Decomposition of multi-energy scan projections using multi-step fitting

1. A method for decomposition of projection data acquired by scanning a set of objects using at least two x-ray spectra, the projection data including low energy projections (P L ) and high energy projections (P H ), said method comprising:

A. solving the projections P L and P H to determine a photoelectric line integral (A p ) component of attenuation and a Compton line integral (A c ) component of attenuation of the set of scanned objects using multi-step fitting; and

B. reconstructing a Compton image I c and a photoelectric image I p from A c and A p .

2. The method of claim 1 , further including, prior to step A, performing a calibration procedure using simulated data or measured data or some combination of simulated and measured data.

3. The method of claim 1 , further including, prior to step A, performing a calibration procedure, wherein said calibration procedure includes generating low energy iso-transmission contours for known values of P L at known values of A p and at known values of A c .

4. The method of claim 3 , wherein said calibration procedure includes, for each of said low energy iso-transmission contours, fitting A p to a polynomial function g L (A c ), wherein g L is a polynomial function that represents the shape of the contour.

5. The method of claim 4 , wherein g L includes a set of coefficients g Li determined at said known values of P L and said calibration procedure includes fitting the values of each coefficient g Li to a polynomial function h Li (P L ).

6. The method of claim 1 , wherein said calibration procedure includes computing minimum and maximum values of P H for each of said low-energy iso-transmission contours as a function of P L .

7. The method of claim 6 , wherein said calibration procedure includes fitting the minimum values of P H to a polynomial function m L (P L ).

8. The method of claim 6 , wherein said calibration procedure includes fitting the maximum values of P H to a polynomial function n L (P L ).

9. The method of claim 1 , wherein said calibration procedure includes generating high energy iso-transmission contours for known values of P H at known values of the A p and at known values of the A c .

10. The method of claim 9 , wherein said calibration procedure includes, for each of said high energy iso-transmission contours, fitting A p to a polynomial function g H (A c ), wherein g H is a polynomial function that represents the shape of the contour for a given P H .

11. The method of claim 10 , wherein said g H includes a set of coefficient g Hi determined at said known values of P H and said calibration procedure includes fitting the values of each coefficient g Hi to a polynomial function h Hi (P H ).

12. The method of claim 1 , wherein step A includes generating a low energy iso-transmission contour corresponding to P L and a high energy iso-transmission contour corresponding to P H .

13. The method of claim 12 , wherein step A includes determining the values of the A p and A c at the intersection of said low energy iso-transmission contour and the high energy iso-transmission contour.

14. The method of claim 13 , wherein the intersection of the low energy iso-transmission contour and the high energy iso-transmission contour is determined by equating a first polynomial function g L (A c ) representing said low energy iso-transmission contour, wherein g L is a polynomial function that represents the shape of the contour of P L , with a second polynomial function g H (A c ) representing said high energy iso-transmission contour, wherein g H is a polynomial function that represents the shape of the contour for a given P H .

15. The method of claim 12 , further including computing a modified value of the input low energy projection data (P Lc ) and a modified value of the input high energy projection data (P Hc ), wherein each of said modified values P Lc and P Hc are clamped to be bounded between two values.

16. The method of claim 15 , including representing the low energy iso-transmission contour with a polynomial function g L and determining a set of coefficients of g L as a function of P Lc .

17. The method of claim 15 , including representing the high energy iso-transmission contour with a polynomial function g H and determining a set of coefficients of g H as a function of P Hc .

18. The method of claim 15 , further including, prior to step A, generating calibration data using P L , wherein P Lc is computed by clamping the value of P L to lie between 0 and the maximum value of P L used to generate said calibration data.

19. The method of claim 15 , wherein the modified value P Hc is determined by clamping the value of P H to lie between a minimum value of P H (P Hmin ) and a maximum value of P H (P Hmax ).

20. The method of claim 19 , including determining P Hmin as a function of P Lc and a polynomial function n L and determining P Hmax as a function of P Lc and a polynomial function m L , wherein m L is a polynomial function that determines P Hmin for a given value of P Lc and wherein n L is a polynomial function that determines P Hmin for a given value of P Lc .

21. The method of claim 1 , wherein step A includes calculating a scaled Compton line integral value (A cs ) as a function of a scale factor s c and A c and calculating a scaled photoelectric line integral value (A ps ) as a function of a scale factor s p and A ps .

