# Patent

## Title
Procedure for deriving a three-dimensional digital mask starting from a series of two-dimensional masks, plus a device for doing this

## Abstract

A derivation procedure for a three-dimensional digital mask from a series of two-dimensional masks in a radiographic device containing a source (S) of X-rays, a means of recording and a volume of interest hat contains the object to be X-rayed located between the source (S) and the means of recording consists of an extrapolation of each mask Mθ2 includes determining a last segment 1fin beyond the limits of the means of recording; and working out a two-dimensional mask Mγ associated with a position Sγ of the source, for any angle γ included in the angular range θ2 to θ1 (a position close to θ2). For every parallel segment 1 located between segment d (or d′ respectively) and segment 1fin, the procedure further includes deriving a three-dimensional mask (21) of the object for each voxel at the intersection of plane P1θ2 and the volume of interest; and projecting the three-dimensional mask onto the segment 1.

## Background
CROSS-REFERENCE TO RELATED APPLICATIONS
This application claims priority under 35 U.S.C. §119(a)-(d) or (f) to prior-filed, co-pending French patent application serial number 0760151, filed on Dec. 20, 2007, which is hereby incorporated by reference in its entirety.
STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT
Not Applicable
NAMES OF PARTIES TO A JOINT RESEARCH AGREEMENT
Not Applicable
REFERENCE TO A SEQUENCE LISTING, A TABLE, OR COMPUTER PROGRAM LISTING APPENDIX SUBMITTED ON COMPACT DISC
Not Applicable
BACKGROUND OF THE INVENTION
1. Field of the Invention
The field of the invention relates to radiographic devices and methods generally. More particularly, the field of the invention is concerned with an extrapolation procedure for a two-dimensional mask and a procedure for working out a three-dimensional mask based on an extrapolated series of two-dimensional masks, these procedures being intended for use in a radiographic device, particularly of the tomosynthesis type.
2. Description of Related Art
In any tomosynthesis application, it can be interesting to be able to extrapolate a two-dimensional mask for an object projected at the level of truncated views of the object, and then to be able to estimate a three-dimensional mask based on a series of two-dimensional masks extrapolated in this way. This extrapolation of the two-dimensional masks is indeed useful when it is necessary to have an estimate of the signal outside the physical limits of the detector during reconstruction. The three-dimensional mask can be used as a priori information in iterative reconstruction methods in order to speed up the reconstruction while focusing only on the voxels that belong to the object under consideration, or in order to eliminate artefacts that the reconstruction has created outside the object under consideration. The principle difficulties in constructing reliable three-dimensional masks derive from the limited angular interval swept out by the source and the truncation of the object under consideration in some projections as a result of the finite dimensions of the detector.
For example, in the case of digital tomosynthesis of the breast, such tomosynthesis is a new imaging technique for the breast using three-dimensional tomography that is restricted in terms of angles. It allows the problem of superposition in the detection of lesions to be largely circumvented. Multiple views projected from different angles of acquisition potentially reduce the number of false positives due to the addition of artefacts, and the number of false negatives due to masking effects by the covering tissues. During the tomosynthesis examination, a series of images is obtained for a variety of angles of acquisition with the aim of reconstructing a three-dimensional representation of the breast. During the reconstruction procedure, knowing for a given voxel ν of the three-dimensional representation whether or not it has information about the breast can help reduce the artefacts as well as the calculation time. This knowledge can be represented with the help of a three-dimensional mask M of the breast: M [ν]=1 if ν belongs to the breast and M [ν]=0 otherwise. As this mask M has been reconstructed using the two-dimensional masks deriving from the series of images acquired earlier, the assumption is made that all the voxels of the region of interest are located within the limits of the detector. However, in reality there are voxels in the region of interest that are not projected within the limits of the detector when certain images within the series are being acquired. This means that the two-dimensional mask for the breast outside the limits of the detector has to be estimated. Without a priori information, the natural approach is to decide that all pixels outside the limits of the detector are part of the breast. This way, the two-dimensional masks (and consequently the three-dimensional ones) never underestimate the shape of the breast, because they are always larger than that shape. As a result, when the three-dimensional mask is constructed, artefacts can appear: staircases. The contour of the resulting mask is therefore a poor estimation of what it really should be, and the shape of the mask is not continuous. In consequence, using the three-dimensional mask obtained this way on the three-dimensional representation produces an unnatural skin line in the final three-dimensional representation, which complicates the diagnoses that the practitioners make based on it.
BRIEF SUMMARY OF THE INVENTION
One aim of an embodiment of the invention is to provide an extrapolation procedure for two-dimensional masks of an object in views with truncated projections (in order that the mask extrapolated using a specific projection will be regular and consistent with the two-dimensional masks made using other projections) and to reconstruct a three-dimensional mask of the object acquired that does not underestimate the object and which is regular, in the sense that the truncation of the views does not introduce any discontinuities in the mask.
To achieve this, the exemplary embodiment of the invention envisages an extrapolation procedure for a two-dimensional mask Mθ2 in a radiographic device of the type containing a mobile X-ray source taking up at least two positions Sθ1 and Sθ2 in space, associated with their respective two-dimensional masks Mθ1 and Mθ2, a means of recording that is in an essentially planar arrangement opposite the source and which contains a limit (d, d′) and a volume of interest consisting of an object that is suitable for radiography, located between the source and the means of recording, the procedure consisting of the following steps:

