Apparatus and method for imaging with wavefields using inverse scattering techniques

Claims

We claim:

1. A method of producing an image of an object from wavefield energy that has been transmitted into and scattered by the object, said image comprising a high resolution map of the scattering potential at all points within the object, said image being stored in a memory of a central processing unit (CPU), and said method comprising the steps of:

(a) electronically transmitting an electric signal at one or more frequencies and transducing said electric signal at each said frequency into wavefield energy propagated toward said object from one or more of transducer transmitter positions;

(b) electronically processing said electric signal to determine from said one or more transmitter positions an incident field corresponding to said propagated wavefield energy, said incident field being stored in the memory of the CPU in the form of digitized electric signals;

(c) detecting at one or more of transducer receiver positions said wavefield energy transmitted into and scattered by said object;

(d) electronically processing said detected wavefield energy so as to transform said detected wavefield energy into one or more digitized electric signals stored in said memory of said CPU and corresponding to a scattered field detected at said one or more transducer receiver positions;

(e) said CPU setting an initialized estimate of the scattering potential for said object at each said frequency and storing each said scattering potential estimate in said memory;

(f) said CPU performing a convergence step at each said frequency comprising the following steps:

(1) said CPU preparing, using a Green's function, an estimate of the internal field of said object derived from:

((a)) said incident field, and

((b)) said scattering potential estimate, where said estimate of said internal field at each said frequency comprises all orders of scattering;

(2) deriving, using a Green's function, a calculated scattered field from:

(a) said scattering potential estimate, and

(b) said estimate of said internal field;

(3) comparing said scatter field detected at said one or more receiver positions to said calculated scattered field determined by said CPU to derive a comparator;

(4) when said comparator is greater than a preselected tolerance, said CPU determining, using a Green's function, and storing in said memory an updated scattering potential from:

((a)) said estimate of said internal field,

((b)) said calculated scattered field determined by said CPU,

((c)) said scattering potential estimate,

((d)) said scattered field detected at said receiver positions; and

((e)) said CPU utilizing the Jacobian of the calculated scattered field with respect to the scattering potential estimate, said Jacobian utilization being implemented exclusively with shift invariant kernels;

and then setting said scattering potential estimate equal to said updated scattering potential;

(g) repeating said CPU convergence step until said comparator is less than or equal to said preselected tolerance, said CPU thereafter using said updated scattering potential to reconstruct and store said image in said CPU memory.

2. A methods as defined in claim 1 wherein said convergence step is a Gauss-Newton step.

3. A method as defined in claim 2 wherein the Gauss-Newton step is computed using conjugate gradients.

4. A method as defined in claim 1 wherein said convergence step is a Ribiere-Polak step.

5. A method as defined in claim 1 wherein said calculated scattered field at each said one or more transducer receiver positions is propagated from within the object using a propagation matrix.

6. A method as defined in claim 1 wherein the transducing of the transmitted electric signal into wavefield energy is modeled by a transducer transfer function, and the detection of said wavefield energy is modeled by a transducer transfer function.

7. A method as defined in claim 1 wherein said shift invariant kernels are implemented by a fast Fourier transform.

8. A method as defined in claim 1 wherein said step of electronically transmitting said electric signal at one or more frequencies further comprises said wavefield energy propagating both into and through said object.

9. A method as defined in claim 1 wherein said step of electronically transmitting said electric signal at one or more frequencies comprises the steps of:

positioning a transducer array adjacent to said object, said array comprising one or more wavefield transmitters and one or more wavefield receivers;

sending said electric signal at a first frequency to each said transmitter so that each said transmitter will in turn propagate wavefield energy at said first frequency; and

thereafter changing the frequency of said signal and sending said electrical signal at said changed frequency to each said transmitter so as to sequentially propagate wavefield energy from each said transmitter at said changed frequency.

