Methods for two-dimensional autofocus in high resolution radar systems

• US 7,663,529 B2
• Filed: 08/15/2007
• Issued: 02/16/2010
• Est. Priority Date: 08/15/2006
• Status: Active Grant

• Current Assignee: Maxar Mission Solutions Incorporated (Galileo Parent, Inc.)
• Original Assignee: General Dynamics Advanced Information Systems Incorporated (General Dynamics Corporation)
• Inventors: Batteh, Elizabeth T., Goodman, Ron S., Gabl, Edward F., Ricoy, Mark A., Connell, Scott D.
• Primary Examiner(s): Tarcza; Thomas H
• Assistant Examiner(s): Bythrow; Peter M
• Application Number:
  US11/889,637
• Publication Number:
  US 20080042893A1
• Time in Patent Office: 916 Days
• Field of Search: 342/25.R, 342/25.A, 342/25.B, 342/25.C, 342/25.D, 342/25.E, 342/25.F
• US Class Current: 342/25.00F
• CPC Class Codes: G01S 13/9011 with frequency domain processing of the SAR signals in azimuth G01S13/9005 takes precedence
• CPC Class Codes: G01S 13/9019 Auto-focussing of the SAR signals

Abstract

Provided are two-dimensional autofocus methods in a synthetic aperture radar (SAR) system which include: (1) two-dimensional pulse pair product algorithm including shear PGA, eigenvector phase history (“EPH”), shear PGA/EPH); (2) two-dimensional optimization algorithms including parametric one-dimensional estimate/two-dimensional correction, parametric two dimensional estimate/two-dimensional correction, unconstrained two-dimensional nonparametric and constrained two-dimensional nonparametric methods; (3) a two-dimensional geometry filter algorithm; (4) a two-dimensional prominent point processing algorithm; (5) a one-dimensional phase estimate of higher order two dimensional phase errors; and, (6) a fast SHARP parametric autofocus algorithm.

Claims

1. A synthetic aperture radar (SAR) system for automatically compensating on-axis and off-axis distortions, the system comprising:

an image formation processor (IFP) configured to generate a complex image;

a processor configured to remove the on-axis and off-axis distortions in the complex image based on a two-dimensional autofocus algorithm to generate a corrected image;

a memory configured to store the algorithm and the corrected image; and

an output interface configured to output the corrected image to an external device,

wherein the complex image is defined in two-dimensions with range information in a vertical direction and azimuth information in a horizontal direction.

2. The SAR system according to claim 1, wherein the IFP is coupled with an Ultra-wideband SAR sensor, or other SAR sensor with a substantially large dwell angle or fractional bandwidth.

3. The SAR system according to claim 1, wherein the external device is an image display device and/or a further processing device configured to further process the corrected image.

4. The SAR system according to claim 1, wherein the complex image is divided into a number of partially overlapping two-dimensional strips of a predetermined size.

5. The SAR system according to claim 4, wherein the two-dimensional strips are stacked to form a third dimension.

6. The SAR system according to claim 5, wherein the two-dimensional strips are manipulated to determine two-dimensional phase error.

7. The SAR system according to claim 6, wherein the algorithm is a two-dimensional pulse pair product algorithm comprising a shear phase gradient autofocus (PGA), an eigenvector phase history (EPH) or a PGA/EPH,

wherein the two-dimensional pulse pair product algorithm divides the complex image into subsets, combines the subsets to extract a common phase error and cancels phase from other sources.

8. The SAR system according to claim 7, wherein the two-dimensional pulse pair product algorithm operates on a modified phase history space obtained by multiplying elements in one azimuth bin by complex conjugate of elements in another azimuth bin,

wherein the azimuth bin is a column of the complex image, comprising a set of phase history samples with a same azimuth sample number.

9. The SAR system according to claim 8, wherein the two-dimensional pulse pair product algorithm is performed using a phase gradient of the dispersed phase history information.

10. The SAR system according to claim 9, wherein each of the two-dimensional strips is center shifted and cumulative effects of different linear phases are at least substantially removed.

11. The SAR system according to claim 10, wherein the shear PGA algorithm comprises summing each pulse pair product over all the strips.

12. The SAR system according to claim 11, wherein the shear PGA algorithm is based on the pulse pair or the phase gradient in the azimuth direction and sub-bands in the range direction.

13. The SAR system according to claim 12, wherein the shear PGA algorithm comprises:

creating the pulse pair products along the azimuth directions;

estimating the phase error gradient of each sub-band by summing over the strips;

summing by discrete integration each sub-band in azimuth;

obtaining a final two-dimensional phase error estimate; and

removing an arbitrary range phase error through a one-dimensional range autofocus process.

14. The SAR system according to claim 13, wherein the creating the pulse pair products comprises:

stacking the strips;

center shifting a brightest target;

uncompressing the strips;

computing and applying SNR weighting to the strips; and

multiplying each azimuth bin by the complex conjugate of its neighboring azimuth bin,

wherein the SNR weighting utilizes a SNR ratio defined as a ratio of coherent power to-non coherent power in each of the strips.

15. The SAR system according to claim 7, wherein the EPH algorithm comprises:

stacking image strips;

center-shifting a brightest target;

uncompressing the strips;

computing and applying SNR weighting to the strips;

serializing samples of each of the strips to convert an M×

N matrix into a 1×

MN vector;

calculating an eigenvector corresponding to a first eigenvalue of a MN-dimensional strip covariance matrix; and

converting the 1×

MN vector back into the M×

N matrix,

wherein the SNR weighting utilizes a SNR ratio defined as a ratio of coherent power to-non coherent power in each of the strips.