    
    
        a) at least partial estimation of a series of two-dimensional masks Mγ associated with a series of positions Sγ for the source, located between the source's positions Sθ1 and Sθ2, starting from the two-dimensional masks Mθ1 and Mθ2 
        b) at least partial evaluation of an intermediate three-dimensional mask for the object, starting from the series of two-dimensional masks Mγ and the two-dimensional masks Mθ1 and Mθ2 
        c) extrapolation of the two-dimensional mask Mθ2 beyond the limits d or d′ of the means of recording, according to a relative position between Sθ1 and Sθ2, starting from the intermediate three-dimensional mask
    
    


A beneficial but optional part of the procedure in accordance with the invention includes at least one of the following features:

    
    
        before step a), if the two-dimensional mask Mθ1 associated with the source position Sθ1 is not available, the two-dimensional mask Mθ1 is extrapolated from the available two-dimensional masks Mθ;
        the extrapolation of step c) involves projection of the intermediate three-dimensional mask based on position Sθ2 onto a plane passing through the means of recording;
        the (at least partial) working out from step a) is carried out at the limit d or d′ of the means of recording, according to the relative position between θ1 and Sθ2;
        for each of the two-dimensional masks Mγ, the (at least partial) estimation from step a) involves a step that determines a point Tγ situated at an edge of the object, projected onto the limit d or d′ of the means of recording, with the source at position Sγ
        the points Tγ are estimated by linear interpolation between the points Tθ1 and Tθ2 located at an edge of the object project onto the limit d or d′ of the means of recording (10), with the source (S) at positions Sθ1 and Sθ2 respectively;
        if the two-dimensional mask Mθ1 associated with source position Sθ1 is not available, point Tθ1 is then extrapolated from the points Tθ that are available;
        the procedure includes a supplementary step of:
        d) applying a closure function to the extrapolated two-dimensional mask Mθ2;
        before step a), the procedure involves a step for determining a limit of extrapolation lfin, that is effectively parallel to the limit d or d′ of the means of recording and that is located outside the limits of the means of recording;
        for every line l that is that is effectively parallel to the limit of extrapolation lfin and located between the limit d or d′ of the means of recording and the limit of extrapolation lfin, step b) involves the following substeps:
        
            b1) working out a plane P1θ2 that passes through position Sθ2 and the line l
            b2) working out an intermediate three-dimensional mask for each voxel (ν) located at the intersection of the plane P1θ2 and the volume of interest; and
        
        
        step c) involves a projection step for every line l that is that is effectively parallel to the limit of extrapolation lfin, and located between the limit d or d′ of the means of recording and the limit of extrapolation lfin, for the intermediate three-dimensional mask.[0011] In accordance with the invention, an extrapolation procedure is also envisaged for a series of two-dimensional masks Mθ in a radiographic device, with the feature that each two-dimensional mask Mθ in the series of two-dimensional masks is extrapolated by a procedure exhibiting at least one of the features described earlier.
    
    


A beneficial but optional part of the procedure in accordance with an embodiment of the invention includes at least one of the following features:

    
    
        the procedure that employs at least one of the features listed earlier is applied iteratively, and
        at each iteration, the means of recording are extended into their adjacent virtual equivalents corresponding to a common part of the two-dimensional masks Mθ already extrapolated.
    
    


To achieve this, an embodiment of the invention envisages a calculation procedure for a three-dimensional mask based on a series of two-dimensional masks Mθ in a radiographic device of the type containing a mobile X-ray source taking up at least two positions Sθ1 and Sθ2 in space, associated with their respective two-dimensional masks Mθ1 and Mθ2, a means of recording that is in an essentially planar arrangement opposite the source and which has a limit d or d′ and a volume of interest consisting of an object that is suitable for radiography, located between the source and the means of recording, with the procedure consisting of the following steps:

    
    
        a) extrapolation of the series of two-dimensional masks Mθ by a procedure featuring at least one of the previously listed characteristics, and
        b) determination of a three-dimensional digital mask based on the extrapolated series of two-dimensional masks Mθ.
    
