Moving gradient zeugmatography

Abstract

Nuclear magnetic resonance phenomena are employed to generate a two dimensional image of a thin planar slice through a body under investigation. The apparatus herein operates to determine the spin density distribution in a planar slab within the body which typically comprises a biological organism. Each pixel in the resulting image is distinguished by applying time-varying magnetic field gradients so that the frequency history of the spins in each pixel is uniquely distinguishable. Additionally, novel radio frequency excitation means assure selective excitation within the planar slab.

The invention claimed is: 
     
       1. A nuclear magnetic resonance apparatus for determining spin density distribution in a thin planar slab of an object under examination containing nuclear spins, said body being oriented with respect to orthogonal x and y coordinate directions defined therein and also to a z coordinate direction orthogonal to said slab and to said x and y coordinates within said slab, said apparatus comprising: excitation means for selectively exciting said nuclear spins in said slab, said excitation means including means for applying to said object for a predetermined time period an excitation magnetic field having a gradient in the z-axis direction, said gradient being defined by a function G(t) not having a DC component, said excitation means also including means for applying a pulse of RF energy to said object during the time that said excitation magnetic field is applied, said pulse having an envelope defined by a function h 1  (t)=h 1  γG(t)f(K/K.sub. 0)(T 0  /K 0 ) wherein h 1  is a constant, T 0  is a point in time when G(t) is approximately zero, and f is a window function, so that said excited nuclear spins undergo a radiative free induction decay following termination of said excitation and so that nuclear spins in other regions of said object are substantially unexcited;   means for applying to said object a spatial differentiation magnetic field H(x,y,t), during at least a portion of the free induction decay of said excited nuclear spins, said magnetic field having the form H 0  +G 1  (t)x+G 2  (t)y;   means for receiving radiated electromagnetic energy produced by said free induction decay and converting said energy to a time-varying electric signal representative of the magnitude of said energy; and   means for operating on said electrical signal to generate therefrom signals representative of the spin density distribution in said slab.   
     
     
       2. The nuclear magnetic resonance apparatus of claim 1 in which G(t)=G 0  ((T 0   2  -t 2 )/T 0   2 )exp[-(t/T 0 ) 2  /2]. 
     
     
       3. The nuclear magnetic resonance apparatus of claim 1 in which said receiving means comprises a coil disposed about said object. 
     
     
       4. The nuclear magnetic resonance apparatus of claim 1 in which said radio frequency pulse means includes a transmission coil disposed about the object. 
     
     
       5. The nuclear magnetic resonance apparatus of claim 4 in which said radio frequency pulse transmission coil is the same as the means for receiving radiated electromagnetic energy. 
     
     
       6. The nuclear magnetic resonance apparatus of claim 4 in which said radio frequency pulse transmission coil is disposed within and axially perpendicular to a coil operating as the means for receiving the radiated electromagnetic energy. 
     
     
       7. The nuclear magnetic resonance apparatus of claim 1 in which said functions G 1  (t) and G 2  (t) are selected to produce K-space trajectories which are Lissajous figures. 
     
     
       8. The nuclear magnetic resonance apparatus of claim 1 in which said functions G 1  (t) and G 2  (t) are selected to produce K-space trajectories which are rosettes. 
     
     
       9. The nuclear magnetic resonance apparatus of claim 1 in which said functions G 1  (t) and G 2  (t) are selected to produce K-space trajectories which are Archimedian spirals. 
     
     
       10. The nuclear magnetic resonance apparatus of claim 1 in which said functions G 1  (t) and G 2  (t) are selected to produce K-space trajectories which are concentric circles. 
     
     
       11. The nuclear magnetic resonance apparatus of claim 1 in which said spatial differentiation magnetic field H(x,y,t) defines a plurality of K-space trajectories, wherein said trajectories provide a complete and uniform coverage of K-space and wherein the dwell time in each region of K-space is approximately equal so that a detailed image of the entire portion of said object lying in the K-space plane is produced. 
     
     
       12. The nuclear magnetic resonance apparatus of claim 1 in which said spatial differentiation magnetic field H(x,y,t) defines a plurality of K-space trajectories, wherein said K-space trajectories concentrate on and have greater dwell time in selected region of K-space, thus enhancing the image contrast in said selected regions. 
     