16. The SAR system according to claim 7, wherein the Shear PGA/EPH algorithm comprises:

stacking image strips;

center-shifting a brightest target;

uncompressing the strips;

computing and applying SNR weighting to the strips;

forming a pulse pair product in the azimuth direction;

serializing samples of each of the strips to convert an M×

N matrix into a 1×

MN vector;

calculating an eigenvector corresponding to a first eigenvalue of a MN-dimensional strip covariance matrix;

converting the 1×

MN vector back into the M×

N matrix; and

removing an arbitrary range phase error through a one-dimensional range autofocus process,

wherein the SNR weighting utilizes a SNR ratio defined as a ratio of coherent power to-non coherent power in each of the strips.

17. A method for automatically compensating on-axis and off-axis distortions in a synthetic aperture radar (SAR) image, the method comprising:

creating a complex image based on information collected by a sensor;

removing the on-axis and off-axis distortions in the complex image based on a two-dimensional autofocus algorithm;

generating a corrected image; and

outputting the corrected image to an external device,

wherein the complex image is defined in two-dimensions with range information in a vertical direction and azimuth dispersed phase history information in a horizontal direction.

18. The method according to claim 17, wherein the sensor is an Ultra-wideband SAR sensor, or other SAR sensor with a substantially large dwell angle or fractional bandwidth.

19. The method according to claim 17, wherein the external device is an image display device and/or a further processing device configured to further process the corrected image.

20. The method according to claim 17, wherein the complex image is divided into a number of partially overlapping two-dimensional strips of a predetermined size.

21. The method according to claim 20, wherein the two-dimensional strips are stacked to form a third dimension.

22. The method according to claim 21, wherein the two-dimensional strips are manipulated to determine two-dimensional phase error.

23. The method according to claim 22, wherein the algorithm is a two-dimensional pulse pair product algorithm comprising a shear phase gradient autofocus (PGA), an eigenvector phase history (EPH) or PGA/EPH,

wherein the two-dimensional pulse pair product algorithm divides the complex image into subsets, combines the subsets to extract a common phase error and cancels phase from other sources.

24. The method according to claim 23, wherein the two-dimensional pulse pair product algorithm operates on a modified phase history space obtained by multiplying elements in one azimuth bin by complex conjugate of elements in another azimuth bin,

wherein the azimuth bin is a column of the complex image, comprising a set of phase history samples with a same azimuth sample number.

25. The method according to claim 24, wherein the two-dimensional pulse pair product method is performed using a phase gradient of the dispersed phase history information.

26. The method according to claim 25, wherein each of the two-dimensional strips is center shifted and cumulative effects of different linear phases are at least substantially removed.

27. The method according to claim 26, wherein the shear PGA algorithm comprises summing each pulse pair product over all the strips.

28. The method according to claim 27, wherein the shear PGA algorithm is based on the pulse pair or the phase gradient in the azimuth direction and sub-bands in the range direction.

29. The method according to claim 28, wherein the shear PGA algorithm comprises:

creating the pulse pair products along the azimuth directions;

estimating the phase error gradient of each sub-band by summing over the strips;

summing by discrete integration each sub-band in azimuth;

obtaining a final two-dimensional phase error estimate; and

removing an arbitrary range phase error through a one-dimensional range autofocus process.

30. The method according to claim 29, wherein the creating the pulse pair products comprises:

stacking image strips;

center shifting a brightest target;

uncompressing the strips;

computing and applying SNR weighting to the strips; and

multiplying each azimuth bin by the complex conjugate of its neighboring azimuth bin,

wherein the SNR weighting utilizes a SNR ratio defined as a ratio of coherent power to non-coherent power in each of the strips.

31. The method according to claim 23, wherein the EPH algorithm comprises:

stacking image strips;

center-shifting a brightest target;

uncompressing the strips;

computing and applying SNR weighting to the strips;

serializing samples of each of the strips to convert an M×

N matrix into a 1×

MN vector;

calculating an eigenvector corresponding to a first eigenvalue of a MN-dimensional strip covariance matrix; and

converting the 1×

MN vector back into the M×

N matrix,

wherein the SNR weighting utilizes a SNR ratio defined as a ratio of coherent power to non-coherent power in each of the strips.

32. The method according to claim 23, wherein the Shear PGA/EPH algorithm comprises:

stacking image strips;

center-shifting a brightest target;

uncompressing the strips;

computing and applying SNR weighting to the strips;

forming a pulse pair product in the azimuth direction;

serializing samples of each of the strips to convert an M×

N matrix into a 1×

MN vector;

calculating an eigenvector corresponding to a first eigenvalue of a MN-dimensional strip covariance matrix;

converting the 1×

MN vector back into the M×

N matrix; and

removing an arbitrary range phase error through a one-dimensional range autofocus process,

wherein the SNR weighting utilizes a SNR ratio defined as a ratio of coherent power to non-coherent power in each of the strips.

33. A computer-readable medium having computer-executable instructions for automatically compensating on-axis and off-axis distortions in a synthetic aperture radar (SAR) image, the computer-executable instructions configured to perform a method comprising:

creating a complex image based on information collected by a sensor;

removing the on-axis and off-axis distortions in the complex image based on a two-dimensional autofocus algorithm;

generating a corrected image; and

outputting the corrected image to an external device,

wherein the complex image is defined in two-dimensions with range information in a vertical direction and azimuth dispersed phase history information in a horizontal direction.

34. The computer-readable medium according to claim 33, wherein the sensor is an Ultra-wideband SAR sensor, or other sensor with a substantially large dwell angle or fractional bandwidth.

35. The computer-readable medium according to claim 34, wherein the external device is an image display device and/or a further processing device configured to further process the corrected image.

36. The computer-readable medium according to claim 35, wherein the image domain is divided into a number of partially overlapping two-dimensional strips of a predetermined size.

37. The computer-readable medium according to claim 36, wherein the two-dimensional strips are stacked to form a third dimension.

