# Patent

## Title
Imaging system calibration

## Abstract

A system includes determination of a first sub-matrix of a projection matrix which describes a geometrical relationship between points of a three-dimensional coordinate system of the imaging system and points of a two-dimensional coordinate system of an image detector, determination of a second sub-matrix of the projection matrix, where the first and second sub-matrixes comprise a decomposition of the projection matrix, conversion of a first point of the two-dimensional coordinate system to a first point of the three-dimensional coordinate system based on the first and second sub-matrixes, determination of an updated first sub-matrix of an updated projection matrix, where the updated projection matrix describes a second geometrical relationship between points of the three-dimensional coordinate system and points of the two-dimensional coordinate system, and conversion of a second point of the two-dimensional coordinate system to a second point of the three-dimensional coordinate system based on the updated first and second sub-matrixes.

## Background
CROSS REFERENCE TO RELATED APPLICATION
The present application claims priority to Chinese Patent Application No. 201410024798.3, filed on Jan. 20, 2014, and entitled “Calibration Method for X-Ray Imaging System”, the entire disclosure of which is incorporated herein by reference.
BACKGROUND
An X-ray imaging system, as shown in FIG. 1, may include an X-ray source 1 and a flat panel detector (FPD) 2 coupled to a gantry 3. Rotation of the gantry 3 causes the X-ray source 1 and the FPD 2 to move in a circular orbit around a patient positioned on a couch 4. The FPD 2 may receive X-rays emitted from the X-ray source 1 after the X-rays have passed through the patient. Because the received X-rays have been attenuated to various degrees by the patient's intervening tissues, the received X-rays may be used to generate a two-dimensional projection image of the tissues.
Two-dimensional projection images may be acquired from different gantry angles, and multiple two-dimensional projection images acquired from different gantry angles may be used collectively to build a three-dimensional reconstructed image of the patient anatomy. Accurate reconstruction of the three-dimensional image requires an accurate expression of a geometric relationship between the three-dimensional volume in which the imaging system resides (e.g., a treatment room) and the two-dimensional images acquired at a given gantry angle and FPD position.
Referring to FIG. 2, the rotation axis of the X-ray source is defined as the International Electrotechnical Commission (IEC) Yf axis. At the vertical gantry position (known as the zero degree position), the axis which passes through the X-ray source focal spot and which is also perpendicular to the IEC Yf axis is defined as IEC Zf axis. The axis perpendicular to the Y-Z plane that passes through the intersection of the Yf axis and Zf axis is defined as the IEC Xf axis. In an ideal trajectory, the X-ray source moves in a circular orbit in the vertical X-Z plane, and the center of this circle coincides with the origin Of of this IEC coordinate system. This point is also known as the machine isocenter. When the gantry is at the zero degree position, the IEC Zf axis points from the machine isocenter Of towards the X-ray source and the IEC Xf axis points towards the right while looking towards the gantry. With reference to FIG. 1, calibrated lasers 5 point to the machine isocenter Of and the crossed lines shown in FIG. 1 indicate axes in different directions.
FIG. 2 further illustrates a two-dimensional pixelized imaging coordinate system defined in the detector plane, defined by the U axis and the V axis. The U axis is the row axis parallel to IEC Xf, and the V axis is the column axis anti parallel to the IEC Yf axis. The pixel coordinate (u=0, v=0) represents the top left corner pixel of the detector. The ideal detector plane is horizontal (perpendicular to the beam axis) and the planar imaging detector is “centered” by IEC Zf axis when the gantry is at the zero degree (vertical) position. Moreover, the pixel coordinate of the center of the detector 2 is (W/2pu, H/2pv), where H (respectively pv) and W (respectively pu), respectively represent the width and height of the detector 2 (respectively width and height of each pixel). Also, the IEC coordinate of the point Of is ideally (0,0,−(f−D)) where f and D represent “source to imager distance” (SID) and “source to axis distance” (SAD), respectively.
The actual trajectory of the X-ray source may differ from the above ideal description. For example, the X-ray source may move in and out of the vertical plane by a marginal amount. Moreover, as shown in FIG. 3, the trajectory of the X-ray source might not follow a perfect circle.
The relationship between the X-ray source and the FPD may also vary from the above-described ideal. For example, the relation between the detector assembly and the gantry might not be very rigid. Therefore, at each gantry angle, and due to gravity, the detector assembly may sag a different amount with respect to the central axis (CAX) of the beam (i.e., the line joining the X-ray source and the imaging isocenter), as shown in FIG. 4. In practice, the CAX meets the detector at a pixel location (u0, v0), which is somewhat different from (W/2pu, H/2pv). This point (W/2pu, H/2pv) is known as the principal point or the optical center. Also, as shown in FIG. 4, with respect to the ideal detector, the actual detector may exhibit (a) an out of plane rotation η about the axis it u=u0 or (b) an out of plane rotation σ about the axis v=v0 or (c) a in plane rotation φ about the point (u0, v0).
However, it has been noted that the out of plane rotations η and σ are quite difficult to determine with reasonable accuracy and these two angles have only a small influence on the image quality compared to other parameters. In practical implementations, these two angles can be kept small (≦10) through good mechanical design and high-accuracy machining. It is therefore reasonable to assume that η=σ=0.
In summary, according to the above model, the plane of the detector is perpendicular to the CAX, but the trajectory of the X-ray source is not a perfect circle and the values of SID f and SAD D in FIG. 2 are not constant for all the gantry angles. Also, the two-dimensional coordinate system comprising the row (U) and column (V) vectors of the image exhibits in-plane rotation and translation about the CAX.
FIG. 5 illustrates these non-idealities. Without loss of generality and applicability to other X-ray imaging systems, the X-ray source of FIG. 5 is a linear accelerator equipped with beam collimation devices, including a multileaf collimator (MLC) that may rotate in a plane that is ideally parallel to that of the corresponding FPD and perpendicular to the CAX. Referring to FIG. 5, the image receptor coordinate axes X—r and Y_r are in the plane of the detector and aligned along X_bld and Y_bld respectively. Due to sag of the detector assembly, the principal point (u0, v0) may differ from the center of the panel. Also, the row (U) and column vectors (V) of the acquired image may exhibit rotation with respect to the projection of the coordinate axis of the MLC (at the zero degree position).
A projection matrix may be used to describe the geometric relationship between any three-dimensional point (xf,yf,zf) in the imaging room and its projection pixel coordinate (u,v) on the two-dimensional FPD. The projection matrix corresponds to a given position of the source/detector (i.e., X-ray source and FPD) pair. For an ideally circular source trajectory, the projection matrix may be written as:




