Design and Analysis of an Euler Transformation Algorithm Applied to Full-Polarimetric ISAR Imagery

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1 Design and Analysis of an Euler Transformation Algorithm Applied to Full-Polarimetric ISAR Imagery Christopher S. Baird Advisor: Robert Giles Submillimeter-Wave Technology Laboratory (STL) Presented in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Physics University of Massachusetts Lowell April 30, 2007

2 Outline I. Introduction to Polarimetry and the Euler Parameters II. Methodology A. Euler Parameter Derivations B. Euler Parameter Ambiguities C. Causes of Euler Parameter Nonpersistence D. Minimizing Euler Parameter Nonpersistence E. Implementation of the full ATR Algorithm III. Results A. Euler Resolution Trends B. Effects of Persistence Optimization on ATR Performance C. Performance of the full ATR algorithm IV. Conclusions

3 I. Introduction to Polarimetry and the Euler Parameters Radar Scattering Matrix, Field Representation Radar Cross Section and Scattering Element: and General Scattering Equation: Scattering Matrix Equation in H-V basis: Scattered Field Propagation Term (Normalized Out) Incident Field Complex Sinclair Scattering Matrix

I. Introduction to Polarimetry and the Euler Parameters Conventional Polarization Basis: H-V Any polarized EM wave has an electric In practice, field

4 I. Introduction to Polarimetry and the Euler Parameters Conventional Polarization Basis: H-V Any polarized EM wave has an electric In practice, field components are field vector that can be broken up into measured one at a time. Here the SVH horizontal and vertical components matrix element is measured.

5 I. Introduction to Polarimetry and the Euler Parameters Two-Dimensional ISAR Imaging

6 I. Introduction to Polarimetry and the Euler Parameters Euler Decomposition PURPOSE: Decompose scattering matrix into parameters that better represent the object's scattering properties METHOD: Transform scattering matrix to the optimalpolarization basis. This is reached when the scattering matrix S is diagonalized. The BSA is used, requiring appropriate conjugations.

7 I. Introduction to Polarimetry and the Euler Parameters Euler Parameter Definitions Unitary Matrix Diagonalized Sinclair Matrix maximum reflectivity orientation angle symmetry angle bounce angle polarizability angle

8 I. Introduction to Polarimetry and the Euler Parameters Meaning of the Euler Parameter

9 I. Introduction to Polarimetry and the Euler Parameters

10 I. Introduction to Polarimetry and the Euler Parameters Euler Parameter Significance and Problems The Euler Parameters represent more fundamental scattering properties of an object than the traditional H-V polarization parameters. Transforming to Euler Parameters should thus improve Automatic Target Recognition (ATR) as well as give more intuitive scattering images. Ambiguities arise in the Euler transformation that limit accuracy. The Euler parameters are also azimuthally nonpersistent. Methods have been developed to remove the ambiguities, improve the azimuthal persistence, and improve ATR using the optimized Euler parameters.

11 Outline I. Introduction to Polarimetry and the Euler Parameters II. Methodology A. Euler Parameter Derivations B. Euler Parameter Ambiguities C. Causes of Euler Parameter Nonpersistence D. Minimizing Euler Parameter Nonpersistence E. Implementation of the full ATR Algorithm III. Results A. Euler Resolution Trends B. Effects of Persistence Optimization on ATR Performance C. Performance of the full ATR algorithm IV. Conclusions

12 II. Methodology Euler Parameter Derivations PURPOSE: Derive the Euler parameters as functions of the known Sinclair matrix METHOD: Invert these Euler definition equations: where Instead of being attempted in this field-matrix representation, switching to the power-matrix representation allows easier inversion:

13 II. Methodology The Power Representation The equations that transform from the Sinclair matrix to the Kennaugh matrix can be found by expanding out both matrix equations and matching up coefficients

14 II. Methodology The Power Representation Before switching to the power matrix, the Sinclair matrix is reduced to 5 meaningful parameters by factoring out an overall phase factor N and using reciprocity to set SHV=SVH. The 5 remaining Sinclair matrix parameters are relabelled a, b, c, d, f: reduce & relabel After matching up coefficients, the Kennaugh power matrix is found in terms of the 5 original Sinclair matrix parameters:

15 II. Methodology The Power Representation To ease notation, the parameters are again relabelled according to Huynen's conventions: The measured Sinclair scattering data is thus represented in succinct, power matrix form and can be expanded when needed into the original parameters. For example: It should be noted that the Kennaugh matrix has 16 elements, but because it was transformed from 5 meaningful parameters, there is much redundant information

16 II. Methodology Euler Parameter Derivation: psi Using the Kennaugh matrix K, the Euler parameters are derived one at at time. Because each Euler parameter is some form of a rotation angle, the dependence of K on each parameter can be rotated out and K is forced to be piecewise diagonal. First the dependence of K on the orientation angle psi is removed by back-rotating: The rotation matrix is found by casting the Euler definition into the power representation:

17 II. Methodology Euler Parameter Derivation: psi The resulting Kennaugh matrix K' is independent of psi and the elements are relabelled: Analysis of the Euler definitions reveals that any Kennaugh matrix that is independent of psi must have H'=0. Forcing this condition on K' yields the solution to psi:

18 II. Methodology Euler Parameter Derivations The same general approach is used to find the solution of the next two Euler parameters. A prime is added to the parameters each time they are relabelled.