    


A beneficial but optional part of the procedure in accordance with an embodiment of the invention includes at least the following features:

    
    
        step b), comprising the following sub-steps:
        
            b1) application of a dilation function followed by a low-pass filter to obtain a membership function μMθ for each two-dimensional mask Mθ
            b2) evaluation of a membership function μM3d based on the membership function μM3d, using a T-norm operator
            b3) determination of the three-dimensional digital mask based on the membership function μM3d, and
        
        
        the t-norm operator is a probabilistic one.
        In accordance with an embodiment of the invention, a radiographic device is envisaged of the type that has:
        a mobile source of X-rays that moves along a circular arc with centre C;
        means of recording, in an essentially planar arrangement opposite the source;
        a volume of interest, containing an object suitable for radiographic examination, located between the source and the means of recording;
        a means of initiating a procedure featuring at least one of the characteristics listed earlier.
    
    


Other characteristics and benefits of embodiments of the invention will appear in the course of the description that follows of a mode of realization of the invention plus variants.



BRIEF DESCRIPTION OF THE DRAWINGS
With regard to the drawings attached:
FIG. 1 gives a three-dimensional diagram of a device and of the procedure in accordance with the invention;
FIG. 2 is a schematic view of the side of the device and of the procedure of FIG. 1 allowing the extrapolation of a two-dimensional mask Mθ2 onto a segment 1;
FIG. 3 is a schematic view from the side similar to FIG. 2 but not allowing the extrapolation of the two-dimensional mask Mθ2 onto the segment 1;
FIG. 4 is a schematic view from the side similar to FIGS. 2 and 3 illustrating the procedure in accordance with the invention for the extrapolation of a two-dimensional mask at the end of the series of two-dimensional masks to be extrapolated by the procedure in accordance with the invention; and
FIG. 5 is a schematic view of a step for determining a three-dimensional digital mask by the procedure in accordance with the invention.



DETAILED DESCRIPTION OF THE INVENTION
With reference to FIG. 1: a radiographic device suitable for producing imagery by three-dimensional tomography contains a means of recording 10 that takes the form of a digital detector that is effectively flat and defines a plane PD. A digital detector such as this comprises a matrix of detectors each of which represents a pixel p, uniformly distributed into lines and columns. The radiographic device also includes an X-ray source opposite the means of recording and which is mobile with respect to this same means of recording. The X-ray source, which will generally be an X-ray generator tube, is suitable for being moved in discrete steps along a trajectory S that is effectively a circle of radius r and centre C. The trajectory S is restricted in terms of angle to an interval of range [θmin; θmax], where 0≦θmin<θmax≦π, and where the angles are measure with respect to the plane PD of the detector. At each discrete source position Sθ along this trajectory S, there are an associated angle θ in the range [θmin; θmax], an image Iθ projected and captured by the detector 10, and a two-dimensional mask Mθ.
Furthermore, there is a volume of interest 20 situated between the X-ray source and the means of recording 10, suitable for containing an objected intended for X-ray examination by the radiographic device. This region of interest 20 is delimited in part by a bottom plane PB and a top plane PT. The planes PB and PT are effectively parallel to the detection plane PD, defined by the means of recording 10. The volume of interest is furthermore delimited by four planes (not shown) passing through each of the edges d, d′, e and e′ of the detector and effectively perpendicular to the planes PB and PT. The region of interest 20 is thus shaped as a parallelepiped rectangle, as illustrated in FIG. 1. In the case of a device for digital tomosynthesis of the breast, the region of interest 20 is delimited by a cushion to support the breast, the upper face of which is in plane PB and the lower surface lies within plane PD, a compression plate of which the lower face lies in plane PT, such that the planes are basically parallel with the patient's torso and pass through the first at last lines (e and e′) of pixels on the detector, and planes effectively perpendicular to the ones just mentioned, passing through the first at last columns (d and d′) of pixels on the detector.
The classical method for describing a three-dimensional digital mask M3d for the object situated within the volume of interest 20 is as follows:




  
    M
    ⁢
    
      
    
    ⁢
    3
    ⁢
    
      d
      ⁡
      
        [
        v
        ]
      
    
  
  =
  
    {
    
      
        
          1
        
        
          
            if
            ⁢
            
              
            
            ⁢
            v
            ⁢
            
              
            
            ⁢
            is
            ⁢
            
              
            
            ⁢
            within
            ⁢
            
              
            
            ⁢
            the
            ⁢
            
              
            
            ⁢
            object
          
        
      
      
        
          0
        
        
          otherwise
        
      
    
  




for all voxels ν within the volume of interest 20. A two-dimensional mask Mθ is thus a projection of this mask M3d corresponding to one position Sθ of the source, as seen on the plane PD of the detector 10, and it is defined by:





  M
  ⁢
  
    
  
  ⁢
  θ
  ⁢
  
    {
    
      
        
          1
        
        
          
            if
            ⁢
            
              
            
            ⁢
            p
            ⁢
            
              
            
            ⁢
            belongs
            ⁢
            
              
            
            ⁢
            to
            ⁢
            
              
            
            ⁢
            the
            ⁢
            
              
            
            ⁢
            area
            ⁢
            
              
            
            ⁢
            in
            ⁢
            
              
            
            ⁢
            which
            ⁢
            
              
            
            ⁢
            the
            ⁢
            
              
            
            ⁢
            object
            ⁢
            
              
            
            ⁢
            is
            ⁢
            
              
            
            ⁢
            projected
          
        
      
      
        
          0
        
        
          otherwise
        
      
    
  




for all pixels p in the plane PD of the detector.

