Scene Representations

A point-scatterer model represents the scene as a collection of $K$ isolated reflectors at positions $\mathbf{p}_{k} \in \mathbb{R}^d$ with complex reflectivities $\alpha_k \in \mathbb{C}$:

$$c(\mathbf{p}) = \sum_{k=1}^{K} \alpha_k\, \delta(\mathbf{p} - \mathbf{p}_{k}).$$

The forward model for Tx $i$, Rx $j$, frequency $f$ is:

$$y(\mathbf{s}_{i}, \mathbf{r}_{j}; f) = \sum_{k=1}^{K} \alpha_k\, \frac{e^{-j\kappa(d(\mathbf{s}_{i}, \mathbf{p}_{k}) + d(\mathbf{p}_{k}, \mathbf{r}_{j}))}}{d(\mathbf{s}_{i}, \mathbf{p}_{k})\,d(\mathbf{p}_{k}, \mathbf{r}_{j})} + w_{i,j,f}.$$

Properties:

• Inherently sparse: $K \ll Q$ (number of grid points), making it natural for compressed sensing.
• Positions $\mathbf{p}_{k}$ are continuous parameters (not grid-restricted), so there is no basis mismatch.
• The forward model is non-linear in $\mathbf{p}_{k}$ (positions appear inside the exponent), unlike the linear model for grid-based representations.
• Best suited for automotive radar (few strong targets), ISAR (isolated scattering centers), and indoor localization.

The reflectivity map discretizes the continuous function $c(\mathbf{p})$ on a regular grid over the target region $\Omega$. For a 2D scene on an $Q_x \times Q_y$ grid with $Q = Q_x Q_y$ pixels:

$$c(\mathbf{p}) \approx \sum_{q=1}^{Q} c_q\, b(\mathbf{p} - \mathbf{p}_{q}),$$

where $c_q \in \mathbb{C}$ is the reflectivity at pixel $q$ and $b(\mathbf{p})$ is the basis function (typically a rect/indicator or sinc interpolation kernel).

The discretized forward model becomes the familiar linear system:

$$\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}, \quad \mathbf{c} = [c_1, \ldots, c_Q]^T.$$

Properties of $c_q$:

• $|c_q|^2$ is proportional to the scattering power (related to RCS).
• $\arg(c_q)$ encodes the sub-grid phase shift: a scatterer between grid points acquires a phase proportional to its offset.
• Grid spacing must satisfy the spatial Nyquist criterion (see [?ch07:s07]): $\Delta x \leq \lambda/(4\sin\theta_{\max})$.

The complex reflectivity $c_q = |c_q|\,e^{j\phi_q}$ encodes two distinct physical quantities:

Amplitude $|c_q|$: Proportional to the scattering strength. For a point target with radar cross-section $\sigma_{\text{RCS}}$:

$$|c_q| \propto \sqrt{\sigma_{\text{RCS}}}.$$

Phase $\phi_q$: Encodes the sub-grid position offset. A scatterer displaced by $\delta\mathbf{p}$ from grid point $\mathbf{p}_{q}$ acquires a phase:

$$\phi_q = -\kappa_{s,r}^{T} \delta\mathbf{p},$$

where $\kappa_{s,r} = \kappa_{s} + \kappa_{r}$ is the combined Tx-Rx wavenumber vector.

This phase encoding is the reason that coherent imaging (preserving phase across measurements) achieves resolution far beyond the grid spacing.

For extended targets (targets whose extent is comparable to or larger than the resolution cell), the reflectivity depends on the viewing angle. The scattering coefficient becomes:

$$c_{i,j,q} = c(\mathbf{p}_{q}, \hat{\mathbf{s}}_i, \hat{\mathbf{r}}_j),$$

where $\hat{\mathbf{s}}_i$ and $\hat{\mathbf{r}}_j$ are the unit direction vectors from the target to the transmitter and receiver.

Physical mechanisms:

• Specular reflection: Strong return only when the surface normal bisects the Tx-target-Rx angle.
• Diffraction: Edge returns that vary with incidence angle (GTD/UTD).
• Creeping waves: Surface waves around curved objects.
• Shadow regions: No return from the shadowed side of a convex target.

The forward model generalizes to:

$$y_{i,j,f} = \sum_{q=1}^{Q} c(\mathbf{p}_{q}, \hat{\mathbf{s}}_i, \hat{\mathbf{r}}_j)\, [\mathbf{A}]_{(i,j,f),q} + w_{i,j,f},$$

and the sensing matrix becomes angle-dependent: $\mathbf{A} \to \mathbf{A}(i,j)$ — different for each Tx-Rx pair.

For penetrating-wave imaging (ground-penetrating radar, medical imaging, non-destructive testing), the scene is represented by the complex permittivity:

$$\varepsilon(\mathbf{p}) = \varepsilon'(\mathbf{p}) - j\varepsilon''(\mathbf{p}),$$

or equivalently the contrast function:

$$\chi(\mathbf{p}) = \varepsilon_r(\mathbf{p}) - 1.$$

Under the Born approximation (Chapter 6), the forward model is:

$$y = \kappa^{2} \int_{\Omega} G(\mathbf{r}, \mathbf{p}')\, \chi(\mathbf{p}')\,u^{\text{inc}}(\mathbf{p}')\,d\mathbf{p}'.$$

Key difference from reflectivity: The contrast $\chi$ is a volumetric quantity (3D), not a surface quantity. For strong scatterers ($|\chi| \gg 1$), the Born approximation breaks and iterative methods (distorted Born, contrast source inversion) are required (Section s05).