SAR Imaging of Dynamic Scenes IPAM SAR Workshop

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1 SAR Imaging of Dynamic Scenes IPAM SAR Workshop Brett Borden, Naval Postgraduate School Margaret Cheney, Rensselaer Polytechnic Institute 7 February, 2012

2 Introduction Need All weather, day/night, sensor system Situational awareness Extend moving target indicator radar (MTI) to scene reconstruction Problem Synthetic aperture radar images assume rigid scenes Moving image elements are mis-located, distorted, or not imaged at all

3 Introduction Need All weather, day/night, sensor system Situational awareness Extend moving target indicator radar (MTI) to scene reconstruction Problem Synthetic aperture radar images assume rigid scenes Moving image elements are mis-located, distorted, or not imaged at all

4 Introduction Need All weather, day/night, sensor system Situational awareness Extend moving target indicator radar (MTI) to scene reconstruction Problem Synthetic aperture radar images assume rigid scenes Moving image elements are mis-located, distorted, or not imaged at all

5 Introduction Need All weather, day/night, sensor system Situational awareness Extend moving target indicator radar (MTI) to scene reconstruction Problem Synthetic aperture radar images assume rigid scenes Moving image elements are mis-located, distorted, or not imaged at all

6 Introduction Need All weather, day/night, sensor system Situational awareness Extend moving target indicator radar (MTI) to scene reconstruction Problem Synthetic aperture radar images assume rigid scenes Moving image elements are mis-located, distorted, or not imaged at all

7 Radar pulse illumination (high range resolution) Spherical pulse radiates at speed c Echo from all reflectors in arc returns to radar at the same time Radar collects energy as function of time: Ä ä E(t) ρ(r)b(r)δ t 2 h 2 + d 2 /c da

8 Radar pulse illumination (high range resolution) Spherical pulse radiates at speed c Echo from all reflectors in arc returns to radar at the same time Radar collects energy as function of time: Ä ä E(t) ρ(r)b(r)δ t 2 h 2 + d 2 /c da

9 Radar pulse illumination (high range resolution) Spherical pulse radiates at speed c Echo from all reflectors in arc returns to radar at the same time Radar collects energy as function of time: Ä ä E(t) ρ(r)b(r)δ t 2 h 2 + d 2 /c da

10 Image from stationary radar A (stationary) radar has no knowledge of angular position within the beam Best image is arcs painted by range intensity values

11 Spotlight SAR

12 SAR imaging of stationary scenes

13 SAR imaging of stationary scenes

14 SAR imaging of stationary scenes

15 SAR imaging of dynamic scenes

16 SAR imaging of dynamic scenes

17 SAR imaging of dynamic scenes

18 Examples Train off the track Ship off its wake

19 Example Image from Sandia National Laboratories

20 Past approaches Don t fix it, exploit it! Phase history analysis Sub-aperture images to isolate motion Ground moving target indicator (GMTI), Space-time adaptive processing (STAP), etc...

21 Past approaches Don t fix it, exploit it! Phase history analysis Sub-aperture images to isolate motion Ground moving target indicator (GMTI), Space-time adaptive processing (STAP), etc...

22 Past approaches Don t fix it, exploit it! Phase history analysis Sub-aperture images to isolate motion Ground moving target indicator (GMTI), Space-time adaptive processing (STAP), etc...

23 Past approaches Don t fix it, exploit it! Phase history analysis Sub-aperture images to isolate motion Ground moving target indicator (GMTI), Space-time adaptive processing (STAP), etc...

24 The radar problem: What does radar data look like? The echo signal at radar receiver s rec(t) = s scatt(t) + n(t) EM radiation decays as R 1, energy as R 2, and radar echo energy as R 4. Signal energy is proportional to field energy. Echo signals compete with system noise n(t) (and are often swamped by it). Raw radar data is usually the result of significant signal processing.

25 The radar problem: What does radar data look like? The echo signal at radar receiver s rec(t) = s scatt(t) + n(t) EM radiation decays as R 1, energy as R 2, and radar echo energy as R 4. Signal energy is proportional to field energy. Echo signals compete with system noise n(t) (and are often swamped by it). Raw radar data is usually the result of significant signal processing.