38. The computer-readable medium according to claim 37, wherein the two-dimensional strips are manipulated to determine two-dimensional phase error.

39. The computer-readable medium according to claim 38, wherein the algorithm is a two-dimensional pulse pair product algorithm comprising a shear phase gradient autofocus (PGA), an eigenvector phase history (EPH) or a PGA/EPH,

wherein the two-dimensional pulse pair product algorithm divides the complex image into subsets, combines the subsets to extract a common phase error and cancels phase from other sources.

40. The computer-readable medium according to claim 39, wherein the two-dimensional pulse pair product algorithm operates on a modified phase history space obtained by multiplying elements in one azimuth bin by complex conjugate of elements in another azimuth bin,

wherein the azimuth bin is a column of the complex image, comprising a set of phase history samples with a same azimuth sample number.

41. The computer-readable medium according to claim 40, wherein the two-dimensional pulse pair product algorithm is performed on a phase gradient of the dispersed phase history information.

42. The computer-readable medium according to claim 41, wherein each of the two-dimensional strips is center shifted and cumulative effects of different linear phases are at least substantially removed.

43. The computer-readable medium according to claim 42, wherein the shear PGA algorithm comprises summing each pulse pair product over all the strips.

44. The computer-readable medium according to claim 43, wherein the shear PGA algorithm is based on the pulse pair or the phase gradient in the azimuth direction and sub-bands in the range direction.

45. The computer-readable medium according to claim 44, wherein the shear PGA algorithm comprises computer-executable instructions for:

creating the pulse pair products along the azimuth directions;

estimating the phase error gradient of each sub-band by summing over the strips;

summing by discrete integration each sub-band in azimuth;

obtaining a final phase error estimate; and

removing an arbitrary range phase error through a one-dimensional range autofocus process.

46. The computer-readable medium according to claim 45, wherein the creating the pulse pair products comprises computer-executable instructions for:

stacking image strips;

center shifting a brightest target;

uncompressing the strips;

computing and applying SNR weighting to the strips; and

multiplying each azimuth bin by the complex conjugate of its neighboring azimuth bin,

wherein the SNR weighting utilizes a SNR ratio defined as a ratio of coherent power to non-coherent power in each of the strips.

47. The computer-readable medium according to claim 39, wherein the EPH algorithm comprises computer-executable instructions for:

stacking image strips;

center-shifting a brightest target;

uncompressing the strips;

computing and applying SNR weighting to the strips;

serializing samples of each of the strips to convert an M×

N matrix into a 1×

MN vector;

calculating an eigenvector corresponding to a first eigenvalue of a MN-dimensional strip covariance matrix; and

converting the 1×

MN vector back into the M×

N matrix,

wherein the SNR weighting utilizes a SNR ratio defined as a ratio of coherent power to non-coherent power in each of the strips.

48. The computer-readable medium according to claim 39, wherein the Shear PGA/EPH algorithm comprises computer-executable instructions for:

stacking image strips;

center-shifting a brightest target;

uncompressing the strips;

computing and applying SNR weighting to the strips;

forming a pulse pair product in the azimuth direction;

serializing samples of each of the strips to convert an M×

N matrix into a 1×

MN vector;

calculating an eigenvector corresponding to a first eigenvalue of a MN-dimensional strip covariance matrix;

converting the 1 xMN vector back into the M×

N matrix; and

removing an arbitrary range phase error through a one-dimensional range autofocus process,

wherein the SNR weighting utilizes a SNR ratio defined as a ratio of coherent power to non-coherent power in each of the strips.

49. A two-dimensional polar geometry model method used in a synthetic aperture radar (SAR) system for removing a two-dimensional phase error in a complex image, the method comprising:

creating the complex image with a sensor;

modeling the two-dimensional phase error as a combination of a range error which is substantially identical from pulse to pulse and a slow-time error;

fitting an unconstrained two-dimensional estimate to the modeled two-dimensional phase error; and

correcting the two-dimensional phase error to generate a corrected image.

50. The method according to claim 49, wherein the modeling the two-dimensional phase error is solely based on azimuth or range.

51. The method according to claim 49, wherein the modeled two-dimensional phase error is represented as two one-dimensional vectors or two sets of parametric basic vectors.

52. The method according to claim 51, wherein a contrast metric is used to determine whether to use a model fit of original data which maximizes the contrast metric.

53. The method according to claim 52, wherein the contrast metric utilizes a line-power normalization of the complex image, a column-power normalization, or both.

55. The computer-readable medium according to claim 51, wherein the modeling the two-dimensional phase error is solely based on azimuth or range.

54. A computer-readable medium having computer-executable instructions for a two-dimensional polar geometry filter method used in a synthetic aperture radar (SAR) system for removing a two-dimensional phase error in a complex image, the computer-executable instructions configured to perform a method comprising:

creating the complex image with a sensor;

modeling the two-dimensional phase error as a combination of a range error which is substantially identical from pulse to pulse and a slow-time error;

fitting an unconstrained two-dimensional estimate to the modeled two-dimensional phase error; and

correcting the two-dimensional phase error to generate a corrected image.

56. The computer-readable medium according to claim 54, wherein the modeled two-dimensional phase error is represented as two one-dimensional vectors or two sets of parametric basic vectors.

57. The method according to claim 56, wherein a contrast metric is used to determine whether to use a model fit of the original data based on which maximizes the contrast metric.

58. The method according to claim 57, wherein the contrast metric utilizes a line-power normalization of the complex image, a column-power normalization, or both.

59. A two-dimensional prominent point processing method used in a synthetic aperture radar (SAR) system for removing a two-dimensional phase error in a complex image, the method comprising:

generating the complex image with a sensor;

identifying a good quality unfocused isolated point target within the complex image;

cropping a target and an immediate area surrounding the target;

taking a two-dimensional inverse Fast Fourier Transform (FFT) to obtain a phase history; and

taking a two-dimensional phase from the phase history of the cropped target as a measurement of the two-dimensional phase error on the image.

