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

Maxwell’s equations —-> scalar wave equation Green’s function + Born approximation

p(t, xr; xs) ∝ Z ρ(y)f 00[t − τ(y, xs) − τ(y, xr)] |y − xs||y − xr| dy

standard (monostatic) SAR:

Mathematical Model p(t, s) ∝ Z ρ(y)f 00 (t − 2Rs,y/c0) R2

s,y

dy

τ(y, x) = |y − x| c0 = |γ(s) − y| c0

y Rs,y xr = xs = γ(s)

figures from Brett Borden, Naval Postgraduate School

Image formation

standard (monostatic) SAR array imaging

I(y) = X

xr,xs

data(t = τ(y, xs) + τ(y, xr), xr; xs)

why does this work?

= X

s

data

• t = 2Rs,y/c0, γ(s)

Imaging from a single viewing position

Imaging from a single view Example with 3 scatterers

Imaging from two views

Imaging from three views synthetic aperture

Frequency domain viewpoint

p(t, s) ∝ Z ρ(y)f 00 (t − 2Rs,y/c0) R2

s,y

dy

time-domain model

P(ω, s) ∝ Z ρ(y)ω2F(ω)e−2iωRs,y/c0 R2

s,y

dy

Z . . . e−iωtdt

Fourier transform in t far-field approximation

|γ| |y|

Rs,y = |γ(s) − y| ≈ |γ| − b γ · y + · · ·

k = ω c0

to form image, invert Fourier transform!

b γ = γ |γ|

P(ω, s) ∝ Z ρ(y)e−2ikb

γ(s)·ydy

Approximating targets by point clouds P(ω, s) ∝ Z ρ(y)e−2ikb

γ·ydy

k large -> use geometrical optics main contributions are from corners, edges, and specular points

Interferometry p(t, s) ∝ Z ρ(y)f 00 (t − 2Rs,y/c0) R2

s,y

dy p(t, s) ∝ Z ρ(y)eiω0(t−2Rs,y/c0) a(t − 2Rs,y/c0) R2

s,y

dy ρ(y) = ˜ ρ(y1, y2)δ(y3 − h(y1, y2)) f(t) = a(t)eiω0t

narrowband slowly varying (complex) amplitude

a =

scattering takes place on surface

yT = (y1, y2, 0) Rs,y = |yT + hˆ e3 − γ| = |yT − γ| | {z }

Rs,yT

+ h(yT )ˆ e3 · \ yT − γ | {z }

d(yT )

y = yT + h(yT )ˆ e3 p(t, s) ≈ Z h ˜ ρ(yT )e2ik0d(yT )i eiω0(t−Rs,yT /c0) a(t − Rs,yT /c0) R2

s,yT

dyT

target phase encodes height information!