  
    
      [
      
        
          
            
              λ
              ⁢
              
                
              
              ⁢
              u
            
          
        
        
          
            
              λ
              ⁢
              
                
              
              ⁢
              v
            
          
        
        
          
            λ
          
        
      
      ]
    
    =
    
      
        
          
            
              [
              
                
                  
                    
                      1
                      /
                      
                        p
                        w
                      
                    
                  
                  
                    0
                  
                  
                    
                      u
                      0
                    
                  
                
                
                  
                    0
                  
                  
                    
                      
                        -
                        1
                      
                      /
                      
                        p
                        h
                      
                    
                  
                  
                    
                      v
                      0
                    
                  
                
                
                  
                    0
                  
                  
                    0
                  
                  
                    1
                  
                
              
              ]
            
            θ
          
          *
          
            
              [
              
                
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      ∅
                    
                  
                  
                    
                      
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                      ⁢
                      
                        
                      
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              ]
            
            θ
          
          *
          
            
              [
              
                
                  
                    
                      -
                      f
                    
                  
                  
                    0
                  
                  
                    0
                  
                  
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                      -
                      f
                    
                  
                  
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              ]
            
            θ
          
          *
          
            [
            
              
                
                  1
                
                
                  0
                
                
                  0
                
                
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                  0
                
                
                  0
                
              
              
                
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                  1
                
                
                  
                    -
                    D
                  
                
              
              
                
                  0
                
                
                  0
                
                
                  0
                
                
                  1
                
              
            
            ]
          
          *
          
            [
            
              
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    θ
                  
                
                
                  0
                
                
                  
                    
                      -
                      sin
                    
                    ⁢
                    
                      
                    
                    ⁢
                    θ
                  
                
                
                  0
                
              
              
                
                  0
                
                
                  1
                
                
                  0
                
                
                  0
                
              
              
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    θ
                  
                
                
                  0
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    θ
                  
                
                
                  0
                
              
              
                
                  0
                
                
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                  1
                
              
            
            ]
          
        
        
          {
          
            P
            θ
          
        
      
      *
      
        [
        
          
            
              
                x
                f
              
            
          
          
            
              
                y
                f
              
            
          
          
            
              
                z
                f
              
            
          
          
            
              1
            
          
        
        ]
      
    
  
  ,




where the symbol “*” denotes multiplication throughout this document.

However, as noted, the source trajectory is typically not a perfect circle of radius D in the vertical XfZf plane. [T]θ and [R]θ are the unknown translation and rotation, respectively, from the frame of reference of the imaging/treatment room to the frame of reference of the FPD. The subscript θ indicates that these transformations change with the gantry angle θ. Therefore, the projection matrix Pθ for a non-ideal source trajectory may be written as:




  
    [
    
      
        
          
            λ
            ⁢
            
              
            
            ⁢
            u
          
        
      
      
        
          
            λ
            ⁢
            
              
            
            ⁢
            v
          
        
      
      
        
          λ
        
      
    
    ]
  
  =
  
    
      
        
          
            [
            
              
                
                  
                    1
                    /
                    
                      p
                      w
                    
                  
                
                
                  0
                
                
                  
                    u
                    0
                  
                
              
              
                
                  0
                
                
                  
                    
                      -
                      1
                    
                    /
                    
                      p
                      h
                    
                  
                
                
                  
                    v
                    0
                  
                
              
              
                
                  0
                
                
                  0
                
                
                  1
                
              
            
            ]
          
          θ
        
        *
        
          
            [
            
              
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    ∅
                  
                
                
                  
                    
                      -
                      sin
                    
                    ⁢
                    
                      
                    
                    ⁢
                    ∅
                  
                
                
                  0
                
              
              
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    ∅
                  
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    ∅
                  
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
                
                  1
                
              
            
            ]
          
          θ
        
        *
        
          
            [
            
              
                
                  
                    -
                    f
                  
                
                
                  0
                
                
                  0
                
                
                  0
                
              
              
                
                  0
                
                
                  
                    -
                    f
                  
                
                
                  0
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
                
                  0
                
                
                  1
                
              
            
            ]
          
          θ
        
        *
        
          
            [
            T
            ]
          
          θ
        
        *
        
          
            [
            R
            ]
          
          θ
        
      
      
        {
        
          P
          θ
        
      
    
    *
    
      [
      
        
          
            
              x
              f
            
          
        
        
          
            
              y
              f
            
          
        
        
          
            
              z
              f
            
          
        
        
          
            1
          
        
      
      ]
    
  




Calibration of an imaging system may include determination of the above projection matrix Pθ for the non-ideal source trajectory. Conventionally, this calibration involves imaging of a geometry calibration phantom. Geometry calibration phantoms typically consist of radio-opaque beads at known three-dimensional locations with respect to some known frame of reference. This reference frame may be within the phantom itself. The elements of the projection matrix are found by solving equations which relate the known three-dimensional locations within the phantom with the detected two-dimensional pixel locations in a projection image.
Others calibration methods that do not involve a phantom. For example, calibration may include an iterative reconstruction method of perturbing a model projection matrix and generating forward projections to best-match observed projections. Another method uses redundant projections over 180′ and optimization to determine misalignment parameters with respect to an ideal projection matrix.
The existing methods for determining a non-ideal projection matrix are inefficient, time-consuming, based on inaccurate assumptions, and/or incompatible with treatment workflow.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a view of an X-ray imaging system;
FIG. 2 illustrates a coordinate system associated with an imaging system;
FIG. 3 illustrates trajectories of an X-ray source;
FIG. 4 illustrates rotation of a flat panel detector about different axes;
FIG. 5 schematically illustrates the row (U) and column vectors (V) of the acquired image with respect to projection of the coordinate axis of the MLC;
FIG. 6 is a flow diagram of a calibration method for an imaging system according to some embodiments;
FIG. 7 illustrates determination of a first sub-matrix of a projection matrix using an MLC pattern according to some embodiments;
FIG. 8 illustrates determination of a first sub-matrix of a projection matrix using an MLC according to some embodiments; and
FIG. 9 is a flow diagram of a calibration method for an imaging system according to some embodiments.