     
       13. The nuclear magnetic resonance apparatus of claim 1 wherein the relation of the maximum absolute value G of said function G(t) to the peak amplitude H 1  of said function h 1  (t) is given by the expression (G)(ΔZ)≈H 1 . .Iadd. 
     
     
       14.  A nuclear magnetic resonance apparatus for determining spin-density distribution in a thin planar slab of an object under examination containing nuclear spins, said object being positioned in a static magnetic field, H 0 , and being oriented with respect to orthogonal x- and y-coordinate directions defined therein and also to a z-coordinate direction orthogonal to said slab and to said x and y coordinates within said slab, said apparatus comprising: excitation means for selectively exciting said nuclear spins in said slab so that said excited nuclear spins undergo a radiative free-induction decay thereby to produce an NMR signal in the form of radiated electromagnetic energy following termination of said excitation and so that nuclear spins in other regions of said object are substantially unexcited;   means for applying to said object a spatial differentiation magnetic field H(x,y,t), during at least a portion of the NMR signal produced by said excited nuclear spins, said spatial differentiation magnetic field having the form H 0  +G 1  (t)x+G 2  (t)y, wherein the magnetic-field gradients G 1  (t) and G 2  (t) are selected to be continuously varying with time during the NMR signal so as to produce a plurality of trajectories in K-space as parameter t varies, said trajectories being selectable to have variable uniformity and dwell times in predetermined regions of K-space;   means for receiving the radiated electromagnetic energy associated with said NMR signal and converting said energy to a time-varying electric signal representative of the magnitude of said energy; and   means for operating on said electrical signal to generate therefrom signals representative of the spin-density distribution in said slab. .Iaddend. .Iadd.15. The nuclear magnetic resonance apparatus of claim 14 in which said excitation means comprise:   means for applying to said object for a predetermined time period an excitation magnetic field directed along said z direction, said excitation magnetic field having an intensity given by H 0  +zG(t); and   means for applying a pulse of electromagnetic, radio-frequency energy to said object during the time that said excitation magnetic field is applied, whereby the nuclear spins in said slab are selectively excited to a higher energy state and said nuclear spins in other regions of said   
     
     
        object are substantially unexcited. .Iaddend. .Iadd.16.  The nuclear magnetic resonance apparatus of claim 14 in which said excitation means operates by applying a time-dependent gradient magnetic field in the z direction together with a radio-frequency pulse. .Iaddend. .Iadd.17. The nuclear magnetic resonance apparatus of claim 14 wherein said excitation means includes means for applying to said object for a predetermined time period an excitation magnetic field having a gradient in the z-coordinate direction, said gradient being defined by a function G(t) not having a DC component, said excitation means also including means for applying a pulse of radio-frequency energy during the time that said excitation magnetic field is applied. .Iaddend. .Iadd.18. The nuclear magnetic resonance apparatus of claim 17 wherein   G(t)=G.sub.0 [(T.sub.0.sup.2 -t.sup.2)/T.sub.0.sup.2 ] exp [-(t/T.sub.0).sup.2 /2]     where G 0  is a magnitude constant,   T 0  is a point in time when G(t) is substantially zero, and   t is a time parameter. .Iaddend. .Iadd.19. The nuclear magnetic resonance apparatus of claim 17 wherein said pulse of radio-frequency energy comprises a pulse having an envelope defined by a function   h.sub.1 (t)=h.sub.1 γG(t)f(K(t)/K.sub.0) (T.sub.0 /K.sub.0)       where h 1  is a constant,   γ is the gyromagnetic ration, and   f is a window function in which   K(t)=γ∫.sup.t G(t')dt'. .Iaddend.      .Iadd.20. The nuclear magnetic resonance apparatus of claim 19 wherein the relation of the maximum absolute value G of said function G(t) to the peak amplitude H 1  of said function h 1  (t) is given by the expression     (G) (ΔZ)≈H.sub.1       
     
