Method and arrangement for medical X-ray imaging

• US 7,068,752 B2
• Filed: 03/30/2004
• Issued: 06/27/2006
• Est. Priority Date: 11/14/2003
• Status: Expired due to Fees

• Current Assignee: Instrumentarium Dental Incorporated (Envista Holdings Corporation)
• Original Assignee: Instrumentarium Dental Incorporated (Envista Holdings Corporation)
• Inventors: Brandt, Sami
• Primary Examiner(s): BRUCE, DAVID VERNON
• Application Number:
  US10/813,563
• Publication Number:
  US 20050105694A1
• Time in Patent Office: 819 Days
• Field of Search: 378/4, 378/8, 378/15, 378/22, 378/62, 378/901
• US Class Current: 378/62
• CPC Class Codes: G01N 2223/419 computed tomograph
• CPC Class Codes: G01N 2223/612 biological material
• CPC Class Codes: G01N 23/046 using tomography, e.g. computed tomography [CT]
• CPC Class Codes: G06T 11/006 Inverse problem, transformation from projection-space into object-space, e.g. transform methods, back-projection, algebraic methods
• CPC Class Codes: G06T 2211/412 Dynamic
• CPC Class Codes: G06T 2211/436 Limited angle
• CPC Class Codes: Y10S 378/901 Computer tomography program or processor

Abstract

A medical X-ray device 5 arrangement for producing three-dimensional information of an object 4 in a medical X-ray imaging comprises an X-ray source 2 for X-radiating the object from at least two different directions; a detector 6 for detecting the X-radiation to form projection data of the object 4; a computational device 15 for modelling the object 4 mathematically utilizing the projection data to solve the imaging geometry and/or the motion of the object, where the solving concerns either some or all parts of the imaging geometry and/or the motion of the object. The computational device 15 utilizes said projection data and said mathematical modelling of the object in Bayesian inversion based on Bayes'"'"' formula

$p\left(x,\theta ❘m\right)=\frac{p_{\mathrm{pr}}\left(\theta \right)p_{\mathrm{pr}}\left(x\right)p\left(m❘x,\theta \right)}{p\left(m\right)}$
to produce three-dimensional information of the object.

Claims

1. A method for producing three-dimensional information of an object (4) in medical X-ray imaging, characterized in that

the object is X-radiated from at least two different directions and the said X-radiation is detected to form projection data of the object (4)

the object is modelled mathematically utilizing the projection data to solve the imaging geometry and/or the motion of the object, where the solving concerns either some or all parts of the imaging geometry and/or the motion of the object

and said projection data and said mathematical modelling of the object are utilized in Bayesian inversion based on Bayes'"'"' formula

$p$

( x , θ

| m ) = p pr ⁡

( θ

) ⁢

p pr ⁡

( x ) ⁢

p ⁡

( m | x , θ

) p ⁡

( m )

 

to produce three-dimensional information of the object, the prior distribution ppr(θ

) representing the prior knowledge of the imaging geometry and/or the motion of the object (4), the prior distribution ppr(x) representing mathematical modelling of the object, x representing the object image vector, which comprises values of the X-ray attenuation coefficient inside the object, θ

representing the parameter vector of the imaging geometry and/or the motion of the object (4), m representing projection data, the likelihood distribution p(m|x,θ

) representing the X-radiation attenuation model between the object image vector x, geometry parameter vector θ

, and projection data m, p(m) being a normalization constant and the posteriori distribution p(x,θ

|m) representing the three-dimensional information of the object (4) and the imaging geometry including the motion of the object.

2. A method according to claim 1, characterized in that the three-dimensional information of the object (4) is one or more two-dimensional images representing X-ray attenuation coefficient along slices through the object.

3. A method according to claim 1, characterized in that the three-dimensional information of the object (4) is a three-dimensional voxel representation of the X-ray attenuation in the object.

4. A method according to claim 1, characterized in that the measurement model is m=Aθ

x+e, where matrix Aθ

contains the weights describing how much each voxel contributes to the X-ray attenuation along the X-ray paths and the noise e is independent of object image vector x and the geometry parameter vector θ

leading to the likelihood distribution

p(m|x,θ

)=p_noise(m−

A_θ

x).