DETAILED DESCRIPTION
The present inventors have recognized that, for many imaging systems, the assembly which mounts the imaging source 1 to the gantry 3 is often quite rigid, and particularly in a case where the X-ray imaging source 1 is a radiation therapy delivery system (i.e., a linac). As a result, even though the trajectory of the X-ray source 1 does not circumscribe a perfect circle, the actual trajectory may be quite stable and reproducible over a substantial time frame (e.g., several months or years). Moreover, the inventors have recognized that the geometric relationship between the gantry and the detector panel is typically not as rigid (e.g., in order to reduce bulk and/or costs) and therefore varies to a greater extent over time.
The inventors have further discovered a system to decompose the non-ideal projection matrix Pθ into a “flexible” projection sub-matrix and a “rigid” projection sub-matrix. The terms “flexible” and “rigid” are used for convenience and are not intended to denote any particular degree of rigidity or flexibility. Decomposition of the projection matrix Pθ into a first projection sub-matrix PθFlexible and a second projection sub-matrix PθRigid according to some embodiments may be illustrated mathematically as shown below:




  
    [
    
      
        
          
            λ
            ⁢
            
              
            
            ⁢
            u
          
        
      
      
        
          
            λ
            ⁢
            
              
            
            ⁢
            v
          
        
      
      
        
          λ
        
      
    
    ]
  
  =
  
    
      
        
          
            
              
                [
                
                  
                    
                      
                        1
                        /
                        
                          p
                          w
                        
                      
                    
                    
                      0
                    
                    
                      
                        u
                        0
                      
                    
                  
                  
                    
                      0
                    
                    
                      
                        
                          -
                          1
                        
                        /
                        
                          p
                          h
                        
                      
                    
                    
                      
                        v
                        0
                      
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                  
                
                ]
              
              θ
            
            *
            
              
                [
                
                  
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        ∅
                      
                    
                    
                      
                        
                          -
                          sin
                        
                        ⁢
                        
                          
                        
                        ⁢
                        ∅
                      
                    
                    
                      0
                    
                  
                  
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        ∅
                      
                    
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        ∅
                      
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                  
                
                ]
              
              θ
            
          
          
            }
            
              P
              θ
              Flexible
            
          
        
        *
        
          
            
              
                [
                
                  
                    
                      
                        -
                        f
                      
                    
                    
                      0
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        -
                        f
                      
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                  
                
                ]
              
              θ
            
            *
            
              
                [
                T
                ]
              
              θ
            
            *
            
              
                [
                R
                ]
              
              θ
            
          
          
            }
            
              P
              θ
              Rigid
            
          
        
      
      
        {
        
          P
          θ
        
      
    
    *
    
      [
      
        
          
            
              x
              f
            
          
        
        
          
            
              y
              f
            
          
        
        
          
            
              z
              f
            
          
        
        
          
            1
          
        
      
      ]
    
  




According to the embodiments described below, the flexible projection sub-matrix PθFlexible and the rigid projection sub-matrix PθRigid are determined, and then used to convert coordinates of a two-dimensional coordinate system of a detector to a three-dimensional coordinate system, using calibrated projection matrix Pθ=PθFlexible*PθRigid. Thereafter, the flexible projection sub-matrix PθFlexible is periodically re-determined according to a first calibration schedule, and the rigid projection sub-matrix PθRigid is periodically re-determined according to a second calibration schedule, which is less frequent than first calibration schedule.
Calibration of the flexible projection sub-matrix PθFlexible may be faster and less-intrusive than conventional calibration of the non-ideal projection matrix Pθ. According to some embodiments, the flexible projection sub-matrix PθFlexible is re-determined (i.e., calibrated) daily or weekly, and the rigid projection sub-matrix PθRigid is calibrated every 6-12 months during scheduled maintenance service. The foregoing may result in increased uptime and cost savings.
According to some embodiments, calibration of the flexible projection sub-matrix PθFlexible does not require a phantom, thereby providing further cost savings. As such, calibration of the flexible projection sub-matrix PθFlexible may occur on-the-fly and, if desired, during all scans.
FIG. 6 is a flow diagram of process 600 for an imaging system according to some embodiments. Process 600 and the other processes described herein may be performed using any suitable combination of hardware, software or manual means. Software embodying these processes may be stored by any non-transitory tangible medium, including a fixed disk, a floppy disk, a CD, a DVD, a Flash drive, or a magnetic tape.
According to some embodiments, most or all of the steps of process 600 are embodied within executable code of a system control program stored in and executed by one or more processing units (e.g., microprocessors, microprocessor cores, execution threads) of control system 10 of FIG. 1. Imaging according to some embodiments includes, but is not limited to, X-ray portal imaging, cone beam computed tomography projection imaging and fan beam computed tomography imaging. Embodiments may also implement any type or construction of flat panel detector that is or becomes known.
Initially, at S610, a first sub-matrix of a projection matrix is determined. The projection matrix describes a geometrical relationship between a point of a three-dimensional coordinate system and a point of a two-dimensional coordinate system. For example, the projection matrix may describes the geometrical relationship between a point of a three-dimensional coordinate system of an imaging room and a point of a two-dimensional coordinate system of a flat panel detector located in the imaging room.
The first sub-matrix may be associated with in-plane rotation φ of the flat panel detector and variation in the principal point (u0, v0) as described above. According to some embodiments, the first sub-matrix represents the first projection sub-matrix PθFlexible described above.
In some embodiments of S610, the first projection sub-matrix PθFlexible is determined as:




  
    
      
        
          P
          θ
          Flexible
        
        =
        
          
            
              [
              
                
                  
                    
                      1
                      /
                      
                        p
                        w
                      
                    
                  
                  
                    0
                  
                  
                    
                      u
                      0
                    
                  
                
                
                  
                    0
                  
                  
                    
                      
                        -
                        1
                      
                      /
                      
                        p
                        h
                      
                    
                  
                  
                    
                      v
                      0
                    
                  
                
                
                  
                    0
                  
                  
                    0
                  
                  
                    1
                  
                
              
              ]
            
            θ
          
          *
          
            
              [
              
                
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      ∅
                    
                  
                  
                    
                      
                        -
                        sin
                      
                      ⁢
                      
                        
                      
                      ⁢
                      ∅
                    
                  
                  
                    0
                  
                
                
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      ∅
                    
                  
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      ∅
                    
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    0
                  
                  
                    1
                  
                
              
              ]
            
            θ
          
        
      
      
        }
        
          P
          θ
          Flexible
        
      
    
    
      {
      
        P
        θ
      
    
  
  .




Accordingly, determination of the first projection sub-matrix PθFlexible according to some embodiments includes determination of pixel dimensions pw, ph, co-plane rotation φ, and principal point (u0, v0). Pixel dimensions pw and ph may be directly obtained from the design specification of the flat panel detector, and several embodiments for determining co-plane rotation φ, and principal point (u0, v0) will be described below.