10. A method as defined in claim 1 wherein said step of electronically transmitting an electric signal at one or more frequencies comprises the steps of:

positioning a transducer array adjacent to said object, said array comprising one or more wavefield transmitters and one or more wavefield receivers;

generating said electric signal in the form of a waveform which is characterized by one or more different frequencies; and

sending said generated waveform in turn to each said transmitter so as to propagate wavefield energy at said one or more frequencies from each said transmitter.

11. A method as defined in claims 9 or 10 wherein said transducer array is configured to encircle said object.

12. A method as defined in claim 1 wherein said step of detecting at said one or more transducer-receiver positions said wavefield energy transmitted into and scattered by said object comprises the steps of:

positioning a transducer array adjacent to said object, said array comprising one or more transmitters and one or more receivers; and

after wavefield energy is transmitted from one of said transmitters, sequencing each said receiver so as to detect said scattered wavefield energy at each said receiver in turn.

13. A method as defined in claim 12 wherein said step of electronically processing said detected wavefield energy comprises the steps of:

transducing the wavefield energy detected by each said receiver transducer into a corresponding electric signal;

amplifying said corresponding electric signal; and thereafter processing each said amplified signal so as to generate two signals which correspond to mathematical real and imaginary parts of a representation of each said amplified signal.

14. A method as defined in claim 13 wherein the step of processing said amplified signal from each said receiver transducer so as to generate said signals corresponding to said mathematical real and imaginary parts of a representation of each said amplified signal comprises the steps of:

inputting each said amplified signal detected at each said receiver transducer to first and second multiplier circuits and multiplying each said amplified signal input to said first multiplier circuit by each said electric signal sent to each said transmitter transducer;

shifting by 90.degree. the phase of an electric signal that is the duplicate of each said electric signal input to said first multiplier circuit, and thereafter multiplying each said amplified signal input to said second multiplier circuit by each said electric signal that is shifted by 90.degree.; and

filtering the output of each said multiplier circuit with a low-pass filter circuit and thereafter integrating and digitizing the output of each said low-pass filter circuit.

15. A method as defined in claim 13 wherein said step of processing each said amplified signal from each said receiver transducer so as to generate said signals corresponding to said mathematical real and imaginary parts of a representation of each said amplified signal comprises the steps of:

inputting each said amplified signal to a high speed analog-to-digital converter so as to digitize each said amplified signal; and

inputting each said digitized signal from said high speed analog-to-digital converter into a parallel processor programmed to take the complex fast Fourier transform of each said digitized signal.

16. A method as defined in claim 12 wherein said transducer array is configured to encircle said object.

17. A method as defined in claim 1 wherein said scattering potential estimate is initialized to zero.

18. A method as defined in claim 1 wherein said scattering potential estimate is initialized to an average value determined by an estimated average of density, wavefield speed and wavefield absorption of said object.

19. A method as defined in claim 1 wherein said incident field is used as an initial estimate of said estimate of the internal field of said object.

20. A method as defined in claim 1 wherein said incident field, said scattered field detected at said one or more transducer-receiver positions, each said estimate of the internal field, and each said scattering potential estimate are obtained separately by said CPU at each said frequency sequentially.

21. A method as defined in claim 20 wherein said updated scattering potential is formulated using a plurality of frequency-independent components represented as a vector .GAMMA. multiplied by a frequency-dependent matrix M, and wherein said CPU updates said vector .GAMMA. so as to determine said updated scattering potential.

22. A method as defined in claim 21 wherein said vector .GAMMA. is updated using a conjugate gradient method.

23. A method as defined in claim 1 wherein said updated scattering potential is determined by said CPU using a conjugate gradient method.

24. A method as defined in claim 1 wherein said step of said CPU preparing said estimate of the internal field of said object uses a biconjugate gradient method.

25. A method as defined in claim 1 wherein said step of said CPU preparing said estimate of the internal field of said object uses a biconjugate gradient stabilized method.