5. A method according to claim 1, characterized in that the said mathematical modelling comprises that X-radiation attenuates when passing the object (4), which means that every image voxel is nonnegative.

6. A method according to claim 1, characterized in that mathematical modelling is expressed by the formula:

$p_{\mathrm{pr}}<br><br>\left(x\right)=\exp <br><br>\left(-\mathrm{\alpha <br><br>}<br><br>\underset{N}{\sum <br><br>}<br><br>U_N<br><br>\left(x\right)\right)$

where the sum is taken over a collection of 3D neighbourhoods N and the value U_N(x) depends only on the values of voxels belonging to the neighborhood N, and a is a positive regularization parameter used to tune the width of the prior distribution.

7. A method according to claim 1, characterized in that the 3D tomographic problem is divided into a stack of 2D tomographic problems and on every such 2D problem, the mathematical modelling is expressed by the formula:

$p_{\mathrm{pr}}<br><br>\left(x\right)=\exp <br><br>\left(-\mathrm{\alpha <br><br>}<br><br>\underset{N}{\sum <br><br>}<br><br>U_N<br><br>\left(x\right)\right)$

where the sum is taken over a collection of 2D neighbourhoods N and the value U_N(x) depends only on the values of pixels belonging to the neighborhood N, and α

is a positive regularization parameter used to tune the width of the prior distribution, and the 2D tomographic problems are related to each other by the formula

pr3D(x(j))=exp(−

γ

Σ

Σ

|x(j)[k,q]−

x(j−

1)[k,q]|),

where the sums are taken over all pixels (k=1, . . . , K, q=1, . . . , Q) and y>

0 is another regularization parameter.

8. A method according to claim 7, characterized in that the neighborhoods comprise two adjacent pixels and U calculates a power of the absolute value of the difference, leading to the formula

$\begin{array}{c}{P}_{\mathrm{pr}}\end{array}$

( x ( j ) ) = ⁢

exp ( - α

( ∑

k = 1 K - 1 ⁢

∑

q = 1 Q ⁢



x ( j ) ⁡

[ k , q ] - x ( j ) ⁡

[ k + 1 , q ] 

8 ++ ⁢

∑

k = 1 K ⁢

∑

q = 1 Q - 1 ⁢



x ( j ) ⁡

[ k , q ] - x ( j ) ⁡

[ k , q + 1 ] 

8 ) )

where s is a positive real number.

9. A method according to claim 8, characterized in that s=1 corresponding to total variation (TV) prior describing objects (4) consisting of different regions with well-defined boundaries.

10. A method according to claim 1, characterized in that mathematical modelling is qualitative structural information of the target where the structural information is encoded in prior distributions that are concentrated around object image vectors x that correspond to the physiological structures of the object (4).

11. A method according to claim 1, characterized in that mathematical modelling comprise a list or probability distribution of possible attenuation coefficient values in the object (4).

12. A method according to claim 1, characterized in that the X-ray imaging geometry, such as X-ray source position, has unknown error modelled in the distribution p(m|x,θ

).

13. A method according to claim 1, characterized in that the X-radiation measurement noise is Poisson distributed.

14. A method according to claim 1, characterized in that the medical X-ray imaging is dental radiography.

15. A method according to claim 1, characterized in that the medical X-ray imaging is surgical C-arm imaging.

16. A method according to claim 1, characterized in that the medical X-ray imaging is mammography.

17. A method according to claim 1, characterized in that three-dimensional information of the object (4) is produced on the basis of the maximum a posteriori estimator (MAP) which is calculated by the equation:

p(y_MAP|m)=maxp(y|m)

m representing projection data and y=(x,θ

) representing the vector composed of the object image vector and the geometry parameter vector and where the maximum on the right hand side of the equation is taken over all y.

