METHODS AND SYSTEMS FOR INVERSE PROBLEM RECONSTRUCTION AND APPLICATION TO ECT RECONSTRUCTION

20150145885

14/374158

February 01, 2013

Methods and systems for obtaining a reconstruction of an object, the reconstruction of the object comprising an inverse problems. The method includes expressing the reconstruction of the object as a optimization problem with total variation regularization, and obtaining the reconstruction of the object from convergence of the iterations, an object being an input/representation of input to a system, the system providing output data resulting from the input, the object being reconstructed from the output data obtained from interaction of the object with the system.

1. A method, implemented utilizing a computer, for obtaining a reconstruction of an object, the reconstruction of the object comprising an inverse problems, the method comprising: expressing the reconstruction of the object as a optimization problem with total variation regularization; expressing solution of the optimization problem as a system of fixed point equations, each of the fixed point equations expressed using projections; performing iterations applying the protections alternatingly; and obtaining the reconstruction of the object from convergence of the iterations; the object being an input to a system, the system providing output data resulting from the input, the object being reconstructed from the output data obtained from interaction of the object with the system; iterations being performed by means of a computer usable medium having computer readable code that causes the computer to perform the iterations.

2. The method of claim 1 wherein at least some of the projections are expressed using proximity operators.

3. The method of claim 1 wherein the reconstruction of the object is expressed as min{−+λ(φ∘B)(ƒ):ƒεRd}. where g is the output data resulting from the input, ƒ is the object, A is the operator representing the system, γ is the background noise, λ is a regularization parameter, and (φ∘B)(ƒ) is a total variation regularization expression; and wherein the fixed point equations are b=(I−proxμ−1φ)(b+σ∇bH(ƒ,b)), ƒ=proxγ(ƒ−τS∇ƒH(ƒ,b)). where I is the identity matrix or operator, prox is the proximity operator, ∇ is the gradient operator with respect to the subscripted variable, H=−+λ(Φ∘B)(ƒ):ƒεRd λ, β and μ are positive numbers, [mathematical expression included] and S is a predetermined matrix including the identity matrix.

4. The method of claim 1 further comprising using a preconditioner in one of the fixed point equations using projections in order to improve convergence while not substantially affecting result of the convergence.

5. The method of claim 3 wherein S is a preconditioner.

6. The method of claim 5 wherein the preconditioner in one iteration is different from the preconditioner in a subsequent iteration.

7. The method of claim 1 wherein the object is an image from emission computed tomography (ECT), output data is scanner projection data; the system being expressed as: g=Aγ+γ where g are expected photon counts, A is a system matrix, ƒ is a expected radiotracer distribution, γ is background photon count.

8. The method of claim 7 wherein reconstruction of the expected radiotracer distribution is expressed as min{−+λ(φ∘B)(ƒ):ƒεRd}. where λ is a regularization parameter, and (φ∘B)(ƒ) is a total variation regularization expression; and wherein the fixed point equations are b=(I−proxμ−1φ)(b+σ∇bH(ƒ,b)), ƒ=proxγ(ƒ−τS∇ƒH(ƒ,b)). where I is the identity matrix or operator, prox is the proximity operator, ∇ is the gradient operator with respect to the subscripted variable, H=−+λ(φ∘B)(ƒ):ƒεRd λ, β and μ are positive numbers, [mathematical expression included] and S is a predetermined matrix including the identity matrix.

9. The method of claim 8 wherein S is a preconditioner.

10. The method of claim 9 wherein S is given by [mathematical expression included] where diag indicates a diagonal matrix, AT is the transpose of the system matrix, I is the unitary matrix.

11. A system comprising: at least one processor; and at least one computer usable medium having computer readable code embodied therein, said computer readable code causing said at least one processor to: receive data from interaction of an object with a system; express the reconstruction of the object as a optimization problem with total variation regularization; express solution of the optimization problem as a system of fixed point equations, each of the fixed point equations expressed using projections; perform iterations applying the protections alternatingly; and obtain the reconstruction of the object from convergence of the iterations; wherein the object is an input to a system, the system provides output data resulting from the input, the object is reconstructed from the output data obtained from interaction of the object with the system.