60. The method according to claim 59, the method further comprising estimating separable one-dimensional azimuth and range errors with a single extracted line and a single extracted column, respectively.

61. The method according to claim 60, wherein the two-dimensional prominent point processing method is carried out by a prominent point detector (PPD), utilizes spatially variant apodization to identify mainlobes and selects the mainlobes above an amplitude threshold with sufficient sidelobe symmetry as targets.

62. The method according to claim 61, wherein the two-dimensional prominent point processing combines a phase measured from multiple targets using a weighted average.

63. The method according to claim 61, wherein the two-dimensional prominent point processing fits a two-dimensional Polar Geometry Model to data to obtain a constrained two-dimensional phase error.

64. A computer-readable medium having computer-executable instructions for a two-dimensional prominent point processing method used in a synthetic aperture radar (SAR) system for removing a two-dimensional phase error in a complex image, the computer-executable instructions configured to perform a method comprising:

generating the complex image with a sensor;

identifying a good quality unfocused isolated point target within the complex image;

cropping a target and an immediate area surrounding the target;

taking a two-dimensional inverse Fast Fourier Transform (FFT) to obtain a phase history; and

taking a two-dimensional phase from the phase history of the cropped target as a measurement of the two-dimensional phase error on the image.

65. The computer-readable medium according to claim 64 having the computer-executable instructions configured to perform the method further comprising:

estimating separable one-dimensional azimuth and range errors with a single extracted line and a single extracted column, respectively.

66. The computer-readable medium according to claim 65, wherein the two-dimensional prominent point processing method is carried out by a prominent point detector (PPD), utilizes spatially variant apodization to identify mainlobes and selects the mainlobes above an amplitude threshold with sufficient sidelobe symmetry as targets.

67. The computer-readable medium according to claim 66, wherein an algorithm combines a phase measured from multiple targets using a weighted average.

68. The computer-readable medium according to claim 66, wherein an algorithm fits a two-dimensional Polar Geometry Model to data to obtain a constrained two-dimensional phase error.

69. A method for slow-time error correction with one-dimensional phase estimate of higher order two-dimensional phase errors in an image generated by a synthetic aperture radar, the method comprising:

taking a portion in range of phase history which extends a full azimuth extent of the phase history to generate a coarse resolution range and fine resolution azimuth sub-band image;

focusing the coarse resolution range and fine resolution azimuth sub-band image with a one-dimensional autofocus algorithm;

mapping a resulting one-dimensional phase error estimate sample spacings as a function of angle, and scaling each of the phase estimate sample spacings to a center frequency using the ratio of a center frequency to an original phase history sample frequency; and

applying to the phase history a two-dimensional correction mapping the one-dimensional phase equation φ

(θ

/θ

_max) to a two-dimensional phase equation as

${\mathrm{\varphi <br><br>}}_{2<br><br><br><br>D}<br><br>\left(u,v\right)=\mathrm{\varphi <br><br>}<br><br>\left({\tan }^{-1}<br><br>\left(\frac{u}{v}\right)/{\mathrm{\theta <br><br>}}_{\max }\right)<br><br>\frac{\sqrt{u^2+v^2}}{f_c},$

wherein φ

denotes the phase, θ

_max denotes a maximum dwell angle of a processed aperture with θ

=0°

at center aperture, ƒ

_c denotes a pulse center frequency, u denotes a polar formatted azimuth frequency for a given phase history sample, and v denotes a polar formatted frequency for the given phase history sample.

70. The method according to claim 69, wherein the phase history is divided in range into multiple sub-bands and the one-dimensional phase error estimates from each of the multiple sub-bands are combined to form a final phase error estimate.

71. The method according to claim 70, wherein the multiple sub-bands are overlapping each other.

72. A method for fast-time error correction with one-dimensional phase estimate of higher order two-dimensional phase errors in an image generated by a synthetic aperture radar, the method comprising:

taking a portion in azimuth of phase history data centered on a set theoretical pulse to generate a coarse resolution azimuth and fine resolution range sub-band image;

focusing the coarse resolution azimuth and fine resolution range image with a one-dimensional autofocus algorithm in a range direction;

mapping a phase estimate sample spacings to a function of frequency;

applying to the phase history a two-dimensional correction mapping a one-dimensional phase equation φ

(f) to a two-dimensional phase equation as

${\mathrm{\varphi <br><br>}}_{2<br><br><br><br>D}<br><br>\left(u,v\right)=\mathrm{\varphi <br><br>}<br><br>\left(\sqrt{u^2+v^2}\right),$

wherein φ

denotes the phase, u denotes a polar formatted azimuth frequency for a given phase history sample, and v denotes a polar formatted range frequency for the given phase history sample.

73. The method according to claim 72, wherein the phase history is divided into multiple sub-bands along theoretical pulses at differing dwell angles, and the one-dimensional phase error estimates from each of the multiple sub-bands are combined to form a final phase error estimate.

74. The method according to claim 73, wherein the multiple sub-bands are overlapping each other.

75. A method for slow-time error correction with one-dimensional amplitude estimate of higher order two-dimensional amplitude errors in an image generated by a synthetic aperture radar, the method comprising:

taking a portion in range of phase history which extends a full azimuth extent of the phase history to generate a coarse resolution range and fine resolution azimuth sub-band image;

focusing the coarse resolution range and fine resolution azimuth sub-band image with a one-dimensional amplitude autofocus algorithm;

mapping a resulting one-dimensional amplitude error estimate sample spacings as a function of angle; and

applying to the phase history a two-dimensional correction mapping the one-dimensional amplitude equation α

(θ

/θ

_max) to a two-dimensional amplitude equation as

α

_2D(u,v)=α

(tan<sup>−

1</sup>(u/v)/θ

_max),

wherein α

denotes the amplitude, θ

_max denotes a maximum dwell angle of a processed aperture with θ

=0°

at center aperture, u denotes a polar formatted azimuth frequency for a given phase history sample, and v denotes a polar formatted frequency for the given phase history sample.

