SAR Image Formation Algorithms

The radar transmits an LFM chirp at each slow-time position $\eta$:

$$s_{\text{tx}}(\tau) = \text{rect}\!\left(\frac{\tau}{T_p}\right) \exp\!\left(j2\pi f_0 \tau + j\pi K_r \tau^2\right),$$

where $T_p$ is the pulse duration and $K_r = B/T_p$ is the chirp rate.

A point scatterer at position $(x_0, R_0)$ with reflectivity $\sigma_0$ produces a round-trip delay $\tau_d(\eta) = 2R(\eta)/c$ where:

$$R(\eta) = \sqrt{R_0^2 + (v_p \eta - x_0)^2}.$$

After demodulation to baseband, the received signal is:

$$s(\tau, \eta) = \sigma_0\,\text{rect}\!\left(\frac{\tau - \tau_d(\eta)}{T_p}\right) \exp\!\left(-j\frac{4\pi f_0}{c}R(\eta)\right) \exp\!\left(j\pi K_r\!\left(\tau - \tau_d(\eta)\right)^2\right).$$

Range compression (matched filtering in fast time) yields:

$$s_{\text{rc}}(\tau, \eta) = \sigma_0\,\text{sinc}\!\left(B\!\left(\tau - \frac{2R(\eta)}{c}\right)\right) \exp\!\left(-j\frac{4\pi f_0}{c}R(\eta)\right).$$

The remaining azimuth phase $\phi_{\text{az}}(\eta) = -(4\pi f_0/c)R(\eta)$ encodes the cross-range position.

Expanding $R(\eta)$ about the closest-approach time $\eta_c = x_0/v_p$:

$$R(\eta) \approx R_0 + \frac{v_p^2(\eta - \eta_c)^2}{2R_0},$$

the azimuth phase becomes a quadratic (chirp) in slow time:

$$\phi_{\text{az}}(\eta) = -\frac{4\pi}{\lambda}\left(R_0 + \frac{v_p^2(\eta - \eta_c)^2}{2R_0}\right).$$

This defines the azimuth FM rate:

$$K_a = -\frac{2v_p^2}{\lambda R_0},$$

and the Doppler bandwidth:

$$B_a = |K_a| T_{\text{sa}} = \frac{2v_p L_{\text{sa}}}{\lambda R_0}.$$

Azimuth compression is therefore a matched filter for a chirp with rate $K_a$.

The $\omega$-$k$ algorithm processes SAR data entirely in the 2D frequency domain, avoiding the need for interpolation-based RCMC.

Key idea: After 2D FFT of the raw data, apply the Stolt interpolation — a coordinate transformation from $(f_\tau, f_\eta)$ to $(k_x, k_R)$ that maps the curved phase history to a rectangular grid in wavenumber space.

Advantages over RDA:

• Exact RCMC without interpolation artifacts.
• Handles wide-bandwidth, high-squint geometries better.

Disadvantage: Requires the Stolt interpolation, which is itself an interpolation step (but in the frequency domain, where the signal is smoother).

The chirp scaling algorithm (CSA) achieves range-variant RCMC without interpolation by applying a phase multiply (chirp scaling) in the range-Doppler domain. This converts the range-dependent range cell migration into a range-independent shift that can be corrected by a bulk phase multiply.

CSA is exact under the quadratic approximation and avoids both the interpolation of RDA's RCMC and the Stolt mapping of the $\omega$-$k$ algorithm, making it computationally attractive for wide-swath modes.

Uncompensated platform motion errors introduce a multiplicative phase error $\phi_e(\eta)$ in the azimuth signal. After range compression, the signal from a point target becomes:

$$s_{\text{rc}}(\tau, \eta) = \sigma_0 \, \text{sinc}(\cdots) \exp(j\pi K_a(\eta - \eta_c)^2) \exp(j\phi_e(\eta)).$$

Phase Gradient Autofocus (PGA) is the standard autofocus algorithm for airborne and spaceborne SAR. It estimates $\phi_e'(\eta)$ from the data and integrates to recover $\phi_e(\eta)$.

Steps:

1. Circular shift: For each range bin, shift the brightest target to the scene center (removes linear phase).
2. Windowing: Apply a window around the dominant target.
3. Phase gradient estimation: $\hat{\phi}_e'(\eta) = \text{Im}\left\{ \sum_k s_k^*(\eta)\,s_k'(\eta) \big/ \sum_k |s_k(\eta)|^2\right\}$.
4. Integration: $\hat{\phi}_e(\eta) = \int \hat{\phi}_e'(\eta)\,d\eta$.
5. Correction: Multiply by $\exp(-j\hat{\phi}_e(\eta))$.
6. Iterate until convergence (typically 3--5 iterations).

PGA converges for arbitrary phase errors as long as there are sufficiently bright point-like targets in the scene.

When the scene lacks bright isolated targets, minimum-entropy autofocus (MEA) provides a robust alternative. A well-focused image has lower entropy than a defocused one:

$$H(\phi_e) = -\sum_{m,n} \frac{|I_{mn}|^2}{E} \ln\!\left(\frac{|I_{mn}|^2}{E}\right), \quad E = \sum_{m,n} |I_{mn}|^2.$$

The optimal phase error minimizes image entropy: $\hat{\phi}_e = \arg\min_{\phi_e} H(\phi_e)$.

This is solved iteratively using gradient descent over the phase error coefficients, parameterized as a polynomial $\phi_e(\eta) = \sum_{p=2}^P c_p \eta^p$.