26 The radar problem: What does radar data look like? The echo signal at radar receiver s rec(t) = s scatt(t) + n(t) EM radiation decays as R 1, energy as R 2, and radar echo energy as R 4. Signal energy is proportional to field energy. Echo signals compete with system noise n(t) (and are often swamped by it). Raw radar data is usually the result of significant signal processing.

27 The radar problem: What does radar data look like? The echo signal at radar receiver s rec(t) = s scatt(t) + n(t) EM radiation decays as R 1, energy as R 2, and radar echo energy as R 4. Signal energy is proportional to field energy. Echo signals compete with system noise n(t) (and are often swamped by it). Raw radar data is usually the result of significant signal processing.

28 Correlation reception Usually, the received signal is compared to the ideal echo signal model s scatt(t) = ρ(τ, ν)s inc(t τ)e iν(t τ) dτ dν where ρ(τ, ν) is the reflectivity of a point isotropic scatterer at distance R = cτ/2 and radial velocity Ṙ = νλ/2. Output of correlation receiver is η(τ, ν) = s rec(t)s inc(t τ)e iν(t τ) dt Local maxima of η(τ, ν) 2 define target s range and radial velocity. The parameters τ and ν define a two-dimensional search space appropriate to single pulses.

29 Correlation reception Usually, the received signal is compared to the ideal echo signal model s scatt(t) = ρ(τ, ν)s inc(t τ)e iν(t τ) dτ dν where ρ(τ, ν) is the reflectivity of a point isotropic scatterer at distance R = cτ/2 and radial velocity Ṙ = νλ/2. Output of correlation receiver is η(τ, ν) = s rec(t)s inc(t τ)e iν(t τ) dt Local maxima of η(τ, ν) 2 define target s range and radial velocity. The parameters τ and ν define a two-dimensional search space appropriate to single pulses.

30 Correlation reception Usually, the received signal is compared to the ideal echo signal model s scatt(t) = ρ(τ, ν)s inc(t τ)e iν(t τ) dτ dν where ρ(τ, ν) is the reflectivity of a point isotropic scatterer at distance R = cτ/2 and radial velocity Ṙ = νλ/2. Output of correlation receiver is η(τ, ν) = s rec(t)s inc(t τ)e iν(t τ) dt Local maxima of η(τ, ν) 2 define target s range and radial velocity. The parameters τ and ν define a two-dimensional search space appropriate to single pulses.

31 Correlation reception Usually, the received signal is compared to the ideal echo signal model s scatt(t) = ρ(τ, ν)s inc(t τ)e iν(t τ) dτ dν where ρ(τ, ν) is the reflectivity of a point isotropic scatterer at distance R = cτ/2 and radial velocity Ṙ = νλ/2. Output of correlation receiver is η(τ, ν) = s rec(t)s inc(t τ)e iν(t τ) dt Local maxima of η(τ, ν) 2 define target s range and radial velocity. The parameters τ and ν define a two-dimensional search space appropriate to single pulses.

32 Ambiguity function If the ideal scattering model is the actual model, the substitution for s rec(t) = s scatt(t) + n(t) yields η(τ, ν) = ρ(τ, ν )χ(τ τ ; ν ν ) dτ dν + correlation noise }{{} small (up to a phase factor in the integrand) The kernel here is the radar ambiguity function: χ(τ; ν) = s inc(t 1 2 τ)s inc(t τ)eiνt dt χ(τ; ν) characterizes the radar s ability to estimate R and Ṙ Radar data" usually means η(τ, ν) = ρ(τ, ν )χ(τ τ ; ν ν ) dτ dν

33 Ambiguity function If the ideal scattering model is the actual model, the substitution for s rec(t) = s scatt(t) + n(t) yields η(τ, ν) = ρ(τ, ν )χ(τ τ ; ν ν ) dτ dν + correlation noise }{{} small (up to a phase factor in the integrand) The kernel here is the radar ambiguity function: χ(τ; ν) = s inc(t 1 2 τ)s inc(t τ)eiνt dt χ(τ; ν) characterizes the radar s ability to estimate R and Ṙ Radar data" usually means η(τ, ν) = ρ(τ, ν )χ(τ τ ; ν ν ) dτ dν

