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Introduction to Radar Imaging Margaret Cheney Outline Mathematical - PowerPoint PPT Presentation

Introduction to Radar Imaging Margaret Cheney Outline Mathematical model Image formation time domain viewpoint frequency domain viewpoint (for small scenes) Approximating targets by point clouds SAR interferometry


  1. Introduction to Radar Imaging Margaret Cheney

  2. Outline • Mathematical model • Image formation • time domain viewpoint • frequency domain viewpoint (for small scenes) • Approximating targets by point clouds • SAR interferometry

  3. Mathematical Model Maxwell’s equations —-> scalar wave equation Green’s function + Born approximation ρ ( y ) f 00 [ t − τ ( y , x s ) − τ ( y , x r )] Z p ( t, x r ; x s ) ∝ d y | y − x s || y − x r | standard (monostatic) SAR: x r = x s = γ ( s ) τ ( y , x ) = | y − x | = | γ ( s ) − y | R s, y c 0 c 0 y ρ ( y ) f 00 ( t − 2 R s, y /c 0 ) Z p ( t, s ) ∝ d y R 2 s, y

  4. figures from Brett Borden, Naval Postgraduate School

  5. Image formation X I ( y ) = data( t = τ ( y , x s ) + τ ( y , x r ) , x r ; x s ) x r , x s array imaging X � � = data t = 2 R s, y /c 0 , γ ( s ) standard (monostatic) SAR s why does this work?

  6. Imaging from a single viewing position

  7. Example with 3 scatterers Imaging from a single view

  8. Imaging from two views

  9. Imaging from three views synthetic aperture

  10. Frequency domain viewpoint ρ ( y ) f 00 ( t − 2 R s, y /c 0 ) Z p ( t, s ) ∝ d y time-domain model R 2 s, y Z . . . e − i ω t dt Fourier transform in t ρ ( y ) ω 2 F ( ω ) e − 2 i ω R s, y /c 0 Z P ( ω , s ) ∝ d y R 2 s, y far-field approximation R s, y = | γ ( s ) − y | ≈ | γ | − b γ · y + · · · | γ | � | y | γ = γ b | γ | Z ρ ( y ) e − 2 ik b γ ( s ) · y d y P ( ω , s ) ∝ k = ω to form image, invert Fourier transform! c 0

  11. Approximating targets by point clouds Z ρ ( y ) e − 2 ik b P ( ω , s ) ∝ γ · y d y k large -> use geometrical optics main contributions are from corners, edges, and specular points

  12. Interferometry ρ ( y ) f 00 ( t − 2 R s, y /c 0 ) Z p ( t, s ) ∝ d y R 2 s, y f ( t ) = a ( t ) e i ω 0 t narrowband slowly varying (complex) amplitude a = ρ ( y ) e i ω 0 ( t − 2 R s, y /c 0 ) a ( t − 2 R s, y /c 0 ) Z p ( t, s ) ∝ d y R 2 s, y scattering takes place on surface ρ ( y ) = ˜ ρ ( y 1 , y 2 ) δ ( y 3 − h ( y 1 , y 2 )) y = y T + h ( y T )ˆ y T = ( y 1 , y 2 , 0) e 3 e 3 · \ R s, y = | y T + h ˆ e 3 − γ | = | y T − γ | + h ( y T )ˆ y T − γ | {z } | {z } R s, y T d ( y T ) e i ω 0 ( t − R s, y T /c 0 ) a ( t − R s, y T /c 0 ) Z h ρ ( y T ) e 2 ik 0 d ( y T ) i p ( t, s ) ≈ ˜ d y T R 2 s, y T target phase encodes height information!

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