Some embodiments for determining co-plane rotation φ, and principal point (u0, v0) (and, as a result, the first projection sub-matrix PθFlexible) may be automated, performed on-the-fly and/or performed without use of a phantom. Advantageously, some on-the-fly calibration procedures may be conducted during acquisition of patient images.
A second sub-matrix of the projection matrix is determined at S620. The first sub-matrix determined at S610 and the second sub-matrix determined at S620 comprise a decomposition of the above-described projection matrix. The second sub-matrix is associated with system translations and rotations other than in-plane rotation φ of the flat panel detector and variation in the principal point (u0, v0)
According to some embodiments, the second sub-matrix comprises the rigid projection sub-matrix PθRigid described above, and is computed at S620 according to:




  
    P
    θ
    Rigid
  
  =
  
    
      
        [
        
          
            
              
                -
                f
              
            
            
              0
            
            
              0
            
            
              0
            
          
          
            
              0
            
            
              
                -
                f
              
            
            
              0
            
            
              0
            
          
          
            
              0
            
            
              0
            
            
              0
            
            
              1
            
          
        
        ]
      
      θ
    
    *
    
      
        [
        T
        ]
      
      θ
    
    *
    
      
        [
        R
        ]
      
      θ
    
  




where [T]θ represents translation quantity, [R]θ represents rotation quantity, and f represents a source to imager distance (SID).

S620 may comprise determination of the complete non-ideal projection matrix Pθ using a fiducial phantom or other techniques known in the art, and determination of the second projection sub-matrix PθRigid based on the projection matrix Pθ and the first projection sub-matrix PθFlexible is as follows: PθRigid=[PθFlexible]−1*Pθ. An example of a conventional technique for determining a non-ideal projection matrix Pθ using a fiducial phantom will be provided below.
Next, at S630, a coordinate of the two-dimensional coordinate system is converted to a coordinate of the three-dimensional coordinate system, based on the first sub-matrix and the second sub-matrix. Since the first sub-matrix and the second sub-matrix comprise a decomposition of the projection matrix between the two coordinate systems, a coordinate of the two-dimensional system (e.g., (u, v)) may be converted to a coordinate of the other system (e.g., (xf, yf, zf)) using the transformation:




  
    [
    
      
        
          
            λ
            ⁢
            
              
            
            ⁢
            u
          
        
      
      
        
          
            λ
            ⁢
            
              
            
            ⁢
            v
          
        
      
      
        
          λ
        
      
    
    ]
  
  =
  
    
      
        
          
            
              
                [
                
                  
                    
                      
                        1
                        /
                        
                          p
                          w
                        
                      
                    
                    
                      0
                    
                    
                      
                        u
                        0
                      
                    
                  
                  
                    
                      0
                    
                    
                      
                        
                          -
                          1
                        
                        /
                        
                          p
                          h
                        
                      
                    
                    
                      
                        v
                        0
                      
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                  
                
                ]
              
              θ
            
            *
            
              
                [
                
                  
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        ∅
                      
                    
                    
                      
                        
                          -
                          sin
                        
                        ⁢
                        
                          
                        
                        ⁢
                        ∅
                      
                    
                    
                      0
                    
                  
                  
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        ∅
                      
                    
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        ∅
                      
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                  
                
                ]
              
              θ
            
          
          
            }
            
              P
              θ
              Flexible
            
          
        
        *
        
          
            
              
                [
                
                  
                    
                      
                        -
                        f
                      
                    
                    
                      0
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        -
                        f
                      
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                  
                
                ]
              
              θ
            
            *
            
              
                [
                T
                ]
              
              θ
            
            *
            
              
                [
                R
                ]
              
              θ
            
          
          
            }
            
              P
              θ
              Rigid
            
          
        
      
      
        {
        
          P
          θ
        
      
    
    *
    
      [
      
        
          
            
              x
              f
            
          
        
        
          
            
              y
              f
            
          
        
        
          
            
              z
              f
            
          
        
        
          
            1
          
        
      
      ]
    
  