Let proj: [0; π]×R3→PD be the application that associates a voxel ν of space R3 containing the volume of interest 20 and an angle θ with the projection onto the plane PD corresponding to position Sθ of the source. We then have:

∀pεPD, ∀ν such that p=proj(θ,ν) M3d[ν]=1Mθ[p]=1

Mθ[p]=0M3d[ν]=0

which means, for a given position S of the source, that all the voxels projected outside a region of the object on plane PD of the detector do not belong to the three-dimensional digital mask for the object in question.

Notation
A reference frame O used hereafter in the description is described by the plane PD of the detector and the plane that is perpendicular to it, passing through the first line of pixels e in the detector and which contains the trajectory of the source. An x-axis is therefore aligned with the column d of pixels in the detector and oriented from the first line e towards the last line e′. A y-axis is located along the first line e and oriented from the last column d towards the first column d′. The origin of the reference frame is located in plane PD such that the centre C of the trajectory of the source has coordinates (0,0, zC).
Within this reference frame, each voxel ν of the space R3 has coordinates (xv, yv, zv).[0029] ∀θε[0; π], Sθ is the position of the source on the trajectory of angle θ with respect to the y-axis. Therefore ∀θε[0; π], xSθ=0; ySθ=r cos(θ), zSθ≦r sin(θ)+zC.
yd is the ordinate of the last column d; yd′ is that of the first column d′.
∀θε[0; π], Tθ is the point located on the final column d of the detector (or the first column d′ respectively), the point corresponding to an edge of the object situated within the volume of interest 20 in the projection associated with angle θ.
∀θε[0; π] points Eθ and Jθ are the intersection points of the line segment SθTθ and the planes PT and PB respectively.
For a column 1 of pixels in the detector, A1=(xe=0, y1, 0) is the point corresponding to the first pixel of the detector in column 1, and B1=(xe′, y1, 0) is the point corresponding to the last pixel of the detector in this column 1. Let P1θ be the plane passing through the three points Sθ, A1 and B1.
Description of a Procedure in Accordance with an Embodiment of the Invention
As the points Tθ are on the edge of the object as projected at the last column d (or the first column d′ respectively), the line segments SθTθ are tangential to the edge of the object in the three-dimensional digital mask M3d. As the volume of interest 20 is delimited by planes PT and PB, this means that the line segments EθJθ are tangential to the said edge of the object. The idea of the procedure in accordance with the invention is to use these segments to define an intermediate three-dimensional mask M ̃ enveloping the object and to project this intermediate three-dimensional mask outside the limits of the detector as represented by the last column D (and the first column d′ respectively) in a manner allowing the two-dimensional masks Mθ to be extrapolated, one column of pixels at a time.
Let us suppose that we want to extrapolate a two-dimensional mask Mθ2, associated with an angle θ2 and a specific source location Sθ2 at the position of a column 1 of pixels that is beyond the limits of the detector. We therefore have y1<yd<0 (and y1>yd′>0 respectively) within the frame of reference O in FIG. 1. Let us take an angle θ1>θ2 (or θ1<θ2 respectively) corresponding to a particular source position Sθ1 close to position Sθ2. So as not to overload this proposition, let us assume that there is only a single point Tθ2 and a single point Tθ1 (the case of several points Tθ will be treated later on). For all angles γ in the interval from θ1 to θ2, we work out the position of the point Tγ on the final column d (or the first column d′ respectively) making use of a linear interpolation between Tθ1 and Tθ2:




   
  
    {
    
      
        
          
            
              x
              
                T
                
                  θ
                  1
                
              
            
            =
            
              
                α
                ⁢
                
                  
                
                ⁢
                
                  θ
                  1
                
              
              +
              β
            
          
        
      
      
        
          
            
              x
              
                T
                
                  θ
                  2
                
              
            
            =
            
              
                α
                ⁢
                
                  
                
                ⁢
                
                  θ
                  2
                
              
              +
              β
            
          
        
      
      
        
          
            
              x
              
                T
                γ
              
            
            =
            
              αγ
              +
              β
            
          
        
      
    
  




where α and β are constants.