19 II. Methodology Euler Parameter Derivations The remaining Kennaugh matrix K is independent of psi, tau, and nu. When K is matched with the definition below, the remaining Euler parameters gamma and m are found.

20 II. Methodology Visual Analysis of Euler Parameters: Slicy

21 II. Methodology Visual Analysis of Euler Parameters: T-72

22 II. Methodology Euler Parameter Ambiguities PURPOSE: Remove Euler parameter ambiguities that degrade the accuracy and usability of the Euler parameters. METHOD: 1. All possible sets of Euler parameters are numerically tabulated. 2. The corresponding Sinclair matrices are calculated for each Euler set using the Euler definitions. 3. Identical Sinclair matrices in the table are grouped and constitute an ambiguity. 4. All ambiguities are shown to be physically meaningless and removable through redefinition.

23 II. Methodology Euler Parameter Ambiguity Example Example: The set of Euler parameters, when substituted into the Euler definition equations, are shown to correspond to the same Sinclair matrix no matter the value of psi: When this Sinclair matrix is encountered in practice, the ambiguity in the Euler transform leads to unstable results: Physically, this Sinclair matrix corresponds to a sphere which has no orientation angle due to its symmetry. This special case can be safely redefined to psi=0

24 II. Methodology Euler Parameter Ambiguity Redefinitions (continued in the paper)

25 II. Methodology Euler Parameter Nonpersistence PURPOSE: Characterize and minimize the Euler parameter azimuthal nonpersistence which degrades Euler parameter accuracy METHOD: Test the Multiple-Scatterer Cell Hypothesis

26 II. Methodology Multiple-Scatterer Cell Hypothesis The azimuthal nonpersistence of the Euler parameters is thought to be caused by interference of the backscattered signals from objects within the same pixel:

27 II. Methodology Predictions of the Multiple-Scatterer Cell Hypothesis 1. The reproducibility error of the angular Euler parameters should diminish for better imaging resolutions. The magnitude parameters should be unaffected. 2. The azimuthal persistence of the angular Euler parameters should improve for better imaging resolutions. The magnitude parameters should be unaffected.

28 II. Methodology Determining Error-vs-Resolution Trends 1. Full 360 radar scattering signatures were obtained twice at several imaging resolutions for three targets - Slicy, the Simulator, and the T-72: 2. The two data sets for each target were formed into ISAR images at each resolution 3. The two image sets for each target at each resolution were compared to find the reproducibility error, which is measured as the Average Percent Difference (APD):

29 II. Methodology Determining Persistence-vs-Resolution Trends 1. The signatures of Slicy, the Simulator, and the T-72 were again used and formed into ISAR images that increment in 0.2 azimuth intervals. 2. Each image in an azimuthal sequence was exactly back-rotated to ensure the target was stationary as the look angle sweeps 3. The persistence of all pixels were measured and averaged

30 II. Methodology The Exact Back-Rotation Method

31 II. Methodology The Persistence-Optimized Euler Parameters The error-vs-resolution and persistence-vs-resolution trends were found to confirm the Multiple-Scatterer Cell Hypothesis (to be shown in the Results section). Therefore, persistence optimization should improve Euler parameter accuracy and target recognition. Persistence optimization was achieved by weighting the reliability of each pixel according to its persistence when measuring the APD s. The effect of persistence optimization was found using a test suite of spatially similar tanks. The tanks were chosen so as to differ mainly in their equipment configurations, thus posing a difficult target recognition test.

32 II. Methodology Testing the Effectiveness of Persistence Optimization Two tanks from the test suite showing the similarity of the tanks chosen in order to create a difficult ATR test T-72 M1 T-72 BK The tanks were obtained as high-quality scaled models through the ERADS program.

33 II. Methodology Testing the Effectiveness of Persistence Optimization Each tank was imaged and compared, and the APD s were formed into probability densities and ROC curves.

34 II. Methodology Implementation of the full ATR Algorithm PURPOSE: Further test the effect of Euler parameters and persistence optimization on ATR using a realistic test environment. METHOD: Create a full ATR algorithm that uses a reference library of pre-rendered ISAR images to match to the unknown target. Again use the test suite of model tanks to test performance.