18. A method according to claim 1, characterized in that three-dimensional information of the object (4) is produced on the basis of the conditional mean estimator (CM), which is calculated by the equation:

y_CM=∫

yp(y|m)dy

where m represents projection data and y=(x,θ

) represents the vector composed of the object image vector and the geometry parameter vector.

19. A medical X-ray device (5) arrangement for producing three-dimensional information of an object (4) in a medical X-ray imaging, characterized in that the medical X-ray device (5) arrangement comprises:

an X-ray source (2) for X-radiating the object from at least two different directions

a detector (6) for detecting the X-radiation to form projection data of the object (4)

means (15) for modelling the object (4) mathematically utilizing the projection data to solve the imaging geometry and/or the motion of the object, where the solving concerns either some or all parts of the imaging geometry and/or the motion of the object

and the medical X-ray device (5) arrangement includes means (15) for utilizing said projection data and said mathematical modelling of the object in Bayesian inversion based on Bayes'"'"' formula

$p<br><br>\left(x,\mathrm{\theta <br><br>}|m\right)=\frac{p_{\mathrm{pr}}<br><br>\left(\mathrm{\theta <br><br>}\right)<br><br>p_{\mathrm{pr}}<br><br>\left(x\right)<br><br>p<br><br>\left(m|x,\mathrm{\theta <br><br>}\right)}{p<br><br>\left(m\right)}$

 

to produce three-dimensional information of the object, the prior distribution p_pr(θ

) representing the prior knowledge of the imaging geometry and/or the motion of the object (4), the prior distribution p_pr(x) representing mathematical modelling of the object, x representing the object image vector, which comprises values of the X-ray attenuation coefficient inside the object, θ

representing the parameter vector of the imaging geometry and/or the motion of the object (4), m representing projection data, the likelihood distribution p(m|x,θ

) representing the X-radiation attenuation model between the object image vector x, geometry parameter vector θ

, and projection data m, p(m) being a normalization constant and the posteriori distribution p(x,θ

|m) representing the three-dimensional information of the object (4) and the imaging geometry including the motion of the object.

20. A medical x-ray device (5) arrangement according to claim 19, characterized in that the three-dimensional information of the object (4) is one or more two-dimensional images representing X-ray attenuation coefficient along slices through the object.

21. A medical x-ray device (5) arrangement according to claim 19, characterized in that the three-dimensional information of the object (4) is a three-dimensional voxel representation of the X-ray attenuation in the object.

22. A medical X-ray device (5) arrangement according to claim 19, characterized in that the medical X-ray device arrangement comprises means (15) for modelling the measurement as

m=A_θ

x+e,

where matrix A_θ

contains the weights describing how much each voxel contributes to the X-ray attenuation along the X-ray paths and the noise e is independent of object image vector x and the geometry parameter vector θ

leading to the likelihood distribution

p(m|x,θ

)=p_noise(m−

A_θ

x).

23. A medical X-ray device (5) arrangement according to claim 19 characterized in that the medical X-ray device arrangement comprises means (15) for modelling the object (4) mathematically so that X-radiation attenuates when passing the object (4), which means that every image voxel is nonnegative.

24. A medical X-ray device (5) arrangement according to claim 19, characterized in that the medical X-ray device arrangement comprises means (15) for modelling the object (4) mathematically by the formula:

$p_{\mathrm{pr}}<br><br>\left(x\right)=\exp <br><br>\left(-\mathrm{\alpha <br><br>}<br><br>\underset{N}{\sum <br><br>}<br><br>U_N<br><br>\left(x\right)\right)$

where the sum is taken over a collection of 3D neighbourhoods N and the value U_N(x) depends only on the values of voxels belonging to the neighborhood N, and α

is a positive regularization parameter used to tune the width of the prior distribution.