This then gives the coordinates of the point Tγ:




  
    T
    γ
  
  =
  
    (
    
      
        
          
            
              
                x
                
                  T
                  
                    θ
                    2
                  
                
              
              -
              
                x
                
                  T
                  
                    θ
                    1
                  
                
              
            
            
              
                θ
                2
              
              -
              
                θ
                1
              
            
          
          ⁢
          γ
        
        +
        
          x
          
            T
            
              θ
              1
            
          
        
        -
        
          
            
              
                x
                
                  T
                  
                    θ
                    2
                  
                
              
              -
              
                x
                
                  T
                  
                    θ
                    1
                  
                
              
            
            
              
                θ
                2
              
              -
              
                θ
                1
              
            
          
          ⁢
          
            θ
            1
          
        
      
      ,
      
        y
        d
      
      ,
      0
    
    )
  




A surface defined by {SγTγIγε[θ1; θ2]} therefore delimits a frontier region of the intermediate three-dimensional mask M ̃ containing, in the manner worked out above, the object in question.
The extrapolation of the two-dimensional mask Mθ2 onto the column of pixels 1 is done by projecting the intermediate three-dimensional mask M ̃ onto the plane PD of the detector 10, taking the source to be located at position Sθ2.
In the case in question, we use a first step to work out the final column d (or the first column d′ respectively) in a two-dimensional mask Mγ associated with the source position Sγ for angle γ, which is between θ1 and θ2:




  
    ∀
    
      p
      ∈
      
        [
        
          
            A
            d
          
          ⁢
          
            B
            d
          
        
        ]
      
    
  
  ,
  
    
      
        M
        γ
      
      ⁡
      
        [
        p
        ]
      
    
    =
    
      {
      
        
          
            1
          
          
            
              
                if
                ⁢
                
                  
                
                ⁢
                xp
              
              ≤
              
                X
                
                  T
                  ⁢
                  
                    
                  
                  ⁢
                  γ
                
              
            
          
        
        
          
            0
          
          
            otherwise
          
        
      
    
  




In a second step, we work out the intersection between a three-dimensional envelope for the object under consideration and a plane P1θ2=Sθ2A1B1, a plane that therefore contains column 1 or pixels onto which the extrapolation is to be performed for the two-dimensional mask Mθ2. With reference to FIGS. 2 and 3: there is a portion 21 of the volume of interest 20 that is “seen” by the means of recording 10 when the source is in position Sθ1 but not when the source is in position Sθ2. For every voxel ν that belongs to the intersection of plane P1θ2 and the volume of interest, there is an angle γ between θ1 and θ2 such that the voxel ν is projected onto the final column d (or the first column d′ respectively) when the source is in position Sγ. Point Sγ is the intersection between a line passing through Ad (or Ad′ respectively) and the point (0, yv, zv) and the locus S of the source. A necessary condition at all times is that the angle γ exists such that all the voxels ν of the intersection between the volume of interest 20 and the plane P1θ2 can be projected onto the last column d (or first column d′ respectively):




  
    y
    t
  
  >
  
    max
    (
    
      
        y
        
          proj
          ⁡
          
            (
            
              
                θ
                2
              
              ,
              
                E
                
                  θ
                  1
                
              
            
            )
          
        
      
      ,
      
        y
        
          proj
          ⁡
          
            (
            
              
                θ
                2
              
              ,
              
                J
                
                  θ
                  1
                
              
            
            )
          
        
      
    
    )
  




and





  
    y
    t
  
  <
  
    min
    (
    
      
        y
        
          proj
          ⁡
          
            (
            
              
                θ
                2
              
              ,
              
                E
                
                  θ
                  1
                
              
            
            )
          
        
      
      ,
      
        y
        
          proj
          ⁡
          
            (
            
              
                θ
                2
              
              ,
              
                J
                
                  θ
                  1
                
              
            
            )
          
        
      
    
    )
  




respectively. All voxels ν of the intersection being considered therefore belong to the portion 21 of the volume of interest 20, as illustrated in FIG. 2.

If this condition is not fulfilled, we are in a situation similar to that illustrated in FIG. 3. Only a portion of the voxels ν in the intersection being considered belong to the portion 21 of the volume of interest 20. Because of this, the intermediate three-dimensional mask M ̃ cannot be worked out for the entirety of the volume of interest 20, since information is missing as a result of the voxels ν in the intersection being considered that are outside the portion 21 of the volume of interest 20 and that have not been “seen” by the means of recording 10 when the source is either at position Sθ2 or at a position Sθ1. There is therefore a column 1fin, outside of which it is not possible to work out the two-dimensional mask Mθ2. This column 1fin, fits a linear equation




  y
  =
  
    max
    (
    
      
        y
        
          proj
          ⁡
          
            (
            
              
                θ
                2
              
              ,
              
                E
                
                  θ
                  1
                
              
            
            )
          
        
      
      ,
      
        y
        
          proj
          ⁡
          
            (
            
              
                θ
                2
              
              ,
              
                J
                
                  θ
                  1
                
              
            
            )
          
        
      
    
    )
  




and





  y
  =
  
    min
    (
    
      
        y
        
          proj
          ⁡
          
            (
            
              
                θ
                2
              
              ,
              
                E
                
                  θ
                  1
                
              
            
            )
          
        
      
      ,
      
        y
        
          proj
          ⁡
          
            (
            
              
                θ
                2
              
              ,
              
                J
                
                  θ
                  1
                
              
            
            )
          
        
      
    
    )
  




respectively.