     
       where ΔZ is the thickness of the excited slab. .Iaddend. .Iadd.21. The nuclear magnetic resonance apparatus of claim 14 in which said receiving means comprises a coil disposed about said object. .Iaddend. .Iadd.22. The nuclear magnetic resonance apparatus of claim 14 in which said excitation means includes a transmission coil disposed about the object. .Iaddend. .Iadd.23. The nuclear magnetic resonance apparatus of claim 22 in which said radio-frequency pulse transmission coil is the same as the means for receiving radiated electromagnetic energy. .Iaddend. .Iadd.24. The nuclear magnetic resonance apparatus of claim 22 in which said radio-frequency pulse transmission coil is disposed within and axially perpendicular to a coil operating as the means for receiving the radiated electromagnetic energy. .Iaddend. .Iadd.25. The nuclear magnetic resonance apparatus of claim 14 wherein said means for applying a spatial differentiation magnetic field includes means for simultaneously applying magnetic-field gradients G 1  (t) and G 2  (t), such that in the sequential applications thereof said gradients produce K-space trajectories which transverse K-space in a substantially uniform pattern, and wherein the dwell time in each region of K-space is approximately equal so that a detailed image of all spatial frequency components, up to a maximum spatial frequency, of said object is produced. .Iaddend. .Iadd.26. The nuclear magnetic resonance apparatus of claim 14 wherein said means for applying a spatial differentiation magnetic field includes means for simultaneously applying magnetic-field gradients G 1  (t) and G 2  (t), such that, in the sequential applications thereof, said gradients produce K-space trajectories which concentrate on and have a greater dwell time in predetermined regions of K-space so as to enhance 
     
     
        predetermined qualities in the reconstructed image. .Iaddend. .Iadd.27. The nuclear magnetic resonance apparatus of claim 14 in which said functions G 1  (t) and G 2  (t) are selected to produce a K-space trajectory which is a Lissajous figure. .Iaddend. .Iadd.28. The nuclear magnetic resonance apparatus of claim 14 in which said functions G 1  (t) and G 2  (t) are selected to produce a K-space trajectory which is a rosette. .Iaddend. .Iadd.29. The nuclear magnetic resonance apparatus of claim 14 in which said functions G 1  (t) and G 2  (t) are selected to produce a K-space trajectory which is an Archimedian spiral. .Iaddend. .Iadd.30. The nuclear magnetic resonance apparatus of claim 14 in which said functions G 1  (t) and G 2  (t) are selected to produce K-space trajectories which are concentric circles. .Iaddend. .Iadd.31. The nuclear magnetic resonance apparatus of claim 14 wherein said means for operating includes means for determining the local nuclear spin density μ R  (x,y) distribution in said slab by evaluating   μ.sup.R (x,y)=∫V(t)f(t)exp|-i[K.sub.1 (t)x+K.sub.2 (t)y+ω.sub.1 (x,y)t]|dt     where V(t) is the voltage induced in a receiver coil sensing the NMR signal,   f(t) is a weighting function,   -i is the square root of -1,   K 1  (t) and K 2  (t) are functions defining a point in K-space, and   ω 1  is the resonant frequency offset at (x,y). .Iaddend.   
     
     
        .Iadd.     A nuclear magnetic resonance apparatus for determining spin-density distribution in a region of an object under examination containing nuclear spins, said region being positioned in a static magnetic field, H 0 , and being oriented with respect to orthogonal x- and y-coordinate directions defined therein and also to a z-coordinate direction orthogonal to said x and y coordinates within said region, said apparatus comprising: excitation means for exciting said nuclear spins in said region so that said excited nuclear spins undergo a radiative free-induction decay thereby to produce an NMR signal in the form of radiated electromagnetic energy following termination of said excitation;   means for applying to said region a spatial differentiation magnetic field H(x,y,t), during at least a portion of the NMR signal produced by said excited nuclear spins, said spatial differentiation magnetic field having the form H 0  +G 1  (t)x+G 2  (t)y, wherein the magnetic field gradients G 1  (t) and G 2  (t) are selected to be continuously varying with time during the NMR signal so as to produce a trajectory in K-space, said trajectory being selectable to have variable uniformity and dwell times in predetermined regions of K-space;   means for receiving the radiated electromagnetic energy associated with said NMR signal and converting said energy to a time-varying electric signal representative of said energy; and   means for operating on said electrical signal to generate therefrom signals representative of the spin-density distribution in said region. .Iaddend.   
     