22. The method of claim 21 , wherein step B includes constructing said I c and said I p as a function of said A cs and said A ps .

23. The method of claim 1 , further including, after step B, determining an image of a basis function X(I X ) and a basis function Y(I Y ), by solving I c and I p on a pixel-by-pixel basis, wherein the basis functions X(I X ) and Y(I Y ) are functions linearly combined to determine the pixel intensities in I c and I p .

24. A method for decomposition of projection data acquired by scanning a set of objects using at least two x-ray spectra, said projection data including low energy projection data (P L ) and high energy projection data (P H ), said method comprising:

A. performing a calibration procedure using at least some simulated data or measured data or a combination of simulated and measured data, including:

i. generating low energy iso-transmission contours for known values of P L and high energy iso-transmission contours for known values of P H ;

ii. generating a polynomial g L that represents the shape of the low energy iso-transmission contour for each P L , wherein g L includes a set of coefficients g Li determined at said known values of P L ;

iii. generating a polynomial g H that represents the shape of the high energy iso-transmission contour for each P H , wherein g H includes a set of coefficients g Hi determined at said known values of P H ;

iv. generating polynomials h L that represents the variation of the coefficients of the polynomial g L as a function of P L ;

v. generating polynomials h H that represents the variation of the coefficients of the polynomial g H as a function of P H ;

vi. determining the minimum and maximum values of P H for each transmission line corresponding each P L ;

vii. generating a polynomial m H that represents the variation of the minimum value of P H as a function of P L ; and

viii. generating a polynomial n H that represents the variation of the maximum value of P H as a function of P L ;

B. solving the projections P L and P H to determine a photoelectric line integral (A p ) component of attenuation and a Compton line integral (A c ) component of attenuation of the set of scanned objects using a multi-step fitting procedure, including:

i. computing the values of each coefficient g Li using a polynomial function h Li (P L ) and computing the values of each coefficient g Hi using a polynomial function h Hi (P H ); and

ii. determining A c and A p as a function of P L and P H , using the coefficients of g L and the coefficients of g H ; and

C. reconstructing a Compton image I c and a photoelectric image I p from A c and A p .

25. The method of claim 24 , further including, after step C, determining an image of a basis function X(I X ) and a basis function Y(I Y ), by solving image I c and image I p on a pixel-by-pixel basis.

26. A system for decomposing projection data for a set of scanned objects acquired using at least two x-ray spectra, said system comprising:

A. media for storing low energy projection data (P L ) and high energy projection data (P H );

B. a decomposition module configured to determine a photoelectric line integral (A p ) component of attenuation and a Compton line integral (A c ) component of attenuation for P L and P H using multi-step fitting; and

C. an image construction module configured to construct a Compton image (I c ) and a photoelectric image (I p ) from the A p and A c .

27. The system of claim 26 , wherein the decomposition module includes:

D. a calibration module configured to calibrate the decomposition module using at least some simulated data or measured data or a combination of simulated data and measured data.

28. The system of claim 27 , wherein the calibration module is configured to generate low energy iso-transmission contours for known values of P L at known values of A p and at known values of A c .

29. The system of claim 28 , wherein the calibration module is configured, for each of said low energy iso-transmission contours, to fit A p to a polynomial function g L (A c ), wherein g L is a polynomial function that represents the shape of the contour.

30. The system of claim 29 , wherein g L includes a set of coefficients g Li determined at said known values of P L and the calibration module is configured to fit said set of coefficients g Li to a polynomial function h Li (P L ).

31. The system of claim 28 , wherein the calibration module is configured to compute the minimum and maximum values of P H for each of the low-energy iso-transmission contours corresponding to P L .

32. The system of claim 28 , wherein the calibration module is configured to fit the minimum values of P H to a polynomial function m L (P L ).

33. The system of claim 28 , wherein the calibration module is configured to fit the maximum values of P H to a polynomial function n L (P L ).

34. The system of claim 27 , wherein the calibration module is configured to generate high energy iso-transmission contours for known values of P H at known values of A p and at known values of A c .

35. The system of claim 34 , wherein the calibration module is configured, for each of said high energy iso-transmission contours, to fit A p to a polynomial function of g H (A c ), wherein g H is a polynomial function that represents the shape of the contour for a given P H .