Then, for every column 1 of pixels contained between the last column d (or the first column d′ respectively) and the column 1fin, whatever the voxel ν belonging to the intersection of plane P1θ2 and the volume of interest 20, the intermediate three-dimensional mask M ̃ is determined by:




  
    M
    ∼
    
      [
      v
      ]
    
  
  =
  
    {
    
      
        
          1
        
        
          
            
              if
              ⁢
              
                
              
              ⁢
              M
              ⁢
              
                
              
              ⁢
              
                γ
                ⁡
                
                  [
                  
                    proj
                    ⁡
                    
                      (
                      
                        γ
                        ⁢
                        
                          
                        
                        ⁢
                        v
                      
                      )
                    
                  
                  ]
                
              
            
            =
            1
          
        
      
      
        
          0
        
        
          otherwise
        
      
    
  




Then, in a third step, we project the three-dimensional mask M ̃ onto the column of pixels 1, making use of the position Sθ2 of the source S:




  
    ∀
    
      ∈
      θ
    
  
  =
  
    {
    
      
        
          
            
              
                
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  if
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ∃
                    
                      v
                      ∈
                      
                        volume
                        ⁢
                        
                          
                        
                        ⁢
                        of
                        ⁢
                        
                          
                        
                        ⁢
                        interest
                        ⁢
                        
                          
                        
                        ⁢
                        
                          (
                          20
                          )
                        
                        ⁢
                        
                          
                        
                        ⁢
                        such
                        ⁢
                        
                          
                        
                        ⁢
                        that
                      
                    
                  
                
              
            
            
              
                
                  
                    
                      proj
                      ⁢
                      
                        
                      
                      ⁢
                      
                        (
                        
                          
                            θ
                            2
                          
                          ,
                          V
                        
                        )
                      
                    
                    ]
                  
                  =
                  
                    
                      p
                      ⁢
                      
                        
                      
                      ⁢
                      and
                      ⁢
                      
                        
                      
                      ⁢
                      M
                      ⁢
                      
                        ~
                      
                      ⁢
                      
                        (
                        v
                        )
                      
                    
                    =
                    1
                  
                
              
            
          
        
      
      
        
          
            0
            ⁢
            
              
            
            ⁢
            otherwise
          
        
      
    
  