     
        .Iadd.33.  The nuclear magnetic resonance apparatus of claim 32 in which said excitation means comprise: means for applying for a predetermined time period to said object an excitation magnetic field directed along said z direction said excitation magnetic field having an intensity given by H 0  +zG(t); and   means for applying a pulse of electromagnetic, radio-frequency energy to said object during the time that said excitation magnetic field is applied, whereby the nuclear spins in said region are selectively excited to a higher energy state and said nuclear spins in other regions of said object are substantially unexcited. .Iaddend. .Iadd.34. The nuclear magnetic resonance apparatus of claim 32 in which said excitation means operates by applying a time-dependent gradient magnetic field in the z   
     
     
        direction together with a radio-frequency pulse. .Iaddend. .Iadd.35.  The nuclear magnetic resonance apparatus of claim 32 wherein said excitation means includes means for applying to said object for a predetermined time period an excitation magnetic field having a gradient in the z-coordinate direction, said gradient being defined by a function G(t) not having a DC component, said excitation means also including means for applying a pulse of radio-frequency energy during the time that said excitation magnetic field is applied. .Iaddend. .Iadd.36. The nuclear magnetic resonance apparatus of claim 35 wherein   G(t)=G.sub.0 [(T.sub.0.sup.2 -t.sup.2)/T.sub.0.sup.2 ] exp [-(t/T.sub.0).sup.2 /2]     where G 0  is a magnitude constant,   T 0  is a point in time when G(t) is substantially zero, and   t is a time parameter. .Iaddend. .Iadd.37. The nuclear magnetic resonance apparatus of claim 36 wherein said pulse of radio-frequency energy comprises a pulse having an envelope defined by a function   h.sub.1 (t)=h.sub.1 γG(t)f(K(t)/K.sub.0) (T.sub.0 /K.sub.0)       where h 1  is a constant,   γ is the gyromagnetic ration, and   f is a window function in which K(t)=γ.sub.∫ t  G(t')dt'. .Iaddend. .Iadd.38. The nuclear magnetic resonance apparatus of claim 37 wherein the relation of the maximum absolute value G of said function G(t) to the peak amplitude H 1  of said function h 1  (t) is given by the expression   (G) (ΔZ)≈H.sub.1       where ΔZ is the thickness of the excited region. .Iaddend. .Iadd.39. The nuclear magnetic resonance apparatus of claim 32 wherein said excitation means includes means for exciting said region a plurality of times so as to produce a corresponding plurality of NMR signals, and wherein said means for applying includes means for applying a spatial differentiation magnetic field during at least a portion of each of said plurality of NMR signals such that a K-space trajectory produced for at least one of said NMR signals differs from a trajectory produced for another one of said NMR signals by at least one trajectory parameter. .Iaddend. .Iadd.40. The nuclear magnetic resonance apparatus of claim 39 wherein said trajectory parameter comprises at least one of uniformity and dwell time. .Iaddend. .Iadd.41. The nuclear magnetic resonance apparatus of claim 39 wherein said trajectory parameter is selected such that collectively said K-space trajectories traverse K-space in a substantially uniform pattern. .Iaddend. .Iadd.42. The nuclear magnetic resonance apparatus of claim 39 wherein said trajectory parameter is selected so as to vary the dwell time of said trajectories in said K-space. .Iaddend. .Iadd.43. The nuclear magnetic resonance apparatus of claim 32 in which said functions G 1  (t) and G 2  (t) are selected to produce a K-space trajectory which is a Lissajous figure. .Iaddend. .Iadd.44. The nuclear magnetic resonance apparatus of claim 32 in which said functions G 1  (t) and G 2  (t) are selected to produce a K-space trajectory which is a rosette. .Iaddend. .Iadd.45. The nuclear magnetic resonance apparatus of claim 32 in which said functions G 1  (t) and G 2  (t) are selected to produce a K-space trajectory which is an Archimedian spiral. .Iaddend. .Iadd.46. The nuclear magnetic resonance apparatus of claim 32 in which said functions G 1  (t) and G 2  (t) are selected to produce K-space trajectories which comprises concentric circles. .Iaddend. .Iadd.47. The nuclear magnetic resonance apparatus of claim 32 wherein said means for