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

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

figures from Brett Borden, Naval Postgraduate School

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?

Imaging from a single viewing position

Example with 3 scatterers Imaging from a single view

Imaging from two views

Imaging from three views synthetic aperture

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

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

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!