This allows the extrapolation of the two-dimensional mask Mθ2 onto pixel column 1 to be obtained.
Extrapolation for Two-Dimensional Masks Mθ2 Associated with Projections of the Extremity of the Source Trajectory S
In this situation, the angle θ2 is equal to either θmax or θmin. There is therefore no angle θ1>θ2=θmax (or θ1<θ2=θmin respectively) corresponding to a discrete position Sθ1 for the source close to position Sθ2 that would allow the procedure according to an embodiment of the invention to be used as described previously.
To alleviate this situation and in order to apply the procedure as described according to the invention previously, we are going to extrapolate a position S′θ1 for the source, associated with the angle θ1>θ2=θmax (or θ1<θ2=θmin respectively). To this end, let us take an angle θ1>θ2=θmax (or θ1<θ2=θmin respectively) corresponding to a discrete source position Sθ3 close to position Sθ2 (see FIG. 4). We are therefore working out the position of point T′θ1 on the final column d (or the first column d′ respectively) making use of a linear interpolation between Tθ2 and Tθ3, as described previously for working out the point Tγ starting from points Tθ1 and Tθ2. It is therefore sufficient to apply the procedure in accordance with the invention as described earlier.
Shapes of real objects likely to be present in the volume of interest 20:
To describe the procedure according to an embodiment of the invention, we have assumed the existence of a single point Tθ on the last column d (or the first column d′ respectively). However, much of the time this is not the case: there are several points Tθ on this column d (or d′ respectively). In order to resolve these cases, the procedure in accordance with the invention described previously is applied considering the points Tθ one at a time, giving an associated intermediate result. If there are three points Tθ, for example, the procedure in accordance with the invention is applied three times and there are three associated intermediate results. Once all the points Tθ have been considered one after the other, the final result involves applying a logical AND to the entire group of associated intermediate results.
In one variant that can be realised, the logical AND is applied during the evaluation of the intersection between the volume of interest 20 and plane P1θ2.
Once all the two-dimensional masks have been extrapolated, the limit lfin, is being removed by considering a virtual detector covering a zone that is common to all the extrapolated two-dimensional masks and by iterating the extrapolation procedure according to the invention as has just been described above.
Once the series of two-dimensional masks Mθ2 has been extrapolated this way, a final step can be applied to these masks in order to by sure that they are as natural as possible with respect to the object being X-rayed within the volume of interest 20. This final step is the application of a morphological closure based on a disc, with a diameter of a predetermined number of pixels. A morphological closure such as this consists of a dilation function based on the said disc followed by an erosion based on the same disc. This is a known method for image handling.
Determination of the Final Three-Dimensional Digital Mask:
Reconstruction of the final three-dimensional digital mask can then be realized based on the series of two-dimensional masks Mθ2 thus extrapolated. This type of reconstruction is well known. In order to avoid sudden transitions between the object and the background, a weighting (also known as “fuzzification”) is always applied, making use of the theory of fuzzy sets.
To do this, member functions are associated with properties of the three-dimensional digital mask M3d (M3d[v]=1 if the voxel ν belongs to the object in question) and the two-dimensional mask Mθ(Mθ[p]=1 if the pixel p belongs to the surface when the object in question is projected onto the plane PD). Let these member functions be μM3d and μMθ respectively. These member functions provide values that indicate how much the element complies with the property associated with the member function: these values range from 0 (the associated property is not the case) and 1 (the associated property is fully confirmed).
In order to determine the member function μMθ, the transition from 1 to 0 (object to background) is used that is present in the two-dimensional masks Mθ extrapolated as described earlier. In the current state of these two-dimensional masks, a precision error can arise at the transitions between object and background in the two-dimensional masks Mθ. It is therefore sensible to balance or blur out the transition between the object and the background. To do this, the two-dimensional masks Mθ are treated as images in which the pixels take values from 0 (background) to 1 (object in question), to which a dilation based on a disc has been applied and then a low-pass filter. The dilation ensures that the kernel of the member function μMθ (when this is equal to 1) covers the whole of the object under consideration, whereas applying a low-pass filter allows smooth transitions between the object and the background to be obtained. The size of the disc used for the dilation should correspond to the size of a kernel used when applying the low-pass filter (for example an average filter) once it has finished. This condition ensures that a result from applying the low-pass filter will be the same as the result in the zones of the two-dimensional mask Mθ “flagged” as belonging to the object under consideration, before the dilation.
As a variant, if the low-pass filter is an infinite impulse response filter (such as a Gaussian filter), the size of the dilation disc should be a value based on which the coefficients of the kernel of the low-pass filter can be disregarded (in a Gaussian filter, the kernel can be small if the deviation type is small and if the coefficients are only represented by a few of the bits, in IT terms).
It should be noted that the precision error can be modelled by the size of the kernel of the low-pass filter: filters with large kernels containing significant values rarely retain the low frequencies and therefore smear out (make fuzzy, balance out) the transitions between the object and the background more.
In order to determine the member function μM3d corresponding to the property associated with the three-dimensional digital mask M3d, the information supplied by the properties associated with the two-dimensional masks Mθ (for each angle θ associated with each two-dimensional mask in the series of extrapolated two-dimensional masks) must be reassembled making use of an aggregation operator (see FIG. 5). An aggregation operator such as this could be a t-norm operator T. This allows the member function μM3d to be calculated as a fuzzy counterpart of the logical AND. Numerous t-norm operators exist: probabilistic, drastic, Zadeh, Lukasiewicz, etc. The procedure has been implemented using the probabilistic t-norm operator.
The member function μM3d is therefore determined as follows: ∀νεvolume of interest (20),




  
    
      μ
      
        M
        
          3
          ⁢
          d
        
      
    
    ⁡
    
      (
      v
      )
    
  
  =
  
    
      T
      θ
    
    ⁢
    
      
        μ
        
          M
          θ
        
      
      ⁡
      
        (
        
          proj
          ⁡
          
            (
            
              θ
              ,
              v
            
            )
          
        
        )
      
    
  




We thus obtain a final weighted three-dimensional digital mask M3d.
There are of course numerous modifications that can be applied to various embodiments of the invention such as that described above without deviating from the framework described for it.
In this document, the terms “procedure” and “method” are used interchangeably.
This written description uses examples to disclose embodiments of the invention, including the best mode, and also to enable any person skilled in the art to make and use the claimed invention. The patentable scope of the invention is defined by the claims, and may include other examples that occur to those skilled in the art. Such other examples are intended to be within the scope of the claims if they have structural elements that do not differ from the literal language of the claims, or if they include equivalent structural elements with insubstantial differences from the literal languages of the claims.
Although specific features of embodiments of the invention are shown in some drawings and not in others, this is for convenience only as each feature may be combined with any or all of the other features in accordance with the invention. The words “including”, “comprising”, “having”, and “with” as used herein are to be interpreted broadly and comprehensively and are not limited to any physical interconnection. Moreover, any embodiments disclosed in the subject application are not to be taken as the only possible embodiments. Other embodiments will occur to those skilled in the art and are within the scope of the following claims.