35 II. Methodology Implementation of the full ATR Algorithm

36 Outline I. Introduction to Polarimetry and the Euler Parameters II. Methodology A. Euler Parameter Derivations B. Euler Parameter Ambiguities C. Causes of Euler Parameter Nonpersistence D. Minimizing Euler Parameter Nonpersistence E. Implementation of the full ATR Algorithm III. Results A. Euler Resolution Trends B. Effects of Persistence Optimization on ATR Performance C. Performance of the full ATR algorithm IV. Conclusions

37 III. Results Error-vs-Resolution Trends T72 The error-vs-resolution trends for Slicy and the Simulator are similar to these T-72 results and match the trends predicted by the Multiple-Scatterer Cell Hypothesis.

38 III. Results Persistence-vs-Resolution Trends - T72 The persistence-vs-resolution trends for Slicy and the Simulator are similar to these T-72 results and match the trends predicted by the Multiple-Scatterer Cell Hypothesis.

39 III. Results Controlled Tests psi Similar results as below were found for all angular Euler parameters

40 III. Results Controlled Tests - m Similar results as below were found for all magnitude parameters

41 III. Results Controlled Tests Conclusions The magnitude parameters generally show less probability curve overlap and thus better target recognition then the angular parameters. Persistence optimization improves ATR performance noticeably yet minutely for the angular parameters Persistence optimization unexpectedly significantly improves ATR performance for the magnitude parameters. This indicates other nonpersistence phenomena besides multiple-scatterer cells that are effected by the optimization. However, further results do not confirm this. The best performing parameter thus far is the persistence-optimized Euler parameter m

42 III. Results Full-ATR Results - Unoptimized

43 III. Results Full-ATR Results Unoptimized (zoomed)

44 III. Results Effect of Persistence Optimization - psi The full-atr test results for psi are similar to all of the angular Euler parameter results

45 III. Results Effect of Persistence Optimization - m The full-atr test results for the m are similar to all of the magnitude parameter results

46 III. Results Full-ATR Tests Conclusions As with the controlled tests, the full-atr algorithm tests showed better target recognition for the magnitude parameters than the angular parameters, with the Euler magnitude parameter m giving the best results As predicted by the Multiple-Scatterer Cell Hypothesis, persistence optimization significantly improves performance for the angular parameters but not for the magnitude parameters.

47 III. Results Full-ATR Score Consolidations The angular parameters perform too poorly to be useful when treated separately. When consolidated, the Euler parameters can improve ATR performance because they specify independent properties of the same scattering object. The consolidated score is defined as the distance in 5-dimensional Euler space between the Euler scores si and some reference point si0. The reference point si0 is found by using a training data set separate from the test data sets, and adjusting the reference point until ATR performance is optimized for the training data.

48 III. Results Full ATR Final Results

49 General Conclusions After ambiguities are removed, scattering objects can be imaged in an intuitive parameter space using the Euler parameters. The presence of multiple scatterers within individual image cells has been shown to be a chief cause of Euler parameter azimuthal nonpersistence and image degradation. The impact of multiple-scatterer-cell degradation was shown to be minimized by optimizing the Euler parameters according to their azimuthal persistence. A full ATR algorithm has been developed that has been shown to better recognize targets than the traditional methods. The best ATR configuration was found to involve the persistence-optimized Euler parameters when consolidated to a score s.

50 End of Presentation (Reference Slides follow)

51 Electromagnetic Scattering Cross Section Unpolarized Cross Section Polarimetric Cross Sections

52 Downrange Imaging

53 Crossrange Imaging Doppler Effect allows crossrange imaging using Inverse Synthetic Aperture Radar (ISAR)

54 Coordinate System Considerations FSA fixes the coordinate BSA fixes the coordinate system on the system on the EM wave. source/detector apparatus. From the wave's viewpoint, the coordinate system flips upon scattering in the BSA, carried out mathematically by a complex conjugation.

55 Coordinate System Considerations Forward Scattering Alignment (FSA) Used in: Optical Scattering Back Scattering Alignment (BSA) Radar Scattering Field Scattering Matrix: Jones Matrix Sinclair Matrix Power Scattering Matrix: Mueller Matrix Kennaugh Matrix Change of Basis: Find U by Solving Eigenvalue Equation:

56 The Power Representation The power scattering representation is defined in terms of the Stokes vector form of an EM wave. Stokes power vector: The power scattering matrix is known as the Mueller matrix M in optics (FSA) and as the Kennaugh matrix K in radar scattering (BSA). The Kennaugh matrix contains that same information as the Sinclair matrix in a different form.