36. The system of claim 34 , wherein g H includes a set of coefficients g Hi determined at said known values of P H and the calibration module is configured to fit said set of coefficients g Hi to a polynomial function h Hi (P H ).

37. The system of claim 26 , wherein the decomposition module is configured to generate a low energy iso-transmission contour corresponding to P L and a high energy iso-transmission contour corresponding to P H .

38. The system of claim 37 , wherein the decomposition module is configured to determine the values of A p and A c at the intersection of said low energy iso-transmission contour and the high energy iso-transmission contour.

39. The method of claim 37 , wherein the decomposition module is configured to compute a modified value of the input low energy projection data (P Lc ) and a modified value of the input high energy projection data (P Hc ), and configured to clamp said modified values P Lc and P Hc between two values.

40. The system of claim 39 , wherein the decomposition module is configured to clamp the values of P L to lie between 0 and the maximum value of P L used to generate a set of calibration data and to compute the modified value P Lc as a function of the clamped values of P L .

41. The system of claim 39 , wherein the decomposition module is configured to clamp P Hc between a minimum value of P H (P Hmin ) and a maximum value of P H (P Hmax ).

42. The system of claim 41 , wherein the decomposition module is configured to determine P Hmin as a function of P Lc and a polynomial n L and to determine P Hmax as a function of P Lc and a polynomial m L , wherein m L is a polynomial function representing the coefficients of P L for the minimum values of P H and wherein n L is a polynomial function representing the coefficients of P L for the maximum values of P H .

43. The system of claim 26 , wherein the decomposition module is configured to calculate a scaled Compton line integral value (A cs ) as a function of a scale factor s c and A c and to calculate a scaled photoelectric line integral value (A ps ) as a function of a scale factor sp and A ps .

44. The system of claim 26 , wherein the image reconstruction module is configured to reconstruct said I c and said I p as a function of said A cs and said A ps .

45. The system of claim 26 , wherein the image reconstruction module is configured to determine an image of a basis function X(I X ) and of a basis function Y(I Y ), by solving I c and I p on a pixel-by-pixel basis, wherein the basis functions X(I X ) and Y(I Y ) are functions linearly combined to determine the pixel intensities in I c and I p .

46. A system for decomposing projection data for a set of scanned objects acquired using at least two x-ray spectra, said system comprising:

A. media for storing low energy projection data (P L ) and high energy projection data (P H );

B. a calibration module configured to calibrate the decomposition module using at least some simulated data or measured data or a combination of simulated and measured data, and configured to:

i. generate a low energy iso-transmission contour corresponding to P L and a high energy iso-transmission contour corresponding to P H ;

ii. generate a polynomial g L that represents the shape of the low energy iso-transmission contour, wherein g L includes a set of coefficients g Li determined at said known values of P L ;

iii. generate a polynomial g H that represents the shape of the high energy iso-transmission contour, wherein g H includes a set of coefficients g Hi determined at said known values of P H ;

iv. generate polynomials h L that represents the variation of the coefficients of the polynomial g L as a function of P L ;

v. generate polynomials h H that represents the variation of the coefficients of the polynomial g H as a function of P H ;

vi. determine the minimum and maximum values of P H for each transmission line corresponding each P L ;

vii. generate a polynomial m H that represents the variation of the minimum value of P H as a function of P L ; and

viii. generate a polynomial n H that represents the variation of the maximum value of P H as a function of P L ;

C. a decomposition module configured to determine a photoelectric line integral (A p ) component of attenuation and a Compton line integral (A c ) component of attenuation for P L and P H using multi-step fitting, and configured to:

i. compute the values of each coefficient g Li using a polynomial function h Li (P L ) and to compute the values of each coefficient g Hi using a polynomial function h Hi (P H ); and

ii. determine A c and A p as a function of P L and P H using the coefficients of g L and the coefficients of g H ; and

D. an image reconstruction module configured to reconstruct a Compton image (I c ) and a photoelectric image (I p ) from the A p and A c .

47. The system of claim 46 , wherein the image construction module is configured to determine an image of a basis function X(I X ) and of a basis function Y(I Y ), by solving image I c and image I p on a pixel-by-pixel basis.