## Claims

1. A computer-implemented method for processing radiographic images by extrapolating a two-dimensional mask Mθ2 from a series of two-dimensional masks, the method comprising:
at a radiographic device comprising an X-ray source and a digital detector that is in planar arrangement opposite the source and which has a limit (d, d′):
a) recording a series of images at a series of positions Sγ for the X-ray source of a volume of interest consisting of an object located between the X-ray source and the digital detector;
b) calculating a series of two-dimensional masks Mγ from the series of images, the series of two-dimensional masks Mγ associated with the series of positions Sγ for the X-ray source, located between a first position Sθ1 and a second position Sθ2 for the X-ray source, starting from a first two-dimensional mask Mθ1 and a second two-dimensional mask Mθ2 ;
c) calculating an intermediate three-dimensional mask for the object, starting from the series of two-dimensional masks Mθ and the first two-dimensional mask Mθ1 and the second two-dimensional mask Mθ2;
d) extrapolating the second two-dimensional mask Mθ2 beyond the limits d or d′ of the digital detector, according to a relative position between the first position Sθ1 and the second position Sθ2, starting from the intermediate three-dimensional mask; and
e) reconstructing a final three-dimensional digital mask from a resulting series of extrapolated two-dimensional masks Mθ2.

2. The computer-implemented method of claim 1, wherein before step b), if the first two-dimensional mask Mθ1 associated with the first position Sθ1 is not available, the method further includes a step for extrapolating the first two-dimensional mask Mθ1 from the available two-dimensional masks Mθ.

3. The computer-implemented method of claim 1, further comprising projecting the intermediate three-dimensional mask based on the second position Sθ2 onto a plane passing through the digital detector.

4. The computer-implemented method of claim 1, wherein the at least partially estimating step b) is carried out at the limit d or d′ of the digital detector, according to the relative position between the first position Sθ1 and the second position Sθ2.

5. The computer-implemented extrapolation method of claim 4, wherein for each of the two-dimensional masks Mγ, the at least partially estimation from step b) involves a step that determines a point Tγ situated at an edge of the object, projected onto the limit d or d′ of the digital recorder, with the source at position Sγ.

6. The method of claim 5, wherein the point Tθare estimated by linear interpolation between a first point Tθ1 and a second point Tθ2 located at an edge of the object projected onto the limit d or d′ of the digital detector, with the source at the first position Sθ1 and the second position Sθ2 respectively.

7. The computer-implemented extrapolation method of claim 6, wherein if the first two-dimensional mask Mθ1 associated with first position Sθ1 is not available, the method includes extrapolating the first point Tθ1 from the points Tθ that are available.

8. The computer-implemented method of claim 1, further comprising applying a closure function to the extrapolated second two-dimensional mask Mθ2.

9. The extrapolation method of claim 1, wherein before step b), the method further comprises a step for determining a limit of extrapolation lfin that is effectively parallel to the limit d or d′ of the digital detector and that is located outside the limits of the digital detector.

10. The computer-implemented method of claim 9, wherein for every line l that is that is effectively parallel to the limit of extrapolation lfin and located between the limit d or d′ of the digital detector and the limit of extrapolation lfin, the step c) further comprises the following steps:
working out a plane P1θ2 that passes through the second position Sθ2 and the line l; and
working out an intermediate three-dimensional mask for each voxel (ν) located at the intersection of the plane P1θ2 and the volume of interest.

11. The computer-implemented method of claim 10, wherein the step further comprises a projection step for every line l that is effectively parallel to the limit of extrapolation lfin and located between the limit d or d′ of the digital detector and the limit of extrapolation lfin for the intermediate three-dimensional mask.

12. A computer-implemented method for processing images radiographic images by calculating a three-dimensional mask based on a series of two-dimensional masks Mθ, the method consisting of the following steps:
at a radiographic device comprising an X-ray source and a digital detector that is in planar arrangement opposite the source and which has a limit (d, d′):
a) recording a series of images at a series of positions Sγ for the X-ray source of a volume of interest consisting of an object located between the X-ray source and the digital detector;
b) extrapolating the a series of two-dimensional masks Mγ by:
b1) estimating a series of two-dimensional masks Mγ associated with a series of positions Sθ for the X-ray source, located between a first position Sθ1 and a second position Sθ2 of the X-ray source, starting from a first two-dimensional mask Mθ1 and a second two-dimensional mask Mθ2;
b2) evaluating an intermediate three-dimensional mask for the object, starting from the series of two-dimensional masks Mγ and the first two-dimensional mask Mθ1 and the second two-dimensional mask Mθ2; and
b3) extrapolating the second two-dimensional mask Mθ2 beyond the limits d or d′ of the digital detector, according to a relative position between the first position Sθ1 and the second position Sθ2, starting from the intermediate three-dimensional mask; and

c) reconstructing a three-dimensional digital mask based on the extrapolated series of two-dimensional masks Mθ.

13. The computer-implemented method of claim 12, wherein step c) includes steps involving:
c1) applying a dilation function followed by a low-pass filter to obtain a membership function μMθ for each two-dimensional mask Mθ;
c2) evaluating a membership function μM3d based on the membership function μMθ, using a T-norm operator; and
c3) determining the three-dimensional digital mask based on the membership function μM3d.

14. The computer-implemented method of claim 13, wherein the t-norm operator is probabilistic.