26. A method as defined in claim 1 wherein said step of said CPU determining and storing said incident field corresponding to said propagated wavefield energy determines said incident field using a direct measurement means.

27. A method as defined in claim 1 wherein said step of said CPU determining and storing said incident field corresponding to said propagated wavefield energy determines said incident field using a free space acoustic Green's function.

28. A method as defined in claim 1 wherein said step of said CPU determining and storing said incident field corresponding to said propagated wavefield energy determines said incident field using a free space elastic Green's function.

29. A method as defined in claim 1 wherein said step of to said CPU determining and storing said incident field corresponding to said propagated wavefield energy determines said incident field using a free space Biot acoustic Green's function.

30. A method as defined in claim 1 wherein said step of said CPU determining and storing said incident field corresponding to said propagated wavefield energy determines said incident field using a free space Biot elastic Green's function.

31. A method as defined in claim 1 wherein said step of said CPU determining and storing said incident field corresponding to said propagated wavefield energy determines said incident field using a layered non-porous elastic Green's function.

32. A method as defined in claim 1 wherein said step of said CPU determining and storing said incident field corresponding to said propagated wavefield energy determines said incident field using a layered Biot elastic Green's function.

33. A method as defined in claim 1 wherein said step of said CPU determining and storing said incident field corresponding to said propagated wavefield energy determines said incident field using a layered Biot acoustic Green's function.

34. A method as defined in claim 1 wherein said step of said CPU determining and storing said incident field corresponding to said propagated wavefield energy determines said incident field using a layered non-porous acoustic Green's function.

35. A method as defined in claim 1 wherein said step of said CPU determining and storing said incident field corresponding to said propagated wavefield energy determines said incident field using a layered electromagnetic Green's function.

36. A method as defined in claim 1 wherein said step of said CPU determining and storing said incident field corresponding to said propagated wavefield energy determines said incident field using a free space electromagnetic Green's function.

37. A method as defined in claim 1 wherein said convergence step derives said estimate of said internal field of said object, said calculated scattered field, and said updated scattering potential using a free space acoustic Green's function.

38. A method as defined in claim 1 wherein said convergence step derives said estimate of said internal field of said object, said calculated scattered field, and said updated scattering potential using a free space elastic Green's function.

39. A method as defined in claim 1 wherein said convergence step derives said estimate of said internal field of said object, said calculated scattered field, and said updated scattering potential using a free space Biot acoustic Green's function.

40. A method as defined in claim 1 wherein said convergence step derives said estimate of said internal field of said object, said calculated scattered field, and said updated scattering potential using a free space Biot elastic Green's function.

41. A method as defined in claim 1 wherein said convergence step derives said estimate of said internal field of said object, said calculated scattered field, and said updated scattering potential using a layered non-porous elastic Green's function.

42. A method as defined in claim 1 wherein said convergence step derives said estimate of said internal field of said object, said calculated scattered field, and said updated scattering potential using a layered Biot elastic Green's function.

43. A method as defined in claim 1 wherein said convergence step derives said estimate of said internal field of said object, said calculated scattered field, and said updated scattering potential using a layered Biot acoustic Green's function.

44. A method as defined in claim 1 wherein said convergence step derives said estimate of said internal field of said object, said calculated scattered field, and said updated scattering potential using a layered non-porous acoustic Green's function.

45. A method as defined in claim 1 wherein said convergence step derives said estimate of said internal field of said object, said calculated scattered field, and said updated scattering potential using a layered electromagnetic Green's function.

46. A method as defined in claim 1 wherein said convergence step derives said estimate of said internal field of said object, said calculated scattered field, and said updated scattering potential using a free space electromagnetic Green's function.

47. A method as defined in claim 1 wherein said step of determining and storing in said memory said updated scattering potential uses at least a part of a Jacobian matrix which is a function of said estimate of said internal field of said object, said scattering potential estimate, and a Green's function.