operating includes means for determining the local nuclear spin density μ R  (x,y) distribution in said region by evaluating   μ.sup.R (x,y)=∫V(t)f(t)exp | -i[K.sub.1 (t)x+K.sub.2 (t)y+ω.sub.1 (x,y)t]|dt       where V(t) is the voltage induced in a receiver coil sensing the NMR signal,   f(t) is a weighting function,   -i is the square root of -1,   K 1  (t) and K 2  (t) are functions defining a point in K-space, and   ω 1  is the resonant frequency offset at (x,y). .Iaddend. .Iadd.48. The nuclear magnetic resonance apparatus of claim 32 wherein said means for applying a spatial differentiation magnetic field includes means for simultaneously applying magnetic-field gradients G 1  (t) and G 2  (t), such that in the sequential applications thereof said spatial differentiation magnetic field produces K-space trajectories which collectively traverse K-space in a substantially uniform pattern, and wherein the dwell time in each region of K-space is approximately equal so that a detailed image of all spatial frequency components, up to a maximum spatial frequency, of said object can be produced. .Iaddend. .Iadd.49. The nuclear magnetic resonance apparatus of claim 32 wherein said means for applying a spatial differentiation magnetic field includes means for simultaneously applying magnetic-field gradients G 1  (t) and G 2  (t), such that, in the sequential applications thereof, said spatial differentiation magnetic field produces K-space trajectories which collectively concentrate on and have a greater dwell time in predetermined regions of K-space so as to enhance predetermined qualities in the reconstructed image. .Iaddend. .Iadd.50. A nuclear magnetic resonance apparatus for determining spin-density distribution in a thin planar slab of an object under examination containing nuclear spins, said object being positioned in a static magnetic field, H 0 , and being oriented with respect to first and second orthogonal directions defined therein, said apparatus comprising:   excitation means for selectively exciting said nuclear spins in said slab so that said excited nuclear spins undergo a radiative free-induction decay thereby to produce an NMR signal in the form of radiated electromagnetic energy following termination of said excitation and so that nuclear spins in other regions of said object are substantially unexcited;   means for applying to said object a spatial differentiation magnetic field H(x,y,t), during at least a portion of the NMR signal produced by said excited nuclear spins, said spatial differentiation magnetic field having the form H 0  +G 1  (t)x+G 2  (t)y, wherein x and y designate, respectively, said first and second directions, and wherein the magnetic-field gradients G 1  (t) and G 2  (t) are selected to be continuously varying with time during the NMR signal so as to produce a trajectory in K-space;   means for receiving the radiated electromagnetic energy associated with said NMR signal and converting said energy to a time-varying electric signal representative of the magnitude of said energy; and   means for operating on said electrical signal to generate therefrom signals representative of the spin-density distribution in said slab. .Iaddend. .Iadd.51. The nuclear magnetic resonance apparatus of claim 50 wherein said means for applying a spatial differentiation magnetic field includes means for simultaneously applying magnetic-field gradients G 1  (t) and G 2  (t), such that in the sequential applications thereof said spatial differentiation magnetic field produces K-space trajectories which collectively traverse K-space in a substantially uniform pattern, and wherein the dwell time in each region of K-space is approximately equal so that a detailed image of all spatial frequency components, up to a maximum spatial frequency, of said object can be produced. .Iaddend. .Iadd.52. The nuclear magnetic resonance apparatus of claim 50 wherein said means for applying a spatial differentiation magnetic field includes means for simultaneously applying magnetic-field gradients G 1  (t) and G 2  (t), such that, in the sequential applications thereof, said spatial differentiation magnetic field produces K-space trajectories which collectively concentrate on and have a greater dwell time in predetermined regions of K-space so as to enhance predetermined qualities in the   
     