76. The method according to claim 75, wherein the phase history is divided in range into multiple sub-bands and the one-dimensional amplitude error estimates from each of the multiple sub-bands are combined to form a final amplitude error estimate.

77. The method according to claim 76, wherein the multiple sub-bands are overlapping each other.

78. A method for fast-time error correction with one-dimensional amplitude estimate of higher order two-dimensional amplitude errors in an image generated by a synthetic aperture radar, the method comprising:

taking a portion in azimuth of phase history data centered on a set theoretical pulse to generate a coarse resolution azimuth and fine resolution range sub-band image;

focusing the coarse resolution azimuth and fine resolution range image with a one-dimensional amplitude autofocus algorithm in a range direction;

mapping a amplitude estimate sample spacings to a function of frequency; and

applying to the phase history a two-dimensional correction mapping a one-dimensional amplitude equation α

(f) to a two-dimensional amplitude equation as

${\mathrm{\alpha <br><br>}}_{2<br><br><br><br>D}<br><br>\left(u,v\right)=\mathrm{\alpha <br><br>}<br><br>\left(\sqrt{u^2+v^2}\right),$

wherein α

denotes the amplitude, u denotes a polar formatted azimuth frequency for a given phase history sample, and v denotes a polar formatted range frequency for the given phase history sample.

79. The method according to claim 78, wherein the phase history is divided into multiple sub-bands along theoretical pulses at differing dwell angles, and the one-dimensional amplitude error estimates from each of the multiple sub-bands are combined to form a final amplitude error estimate.

80. The method according to claim 79, wherein the multiple sub-bands are overlapping each other.

81. A computer-readable medium having computer-executable instructions for slow-time error correction with one-dimensional phase estimate of higher order two-dimensional phase errors in an image generated by a synthetic aperture radar, the computer-executable instructions configured to perform a method comprising:

taking a portion in range of phase history which extends a full azimuth extent of the phase history to generate a coarse resolution range and fine resolution azimuth sub-band image;

focusing the coarse resolution range and fine resolution azimuth sub-band image with a one-dimensional autofocus algorithm;

mapping a resulting one-dimensional phase error estimate sample spacings as a function of angle, and scaling each of the phase estimate sample spacings to a center frequency using the ratio of a center frequency to an original phase history sample frequency; and

applying to the phase history a two-dimensional correction mapping the one-dimensional phase equation φ

(θ

/θ

_max) to a two-dimensional phase equation as

${\mathrm{\varphi <br><br>}}_{2<br><br><br><br>D}<br><br>\left(u,v\right)=\mathrm{\varphi <br><br>}<br><br>\left({\tan }^{-1}<br><br>\left(\frac{u}{v}\right)/{\mathrm{\theta <br><br>}}_{\max }\right)<br><br>\frac{\sqrt{u^2+v^2}}{f_c},$

wherein φ

denotes the phase, θ

_max denotes a maximum dwell angle of a processed aperture with θ

=0°

at center aperture, ƒ

_c denotes a pulse center frequency, u denotes a polar formatted azimuth frequency for a given phase history sample, and v denotes a polar formatted frequency for the given phase history sample.

82. The computer-readable medium according to claim 81, wherein the phase history is divided in range into multiple sub-bands and the one-dimensional phase error estimates from each of the multiple sub-bands are combined to form a final phase error estimate.

83. The computer-readable medium according to claim 82, wherein the multiple sub-bands are overlapping each other.

84. A computer-readable medium having computer-executable instructions for fast-time error correction with one-dimensional phase estimate of higher order two-dimensional phase errors in an image generated by a synthetic aperture radar, the computer-executable instructions configured to perform a method comprising:

taking a portion in azimuth of phase history data centered on a set theoretical pulse to generate a coarse resolution azimuth and fine resolution range sub-band image;

focusing coarse resolution azimuth and fine resolution range image with a one-dimensional autofocus algorithm in a range direction;

mapping a phase estimate sample spacings to a function of frequency; and

applying to the phase history a two-dimensional correction mapping a one-dimensional phase equation φ

(f) to a two-dimensional phase equation as

${\mathrm{\varphi <br><br>}}_{2<br><br><br><br>D}<br><br>\left(u,v\right)=\mathrm{\varphi <br><br>}<br><br>\left(\sqrt{u^2+v^2}\right),$

wherein φ

denotes the phase, u denotes a polar formatted azimuth frequency for a given phase history sample, and v denotes a polar formatted range frequency for the given phase history sample.

85. The computer-readable medium according to claim 84, wherein the phase history is divided into multiple sub-bands along theoretical pulses at differing dwell angles, and the one-dimensional phase error estimates from each of the multiple sub-bands are combined to form a final phase error estimate.

86. The computer-readable medium according to claim 85, wherein the multiple sub-bands are overlapping each other.

87. A computer-readable medium having computer-executable instructions for slow-time error correction with one-dimensional amplitude estimate of higher order two-dimensional amplitude errors in an image generated by a synthetic aperture radar, the computer-executable instructions configured to perform a method comprising:

taking a portion in range of phase history which extends a full azimuth extent of the phase history to generate a coarse resolution range and fine resolution azimuth sub-band image;

focusing the coarse resolution range and fine resolution azimuth sub-band image with a one-dimensional amplitude autofocus algorithm;

mapping a resulting one-dimensional amplitude error estimate sample spacings as a function of angle; and

applying to the phase history a two-dimensional correction mapping the one-dimensional amplitude equation α

(θ

/θ

_max) to a two-dimensional amplitude equation as

α

_2D(u,v)=α

(tan<sup>−

1</sup>(u/v)/θ

_max),

wherein α

denotes the amplitude, θ

_max denotes a maximum dwell angle of a processed aperture with θ

=0°

at center aperture, u denotes a polar formatted azimuth frequency for a given phase history sample, and v denotes a polar formatted frequency for the given phase history sample.