48. A method of reconstructing an image of an object using a central processing unit (CPU) programmed to process data derived from wavefield energy that has been transmitted at one or more frequencies and scattered by said object, said method comprising the steps of:

(a) propagating wavefield energy waves, each said wavefield energy wave having one or more frequencies, toward said object from one or more transmitter positions;

(b) electronically storing one or more digitized electronic signals derived by said CPU from said propagated wavefield energy and said one or more transmitter positions, said digitized electronic signals representing an incident field obtained at each said frequency;

(c) detecting at one or more receivers wavefield energy waves, each said wavefield energy wave having one or more frequencies, scattered by said object;

(d) electronically storing one or more digitized electronic signals representing all orders scattering obtained at each said frequency and derived by said CPU from the scattered wavefield energy waves detected at said one or more receivers;

(e) said CPU setting an initialized estimate of the scattering potential for said object at each said frequency, and storing each said scattering potential estimate;

(f) said CPU performing a convergence step at each said frequency comprising the following steps:

(1) said CPU preparing, using a Green's function, an estimate of an internal field at each said frequency and at each scattering point within said object derived from:

((a)) said incident field, and

((b)) said scattering potential estimate, where said estimate of said internal field at each said frequency comprises all orders of scattering;

(2) deriving, using a Green's function, a calculated scattered field from:

(a) said scattering potential estimate, and

(b) said estimate of said internal field;

(3) comparing said scatter field detected at said one or more receiver positions to said calculated scattered field determined by said CPU to derive a comparator;

(4) when said comparator is greater than a preselected tolerance, said CPU determining, using a Green's function, and storingan updated scattering potential from:

((a)) said estimate of said internal field,

((b)) said calculated scattered field determined by said CPU,

((c)) said scattering potential estimate,

((d)) said scattered field detected at said receiver positions;

((e)) said CPU utilizing the Jacobian of the calculated scattered field with respect to the scattering potential estimate, said Jacobian utilization being implemented exclusively with shift invariant kernels;

and then setting said scattering potential estimate equal to said updated scattering potential;

(g) repeating said CPU convergence step until said comparator is less than or equal to said preselected tolerance, said CPU thereafter reconstructing from said updated scattering potential said image of said object; and outputting at an output device a visually perceptible display of said image.

49. A method as defined in claim 48 wherein said convergence step is a Gauss-Newton step.

50. A method as defined in claim 49 wherein the Gauss-Newton step is computed using conjugate gradients.

51. A method as defined in claim 48 wherein said convergence step is a Ribiere-Polak step.

52. A method as defined in claim 48 wherein said calculated scattered field at each said one or more transducer receiver positions is propagated from within the object using a propagation matrix.

53. A method as defined in claim 48 wherein said step of using said one more shift invariant kernels is implemented by a fast Fourier transform.

54. A method as defined in claim 48 wherein said step of electronically transmitting said electric signal at one or more frequencies further comprises said wavefield energy propagating both into and through said object.

55. A method as defined in claim 48 wherein said updated scattering potential is formulated using a plurality of frequency-independent components represented as a vector .GAMMA. multiplied by a frequency-dependent matrix M, and wherein said CPU updates said vector .GAMMA. so as to determine said updated scattering potential.

56. A method as defined in claim 55 wherein said vector .GAMMA. is updated using a conjugate gradient method.

57. A method as defined in claim 48 wherein said step of propagating said one or more wavefield energy waves comprises the step of electronically transmitting an electric signal at one or more frequencies and transducing said electric signal at each said frequency into said wavefield energy waves.

58. A method as defined in claim 57 wherein said step of electronically transmitting said electric signal at one or more frequencies comprises the steps of:

positioning a transducer array adjacent to said object, said array comprising one or more wavefield transmitters and one or more wavefield receivers;

sending said electric signal at a first frequency to each said transmitter so that each said transmitter will in turn propagate wavefield energy at said first frequency; and

thereafter changing the frequency of said signal and sending said electrical signal at said changed frequency to each said transmitter so as to sequentially propagate wavefield energy from each said transmitter at said changed frequency.

