Method of cancelling ghosts from NMR images

Abstract

A method of cancelling ghosts from NMR images. The method involves estimating a phase difference function Δ (n 1 , n 2 ) and using that function to solve a linear system of equations to find the magnitudes of the true object densities at the true image and ghost locations x(n 2 ,n 2 ) and x(n 1 ,n 2 +N/2), respectively, where the image has dimensions N×N s . Experimental values of Δ (n 1 , n 2 ) for a variety of objects indicate that its variation along n 1 is considerably larger than along n 2 . Thus, for each column n 1 , the phase difference function Δ (n 1 , n 2 ) can be modelled as a one-dimensional function of n 2 with two parameters α (n 1 ) and β (n 1 ), which are estimated from the pixels in the 2-D FFT processed reconstructed image Y(n 1 ,n 2 ). These parameters are then used to estimate Δ (n 1 , n 2 ), which is ultimately used to de-ghost the image.

What is claimed is: 
     
       1. A method of cancelling .[.ghosts.]. .Iadd.a ghost .Iaddend.from .Iadd.a .Iaddend.NMR .[.images.]. .Iadd.image.Iaddend., comprising the steps of: (a) taking a two dimensional inverse Fourier transform of a raw NMR signal to obtain a ghosted image Y(n 1 ,n 2 );   (b) computing the signal energy for each column of said Fourier transformed signal using ##EQU17## (c) discarding the columns whose signal energy level are below a predetermined threshold;   (d) .[.estimating α (n 1 ) and β (n 1 ) for.]. .Iadd.identifying ghosting pixels in .Iaddend.each remaining column of data; i.e. n 1  =0, . . . N s  -1, .[.by:.].   .[.(i).]. .Iadd.(e) .Iaddend.finding the .Iadd.actual .Iaddend.phase difference .[ƒ]. Δ.Iadd. actual  .Iaddend.(n 1 ,n 2 ) for .[.all.]. .Iadd.the .Iaddend.ghosting pixels .[.of the column; and.].   .[.(ii).]. .Iadd.(f) .Iaddend.solving the following simultaneous equations to find linear least square estimates of α (n 1 ) and β (n 1 ): ##EQU18## .[.(e).]. .Iadd.(g) .Iaddend.using .Iadd.the linear least square estimates of .Iaddend.α (n 1 ) and β (n 1 ) .[.in the above equation.]. to find .[.the.]. .Iadd.an estimated .Iaddend.phase difference .Iadd.function .Iaddend.Δ (n 1 ,n 2 ) for 0≦n 2  <N.[.; and.]. .Iadd.using .Iaddend. ##EQU19## .[.(f).]. .Iadd.(h) .Iaddend.using .Iadd.the estimated phase difference function .Iaddend.Δ (n 1 ,n 2 ) in ##EQU20## to find A(n 1 ,n 2 ) and B(n 1 ,n 2 ) for 0≦n 2  <N, where the dimensions of the .[.reconstructed.]. image .Iadd.data x(n 1 ,n 2 ) corresponding to the NMR image .Iaddend.are N×N s  .Iadd.and where .Iaddend. ##EQU21##   
     
     
       2. The method of claim 1, wherein the ratio of the magnitude of even and odd parts of a pixel at location (n 1 ,n 2 ) is used to determine whether a pixel is a true image or a ghost. 
     
     
       3. The method of claim 2, wherein the pixel is classified as a ghost if said ratio is approximately equal to one. 
     
     
       4. The method of claim 2, wherein the following ratio is computed, indicating that a pixel near the center of the magnetic cord is a ghost when the ratio extends to a predetermined value. 
     
     
       5. The method of claim 1, wherein columns with small signal components are not processed. 
     
     
       6. The method of claim 1, wherein pixels at location (n 1 , n 2 ) whose linear base squares mean square error is larger than a predetermined amount are discarded when determining the phase difference function. .Iadd.7. The method of claim 1 wherein the actual phase difference Δ actual  (n 1 ,n 2 ) is calculated using: Δ actual  (n 1 ,n 2 )=phase (Y even     
     
     
        (n 1 ,n 2 ))-phase (Y odd  (n 1 ,n 2 ))..Iaddend..Iadd.8. A method of producing a magnetic resonance image, comprising the steps of: applying a bipolar readout gradient field;   sampling NMR data by scanning N lines of the k x  -k y  space in connection with the application of the bipolar readout gradient field, with reversal of the direction of data sampling on odd and even lines;   processing the NMR data to generate image data representative of a magnetic resonance image such that there is substantially no phase difference between even and odd parts of the image data..Iaddend..Iadd.9. The method of claim 1 wherein the phase difference between even and odd parts of the transform data is a function that varies in at least one of the x and y directions..Iaddend..Iadd.10. The method of claim 8 wherein the step of sampling NMR data comprises introducing a time delay in sampling the data, the time delay being selected to provide substantially no phase difference between even and odd parts of the image data..Iaddend..Iadd.11. The method of claim 10 wherein the step of sampling NMR data comprises introducing a time delay in sampling the data, the time delay being selected to provide substantially no first order phase difference between even and odd parts of the image data..Iaddend..Iadd.12. The method of claim 8 wherein the step of processing the NMR data comprises transforming the NMR data into transform data..Iaddend..Iadd.13. The method of claim 12 wherein the step of processing the NMR data comprises determining a function representative of the phase difference between even and odd parts of the transform data..Iaddend..Iadd.14. The method of claim 12 wherein the step of transforming the NMR data comprises taking an inverse Fourier transform of the NMR data..Iaddend..Iadd.15. The method of claim 14 wherein the step of transforming the NMR data comprises taking a two-dimensional inverse Fourier transform of the NMR data..Iaddend..Iadd.16. The method of claim 12 wherein the transform data are denoted by Y(n 1 ,n 2 ) and where: ##EQU22##.Iadd.17. The method of claim 16 further comprising the step of computing the signal energy for each column of the transform data using .Iadd.18. The method of claim 17 further comprising the step of discarding columns   
     