57 The Power Representation The equations that transform from the Sinclair matrix to the Kennaugh matrix can be found by expanding out both matrix equations and matching up coefficients

58 Euler Parameter Derivation: tau The same general approach as for psi is used to find the solution to all of the Euler parameters. The dependence of K' on the symmetry angle tau is removed by back-rotating: The rotation matrix is again found by casting the definition into the power representation:

59 Euler Parameter Derivation: tau The resulting Kennaugh matrix K'' is independent of psi and tau and the elements are relabelled: Again, analysis of the Euler definitions reveals that any Kennaugh matrix that is independent of psi and tau must have E''=0, F''=0, G''=0. Forcing this condition on K'' yields the solution to tau:

60 Euler Parameter Derivation: nu The dependence of K'' on the bounce angle nu is now removed by back-rotating: The rotation matrix is again found by casting the definition into the power representation:

61 Euler Parameter Derivation: nu The resulting Kennaugh matrix K''' is independent of psi, tau, and nu and the elements are relabelled: Again, analysis of the Euler definitions reveals that any Kennaugh matrix that is independent of psi, tau, and nu must have D'''=0. Forcing this condition on K''' yields the solution to nu:

62 Euler Parameter Derivation: m and gamma Casting the Euler definitions for m and gamma into the power representation gives us the form expected for K''': By comparing the expected form of K''' with the actual K''' found in the last slide, the solutions for m and gamma can be found.

63 Euler Parameter Derivation: m and gamma Upon matching up the expected K''' and the actual K''' and solving for m and gamma, the final solutions are derived: Thus the explicit solutions to the 5 Euler parameters have been found in terms of the original known Sinclair matrix elements, although a large amount of intermediary parameters had to be used.

64 Predictions of the Multiple-Scatterer Cell Hypothesis

65 Scale Considerations of Nonpersistence The azimuthal nonpersistence caused by multiple-scatterer cells is on a different order of magnitude than natural nonpersitence. Experimental data of simple shapes confirms this:

66 Determining Persistence-vs-Resolution Trends The persistence of each pixel is found by plotting the pixel's Euler parameter value as a function of azimuthal look angle. The curve is divided into persistent swaths for which the pixel value stays within 15% of its original value.

67 The Exact Back-Rotation Method Traditional method: 2D discrete Fourier transform followed by a rotation Exact method: rotation done and then the Fourier transform

68 The ROC Curve Definition To construct a ROC curve, a decision threshold is scanned across the group of probability density curves. At each possible threshold, the percent of all comparisons below the threshold that are true-positive (TP) and false-positive (FP) identifications is computed.

69 The ROC Curve Definition

70 Azimuth Identification Module Threshold and flatten image so that target details are lost and only rectangular outline of the tank remains Rotate the image until the tank's front is parallel to the image bottom The tank's front (shortest side of the rectangular outline) is found to be parallel when the projection of the image on the x-axis gives the narrowest profile The rotation angle needed to align the tank is outputted as the azimuthal look angle between the radar and target Because the Azimuth Identification module has limited accuracy, several possible azimuths are outputted to be tested separately

71 II. Methodology Azimuth Identification Module

72 Target Registration Module Before two images can be compared, they must be registered (aligned) or the comparison is meaningless. Unlike the controlled ATR tests done previously, the full ATR algorithm registers the images automatically without human intervention. The Registration Module takes the autocorrelation of the two images to find the pixel shift that best aligns both images. After aligned by the autocorrelation, pixel shifts of a few pixels are applied to ensure exact registration. The shift with the lowest APD is considered the best shift.

73 Error-vs-Resolutions Trends Simulated Target Expected Trend

74 Error-vs-Resolution Trends - Slicy

75 Error-vs-Resolution Trends - Simulator

76 Pers.-vs-Resolution Trends Simulated Target Expected Trend

77 Persistence-vs-Resolution Trends - Slicy

78 Persistence-vs-Res. Trends Simulator

79 Controlled Tests - gamma

80 Controlled Tests - tau

81 Controlled Tests - nu

82 Controlled Tests - HH

83 Controlled Tests - HV

84 Controlled Tests - VH

85 Controlled Tests - VV

86 Effect of Persistence Optimization Full-ATR Test Results for the psi, nu

87 Effect of Persistence Optimization Full-ATR Test Results for gamma, tau

88 Effect of Persistence Optimization Full-ATR Test Results for HH, VV (zoomed)

89 Effect of Persistence Optimization Full-ATR Test Results for HV, VH

90 Full-ATR Score Consolidations

Development and assessment of a complete ATR algorithm based on ISAR Euler imagery

Development and assessment of a complete ATR algorithm based on ISAR Euler imagery Baird* a, R. Giles a, W. E. Nixon b a University of Massachusetts Lowell, Submillimeter-Wave Technology Laboratory (STL)