Operation at S630 may comprise any one or more uses of the projection matrix consisting of the first sub-matrix and the second sub-matrix. For example, S630 may comprise reconstruction of a three-dimensional image based on two-dimensional projection images acquired by the imaging system at various gantry angles, image-assisted patient positioning procedures, radiation field verification, etc. S630 may span several treatment fractions of one or more patients, and any period of time.
S640 and S660 comprise monitoring agents of process 600. Specifically, execution continues at S630 to convert coordinates as desired until it is determined at S640 that the first sub-matrix is to be re-determined. The determination at S640 is based on a first calibration schedule associated with the first sub-matrix. For example, S640 may comprise determining whether a particular amount of time (e.g., one day) has passed since a last determination of the first sub-matrix. The schedule may be event-based. In one example, the first sub-matrix is to be re-determined prior to imaging each patient. The first calibration schedule may comprise a combination of time- and event-based rules.
Flow proceeds to S650 if it is determined at S640 that the first sub-matrix is to be re-determined. The first sub-matrix is re-determined at S650 as described with respect to S610 and flow returns to S630. Accordingly, subsequent coordinate conversion at S630 uses the newly-determined first sub-matrix and the existing second sub-matrix (i.e., Pθ=PθFlexiblenew*PθRigidcurrent).
Similarly, execution continues at S630 until it is determined at S660 that the second sub-matrix is to be re-determined. The determination at S650 is based on a second calibration schedule associated with the second sub-matrix. The second calibration schedule may be less frequent than the first calibration schedule according to some embodiments. The second calibration schedule may indicate a re-determination of the second sub-matrix every six months, at each service appointment, or any combination thereof.
If it is determined at S660 to re-determine the second sub-matrix, flow returns to S620 to re-determine the second sub-matrix and then proceeds as described above. In this regard, subsequent coordinate conversions at S630 use the currently-existing first sub-matrix and the newly-determined second sub-matrix (i.e., Pθ−PθFlexiblecurrent*PθRigidnew).
Some embodiments for determining the co-plane rotation φ, and the principal point (u0, v0) for determination of the first projection sub-matrix at S610 will now be described in more detail. Some of the described embodiments provide automated phantom-less on-the-fly determination of the first projection sub-matrix.
Initially, a MLC of the imaging device is set to a predefined leaf configuration. The predefined configuration may be symmetric about the Y axis. Next, at a single gantry angle, a first image is acquired by the detector while the collimator is at a 0 degree rotational position and a second image is acquired by the detector while the collimator is at a 90 degree rotational position. The two images are superimposed to obtain a rectangular-shaped MLC pattern.
FIG. 7 illustrates an “ideal” MLC pattern 70 for a given leaf configuration. The ideal pattern 70 represents a scenario in which the detector exhibits no co-plane rotation or translation. The actually-acquired MLC pattern 75 is also shown, which exhibits both co-plane rotation and translation with respect to the ideal pattern 70. The co-plane rotation φ and the translation vector can Ŝ be computed by registering the “ideal” MLC pattern 70 with the acquired MLC pattern 75 and then computing the principal point (u0, v0) from the translation vector Ŝ.
According to some embodiments, only one projection image is acquired, while the collimator is at 0 degrees, to determine the co-plane rotation φ, the translation vector Ŝ, and the principal point (u0, v0). FIG. 8 illustrates an example of a projection image 810 acquired on detector surface 800, having a imaging center point 805. As shown, the MLC is configured to allow four predetermined leaves 811-814 to enter the image field. The lateral edges of the leaves are parallel to the Xr axis 815 and each lateral edge is at a known distance from the Xr axis. Therefore, even one lateral edge of a leaf in the acquired image can be used to identify the Xr axis. Similarly, the longitudinal end edges of the leaves are parallel to the Yr axis and each longitudinal edge is at a known distance from the Yr axis. Therefore, even one longitudinal edge of a leaf can be used to identify Yr axis.
According to some embodiments, and also illustrated in FIG. 9, the edges of predetermined leaves are symmetrically placed about the Xr and Yr axes. Therefore, after identifying the lateral edges and lateral lines defined thereby such as lines 817 and 818, a line symmetrical to the lateral lines may be determined as the Xr axis 815. Similarly, after identifying the longitudinal end edges of equally-extended leaves from opposing banks of leaves and longitudinal lines defined thereby such as lines 819 and 820, a line symmetrical to the longitudinal lines may be determined as the the Yr axis 816.
As described above, the planned and measured positions of the end (longitudinal) edges of the leaves may be used to identify the Yr axis. However, since the leaves are typically prevented from significantly restricting the open-field, the end positions of these protruded leaves lie near the boundary. While the lateral edges can be detected with better accuracy, the effect of beam penumbra near the boundary may affect the accuracy of detecting the end (i.e., longitudinal) edge of the leaves if those edges are near the boundary. Therefore, in another variation of on-the-fly determination of the first sub-matrix, the MLC may use Y-jaws with a center “notch” such as the notches of FIG. 8. The Yr axis 816 may then be identified by identifying the notches 830 and 835 in the projection image 810 and joining them with a straight line.
A notch as described above may be simulated by positioning an opposing pair of leaves near the upper Y boundary close together, leaving a narrow opening in the center between the leaves, and similarly positioning an opposing pair of leaves near the lower Y boundary. The Yr axis may then be identified by identifying the openings in the projection image and joining them with a straight line.
The following is a description of a conventional technique for determining the non-ideal projection matrix Pθ. As described above, the non-ideal projection matrix Pθ may be used in some embodiments, in conjunction with the first sub-matrix determined at S610, to determine the second sub-matrix at S620.
Initially, a geometry phantom is placed approximately at the imaging isocenter of the imaging system, with the axis of the phantom being approximately aligned with the expected IEC axis. Images of the phantom are acquired at each of many gantry angles (e.g., at 0.5 or 1 degree intervals). Fiducials are detected in the images and linear equations are formulated which relate three-dimensional locations of the fiducials and the corresponding two-dimensional projections, with the parameters of the projection matrix Pθ as unknowns.
The overdetermined system of linear equations is solved to obtain Pθphantom in the phantom coordinate system, and the location of the imaging source is determined in the phantom coordinate system. Next, a circle is fit through the source points, the center of rotation and plane of rotation are determined, and the isocenter is located at the center of rotation. The IEC Y axis is determined to be normal to the plane of rotation and the source location is projected at gantry 0 degree position on the vertical plane. The IEC Z axis is the unit vector from the center to the projected source location and the IEC X axis is the cross product of the Y and Z axis. Using the above information, Pθphantom is transformed to the projection matrix Pθ.
FIG. 9 is a flow diagram of process 900 according to some embodiments. Process 900 is similar to process 600, but instead of determining whether or not to redetermine the second sub-matrix according to a second calibration schedule, process 900 includes (at S960) a determination of whether to execute a quality assurance procedure to determine whether the currently-determined second sub-matrix (e.g., PθRigid) is acceptably valid, and (at S970) the quality assurance procedure itself. Accordingly, re-determination of the second sub-matrix may be avoided if the current second sub-matrix is determined to be acceptable at S970.
Avoidance of unnecessary determination of the second sub-matrix may be desirable because this determination (as described above) requires the use of phantoms with embedded fiducials at known three-dimensional locations with respect to a frame of reference. A quality assurance procedure according to some embodiments only requires assurance phantoms at static locations in the imaging room. Moreover, knowledge of the three-dimensional locations of the assurance phantoms is not required. The quality assurance procedure may therefore be performed on-the-fly and with significantly less disruption than re-determination of the second sub-matrix.
A quality assurance procedure according to some embodiments is intended to determine whether the current source trajectory suitably matches the trajectory modeled by the currently-determined second sub-matrix. According to some embodiments, it is first ensured that the first sub-matrix (e.g., PθFlexible) is current for all gantry angles. Next, spherical radio-opaque beads (e.g., ball bearings) are attached to the couch, projections images of the beads are acquired (with or without a patient positioned on the couch), and a three-dimensional image of the beads is reconstructed from the projection images, using the projection matrix composed of the current first sub-matrix and the current second sub-matrix.
Next, the dimensions and/or blur of the ball bearings are determined in three-dimensional space from the three-dimensional image. If the determined dimensions are within a suitable threshold of the actual dimensions, and/or if the detected blur is not significant blur, it is determined that the current second sub-matrix does not require redetermination. Notably, the foregoing procedure may be performed quickly and/or during patient imaging, and may eliminate unnecessary determination of the second sub-matrix (i.e., PθRigid).
An alternative quality assurance procedure attempts to match computed locations of fiducials from different pairs of gantry angles. After ensuring that the first sub-matrix (e.g., PθFlexible) is current for all gantry angles, two-dimensional projections (u1, v1), . . . , (uK, vK) of spherical radio-opaque beads are acquired from gantry angles θ1, . . . , θK, respectively. The beads are attached to the couch in locations (possibly near the edge of open field) such that their projection is visible even while acquiring portal images or CBCT projection images of the patient.
Assuming the unknown three-dimensional location of such a bead to be (xf,yf,zf), and if the calibration of the rigid component is still holding properly,




  
    [
    
      
        
          
            
              λ
              i
            
            ⁢
            
              u
              i
            
          
        