59. A method as defined in claim 57 wherein said step of electronically transmitting an electric signal at one or more frequencies comprises the steps of:

positioning a transducer array adjacent to said object, said array comprising one or more wavefield transmitters and one or more wavefield receivers;

generating said electric signal in the form of a waveform which is characterized by one or more different frequencies; and

sending said generated waveform in turn to each said transmitter so as to propagate wavefield energy at said one or more frequencies from each said transmitter.

60. A method as defined in claims 57 or 58 wherein said transducer array is configured to encircle said object.

61. A method as defined in claim 48 wherein said step of detecting at said one or more receivers said wavefield energy waves scattered by said object comprises the steps of:

positioning a transducer array adjacent to said object, said array comprising one or more wavefield receivers;

sequencing each said receiver so as to detect said scattered wavefield energy waves at each said receiver in turn; and

electronically processing said detected wavefield energy waves so as to transform said detected wavefield energy waves into one or more digitized electric signals.

62. A method as defined in claim 61 wherein said step of electronically processing said detected wavefield energy waves comprises the steps of:

transducing said wavefield energy waves detected by each said receiver into a corresponding electric signal;

amplifying said corresponding electric signal; and

thereafter processing each said amplified signal so as to generate two signals which correspond to mathematical real and imaginary parts of a representation of each said amplified signal.

63. A method as defined in claim 62 wherein said step of processing each said amplified signal so as to generate said signals corresponding to said mathematical real and imaginary parts of a representation of said amplified signal comprises the steps of:

inputting each said amplified signal to first and second multiplier circuits and multiplying each said amplified signal input to said first multiplier circuit by each said electric signal sent to each said transmitter transducer;

shifting by 90.degree. the phase of an electric signal that is the duplicate of each said electric signal input to said first multiplier circuit, and thereafter multiplying each said amplified signal input to said second multiplier circuit by each said electric signal that is shifted by 90.degree.; and

filtering the output of each said multiplier circuit with a low-pass filter circuit and thereafter integrating and digitizing the output of each said low-pass filter circuit.

64. A method as defined in claim 62 wherein said step of processing each said amplified signal so as to generate said signals corresponding to said mathematical real and imaginary parts of a representation of each said amplified signal comprises the steps of:

inputting each said amplified signal to a high speed analog-to-digital converter so as to digitize each said amplified signal; and

inputting each said digitized signal from said high speed analog-to-digital converter into a parallel processor programmed to take the complex fast Fourier transform of each said digitized signal.

65. A method as defined in claim 61 wherein said transducer array is configured to encircle said object.

66. A method as defined in claim 48 wherein said step of said CPU deriving and storing said incident field corresponding to said propagated wavefield energy determines said incident field using a direct measurement means.

67. A method as defined in claim 48 wherein said step of said CPU deriving and storing said incident field corresponding to said propagated wavefield energy determines said incident field using a free space acoustic Green's function.

68. A method as defined in claim 48 wherein said step of said CPU deriving and storing said incident field corresponding to said propagated wavefield energy determines said incident field using a free space elastic Green's function.

69. A method as defined in claim 48 wherein said step of said CPU deriving and storing said incident field corresponding to said propagated wavefield energy determines said incident field using a free space Biot acoustic Green's function.

70. A method as defined in claim 48 wherein said step of said CPU deriving and storing said incident field corresponding to said propagated wavefield energy determines said incident field using a free space Biot elastic Green's function.

71. A method as defined in claim 48 wherein said step of said CPU deriving and storing said incident field corresponding to said propagated wavefield energy determines said incident field using a layered non-porous elastic Green's function.

72. A method as defined in claim 48 wherein said step of said CPU deriving and storing said incident field corresponding to said propagated wavefield energy determines said incident field using a free space electromagnetic Green's function.

