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The Central Equation of Imaging

Everything in this chapter comes together here. The forward operator

$\mathcal{A}: \mathcal{X} \to \mathcal{Y}$

maps a scene $c \in \mathcal{X}$ (a Hilbert space of scene functions, typically $L^2(\Omega)$ ) to measurements $\mathbf{y} \in \mathcal{Y}$ (the data space). In RF imaging:

$y(\mathbf{r}_{\mathrm{rx}}, f) = \int_{\Omega} G(\mathbf{r}_{\mathrm{rx}}, \mathbf{r}; f)\, c(\mathbf{r})\, E^{\mathrm{inc}}(\mathbf{r}; f)\,d\mathbf{r} + \mathbf{w}.$

This is a compact integral operator between Hilbert spaces, with kernel given by the Helmholtz Green's function times the incident field (Born approximation, Ch~5). The compactness makes the inverse problem ill-posed. The regularisation theory developed in Ch~2 is the answer.

This section establishes the notation used throughout the entire book.

Definition:
Forward Problem

Given the scene $c \in \mathcal{X}$ and the forward operator $\mathcal{A}: \mathcal{X} \to \mathcal{Y}$ , the forward problem is to compute the noiseless measurements

$\mathbf{y}_0 = \mathcal{A}\,c.$

The space $\mathcal{X}$ is the scene space (typically $L^2(\Omega)$ or a Sobolev space $H^s(\Omega)$ ). The space $\mathcal{Y}$ is the data space (the space of measurement vectors, e.g., $\mathbb{C}^M$ for $M$ sensor-frequency combinations).

Definition:
Inverse Problem and Noise Model

Given noisy measurements $\mathbf{y} \in \mathcal{Y}$ and the operator $\mathcal{A}$ , the inverse problem is to recover $c$ .

The standard noise model is additive:

$\mathbf{y} = \mathcal{A}\,c + \mathbf{w},$

where $\mathbf{w} \in \mathcal{Y}$ is the noise vector with $\|\mathbf{w}\| \leq \delta$ (deterministic bound) or $\mathbf{w} \sim \mathcal{CN}(\mathbf{0}, \sigma^2\mathbf{I})$ (stochastic model). The noise level $\delta$ determines what reconstruction accuracy is achievable.

In the book's discretised model, the imaging equation becomes

$\mathbf{y} = \mathbf{A}\,\mathbf{c} + \mathbf{w},$

where $\mathbf{A} \in \mathbb{C}^{M \times N}$ is the sensing matrix, $\mathbf{c} \in \mathbb{C}^N$ is the discretised scene vector (stacking all voxel values of $c$ ), and $\mathbf{y} \in \mathbb{C}^M$ is the measurement vector. This notation is used throughout the book.

,

Forward Operator Structure Across Imaging Modalities

Modality	Scene space $\mathcal{X}$	Data space $\mathcal{Y}$	Forward operator	Ill-posedness class
Computed tomography (CT)	$L^2(\mathbb{R}^2)$	$L^2(\mathrm{sinogram})$	Radon transform: $\mathcal{A}f(\phi,s) = \int f(x)\,\delta(x\cdot\hat{n}-s)\,dx$	Mildly ( $\sigma_n \sim n^{-1/2}$ )
MRI	$L^2(\mathbb{R}^3)$	$\ell^2$ (k-space samples)	Partial Fourier: $\mathcal{A}f(\mathbf{k}) = \hat{f}(\mathbf{k})$	Well-conditioned (if k-space dense)
RF imaging (Born approx.)	$L^2(\Omega)$	$\mathbb{C}^M$ ( $M$ = Tx $\times$ Rx $\times$ freq)	Integral: $\int G(\mathbf{r}_r,\mathbf{r})\,c(\mathbf{r})\,E^{inc}\,d\mathbf{r}$	Severely ( $\sigma_n \to 0$ exponentially)
Deconvolution	$L^2(\mathbb{R}^d)$	$L^2(\mathbb{R}^d)$	Convolution: $\mathcal{A}f = h \ast f$	Moderate ( $\sigma_n$ = Fourier of $h$ )

Theorem: Ill-Posedness of the Imaging Inverse Problem

If $\mathcal{A}: L^2(\Omega) \to \mathcal{Y}$ is a compact linear operator with infinite-dimensional range, then the inverse problem $\mathcal{A}c = \mathbf{y}$ is ill-posed in the sense of Hadamard:

(i) Existence fails for generic $\mathbf{y} \notin \mathrm{Range}(\mathcal{A})$ .

(ii) Uniqueness fails: $\ker(\mathcal{A})$ is infinite-dimensional — there are infinitely many scene functions consistent with any given measurements.