      
      
        
          
            
              λ
              i
            
            ⁢
            
              v
              i
            
          
        
      
      
        
          
            λ
            i
          
        
      
    
    ]
  
  =
  
    
      P
      
        0
        i
      
    
    *
    
      [
      
        
          
            
              x
              f
            
          
        
        
          
            
              y
              f
            
          
        
        
          
            
              z
              f
            
          
        
        
          
            1
          
        
      
      ]
    
  




is obtained for 1≦1≦N;

wherein in this equation, (u1,v1), Pθ1 are known but (xf,yf,zf) and λ1 are unknown.
Next, a known three-dimensional point (xfknown,yfknown,zfknown) is chosen, and its expected two-dimensional projection is computed for the gantry angles θ1, . . . , θK under the assumption that the calibration of the rigid component is still valid; and as a result




  
    [
    
      
        
          
            
              w
              i
            
            ⁢
            
              u
              
                i
                known
              
            
          
        
      
      
        
          
            
              w
              i
            
            ⁢
            
              v
              
                i
                known
              
            
          
        
      
      
        
          
            w
            
              i
              known
            
          
        
      
    
    ]
  
  =
  
    
      P
      
        θ
        i
      
    
    *
    
      [
      
        
          
            
              x
              f
              known
            
          
        
        
          
            
              y
              f
              known
            
          
        
        
          
            
              z
              f
              known
            
          
        
        
          
            1
          
        
      
      ]
    
  




is obtained, for 1≦1≦N, where in this equation (u1known, v1known), W1 are computed from known values of (xfknown,yfknown,zfknown) and Pθ1; and

Corresponding equations are subtracted to obtain:




  
    
      [
      
        
          
            
              
                
                  w
                  i
                
                ⁢
                
                  u
                  
                    i
                    known
                  
                
              
              -
              
                
                  λ
                  i
                
                ⁢
                
                  u
                  i
                
              
            
          
        
        
          
            
              
                
                  w
                  i
                
                ⁢
                
                  v
                  
                    i
                    known
                  
                
              
              -
              
                
                  λ
                  i
                
                ⁢
                
                  v
                  i
                
              
            
          
        
        
          
            
              
                w
                
                  i
                  known
                
              
              -
              
                λ
                i
              
            
          
        
      
      ]
    
    =
    
      
        
          P
          
            θ
            i
          
        
        *
        
          [
          
            
              
                
                  
                    x
                    f
                    known
                  
                  -
                  
                    x
                    f
                  
                
              
            
            
              
                
                  
                    y
                    f
                    known
                  
                  -
                  
                    y
                    f
                  
                
              
            
            
              
                
                  
                    z
                    f
                    known
                  
                  -
                  
                    z
                    f
                  
                
              
            
            
              
                
                  1
                  -
                  1
                
              
            
          
          ]
        
      
      =
      
        
          P
          
            θ
            i
          
        
        *
        
          [
          
            
              
                
                  k
                  x
                
              
            
            
              
                
                  k
                  y
                
              
            
            
              
                
                  k
                  z
                
              
            
            
              
                0
              
            
          
          ]
        
      
    
  
  ,




where (kz,ky,kz) is the unknown constant three-dimensional vector from the unknown bead location (xf,yf,zf) to the known three-dimensional point (xfknown,yfknown,zfknown).

For a given gantry angle θi, there are three equations and four unknowns, λ1,kx,ky,kz. However, the three-dimensional unknown vector (kx,ky,kz) is common to all gantry angles. Therefore, under the assumption that the calibration of the rigid component is still holding properly, six equations from two gantry angles (θp,θq) can be used to find five unknowns values (λp, λq, kx,ky,kz).
Thus, the assumption of whether the second sub matrix (i.e., the calibration of rigid components) is sufficiently valid can now be checked. From the projections corresponding to gantry angles θ1, . . . , θK many pairs of projections, preferably orthogonal to one another, are obtained. (kz,ky,kz) is computed from each such pair. In other words, the unknown location, (xf,yf,zf) of each bead can be computed from each such projection pair assuming that the second sub-matrix is still valid. If sufficient agreements in the computed values of the unknown location of the beads are found, it can be concluded that the second sub-matrix is valid.
It should be understood that the embodiments presented above are examples rather than limitations. Those in the art can modify and vary the embodiments without departing from their spirit and scope.

## Claims

1. A calibration method for an imaging system comprising an imaging source and an image detector, comprising:
determining a first sub-matrix of a projection matrix, the projection matrix describing a geometrical relationship between points of a three-dimensional coordinate system of the imaging system and points of a two-dimensional coordinate system of the image detector;
determining a second sub-matrix of the projection matrix, where the first sub-matrix and the second sub-matrix comprise a decomposition of the projection matrix;
converting a first point of the two-dimensional coordinate system to a first point of the three-dimensional coordinate system based on the first sub-matrix and the second sub-matrix;
determining an updated first sub-matrix of an updated projection matrix according to a first calibration schedule, where the updated projection matrix describes a second geometrical relationship between points of the three-dimensional coordinate system and points of the two-dimensional coordinate system, and where the updated first sub-matrix and the second sub-matrix comprise a decomposition of the updated projection matrix; and
converting a second point of the two-dimensional coordinate system to a second point of the three-dimensional coordinate system based on the updated first sub-matrix and the second sub-matrix.

2. The method according to claim 1, further comprising:
determining an updated second sub-matrix of a second updated projection matrix according to a second calibration schedule, where the second calibration schedule is different from the first calibration schedule, where the second updated projection matrix describes a third geometrical relationship between points of the three-dimensional coordinate system and points of the two-dimensional coordinate system, and where the updated first sub-matrix and the updated second sub-matrix comprise a decomposition of the second updated projection matrix; and
converting a third point of the two-dimensional coordinate system to a third point of the three-dimensional coordinate system based on the updated first sub-matrix and the updated second sub-matrix.

3. A method according to claim 1,
wherein determining the first sub-matrix comprises determining in-plane rotation φ of the image detector and determining a point of the two-dimensional coordinate system which is intercepted by a beam axis of the imaging source.

4. The method according to claim 3, wherein determining the first sub-matrix comprises:
acquiring a projection image of a predefined collimator leaf pattern;
determining one or more differences between the projection image of the predefined collimator leaf pattern and an expected projection image of the predefined collimator leaf pattern; and
determining in-plane rotation φ of the image detector and a point of the two-dimensional coordinate system which is intercepted by a beam axis of the imaging source based on the one or more differences.