73. A method as defined in claim 48 wherein said step of said CPU deriving and storing said incident field corresponding to said propagated wavefield energy determines said incident field using a layered electromagnetic Green's function.

74. A method as defined in claim 48 wherein said step of said CPU deriving and storing said incident field corresponding to said propagated wavefield energy determines said incident field using a layered Biot elastic Green's function.

75. A method as defined in claim 48 wherein said step of said CPU deriving and storing said incident field corresponding to said propagated wavefield energy determines said incident field using a layered Biot acoustic Green's function.

76. A method as defined in claim 48 wherein said step of said CPU deriving and storing said incident field corresponding to said propagated wavefield energy determines said incident field using a layered non-porous acoustic Green's function.

77. A method as defined in claim 48 wherein said convergence step derives said estimate of said internal field, said calculated scattered field, and said updated scattering potential using a free space acoustic Green's function.

78. A method as defined in claim 48 wherein said convergence step derives said estimate of said internal field, said calculated scattered field, and said updated scattering potential using a free space elastic Green's function.

79. A method as defined in claim 48 wherein said convergence step derives said estimate of said internal field, said calculated scattered field, and said updated scattering potential using a free space Biot acoustic Green's function.

80. A method as defined in claim 48 wherein said convergence step derives said estimate of said internal field, said calculated scattered field, and said updated scattering potential using a free space Biot elastic Green's function.

81. A method as defined in claim 48 wherein said convergence step derives said estimate of said internal field, said calculated scattered field, and said updated scattering potential using a layered non-porous elastic Green's function.

82. A method as defined in claim 48 wherein said convergence step derives said estimate of said internal field, said calculated scattered field, and said updated scattering potential using a free space electromagnetic Green's function.

83. A method as defined in claim 48 wherein said convergence step derives said estimate of said internal field, said calculated scattered field, and said updated scattering potential using a layered electromagnetic Green's function.

84. A method as defined in claim 48 wherein said convergence step derives said estimate of said internal field, said calculated scattered field, and said updated scattering potential using a layered Biot elastic Green's function.

85. A method as defined in claim 48 wherein said convergence step derives said estimate of said internal field, said calculated scattered field, and said updated scattering potential using a layered Biot acoustic Green's function.

86. A method as defined in claim 48 wherein said convergence step derives said estimate of said internal field, said calculated scattered field, and said updated scattering potential using a layered non-porous acoustic Green's function.

87. A method as defined in claim 48 wherein said step of determining and storing said updated scattering potential uses at least a part of a Jacobian matrix which is a function of said estimate of said internal field, said scattering potential estimate, and a Green's function.

88. A method of producing an image of an object from wavefield energy that has been transmitted into and scattered by the object, said image comprising a high resolution map of the scattering potential at all points within the object, said image being stored in a memory of a central processing unit (CPU), and said method comprising the steps of:

(a) electronically transmitting an electric signal to a laser means, said laser means transducing said electric signal into wavefield energy at one or more frequencies which are propagated toward said object from one or more of transducer transmitter positions;

(b) electronically processing said electric signal to determine an incident field corresponding to said propagated wavefield energy, said incident field being stored in the memory of the CPU in the form of digitized electric signals;

(c) detecting at an intensity measurement means said wavefield energy transmitted into and scattered by said object;

(d) deriving from said wavefield energy detection at said intensity measurement means the amplitude and the phase of said detected wavefield energy;

(e) deriving a detected wavefield energy from the amplitude and the phase;

(f) electronically processing said detected wavefield energy so as to transform said detected wavefield energy into one or more digitized electric signals stored in said memory of said CPU and corresponding to a scattered field detected;

(g) said CPU setting an initialized estimate of the scattering potential for said object at each said frequency and storing each said scattering potential estimate in said memory; and

said CPU computing an updated scattering potential from said incident field and said scattered field detected.