34 Ambiguity function If the ideal scattering model is the actual model, the substitution for s rec(t) = s scatt(t) + n(t) yields η(τ, ν) = ρ(τ, ν )χ(τ τ ; ν ν ) dτ dν + correlation noise }{{} small (up to a phase factor in the integrand) The kernel here is the radar ambiguity function: χ(τ; ν) = s inc(t 1 2 τ)s inc(t τ)eiνt dt χ(τ; ν) characterizes the radar s ability to estimate R and Ṙ Radar data" usually means η(τ, ν) = ρ(τ, ν )χ(τ τ ; ν ν ) dτ dν

35 Ambiguity function If the ideal scattering model is the actual model, the substitution for s rec(t) = s scatt(t) + n(t) yields η(τ, ν) = ρ(τ, ν )χ(τ τ ; ν ν ) dτ dν + correlation noise }{{} small (up to a phase factor in the integrand) The kernel here is the radar ambiguity function: χ(τ; ν) = s inc(t 1 2 τ)s inc(t τ)eiνt dt χ(τ; ν) characterizes the radar s ability to estimate R and Ṙ Radar data" usually means η(τ, ν) = ρ(τ, ν )χ(τ τ ; ν ν ) dτ dν

36 SAR imaging Coordinate change from radar-centric to geo-centric coordinates r. If γ(θ) denotes radar flight path, then d γ(θ) r χ(τ; ν) χ θ (τ(r); ν(r)) η(τ, ν) η θ (τ(r), ν(r)) = ρ(r )χ θ (2(r r )/c; 0) dx dy The image is formed as I(r) = η θ (τ(r), ν(r)) dθ = ρ(r )χ θ (2ˆd (r r )/c; 0) dx dy dθ

37 SAR imaging Coordinate change from radar-centric to geo-centric coordinates r. If γ(θ) denotes radar flight path, then d γ(θ) r χ(τ; ν) χ θ (τ(r); ν(r)) η(τ, ν) η θ (τ(r), ν(r)) = ρ(r )χ θ (2(r r )/c; 0) dx dy The image is formed as I(r) = η θ (τ(r), ν(r)) dθ = ρ(r )χ θ (2ˆd (r r )/c; 0) dx dy dθ

38 Imaging via filtered adjoint SAR correlates echo field with model at each step of the data collection process. What if we wait until the end? Point scattering model: s (θ) scatt (t) = s(θ) inc (t τ θ)e iν(t τ θ) where τ θ,r = 2 γ(θ) r /c If s (θ) rec (t) denotes scattered field measurements along γ(θ) then I(r) = s (θ) scatt (t )s (θ) rec (t ) dt dθ = = ρ(r )χ θ (2ˆd (r r )/c, 0) dx dy dθ (same result)

39 Imaging via filtered adjoint SAR correlates echo field with model at each step of the data collection process. What if we wait until the end? Point scattering model: s (θ) scatt (t) = s(θ) inc (t τ θ)e iν(t τ θ) where τ θ,r = 2 γ(θ) r /c If s (θ) rec (t) denotes scattered field measurements along γ(θ) then I(r) = s (θ) scatt (t )s (θ) rec (t ) dt dθ = = ρ(r )χ θ (2ˆd (r r )/c, 0) dx dy dθ (same result)

40 Imaging via filtered adjoint SAR correlates echo field with model at each step of the data collection process. What if we wait until the end? Point scattering model: s (θ) scatt (t) = s(θ) inc (t τ θ)e iν(t τ θ) where τ θ,r = 2 γ(θ) r /c If s (θ) rec (t) denotes scattered field measurements along γ(θ) then I(r) = s (θ) scatt (t )s (θ) rec (t ) dt dθ = = ρ(r )χ θ (2ˆd (r r )/c, 0) dx dy dθ (same result)