25. A medical x-ray device (5) arrangement according to claim 19, characterized in that the 3D tomographic problem is divided into a stack of 2D tomographic problems and on every such 2D problem, the medical X-ray device arrangement comprises means (15) for modelling the object (4) mathematically by the formula:

$p_{\mathrm{pr}}<br><br>\left(x\right)=\exp <br><br>\left(-\mathrm{\alpha <br><br>}<br><br>\underset{N}{\sum <br><br>}<br><br>U_N<br><br>\left(x\right)\right)$

where the sum is taken over a collection of 2D neighbourhoods N and the value U_N(x) depends only on the values of pixels belonging to the neighborhood N, and α

is a positive regularization parameter used to tune the width of the prior distribution, and the 2D tomographic problems are related to each other by the formula

pr3D(x(j))=exp(−

γ

Σ

Σ

|x(j)[k,q]−

x(j−

1)[k,q]|),

where the sums are taken over all pixels (k=1, . . . , K, q=1, . . . , Q) and γ

>

0 is another regularization parameter.

26. A medical X-ray device (5) arrangement according to claim 25, characterized in that the neighborhood systems comprise two neighboring pixels xj, xk or voxels xj, xk and U_N(x) calculates a power of the

$p_{\mathrm{pr}}$

( x ( j ) ) = exp ⁡

( - α

⁡

( ∑

k = 1 K - 1 ⁢

⁢

∑

q = 1 Q ⁢

⁢



x ( j ) ⁡

[ k , q ] - x ( j ) ⁡

[ k + 1 , q ] 

8 ++ ⁢

⁢

⁢

∑

k = 1 K ⁢

⁢

∑

q = 1 Q - 1 ⁢

⁢



x ( j ) ⁡

[ k , q ] - x ( j ) ⁡

[ k , q + 1 ] 

8 ) )

absolute value of the difference, leading to the formula where s is a positive real number and α

is a regularization parameter used to tune the width of the prior distribution.

27. A medical X-ray device (5) arrangement according to claim 26, characterized in that the medical X-ray device arrangement comprises means (15) for modelling the object (4) mathematically by setting s=1 corresponding to total variation (TV) prior describing objects consisting of different regions with well-defined boundaries.

28. A medical X-ray device (5) arrangement according to claim 19, characterized in that the medical X-ray device arrangement comprises means (15) for modelling the object (4) mathematically by assuming that mathematical modelling is qualitative structural information of the target where the structural information is encoded in prior distributions that are concentrated around image vectors x that correspond to the physiological structures of the target.

29. A medical X-ray device (5) arrangement according to claim 19, characterized in that the medical X-ray device arrangement comprises means (15) for modelling the object (4) mathematically by assuming that mathematical modelling comprises a list of possible attenuation coefficient values in the object.

30. A medical X-ray device (5) arrangement according to claim 19, characterized in that the medical X-ray device arrangement comprises means (15) for modelling the object (4) mathematically by assuming that the X-ray imaging geometry, such as X-ray source position, has unknown error modelled in the distribution p(m|x,θ

).

31. A medical X-ray device (5) arrangement according to claim 19, characterized in that the medical X-ray device arrangement comprises means (15) for modelling the object (4) mathematically by assuming that X-radiation measurement noise is Poisson distributed.

32. A medical X-ray device (5) arrangement according to claim 19, characterized in that the medical X-ray imaging is dental radiography.

33. A medical X-ray device (5) arrangement according to claim 19, characterized in that the medical X-ray imaging is surgical C-arm imaging.

34. A medical X-ray device (5) arrangement according to claim 19, characterized in that the medical X-ray imaging is mammography.

35. A medical X-ray device (5) arrangement according to claim 19, characterized in that the medical X-ray device arrangement comprises means (15) for producing three-dimensional information of the object (4) on the basis of the maximum a posteriori estimator (MAP), which is calculated by the equation:

p(y_MAP|m)=maxp(y|m)

m representing projection data and y=(x,θ

) representing the vector composed of the object image vector and the geometry parameter vector, and where the maximum on the right hand side of the equation is taken over all y.

36. A medical X-ray device (5) arrangement according to claim 19, characterized in that the medical X-ray device arrangement comprises means (15) for producing three-dimensional information of the object (4) on the basis of the conditional mean estimator (CM), which is calculated by the equation

y_CM=∫

yp(y|m)dy

where m represents projection data and y=(x,θ

) represents the vector composed of the object image vector and the geometry parameter vector.