     
        reconstructed image. .Iaddend. .Iadd.53.  The nuclear magnetic resonance apparatus of claim 50 wherein said excitation means includes means for exciting said region a plurality of times so as to produce a corresponding plurality of NMR signals, and wherein said means for applying includes means for applying a spatial differentiation magnetic field during at least a portion of each of said plurality of NMR signals such that a K-space trajectory produced for at least one of said NMR signals differs from a trajectory produced for another one of said NMR signals by at least one trajectory parameter. .Iaddend. .Iadd.54. The nuclear magnetic resonance apparatus of claim 53 wherein said trajectory parameter comprises at least one of uniformity and dwell time. .Iaddend. .Iadd.55. The nuclear magnetic resonance apparatus of claim 53 wherein said trajectory parameter is selected such that collectively said K-space trajectories traverse K-space in a substantially uniform pattern. .Iaddend. .Iadd.56. The nuclear magnetic resonance apparatus of claim 53 wherein said trajectory parameter is selected so as to vary the dwell time of said trajectories in said K-space. .Iaddend. .Iadd.57. The nuclear magnetic resonance apparatus of claim 50 in which said functions G 1  (t) and G 2  (t) are selected to produce a K-space trajectory which is a Lissajous figure. .Iaddend. .Iadd.58. The nuclear magnetic resonance apparatus of claim 50 in which said functions G 1  (t) are selected to produce a K-space trajectory which is a rosette. .Iaddend. .Iadd.59. The nuclear magnetic resonance apparatus of claim 50 in which said functions G 1  (t) and G 2  (t) are selected to produce a K-space trajectory which is an Archimedian spiral. .Iaddend. .Iadd.60. The nuclear magnetic resonance apparatus of claim 50 in which said functions G 1  (t) and G 2  (t) are selected to produce K-space trajectories which comprises 
     
     
        concentric circles. .Iaddend. .Iadd.61.  The nuclear magnetic resonance apparatus of claim 50 wherein said means for operating includes means for determining the local nuclear spin density μ R  (x,y) distribution in said slab by evaluating   μ.sup.R (x,y)=∫V(t)f(t)exp |-i[K.sub.1 (t)x+K.sub.2 (t)y+ω.sub.1 (x,y)t]|dt     where V(t) is the voltage induced in a receiver coil sensing the NMR signal,   f(t) is a weighting funcion,   -i is the square root of -1,   K 1  (t) and K 2  (t) are functions defining a point in K-space, and   ω 1  is the resonant frequency offset at (x,y). .Iadd.62. A nuclear magnetic resonance apparatus for determining spin-density distribution in a region of an object under examination containing nuclear spins, said region being positioned in a static magnetic field, H 0 , and being oriented with respect to first and second orthogonal directions defined therein, said apparatus comprising:   excitation means for exciting said nuclear spins in said region so that said excited nuclear spins undergo a radiative free-induction decay thereby to produce an NMR signal in the form of radiated electromagnetic energy following termination of said excitation;   means for applying to said region a spatial differentiation magnetic field H(x,y,t), during at least a portion of the NMR signal produced by said excited nuclear spins, said magnetic field having the form Ho+G 1  (t)x+G 2  (t)y, wherein x and y designate, respectively, said first and second directions, and wherein the magnetic field gradients G 1  (t) and G 2  (t) are selected to be continuously varying with time during the NMR signal so as to produce a trajectory in K-space;   means for receiving the radiated electromagnetic energy associated with said NMR signal and coverting said energy to a time-varying electric signal representative of said energy; and   means for operating on said electrical signal to generate therefrom signals representative of the spin-density distribution in said region. .Iaddend.   
     