41 Revised scattering model The two parameter model can be modified to include any image-dependent parameters. Target-specific velocities: τ θ τ θ,r,v = 2 γ(θ) r vt /c ρ(r) ρ v(r) χ θ (τ(r), ν(r)) χ θ (τ(r, v); ν(r, v)) Moving point scattering model: s (θ,r,v) scatt (t) = s (θ) inc (t τ θ,r,v)e iν(t τ θ,r,v)

42 Revised scattering model The two parameter model can be modified to include any image-dependent parameters. Target-specific velocities: τ θ τ θ,r,v = 2 γ(θ) r vt /c ρ(r) ρ v(r) χ θ (τ(r), ν(r)) χ θ (τ(r, v); ν(r, v)) Moving point scattering model: s (θ,r,v) scatt (t) = s (θ) inc (t τ θ,r,v)e iν(t τ θ,r,v)

43 Revised scattering model The two parameter model can be modified to include any image-dependent parameters. Target-specific velocities: τ θ τ θ,r,v = 2 γ(θ) r vt /c ρ(r) ρ v(r) χ θ (τ(r), ν(r)) χ θ (τ(r, v); ν(r, v)) Moving point scattering model: s (θ,r,v) scatt (t) = s (θ) inc (t τ θ,r,v)e iν(t τ θ,r,v)

44 Extended image generation (Hyper) image as: I(r; v) = s (θ,r,v) scatt (t )s (θ) (t ) dt dθ rec Define the imaging kernel K(r, r, v, v ) s (θ) inc (t τ θ,r,v)e iν(t τθ,r,v) s (θ) inc (t τ τ θ,r,v )e iν(t θ,r,v ) dθ Can show (slow mover approximation) K(r, r, v, v ) = = χ θ (2ˆd (r r )/c, 2ˆd (v v )/λ) dθ I(r; v) = ρ v(r ) K(r r, v v ) dx dy dv x dv y

45 Extended image generation (Hyper) image as: I(r; v) = s (θ,r,v) scatt (t )s (θ) (t ) dt dθ rec Define the imaging kernel K(r, r, v, v ) s (θ) inc (t τ θ,r,v)e iν(t τθ,r,v) s (θ) inc (t τ τ θ,r,v )e iν(t θ,r,v ) dθ Can show (slow mover approximation) K(r, r, v, v ) = = χ θ (2ˆd (r r )/c, 2ˆd (v v )/λ) dθ I(r; v) = ρ v(r ) K(r r, v v ) dx dy dv x dv y

46 Extended image generation (Hyper) image as: I(r; v) = s (θ,r,v) scatt (t )s (θ) (t ) dt dθ rec Define the imaging kernel K(r, r, v, v ) s (θ) inc (t τ θ,r,v)e iν(t τθ,r,v) s (θ) inc (t τ τ θ,r,v )e iν(t θ,r,v ) dθ Can show (slow mover approximation) K(r, r, v, v ) = = χ θ (2ˆd (r r )/c, 2ˆd (v v )/λ) dθ I(r; v) = ρ v(r ) K(r r, v v ) dx dy dv x dv y

47 Implementation: Choice of s inc (t)? Impulse radar uses s inc(t) δ(t) and so χ(τ, ν) δ(τ) May not be the best choice (especially for slow movers). Same appears to be true for high Doppler resolution waveforms (for which χ(τ, ν) δ(ν)) Ideally, we want K(r, r, v, v ) = δ(r r, v v ) But new (extended) imaging kernel is no longer even guaranteed to be shift-invariant. Design depends on s inc(t) as well as on sensor geometry γ(θ)

48 Implementation: Choice of s inc (t)? Impulse radar uses s inc(t) δ(t) and so χ(τ, ν) δ(τ) May not be the best choice (especially for slow movers). Same appears to be true for high Doppler resolution waveforms (for which χ(τ, ν) δ(ν)) Ideally, we want K(r, r, v, v ) = δ(r r, v v ) But new (extended) imaging kernel is no longer even guaranteed to be shift-invariant. Design depends on s inc(t) as well as on sensor geometry γ(θ)

