                    )                                                                    ⁢                                              xe2x80x83                                            ⁢                                              ⅆ                                                  x                          xe2x80x2                                                                                                                                                                                            =                                  xe2x80x83                                ⁢                                                      1                    P                                    ⁢                                      xe2x80x83                                    ⁢                                                            F                      grid                                        ⁡                                          (                      x                      )                                                        *                                      R                    ⁡                                          (                                              x                        P                                            )                                                                                                                                                                R                  ⁡                                      (                    x                    )                                                  ≡                                  xe2x80x83                                ⁢                                  {                                                                                    1                                                                                                                          "LeftBracketingBar"                            x                            "RightBracketingBar"                                                     less than                           0.5                                                                                                                                    0                                                                    otherwise                                                                                                                                                      (          5          )                     If we combine equations (5) and (3), we get                                           G            grid                    ⁡                      (            x            )                          =                              1            P                    ⁢                      (                                                            T                  grid                                ⁡                                  (                  x                  )                                            *                              (                                                      i                    ⁡                                          (                                                                        t                          c                                                ⁡                                                  (                          x                          )                                                                    )                                                                            v                    ⁡                                          (                      x                      )                                                                      )                                      )                    *                      R            ⁡                          (                              x                P                            )                                                          (        6        )             One property of convolutions is that the order in which they are evaluated can be changed without affecting the final result, so   "AutoLeftMatch"                                                                                                              G                    grid                                    ⁡                                      (                    x                    )                                                  =                                  xe2x80x83                                ⁢                                                                            T                      grid                                        ⁡                                          (                      x                      )                                                        *                                                            i                      xe2x80x2                                        ⁡                                          (                      x                      )                                                                                                                                                                                    i                    xe2x80x2                                    ⁡                                      (                    x                    )                                                  ≡                                  xe2x80x83                                ⁢                                                      1                    P                                    ⁢                                      (                                                                  i                        ⁡                                                  (                                                                                    t                              c                                                        ⁡                                                          (                              x                              )                                                                                )                                                                                            v                        ⁡                                                  (                          x                          )                                                                                      )                                    *                                      R                    ⁡                                          (                                              x                        P                                            )                                                                                                                                (          7          )                     Therefore comparing Equations (3) and (7), we see that when the tube current is described by   "AutoLeftMatch"                                                                                          i                  ⁡                                      (                    t                    )                                                  =                                  xe2x80x83                                ⁢                                                      v                    ⁡                                          (                      t                      )                                                        ⁢                                      h                    ⁡                                          (                                                                        x                          c                                                ⁡                                                  (                          t                          )                                                                    )                                                                                                                                                                h                  ⁡                                      (                    x                    )                                                  ≡                                  xe2x80x83                                ⁢                                                      1                    P                                    ⁢                                      (                                                                  g                        ⁡                                                  (                          x                          )                                                                                            v                        ⁡                                                  (                          x                          )                                                                                      )                                    *                                      R                    ⁡                                          (                                              x                        P                                            )                                                                                                                                (          8          )                     for some arbitrary function g(x), then Ggrid(x) is the moving grid transmission function Fgrid. Since Ggrid is independent of x, the image is free of gridline artifacts. When the grid velocity is constant, then xc(t)=vt and equation (8) simplifies to                               i          ⁡                      (            t            )                          =                              1            P                    ⁢                      xe2x80x83                    ⁢                      g            ⁡                          (              vt              )                                *                      R            ⁡                          (                              vt                P                            )                                                          (        9        )             By applying a Fourier transform to either side of Equation (8), we can show a more fundamental constraint on the function h(x):   "AutoLeftMatch"                                                                                          H                  ⁡                                      (                    k                    )                                                  =                                  xe2x80x83                                ⁢                                                      FT                    ⁡                                          (                                                                        g                          ⁡                                                      (                            x                            )                                                                                                    v                          ⁡                                                      (                            x                            )                                                                                              )                                                        ⁢                  sin                  ⁢                                      xe2x80x83                                    ⁢                                      c                    ⁡                                          (                                              π                        ⁢                                                  xe2x80x83                                                ⁢                        k                        ⁢                                                  xe2x80x83                                                ⁢                        P                                            )                                                                                                                                                                H                  ⁡                                      (                    k                    )                                                  ≡                                  xe2x80x83                                ⁢                                  FT                  ⁡                                      (                                          h                      ⁡                                              (                        x                        )                                                              )                                                                                                            (          10          )                     The sinc function will be equal to zero at all frequencies that are multiple integrals of 1/P. In general, the gridline artifacts will be completely suppressed if the tube current is exactly equal to the function v(t) h(xc(t)), where the Fourier transform H(k) of h(x) has zeroes at all frequencies k that are proportional to the reciprocal of the grid pitch P.