88. The computer-readable medium according to claim 87, wherein the phase history is divided in range into multiple sub-bands and the one-dimensional amplitude error estimates from each of the multiple sub-bands are combined to form a final amplitude error estimate.

89. The computer-readable medium according to claim 88, wherein the multiple sub-bands are overlapping each other.

90. A computer-readable medium having computer-executable instructions for fast-time error correction with one-dimensional amplitude estimate of higher order two-dimensional amplitude errors in an image generated by a synthetic aperture radar, the computer-executable instructions configured to perform a method comprising:

taking a portion in azimuth of phase history data centered on a set theoretical pulse to generate a coarse resolution azimuth and fine resolution range sub-band image;

focusing the coarse resolution azimuth and fine resolution range image with a one-dimensional amplitude autofocus algorithm in a range direction;

mapping a amplitude estimate sample spacings to a function of frequency; and

applying to the phase history a two-dimensional correction mapping a one-dimensional amplitude equation α

(f) to a two-dimensional amplitude equation as

${\mathrm{\alpha <br><br>}}_{2<br><br><br><br>D}<br><br>\left(u,v\right)=\mathrm{\alpha <br><br>}<br><br>\left(\sqrt{u^2+v^2}\right),$

wherein α

denotes the amplitude, u denotes a polar formatted azimuth frequency for a given phase history sample, and v denotes a polar formatted range frequency for the given phase history sample.

91. The computer-readable medium according to claim 90, wherein the phase history is divided into multiple sub-bands along theoretical pulses at differing dwell angles, and the one-dimensional amplitude error estimates from each of the multiple sub-bands are combined to form a final amplitude error estimate.

92. The computer-readable medium according to claim 91, wherein the multiple sub-bands are overlapping each other.

93. A fast SHARP parametric autofocus method used in a synthetic aperture radar (SAR) system for removing a one-dimensional azimuth phase error in an image generated by the SAR system, the method comprising:

choosing a basis function from an orthonormal set;

defining a positive mask and a negative mask based on a sign of the basis function derivative defined as a function of azimuth sample scaled to the interval [−

1 1];

splitting an azimuth dispersed, range compressed phase history aperture into two sub-apertures using the positive mask and the negative mask;

collapsing the two sub-apertures and removing zeros;

multiplying element-by-element each of the collapsed sub-apertures by a complex conjugate of the other sub-aperture;

performing a one-dimensional azimuth fast fourier transform (FFT) to form a complex cross correlate;

detecting the complex cross correlate;

summing cross correlate samples in a range direction and forming a single vector; and

defining an estimated coefficient of the basis function based on a peak location offset from a DC sample,

wherein the DC sample is a 0^th order Fourier Series component.

94. The method according to claim 93, wherein the one dimensional phase error is estimated in the azimuth direction or the range direction by swapping the range and azimuth dimensions.

95. The method according to claim 94, wherein Chebyshev polynomials are used to split the aperture.

96. The method according to claim 94, wherein phase correction is applied and multiple iterations of the method are applied to obtain a final result.

97. A fast SHARP parametric autofocus method used in a synthetic aperture radar (SAR) system for removing a two-dimensional slow-time phase error in an image generated by the SAR system, the method comprising:

choosing a basis function from an orthonormal set;

defining a positive mask and a negative mask based on the sign of the basis function derivative defined as a function of slow-time angle scaled to a range interval from negative one to positive one;

splitting a fully dispersed phase history aperture into two sub-apertures using the positive mask and the negative mask;

performing a one-dimensional range fast fourier transform (FFT) for each sub-aperture;

collapsing the two sub-apertures and removing zeros;

multiplying element-by-element each of the collapsed sub-apertures by a complex conjugate of the other sub-aperture;

performing a one-dimensional azimuth FFT to form a complex cross correlate;

detecting the complex cross correlate;

summing cross correlate samples in a range direction and forming a single vector;

defining an estimated coefficient of the basis function based on a peak location offset from a DC sample;

defining a final two-dimensional phase error using the basis function defined as a function of slow-time angle scaled to the range interval from negative one to positive one;

scaling the final two-dimensional phase error by a ratio of fast-time frequency to center frequency; and

multiplying the scaled final two-dimensional phase error by the estimated coefficient to obtain a final error,

wherein the DC sample is a 0^th order Fourier Series component.

98. The method according to claim 97, wherein Chebyshev polynomials are used to split the aperture.

99. The method according to claim 97, wherein phase correction is applied and multiple iterations of the method are applied to obtain a final result.

100. A fast SHARP parametric autofocus method used in a synthetic aperture radar (SAR) system for removing a two-dimensional fast-time phase error in an image generated by the SAR system, the method comprising:

choosing a basis function from an orthonormal set;

defining a positive mask and a negative mask based on the sign of the basis function derivative defined as a function of fast-time frequency scaled to a range interval from negative one to positive one.

splitting a fully dispersed phase history aperture into two sub-apertures using the positive mask and the negative mask;

performing a one-dimensional range fast fourier transform (FFT) for each sub-aperture;

collapsing the two sub-apertures and removing zeros;

multiplying element-by-element each of the collapsed sub-apertures by a complex conjugate of the other sub-aperture;

performing a one-dimensional range fast fourier transform (FFT) for each sub-aperture;

detecting the complex cross correlate;

summing cross correlate samples in a range direction and forming a single vector;

defining an estimated coefficient of the basis function based on a peak location offset from a DC sample;

defining a final two-dimensional phase error using the basis function defined as a function of fast-time frequency scaled to the range interval from negative one to positive one; and

multiplying the scaled final two-dimensional phase error by the estimated coefficient to obtain a final error,

wherein the DC sample is a 0^th order Fourier Series component.