     
        .Iadd.63.  The nuclear magnetic resonance apparatus of claim 62 wherein said means for applying a spatial differentiation magnetic includes means for simultaneously applying magnetic-field gradients G 1  (t) and G 2  (t), such that in the sequential applications thereof said spatial differentiation magnetic field produces K-space trajectories which traverse K-space in a substantially uniform pattern, and wherein the dwell time in each region of K-space is approximately equal so that a detailed image of all spatial frequency components, up to a maximum spatial frequency, of said object can be produced. .Iaddend. .Iadd.64. The nuclear magnetic resonance apparatus of claim 62 wherein said means for applying a spatial differentiation magnetic field includes means for simultaneously applying magnetic-field gradients G 1  (t) and G 2  (t), such that, in the sequential applications thereof, said spatial differentiation magnetic field produces K-space trajectories which concentrate on and have a greater dwell time in predetermined regions of K-space so as to enhance predetermined qualities in the reconstructed image. .Iaddend. .Iadd.65. The nuclear magnetic resonance apparatus of claim 62 wherein said excitation means includes means for exciting said region a plurality of times so as to produce a corresponding plurality of NMR signals, and wherein said means for applying includes means for applying a spatial differentiation magnetic field during at least a portion of each of said plurality of NMR signals such that a K-space trajectory produced for at least one of said NMR signals differs from a trajectory produced for another one of said NMR signals by at least one trajectory parameter. .Iaddend. .Iadd.66. The nuclear magnetic resonance apparatus of claim 65 wherein said trajectory parameter comprises at least one of uniformity and dwell time. .Iaddend. .Iadd.67. The nuclear magnetic resonance apparatus of claim 65 wherein said trajectory parameter is selected such that collectively said K-space trajectories traverse K-space in a substantially 
     
     
        uniform patterne. .Iaddend. .Iadd.68.  The nuclear magnetic resonance apparatus of claim 65 wherein said trajectory parameter is selected so as to vary the dwell time of said trajectories in said K-space. .Iaddend. .Iadd.69. The nuclear magnetic resonance apparatus of claim 62 in which said functions G 1  (t) and G 2  (t) are selected to produce a K-space trajectory which is a Lissajous figure. .Iaddend. .Iadd.70. The nuclear magnetic resonance apparatus of claim 62 in which said functions G 1  (t) and G 2  (t) are selected to produce a K-space trajectory which is a rosette. .Iaddend. .Iadd.71. The nuclear magnetic resonance apparatus of claim 62 in which said functions G 1  (t) and G 2  (t) are selected to produce a K-space trajectory which is an Archimedian spiral. .Iaddend. .Iadd.72. The nuclear magnetic resonance apparatus of claim 62 in which said functions G 1  (t) and G 2  (t) are selected to produce K-space trajectories which are concentric circles. .Iaddend. .Iadd.73. The nuclear magnetic resonance apparatus of claim 62 wherein said means for operating includes means for determining the local nuclear spin density μ R  (x,y)distribution in said slab by evaluating   μ.sup.R (x,y)=V(t)f(t)exp | -i[K.sub.1 (t)x+K.sub.2 (t)y+ω.sub.1 (x,y)t]|dt     where V(t) is the voltage induced in a receiver coil sensing the NMR signal,   f(t) is a weighting funcion,   -i is the square root of -1,   K 1  (t) and K 2  (t) are functions defining a point in K-space, and   ω 1  is the resonant frequency offset at (x,y). .Iaddend. .Iadd.74. A nuclear magnetic resonance apparatus for determining spin-density distribution in a region of an object under examination containing nuclear spins, said object being positioned in a static magnetic field, H 0 , and being oriented with respect to first and second orthogonal directions defined therein, said apparatus comprising:   excitation means for applying to said object a time-dependent gradient and a radio frequency pulse, said radio frequency pulse being applied in the presence of said time-dependent gradient, and modulated so as to selectively excite said nuclear spins in said region so that said excited nuclear spins undergo a radiative free-induction decay thereby to produce an NMR signal in the form of radiated electromagnetic energy following termination of said excitation and so that nuclear spins in other regions of said object are substantially unexcited;   means for applying to said object a spatial differentiation magnetic field during at least a portion of the NMR signal produced by said excited nuclear spins so as to encode in said NMR signal information of the spin-density distribution of said nuclear spins in said region;   means for receiving the radiated electromagnetic energy associated with said NMR signal and converting said energy to a time-varying electric signal representative of the magnitude of said energy; and   means for operating on said electrical signal to generate therefrom signals representative of the spin-density distribution in said region. .Iaddend. .Iadd.75. The nuclear magnetic resonance apparatus of claim 74 wherein said first and second orthogonal directions define, respectively, x- and y-coordinate directions, and further including a z-coordinate direction orthogonal to said x and y coordinates, wherein said excitation means comprises:   means for applying to said object for a predetermined time period, an excitation magnetic field directed along said z direction, said excitation magnetic field having an intensity given by H 0  +zG(t); and   means for applying a pulse of electromagnetic, radio-frequency energy to said object during the time that said excitation magnetic field is applied, whereby the nuclear spins in said region are selectively excited to a higher energy state and said nuclear spins in other regions of said   
     