89. The method as defined in claim 88 wherein said updated scattering potential is computed using an inverse scattering method.

90. The method as defined in claim 88 wherein said updated scattering potential is derived by a step comprising:

(a) said C.P.U. performing a convergence step comprising the following steps:

(1) said CPU preparing an estimate of the internal field of said object derived from:

((a)) said incident field, and

((b)) said scattering potential estimate, where said estimate of said internal field at each said frequency comprises all orders of scattering;

(2) deriving a calculated scattered field from:

(a) said scattering potential estimate, and

(b) said estimate of said internal field;

(3) comparing said scatter field detected at said one or more receiver position to said calculated scattered field determined by said CPU to derive a comparator;

(4) when said comparator is greater than a preselected tolerance, said CPU determining and storing in said memory said updated scattering potential from:

((a)) said estimate of said internal field,

((b)) said calculated scattered field determined by said CPU,

((c)) said scattering potential estimate, and

((d)) said scattered field detected at said receiver positions;

and then setting said scattering potential estimate equal to said updated scattering potential;

(b) repeating said CPU convergence step until said comparator is less than or equal to said preselected tolerance, said CPU thereafter using said updated scattering potential to reconstruct and store said image in said CPU memory.

91. A method as defined in claim 90 wherein said calculated scattered field at each said one or more transducer receiver positions is propagated from within the object using a propagation matrix.

92. A method as defined in claim 90 wherein said convergence step is a Gauss-Newton step.

93. A method as defined in claim 92 wherein the Gauss-Newton step is computed using conjugate gradients.

94. A method as defined in claim 90 wherein said convergence step is a Ribiere-Polak step.

95. The method as defined in claim 88 wherein said amplitude and said phase are derived by an interferometer.

96. The method as defined in claim 95 wherein said interferometer is a Mach-Zehnder interferometer.

97. The method as defined in claim 88 wherein said intensity measurement means is an electronic camera having one or more intensity measurement components.

98. The method as defined in claim 88 wherein said laser means is an X-ray laser.

99. The method as defined in claim 88 wherein said laser means and said intensity measurement means are operational components of an inverse scattering microscope.

100. A method as defined in claim 90 wherein said convergence step derives said estimate of said internal field of said object using a Green's function.

101. A method as defined in claim 90 wherein said convergence step derives said estimate of said calculated scattered field using a Green's function.

102. A method as defined in claim 90 wherein said convergence step derives said updated scattering potential uses a Green's function.

103. A method as defined in claim 100 or 101 or 102 wherein said Green's function is implemented with one or more shift invariant kernels.

104. A method as defined in claim 103 wherein said shift invariant kernels are implemented by a fast Fourier transform.

105. A method as defined in claim 90 wherein convergence step derives one or more of said incident field, said estimate of said internal field, said calculated scattered field, and said updated scattering potential using a free space electromagnetic Green's function.

106. A method as defined in claim 90 wherein convergence step derives one or more of said incident field, said estimate of said internal field, said calculated scattered field, and said updated scattering potential using a layered electromagnetic Green's function.

107. A method of producing an image of an object from wavefield energy that has been transmitted into and scattered by the object, said image comprising a high resolution map of the scattering potential at all points within the object, said image being stored in a memory of a central processing unit (CPU), and said method comprising the steps of:

(a) electronically transmitting an electric signal at one or more frequencies and transducing said electric signal at each said frequency into wavefield energy propagated toward said object from one or more of antennae transmitter positions;

(b) electronically processing said electric signal to determine an incident field corresponding to said propagated wavefield energy, said incident field being stored in the memory of the CPU in the form of digitized electric signals;

(c) detecting at one or more of antennae receiver positions said wavefield energy transmitted into and scattered by said object;

(d) electronically processing said detected wavefield energy so as to transform said detected wavefield energy into one or more digitized electric signals stored in said memory of said CPU and corresponding to a scattered field detected at said one or more antennae receiver positions;

(e) said CPU setting an initialized estimate of the scattering potential for said object at each said frequency and storing each said scattering potential estimate in said memory: and

(f) said CPU computing an updated scattering potential from said incident field and said scattered field detected.