5. The method according to claim 3, wherein determining the first sub-matrix comprises:
acquiring a first projection image of a predefined collimator leaf pattern with the collimator at a 0 degree position;
acquiring a second projection image of the predefined collimator leaf pattern with the collimator at a 90 degree position;
combining the first projection image and the second projection image to generate a superimposed image; and
determining in-plane rotation φ of the image detector and a point of the two-dimensional coordinate system which is intercepted by a beam axis of the imaging source based on the superimposed image.

6. The method according to claim 3, wherein determining the first sub-matrix comprises:
acquiring a projection image of a predefined collimator leaf pattern;
identifying an Xr axis based on one or more locations of lateral edges of one or more leaves in the predefined collimator leaf pattern; and
identifying a Yr axis based on one or more locations of longitudinal edges of one or more leaves in the predefined collimator leaf pattern.

7. The method according to claim 3, wherein determining the first sub-matrix comprises:
acquiring a projection image of a predefined collimator leaf pattern in which edges of one or more leaves are symmetrically located about the Xr and Yr axes;
identifying a first one or more lines defined by lateral edges of one or more leaves in the projection image;
identifying a first one or more lines defined by longitudinal edges of one or more leaves in the projection image;
identifying an axis of symmetry of the first one or more lines as the Xr axis; and
identifying an axis of symmetry of the second one or more lines as the Yr axis.

8. The method according to claim 3, wherein determining the first sub-matrix comprises:
acquiring a projection image of a collimator comprising two opposing Y-jaws, in which each Y-jaw defines a notch at its lateral center;
identifying each notch in the projection image; and
identifying the Yr axis as a line joining the notches.

9. The method according to claim 3, wherein determining the first sub-matrix comprises:
acquiring a projection image of a collimator in which a first pair of opposing leaves located at an upper Y boundary of the collimator define a first narrow gap, and a second pair of opposing leaves located at a lower Y boundary of the collimator define a second narrow gap;
identifying the first narrow gap and the second narrow gap in the projection image; and
identifying the Yr axis as a line joining the first narrow gap and the second narrow gap.

10. The method according to claim 1, wherein the first sub-matrix is determined as




  
    
      
        
          P
          θ
          Flexible
        
        =
        
          
            ⌈
            
              
                
                  
                    1
                    
                      p
                      w
                    
                  
                
                
                  0
                
                
                  
                    u
                    0
                  
                
              
              
                
                  0
                
                
                  
                    
                      -
                      1
                    
                    
                      p
                      h
                    
                  
                
                
                  
                    v
                    0
                  
                
              
              
                
                  0
                
                
                  0
                
                
                  
                    -
                    1
                  
                
              
            
            ⌉
          
          *
          
            
              [
              
                
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      ∅
                    
                  
                  
                    
                      
                        -
                        sin
                      
                      ⁢
                      
                        
                      
                      ⁢
                      ∅
                    
                  
                  
                    0
                  
                
                
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      ∅
                    
                  
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      ∅
                    
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    0
                  
                  
                    1
                  
                
              
              ]
            
            θ
          
        
      
      
        }
        
          P
          θ
          Flexible
        
      
    
    
      {
      
        P
        θ
      
    
  
  ,




wherein the second sub-matrix is determined as




  
    P
    θ
    Rigid
  
  =
  
    
      
        [
        
          
            
              
                -
                f
              
            
            
              0
            
            
              0
            
            
              0
            
          
          
            
              0
            
            
              
                -
                f
              
            
            
              0
            
            
              0
            
          
          
            
              0
            
            
              0
            
            
              0
            
            
              1
            
          
        
        ]
      
      θ
    
    *
    
      
        [
        T
        ]
      
      θ
    
    *
    
      
        [
        R
        ]
      
      θ
    
  




and
wherein the projection matrix is Pθ=PθFlexible*PθRigid, where Pw and Ph represent pixel dimensions on the FPD, θ represents a gantry angle, [T]θ represents translation quantity, [R]θ represents rotation quantity, and f represents a distance between the imaging source and the image detector.

11. The method according to claim 1, further comprising:
acquiring a plurality of projection images of a plurality of fiducials at respective different gantry angles;
generating a three-dimensional image of the plurality of fiducials based on the plurality of projection images;
determining respective dimensions of the plurality of fiducials based on the three-dimensional image;
determining whether the determined dimensions differ from actual sizes of the plurality of fiducials by more than a threshold; and
if the determined sizes differ from actual dimensions of the plurality of fiducials by more than the threshold:
determining an updated second sub-matrix of a second updated projection matrix, where the second updated projection matrix describes a third geometrical relationship between points of the three-dimensional coordinate system and points of the two-dimensional coordinate system, and where the updated first sub-matrix and the updated second sub-matrix comprise a decomposition of the second updated projection matrix; and
converting a third point of the two-dimensional coordinate system to a third point of the three-dimensional coordinate system based on the updated first sub-matrix and the updated second sub-matrix.

12. The method according to claim 11, wherein the plurality of fiducials comprise a plurality of spherical radio-opaque ball bearings attached to a treatment couch, and
wherein determining respective dimensions of the plurality of fiducials comprises determining blur of each of the plurality of fiducials based on the three-dimensional image.

13. An imaging system comprising: an imaging source to emit radiation;
an image detector to receive the radiation and to generate images based on the received radiation; and
a processing unit configured to execute program code stored on a non-transitory computer readable medium to cause the system to:
determine a first sub-matrix of a projection matrix, the projection matrix describing a geometrical relationship between points of a three-dimensional coordinate system of the imaging system and points of a two-dimensional coordinate system of the image detector;
determine a second sub-matrix of the projection matrix, where the first sub-matrix and the second sub-matrix comprise a decomposition of the projection matrix;
convert a first point of the two-dimensional coordinate system to a first point of the three-dimensional coordinate system based on the first sub-matrix and the second submatrix;
determine an updated first sub-matrix of an updated projection matrix according to a first calibration schedule, where the updated projection matrix describes a second geometrical relationship between points of the three-dimensional coordinate system and points of the two-dimensional coordinate system, and where the updated first sub-matrix and the second sub-matrix comprise a decomposition of the updated projection matrix; and
convert a second point of the two-dimensional system to a second point of the three-dimensional coordinate system based on the updated first sub-matrix and the second sub-matrix.

14. The system according to claim 13, the processing unit to further execute program code stored on a non-transitory computer readable medium to cause the system to:
determine an updated second sub-matrix of a second updated projection matrix according to a second calibration schedule, where the second calibration schedule is different from the first calibration schedule, where the second updated projection matrix describes a third geometrical relationship between points of the three-dimensional coordinate system and points of the two-dimensional coordinate system, and where the updated first sub-matrix and the updated second sub-matrix comprise a decomposition of the second updated projection matrix; and
convert a third point of the two-dimensional coordinate system to a third point of the three-dimensional coordinate system based on the updated first sub-matrix and the updated second sub-matrix.