     
        with a signal energy below a predetermined threshold..Iaddend..Iadd.19. The method of claim 16 wherein there is an actual phase difference Δ actual  (n 1 ,n 2 ) between Y even  (n 1 ,n 2 ) and Y odd  (n 1 ,n 2 )..Iaddend..Iadd.20. The method of claim 19 wherein the actual phase difference Δ actual  (n 1 ,n 2 ) is calculated using: Δ actual  (n 1 ,n 2 )=phase (Y even  (n 1 ,n 2 ))-phase (Y odd  (n 1 ,n 2 ))..Iaddend..Iadd.21. The method of claim 19 wherein the step of processing the NMR data comprises using the transform data Y(n 1 ,n 2 ) and the actual phase difference Δ actual  (n 1 ,n 2 ) to calculate the image data, wherein the image data are denoted by x(n 1 ,n 2 )..Iaddend..Iadd.22. The method of claim 21 wherein the step of processing the NMR data comprises estimating a phase difference function Δ(n 1 ,n 2 ) from the actual phase difference Δ actual  (n 1 ,n 2 )..Iaddend..Iadd.23. The method of claim 22 wherein the estimated phase difference function Δ(n 1 ,n 2 ) is determined by (i) calculating a plurality of values of the actual phase difference Δ actual  (n 1 ,n 2 );   (ii) using the plurality of values of the actual phase difference Δ actual  (n 1 ,n 2 ) to find linear least square estimates of α(n 1 ) and β(n 1 ) using ##EQU23## (ii) using the linear least square estimates of α(n 1 ) and β(n 1 ) to find the estimated phase difference function Δ(n 1 ,n 2 ) for 0≦n 2  <N using ##EQU24##   
     
     
       .Iadd.24. The method of claim 22 wherein the step of processing the NMR data comprises using the transform data Y(n 1 ,n 2 ) and the estimated phase difference function Δ(n 1 ,n 2 ) to calculate the image data x(n 1 ,n 2 )..Iaddend..Iadd.25. The method of claim 24 wherein the image data x(n 1 ,n 2 ) are calculated by: (i) using the estimated phase difference function Δ(n 1 ,n 2 ), Y even  (n 1 ,n 2 ), and Y odd  (n 1 ,n 2 ) to find A(n 1 ,n 2 ) and B(n 1 ,n 2 ) using ##EQU25## where ##EQU26## (ii) using the respective magnitude of A(n 1 ,n 2 ) and B(n 1 ,n 2 ) to find x(n 1 ,n 2 ) and x(n 1 ,n 2   
     
     
        +N/2)..Iaddend..Iadd.26.  A method of reducing a ghost from a NMR image, comprising the steps of: (a) taking a two dimensional inverse Fourier transform of a raw NMR signal to obtain a ghosted image Y(n 1 ,n 2 );   (b) identifying ghosting pixels in columns of Y(n 1 ,n 2 ); i.e. n 1  =0 . . . N s  -1,   (c) finding the actual phase difference Δ actual  (n 1 ,n 2 ) for the ghosting pixels   (d) solving the following simultaneous equations to find linear least square estimates of α(n 1 ) and β(n 1 ); ##EQU27## (e) using the linear least square estimates of α(n 1 ) and β(n 1 ) to find an estimated phase difference function Δ(n 1 ,n 2 ) for 0≦n 2  <N using ##EQU28## (f) using the estimated phase difference function Δ(n 1 ,n 2 ) in ##EQU29## to find A(n 1 ,n 2 ) and B(n 1 ,n 2 ) for 0≦n 2  <N, where the dimensions of the image data x(n 1 ,n 2 ) corresponding to the NMR image are N×N s  and where ##EQU30##   
     
     
       .Iadd.27. A method of reducing a ghost from a NMR image, comprising the steps of: (a) taking a two dimensional inverse Fourier transform of a raw NMR signal to obtain a ghosted image Y(n 1 ,n 2 ); (b) calculating Y even  (n 1 ,n 2 ) and Y odd  (n 1 ,n 2 ) using ##EQU31## (c) estimating a phase difference function Δ(n 1 ,n 2 ) between Y even  (n 1 ,n 2 ) and Y odd  (n 1 ,n 2 ); and   (d) using the estimated phase difference function Δ(n 1 ,n 2 ) in ##EQU32## to find A(n 1 ,n 2 ) and B(n 1 ,n 2 ) for 0≦n 2  <N, where the image data corresponding to the NMR image are denoted by x(n 1 ,n 2 ), and where ##EQU33##   
     
     
       .Iadd.28. The method of claim 27 wherein the estimated phase difference function is predetermined..Iaddend..Iadd.29. The method of claim 28 wherein the estimated phase difference function is estimated from at least a portion of a Fourier transform of the NMR signal..Iaddend..Iadd.30. The method of claim 29 wherein the estimated phase difference function is estimated from the ghosted image Y(n 1 ,n 2 )..Iaddend.