     
        object are substantially unexcited. .Iaddend. .Iadd.76.  The nuclear magnetic resonance apparatus of claim 75 wherein said excitation means includes means for applying to said object for a predetermined time period an excitation magnetic field having a gradient in the z-coordinate direction, said gradient being defined by a function G(t) not having a DC component, said excitation means also including means for applying a pulse of radio-frequency energy during the time that said excitation magnetic field is applied. .Iaddend. .Iadd.77. The nuclear magnetic resonance apparatus of claim 76 wherein   G(t)=G.sub.0 [(T.sub.0.sup.2 -t.sup.2)/T.sub.0.sup.2 ] exp [-(t/T.sub.0).sup.2 /2]     where G 0  is a magnitude constant,   T 0  is a point in time when G(t) is substantially zero, and   t is a time parameter. .Iaddend. .Iadd.78. The nuclear magnetic resonance apparatus of claim 75 wherein said pulse of radio-frequency energy comprises a pulse having an envelope defined by a function   h.sub.1 (t)=h.sub.1 γG(t)f(K(t)/K.sub.0) (T.sub.0 /K.sub.0)       where h 1  is a constant,   γ is the gyromagnetic ration, and   f is a window function in which K(t)= ∫tG(t')dt'. .Iaddend. .Iadd.79. The nuclear magnetic resonance apparatus of claim 78 wherein the relation of the maximum absolute value G of said function G(t) to the peak amplitude H 1  of said function h 1  (t) is given by the expression   (G) (ΔZ)≈H.sub.1       where ΔZ is the thickness of the excited slab. .Iaddend. .Iadd.80. The nuclear magnetic resonance apparatus of claim 78 wherein said means for operating includes means for determining the local nuclear spin density μ R  (x,y) distribution in said region is given by   μ.sup.R (x,y)=V(t)f(t)exp | -i[K.sub.1 (t)x+K.sub.2 (t)y+ω.sub.1 (x,y)t]|dt       where V(t) is the voltage induced in a receiver coil sensing the NMR signal,   f(t) is a weighting function,   -i is the square root of -1,   K 1  (t) and K 2  (t) are functions defining a point in k-space, and   ω 1  is the resonant frequency offset at (x,y). .Iaddend.   
     
     
        .Iadd.     A nuclear magnetic resonance apparatus for determining spin density distribution in a region of an object under examination containing nuclear spins, said region being positioned in a static magnetic field, H 0 , and being oriented with respect to orthogonal x and y coordinate directions defined therein and also to a z coordinate direction orthogonal to said x and y coordinates with said region, said apparatus comprising: excitation means for exciting said nuclear spins in said region so that said excited nuclear spins undergo a radiative free-induction decay thereby to produce an NMR signal in the form of radiated electromagnetic energy following termination of said excitation;   means for applying to said region a spatial differentiation magnetic field H(x,y,t), during at least a portion of the NMR signal produced by said excited nuclear spins, said magnetic field having the form H 0  +G 1  (t) x  +G 2  (t) y , wherein at least one of the magnetic field gradients G 1  (t) and G 2  (t) is chosen to be varying with time during the NMR signal so as to produce a trajectory in K-space, said K-space being the Fourier transform space within which the spatial frequency of the distribution related to the spin density within said region under examination can be specified, and wherein said spatial differentiation magnetic field is chosen such that the frequency history of the spins at each point of said region is distinguishably different from that of every other point;   means for receiving the radiated electromagnetic energy associated with said NMR signal and converting said energy to a time-varying electric signal representative of said energy; and   means for operating on said electrical signal to generate therefrom, based on the frequency history of the spins, signals representative of the spin   
     
     
        density distribution in said region. .Iaddend. .Iadd.82.  The apparatus of claim 81 wherein both gradients G 1  (t) and G 2  (t) are varying with time. .Iaddend.