12. The system of claim 11 wherein alt least some of the projections are expressed using proximity operators.

13. The system of claim 11 wherein the reconstruction of the object is expressed as min{−+λ(φ∘B)(ƒ):ƒεRd}. where g is the output data resulting from the input, ƒ is the object, A is the operator representing the system, γ is the background noise, λ is a regularization parameter, and (φ∘B)(ƒ) is a total variation regularization expression; and wherein the fixed point equations are b=(I−proxμ−1φ)(b+σ∇bH(ƒ,b)), ƒ=proxγ(ƒ−τS∇ƒH(ƒ,b)). where I is the identity matrix or operator, prox is the proximity operator, ∇ is the gradient operator with respect to the subscripted variable, H=−+λ(φ∘B)(ƒ):ƒεRd λ, β and μ are positive numbers, [mathematical expression included] and S is a predetermined matrix including the identity matrix.

14. The system of claim 11 wherein said computer readable code is also capable of causing said at least one processor to use a preconditioner in one of the fixed point equations in order to improve convergence while not substantially affecting result of the convergence.

15. The system of claim 13 wherein S is a preconditioner.

16. The system of claim 15 wherein the preconditioner in one iteration is different from the preconditioner in a subsequent iteration.

17. The system of claim 11 wherein the object is an image from emission computed tomography (ECT), output data is scanner projection data; the system being expressed as: g=Aƒ+γ where g are expected photon counts, A is a system matrix, ƒ is a expected radiotracer distribution, γ is background photon count.

18. The system of claim 17 wherein reconstruction of the expected radiotracer distribution is expressed as min{−+λ(φ∘B)(ƒ):ƒεRd}. where λ is a regularization parameter, and (φ∘B)(ƒ) is a total variation regularization expression; and wherein the fixed point equations are b=(I−proxμ−1φ)(b+σ∇bH(ƒ,b)) ƒ=proxγ(ƒ−τS∇ƒH(ƒ,b)). where I is the identity matrix or operator, prox is the proximity operator, ∇ is the gradient operator with respect to the subscripted variable, H=−+λ(φ∘B)(ƒ): ƒεRd λ, β and μ are positive numbers, [mathematical expression included] and S is a predetermined matrix including the identity matrix.

19. The system of claim 18 wherein S is a preconditioner.

20. The system of claim 19 wherein S is given by [mathematical expression included] where diag indicates a diagonal matrix, AT is the transpose of the system matrix, I is the unitary matrix.

21. A computer program product comprising: a non-transitory computer usable medium having computer readable code embodied therein, said computer readable code causing at least one processor to: receive data from interaction of an object with a system; express the reconstruction of the object as a optimization problem with total variation regularization; express solution of the optimization problem as a system of fixed point equations, each of the fixed point equations expressed using projections; perform iterations applying the protections alternatingly; and obtain the reconstruction of the object from convergence of the iterations; wherein the object is an input to a system, the system provides output data resulting from the input, the object is reconstructed from the output data obtained from interaction of the object with the system.

22. The computer program product of claim 21 wherein at least some of the projections are expressed using proximity operators.

23. The computer program product of claim 21 wherein the reconstruction of the object is expressed as min{−+λ(φ∘B)(ƒ):ƒεRd}. where g is the output data resulting from the input, ƒ is the object, A is the operator representing the system, γ is the background noise, λ is a regularization parameter, and (φ∘B)(ƒ) is a total variation regularization expression; and wherein the fixed point equations are b=(I−proxμ−1φ)(b+σ∇bH(ƒ,b)), ƒ=proxγ(ƒ−τS∇ƒH(ƒ,b)). where I is the identity matrix or operator, prox is the proximity operator, ∇ is the gradient operator with respect to the subscripted variable, H=−+λ(φ∘B)(ƒ):ƒεRd λ, β and μ are positive numbers, [mathematical expression included] and S is a predetermined matrix including the identity matrix.

24. The computer program product of claim 21 wherein the computer readable code also causes the at least one processor to use a preconditioner in one of the fixed point equations using projections in order to improve convergence while not substantially affecting result of the convergence.

25. The computer program product of claim 21 wherein S is a preconditioner, the preconditioner improving convergence while not substantially affecting result of the convergence.

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edspap.20150145885