108. The method as defined in claim 107 wherein said updated scattering potential is derived by a step comprising:

(a) said CPU performing a convergence step comprising the following steps:

(1) said CPU preparing an estimate of the internal field of said object derived from:

((a)) said incident field, and

((b)) said scattering potential estimate, where said estimate of said internal field at each said frequency comprises all orders of scattering;

(2) deriving a calculated scattered field from:

(a) said scattering potential estimate, and

(b) said estimate of said internal field;

(3) comparing said scatter field detected at said one or more receiver positions to said calculated scattered field determined by said CPU to derive a comparator;

(4) when said comparator is greater than a preselected tolerance, said CPU determining and storing in said memory said updated scattering potential from:

((a)) said estimate of said internal field,

((b)) said calculated scattered field determined by said CPU,

((c)) said scattering potential estimate, and

((d)) said scattered field detected at said receiver positions;

and then setting said scattering potential estimate equal to said updated scattering potential;

(b) repeating said CPU convergence step until said comparator is less than or equal to said preselected tolerance, said CPU thereafter using said updated scattering potential to reconstruct and store said image in said memory.

109. A method as defined in claim 108 wherein said convergence step is a Gauss-Newton step.

110. A method as defined in claim 109 wherein the Gauss-Newton step is computed using conjugate gradients.

111. A method as defined in claim 108 wherein said convergence step is a Ribiere-Polak step.

112. The method as defined in claim 107 wherein said updated scattering potential is computed using an inverse scattering method.

113. The method as defined in claim 107 wherein said wavefield energy is electromagnetic radiation.

114. A method as defined in claim 108 wherein said convergence step derives one or more of said incident field, said estimate of said internal field, said calculated scattered field, and said updated scattering potential using a free space electromagnetic Green's function.

115. A method as defined in claim 108 wherein said convergence step derives one or more of said incident field, said estimate of said internal field, said calculated scattered field, and said updated scattering potential using a layered electromagnetic Green's function.

116. A method as defined in claim 108 wherein said convergence step is a Gauss-Newton step.

117. A method as defined in claim 116 wherein the Gauss-Newton step is computed using conjugate gradients.

118. A method as defined in claim 108 wherein said convergence step is a Ribiere-Polak step.

119. A method as defined in claim 108 wherein said calculated scattered field at each said one or more transducer receiver positions is propagated from within the object using a propagation matrix.

120. A method of producing an image of an object from wavefield energy that has been transmitted into and scattered by the object, said image comprising a high resolution map of the scattering potential at all points within the object, said image being stored in a memory of a central processing unit (CPU), and said method comprising the steps of:

(a) electronically transmitting an electric signal at one or more frequencies and transducing said electric signal at each said frequency into wavefield energy propagated toward said object from one or more of transducer transmitter positions;

(b) electronically processing said electric signal to determine from said one or more transmitter positions an incident field corresponding to said propagated wavefield energy, said incident field being stored in the memory of the CPU in the form of transmission stored signals;

(c) detecting at one or more of transducer receiver positions said wavefield energy transmitted into and scattered by said object;

(d) electronically processing said detected wavefield energy so as to transform said detected wavefield energy into one or more reception stored signals stored in said memory of said CPU and corresponding to a scattered field detected at said one or more transducer receiver positions;

(e) said CPU setting, an initialized estimate of the scattering potential for said object at each said frequency and storing each said scattering potential estimate in said memory;

(f) said CPU performing a convergence step at each said frequency comprising the following steps:

(1) said CPU preparing, using a Green's function, an estimate of the internal field of said object derived from:

((a)) said incident field, and

((b)) said scattering potential estimate, where said estimate of