49 Implementation: Choice of s inc (t)? Impulse radar uses s inc(t) δ(t) and so χ(τ, ν) δ(τ) May not be the best choice (especially for slow movers). Same appears to be true for high Doppler resolution waveforms (for which χ(τ, ν) δ(ν)) Ideally, we want K(r, r, v, v ) = δ(r r, v v ) But new (extended) imaging kernel is no longer even guaranteed to be shift-invariant. Design depends on s inc(t) as well as on sensor geometry γ(θ)

50 Implementation: Choice of s inc (t)? Impulse radar uses s inc(t) δ(t) and so χ(τ, ν) δ(τ) May not be the best choice (especially for slow movers). Same appears to be true for high Doppler resolution waveforms (for which χ(τ, ν) δ(ν)) Ideally, we want K(r, r, v, v ) = δ(r r, v v ) But new (extended) imaging kernel is no longer even guaranteed to be shift-invariant. Design depends on s inc(t) as well as on sensor geometry γ(θ)

51 Implementation: Choice of s inc (t)? Impulse radar uses s inc(t) δ(t) and so χ(τ, ν) δ(τ) May not be the best choice (especially for slow movers). Same appears to be true for high Doppler resolution waveforms (for which χ(τ, ν) δ(ν)) Ideally, we want K(r, r, v, v ) = δ(r r, v v ) But new (extended) imaging kernel is no longer even guaranteed to be shift-invariant. Design depends on s inc(t) as well as on sensor geometry γ(θ)

52 Other applications: Start-Stop error Scheme allows for straightforward introduction of a variety of complex imaging configurations Start-stop accuracy? Figure: Error as a function of squint and resolution for a target traveling 100km/hr in the direction opposite to the platform velocity (8 km/sec). (X-band illumination.)

53 Other applications: Multistatic imaging Figure: The geometry (not to scale) for a linear array 11 transmitters and a single receiver and the corresponding combined point-spread function.

54 Future work Understand K(r, r, v, v ) Choice of s inc (t) vs. γ(θ) Nature of its shift-variance. Expect that K will also characterize effects of non-uniqueness of solutions. Extend to accelerating image elements Can other target-specific parameters be introduced? ρ v(r) ρ v(r, θ,...) What about multiple scattering?

55 Future work Understand K(r, r, v, v ) Choice of s inc (t) vs. γ(θ) Nature of its shift-variance. Expect that K will also characterize effects of non-uniqueness of solutions. Extend to accelerating image elements Can other target-specific parameters be introduced? ρ v(r) ρ v(r, θ,...) What about multiple scattering?

56 Future work Understand K(r, r, v, v ) Choice of s inc (t) vs. γ(θ) Nature of its shift-variance. Expect that K will also characterize effects of non-uniqueness of solutions. Extend to accelerating image elements Can other target-specific parameters be introduced? ρ v(r) ρ v(r, θ,...) What about multiple scattering?

57 Future work Understand K(r, r, v, v ) Choice of s inc (t) vs. γ(θ) Nature of its shift-variance. Expect that K will also characterize effects of non-uniqueness of solutions. Extend to accelerating image elements Can other target-specific parameters be introduced? ρ v(r) ρ v(r, θ,...) What about multiple scattering?

58 Future work Understand K(r, r, v, v ) Choice of s inc (t) vs. γ(θ) Nature of its shift-variance. Expect that K will also characterize effects of non-uniqueness of solutions. Extend to accelerating image elements Can other target-specific parameters be introduced? ρ v(r) ρ v(r, θ,...) What about multiple scattering?

59 Future work Understand K(r, r, v, v ) Choice of s inc (t) vs. γ(θ) Nature of its shift-variance. Expect that K will also characterize effects of non-uniqueness of solutions. Extend to accelerating image elements Can other target-specific parameters be introduced? ρ v(r) ρ v(r, θ,...) What about multiple scattering?

Spatial, Temporal, and Spectral Aspects of Far-Field Radar Data

Spatial, Temporal, and Spectral Aspects of Far-Field Radar Data Margaret Cheney 1 and Ling Wang 2 1 Departments of Mathematical Sciences 2 Department of Electrical, Computer, and Systems Engineering Rensselaer