101. The method according to claim 100, wherein Chebyshev polynomials are used to split the aperture.

102. The method according to claim 100, wherein the phase correction is applied and multiple iterations of the algorithm are applied to obtain the final result.

103. A computer-readable medium having computer-executable instructions for a fast SHARP parametric autofocus method used in a synthetic aperture radar (SAR) system for removing a one-dimensional azimuth phase error in an image generated by the SAR system, the computer-executable instructions configured to perform a method comprising:

choosing a basis function from an orthonormal set;

defining a positive mask and a negative mask based on a sign of the basis function derivative defined as a function of azimuth sample scaled to a range interval from negative one to positive one;

splitting an azimuth dispersed, range compressed phase history aperture into two sub-apertures using the positive mask and the negative mask;

collapsing the two sub-apertures and removing zeros;

multiplying element-by-element each of the collapsed sub-apertures by a complex conjugate of the other sub-aperture;

performing a one-dimensional azimuth fast fourier transform (FFT) to form a complex cross correlate;

detecting the complex cross correlate;

summing cross correlate samples in a range direction and forming a single vector; and

defining an estimated coefficient of the basis function based on a peak location offset from a DC sample,

wherein the DC sample is a 0^th order Fourier Series component.

104. The computer-readable medium according to claim 103, wherein the one dimensional phase error is estimated in the azimuth direction or the range direction by swapping the range and azimuth dimensions.

105. The computer-readable medium according to claim 103, wherein the phase correction is applied and multiple iterations of the algorithm are applied to obtain the final result.

106. The computer-readable medium according to claim 104, wherein Chebyshev polynomials are used to split the aperture.

107. A computer-readable medium having computer-executable instructions for a fast SHARP parametric autofocus algorithm used in a synthetic aperture radar (SAR) system for removing a two-dimensional slow-time phase error in an image generated by the SAR system, the computer-executable instructions configured to perform a method comprising:

choosing a basis function from an orthonormal set;

defining a positive mask and a negative mask based on the sign of the basis function derivative defined as a function of slow-time angle scaled to a range interval from negative one to positive one;

splitting a fully dispersed phase history aperture into two sub-apertures using the positive mask and the negative mask;

performing a one-dimensional range fast fourier transform (FFT) for each sub-aperture;

collapsing the two sub-apertures and removing zeros;

multiplying element-by-element each of the collapsed sub-apertures by a complex conjugate of the other sub-aperture;

performing a one-dimensional azimuth FFT to form a complex cross correlate;

detecting the complex cross correlate;

summing cross correlate samples in a range direction and forming a single vector;

defining an estimated coefficient of the basis function based on a peak location offset from a DC sample;

defining a final two-dimensional phase error using the basis function defined as a function of slow-time angle scaled to the range interval from negative one to positive one; and

scaling the final two-dimensional phase error by a ratio of fast-time frequency to center frequency; and

multiplying the scaled final two-dimensional phase error by the estimated coefficient to obtain a final error,

wherein the DC sample is a 0^th order Fourier Series component.

108. The computer-readable medium according to claim 107, wherein Chebyshev polynomials are used to split the aperture.

109. The computer-readable medium according to claim 107, wherein the phase correction is applied and multiple iterations of the algorithm are applied to obtain the final result.

110. A computer-readable medium having computer-executable instructions for a fast SHARP parametric autofocus algorithm used in a synthetic aperture radar (SAR) system for removing a two-dimensional fast-time phase error in an image generated by the SAR system, the computer-executable instructions configured to perform a method comprising:

choosing a basis function from an orthonormal set;

defining a positive mask and a negative mask based on the sign of the basis function derivative defined as a function of fast-time frequency scaled to a range interval from negative one to positive one;

splitting a fully dispersed phase history aperture into two sub-apertures using the positive mask and the negative mask;

performing a one-dimensional range fast fourier transform (FFT) for each sub-aperture;

collapsing the two sub-apertures and removing zeros;

multiplying element-by-element each of the collapsed sub-apertures by a complex conjugate of the other sub-aperture;

performing a one-dimensional range fast fourier transform (FFT) for each sub-aperture;

detecting the complex cross correlate;

summing cross correlate samples in a range direction and forming a single vector;

defining an estimated coefficient of the basis function based on a peak location offset from a DC sample;

defining a final two-dimensional phase error using the basis function defined as a function of fast-time frequency scaled to the range interval from negative one to positive one;

multiplying the scaled final two-dimensional phase error by the estimated coefficient to obtain a final error,

wherein the DC sample is a 0^th order Fourier Series component.

111. The computer-readable medium according to claim 110, wherein Chebyshev polynomials are used to split the aperture.

112. The computer-readable medium according to claim 110, wherein the phase correction is applied and multiple iterations of the algorithm are applied to obtain the final result.

113. A two-dimensional optimization method used in a synthetic aperture radar, the optimization method comprising:

generating a complex image with a sensor;

defining a relative quality of a focus of the complex image with an optimization metric;

searching for a phase error correction best suited to correct a phase error in the complex image with an algorithm; and

correcting the phase error based on the phase error correction,

wherein the phase error correction optimizes the optimization metric.

114. The method according to claim 113, wherein the algorithm utilizes a parametric form to define the phase error correction.