15. The system according to claim 13,
wherein determination of the first sub-matrix comprises determination of in-plane rotation φ of the image detector and determination of a point of the two-dimensional coordinate system which is intercepted by a beam axis of the imaging source.

16. The system according to claim 15, further comprising a collimator, and wherein determination of the first sub-matrix comprises:
operation of the imaging source and the imaging detector to acquire a projection image of a predefined leaf pattern of the collimator;
determination of one or more difference between the projection image of the predefined leaf pattern and an expected projection image of the predefined leaf pattern; and
determination of in-plane rotation φ of the image detector and a point of the two-dimensional coordinate system which is intercepted by a beam axis of the imaging source based on the one or more differences.

17. The system according to claim 15, further comprising a collimator, and wherein determination of the first sub-matrix comprises:
operation of the imaging source and the imaging detector to acquire a first projection image of a predefined leaf pattern of the collimator with the collimator at a 0 degree position;
operation of the imaging source and the imaging detector to acquire a second projection image of a predefined leaf pattern of the collimator with the collimator at a 90 degree position;
combination of the first projection image and the second projection image to generate a superimposed image; and
determination of in-plane rotation φ of the image detector and a point of the two-dimensional coordinate system which is intercepted by a beam axis of the imaging source based on the superimposed image.

18. The system according to claim 15, further comprising a collimator, and wherein determination of the first sub-matrix comprises:
operation of the imaging source and the imaging detector to acquire a projection image of a predefined leaf pattern of the collimator;
identification of an Xr axis based on one or more locations of lateral edges of one or more leaves in the predefined leaf pattern; and
identification of a Yr axis based on one or more locations of longitudinal edges of one more leaves in the predefined leaf pattern.

19. The system according to claim 15, further comprising a collimator, and wherein determination of the first sub-matrix comprises:
operation of the imaging source and the imaging detector to acquire a projection image of a predefined leaf pattern of the collimator in which edges of one or more leaves are symmetrically located about the Xr and Yr axes;
identification of a first one or more lines defined by lateral edges of one or more leaves in the projection image;
identification of a first one or more lines defined by longitudinal edges of one or more leaves in the projection image;
identification of an axis of symmetry of the first one or more lines as the Xr axis; and
identification of an axis of symmetry of the second one or more lines as the Yr axis.

20. The system according to claim 15, further comprising a collimator comprising two opposing Y-jaws, in which each Y-jaw defines a notch at its lateral center, and wherein determination of the first sub-matrix comprises:
operation of the imaging source and the imaging detector to acquire a projection image of the collimator;
identification of each notch in the projection image; and
identification of the Yr axis as a line joining the notches.

21. The system according to the claim 15, further comprising a collimator in which a first pair of opposing leaves located at an upper Y boundary of the collimator define a first narrow gap, and a second pair of opposing leaves located at a lower Y boundary of the collimator define a second narrow gap, and wherein determination of the first sub-matrix comprises:
operation of the imaging source and the imaging detector to a projection image of a collimator;
identification of the first narrow gap and the second narrow gap in the projection image; and
identification of the Yr axis as a line joining the first narrow gap and the second narrow gap.

22. The system according to claim 13, wherein the first sub-matrix is determined as




  
    
      
        
          P
          θ
          Flexible
        
        =
        
          
            ⌈
            
              
                
                  
                    1
                    
                      p
                      w
                    
                  
                
                
                  0
                
                
                  
                    u
                    0
                  
                
              
              
                
                  0
                
                
                  
                    
                      -
                      1
                    
                    
                      p
                      h
                    
                  
                
                
                  
                    v
                    0
                  
                
              
              
                
                  0
                
                
                  0
                
                
                  
                    -
                    1
                  
                
              
            
            ⌉
          
          *
          
            
              [
              
                
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      ∅
                    
                  
                  
                    
                      
                        -
                        sin
                      
                      ⁢
                      
                        
                      
                      ⁢
                      ∅
                    
                  
                  
                    0
                  
                
                
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      ∅
                    
                  
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      ∅
                    
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    0
                  
                  
                    1
                  
                
              
              ]
            
            θ
          
        
      
      
        }
        
          P
          θ
          Flexible
        
      
    
    
      {
      
        P
        θ
      
    
  
  ,




wherein the second sub-matrix is determined as




  
    P
    θ
    Rigid
  
  =
  
    
      
        [
        
          
            
              
                -
                f
              
            
            
              0
            
            
              0
            
            
              0
            
          
          
            
              0
            
            
              
                -
                f
              
            
            
              0
            
            
              0
            
          
          
            
              0
            
            
              0
            
            
              0
            
            
              1
            
          
        
        ]
      
      θ
    
    *
    
      
        [
        T
        ]
      
      θ
    
    *
    
      
        [
        R
        ]
      
      θ
    
  




and
wherein the projection matrix is Pθ=PθFlexible*PθRigid, where Pw and Ph represent pixel dimensions on the FPD, θ represents a gantry angle, [T]θ represents translation quantity, [R]θ represents rotation quantity, and f represents a distance between the imaging source and the image detector.

23. The system according to claim 13, the processing unit to further execute
program code stored on a non-transitory computer readable medium to cause the system to:
operate the imaging source and the imaging detector to acquire a plurality of projection images of a plurality of fiducials at respective different gantry angles;
generate a three-dimensional image of the plurality of fiducials based on the plurality of projection images;
determine respective dimensions of the plurality of fiducials based on the three-dimensional images;
determine whether the determined dimensions differ from actual sizes of the plurality of fiducials by more than a threshold; and
if the determined sizes differ from actual dimensions of the plurality of fiducials by more than the threshold:
determining an updated second sub-matrix of a second updated projection matrix, where the second updated projection matrix describes a third geometrical relationship between points of the three-dimensional coordinate system and points of the two-dimensional coordinate system, and where the updated first sub-matrix and the updated second sub-matrix comprise a decomposition of the second updated projection matrix; and
converting a third point of the two-dimensional coordinate system to a third point of the three-dimensional coordinate system based on the updated first sub-matrix and the updated second sub-matrix.

24. The system according to claim 23, wherein the plurality of fiducials comprise a plurality of spherical radio-opaque ball bearings attached to a treatment couch, and
wherein determining respective dimensions of the plurality of fiducials comprises determining of blur of each of the plurality of fiducials based on the three-dimensional image.