115. The method according to claim 114, wherein the correcting the phase error comprises optimizing coefficients of phase functions which are n^th order Legendre polynomials.

116. The method according to claim 114, wherein the algorithm utilizes a Golden Section Search.

117. The method according to claim 114, wherein the correcting the phase error comprises:

estimating the phase error using a one-dimensional technique to generate estimated parametric coefficients;

scaling the estimated parametric coefficients such that they are defined over an interval spanned by a full polar formatted dwell angle; and

applying the phase error in two dimensions.

118. The method according to claim 117, wherein the applying the phase error in two dimensions entails applying two full two-dimensional Fast Fourier Transforms.

119. The method according to claim 114, wherein the correcting the phase error comprises:

assuming a parametric phase function to be a function of a slow-time polar angle;

assuming the parametric phase function for any of the slow-time polar angle scales with a fast-time frequency; and

calculating and applying a phase function correction for each samples in two-dimensional phase history.

120. The method according to claim 119, wherein the applying the phase error in two dimensions entails applying two full two-dimensional Fast Fourier Transforms.

121. The method according to claim 114, wherein the correcting the phase error comprises:

assuming a parametric phase function to be a function of a fast-time frequency;

assuming the parametric phase function for any of the fast-time frequency is consistent from pulse to pulse; and

calculating and applying a phase function correction for each samples in two-dimensional phase history.

122. The method according to claim 121, wherein the applying the phase error in two dimensions entails applying two full two-dimensional Fast Fourier Transforms.

123. The method according to claim 113, wherein the phase error correction utilizes a non-parametric technique.

124. The method according to claim 123, wherein the phase error correction utilizes a metric gradient-based optimization for improved efficiency.

125. The method according to claim 124, wherein the phase error correction utilizes a L-BFGS-B search technique.

126. The method according to claim 123 wherein the non-parametric technique is unconstrained or constrained technique.

127. The method according to claim 126, wherein the unconstrained technique produces an independent phase error estimate for each phase history sample in a two-dimensional sample grid.

128. The method according to claim 127, wherein the unconstrained technique optimizes the optimization metric simultaneously over several overlapping segments of the complex image.

129. The method according to claim 126, wherein the constrained technique imposes constraints on two-dimensional phase error estimates.

130. The method according to claim 129, wherein the phase error is assumed to be a two-dimensional slow-time phase error and/or fast-time phase error,

wherein the fast-time phase error is along pulse.

131. The method according to claim 130, wherein the slow-time phase error is constrained to scale in fast-time with frequency.

132. A computer-readable medium having computer-executable instructions for a two-dimensional optimization method used in a synthetic aperture radar, the computer-executable instructions configured to perform a method comprising:

generating a complex image with a sensor;

defining a relative quality of a focus of the complex image with an optimization metric;

searching for a phase error correction best suited to correct a phase error in the complex image with an algorithm; and

correcting the phase error based on the phase error correction,

wherein the phase error correction optimizes the optimization metric.

133. The computer-readable medium according to claim 132, wherein the algorithm utilizes a parametric form to define the phase error correction.

134. The computer-readable medium according to claim 133, wherein the correcting the phase error comprises optimizing coefficients of phase functions which are n^th order Legendre polynomials.

135. The computer-readable medium according to claim 133, wherein the phase error correction algorithm utilizes a Golden Section Search.

136. The computer-readable medium according to claim 133, wherein the correcting the phase error comprises:

estimating the phase error using a one-dimensional technique to generate estimated parametric coefficients;

scaling the estimated parametric coefficients such that they are defined over an interval spanned by a full polar formatted dwell angle; and

applying the phase error in two dimensions.

137. The computer-readable medium according to claim 136, wherein the applying the phase error in two dimensions entails applying two full two-dimensional Fast Fourier Transforms.

138. The computer-readable medium according to claim 132, wherein the correcting the phase error comprises:

assuming a parametric phase function to be a function of a slow-time polar angle;

assuming the parametric phase function for any of the slow-time polar angle scales with a fast-time frequency; and

calculating and applying a phase function correction for each samples in two dimensional phase history.

computer-executable instructions for applying the phase error in two dimensions.

139. The computer-readable medium according to claim 138, wherein the applying the phase error in two dimensions entails applying two full two-dimensional Fast Fourier Transforms.

140. The computer-readable medium according to claim 132, wherein the correcting the phase error comprises:

assuming a parametric phase function to be a function of a fast-time frequency;

assuming the parametric phase function for any of the fast-time frequency is consistent from pulse to pulse; and

calculating and applying a phase function correction for each samples in two dimensional phase history,

computer-executable instructions for applying the phase error in two dimensions.

141. The computer-readable medium according to claim 140, wherein the applying the phase error in two dimensions entails applying two full two-dimensional Fast Fourier Transforms.

142. The computer-readable medium according to claim 132, wherein the phase error correction utilizes a non-parametric technique.

143. The computer-readable medium according to claim 142, wherein the phase error correction utilizes a metric gradient-based optimization for improved efficiency.

144. The computer-readable medium according to claim 143, wherein the phase error correction utilizes a L-BFGS-B search technique.

145. The computer-readable medium according to claim 142, wherein the non-parametric technique is unconstrained or constrained technique.

146. The computer-readable medium according to claim 145, wherein the unconstrained technique produces an independent phase error estimate for each phase history sample in a two-dimensional sample grid.

147. The computer-readable medium according to claim 146, wherein the unconstrained technique optimizes the optimization metric simultaneously over several overlapping segments of the complex image.

148. The computer-readable medium according to claim 145, wherein the constrained technique imposes constraints on two-dimensional phase error estimates.

149. The computer-readable medium according to claim 148, wherein the phase error is assumed to be a two-dimensional slow-time phase error and/or fast-time phase error,

wherein the fast-time phase error is along pulse.