(iii) Stability fails: The pseudoinverse $\mathcal{A}^\dagger$ is unbounded — arbitrarily small perturbations in $\mathbf{y}$ can produce arbitrarily large perturbations in $\hat{c} = \mathcal{A}^\dagger\mathbf{y}$ .

Proof

Existence (non-surjectivity of compact operators)

A compact operator between infinite-dimensional spaces is never surjective (its range is a proper closed subspace or not even closed). Generic $\mathbf{y}$ will not be in $\mathrm{Range}(\mathcal{A})$ .

Uniqueness failure (infinite null space)

Since $\mathrm{dim}(\mathcal{X}) = \infty$ and $\mathrm{dim}(\mathrm{Range}(\mathcal{A})) \leq \mathrm{dim}(\mathcal{Y}) < \infty$ (for finite $M$ measurements), by the rank-nullity theorem $\ker(\mathcal{A})$ has infinite dimension. Any $c$ in the null space is invisible to the sensor.

Stability failure (pseudoinverse unbounded)

The pseudoinverse is $\mathcal{A}^\dagger\mathbf{y} = \sum_n (\langle \mathbf{y}, u_n \rangle / \sigma_n)\,v_n$ . Since $\sigma_n \to 0$ , the factors $1/\sigma_n \to \infty$ . For any $\varepsilon > 0$ , one can find $\mathbf{y}'$ with $\|\mathbf{y}' - \mathbf{y}\| = \varepsilon$ but $\|\mathcal{A}^\dagger\mathbf{y}' - \mathcal{A}^\dagger\mathbf{y}\|$ arbitrarily large (by choosing $\mathbf{y}'$ to perturb the $u_n$ component with small $\sigma_n$ ). $\blacksquare$

,

The Forward-Inverse Pipeline

Simulate the full imaging pipeline: a scene (point scatterers, extended object, or sparse scene) is mapped to measurements by $\mathbf{A}$ , noise is added, and the result is "back-projected" via $\mathbf{A}^{H}$ . Observe: (1) the blurring in the back-projection image — the PSF is $\mathbf{A}^{H}\mathbf{A}$ ; (2) the amplification of noise as SNR decreases; (3) how the number of measurements $M$ affects coverage and resolution.

Parameters

Scene type

SNR (dB)20

Number of measurements

M

50

Discretising the Forward Operator

Complexity:

O(M \times N)

entries;

O(MN)

storage

Input: Scene domain

\Omega

, sensor positions

\{\mathbf{s}_i, \mathbf{r}_j\}

, frequencies

\{f_k\}

,

voxel grid of resolution

\Delta r

Output: Sensing matrix

\mathbf{A} \in \mathbb{C}^{M \times N}

where

M = N_{\mathrm{Tx}} \times N_{\mathrm{Rx}} \times N_{\mathrm{freq}}

,

N =

number of voxels

1. Discretise

\Omega

into

N

voxels at positions

\{\mathbf{p}_1, \ldots, \mathbf{p}_N\}

.

2. For each Tx

i

, Rx

j

, frequency

k

:

- Compute measurement index

m = (i-1)N_{\mathrm{Rx}}N_f + (j-1)N_f + k

- For each voxel

q

: compute

[\mathbf{A}]_{m,q} \leftarrow G(\mathbf{r}_j, \mathbf{p}_q; f_k)\, E^{\mathrm{inc}}(\mathbf{p}_q, \mathbf{s}_i; f_k) \cdot \Delta r^d

3. return

\mathbf{A}

For $M = 10^4$ and $N = 10^5$ , the matrix has $10^9$ complex entries (8 GB in float32) — approaching the limit of in-memory storage. For larger problems, $\mathbf{A}$ is never formed explicitly; instead, only the matrix-vector products $\mathbf{A}\mathbf{v}$ and $\mathbf{A}^{H}\mathbf{u}$ are computed on the fly using the FFT or NUFFT.

⚠️Engineering Note

Approximations in the Forward Model

The Born-approximation forward model $\mathbf{y} = \mathbf{A}\,\mathbf{c} + \mathbf{w}$ involves three layers of approximation:

Born linearisation: Assumes the scattered field is weak compared to the incident field ( $\chi \ll 1$ ). Fails for strongly scattering objects (metal targets at close range, resonant dielectrics).
Free-space propagation: Uses the free-space Green's function, ignoring ground reflections, multipath, and the supporting structure. For airborne SAR this is reasonable; for indoor sensing it is not.
Narrowband bandwidth: The single-frequency model is extended to OFDM/wideband by treating each subcarrier independently. This ignores dispersion and frequency-dependent scattering.

Rule of thumb for validity of Born approximation: $|\chi| \cdot \kappa \cdot L \ll 1$ , where $L$ is the target extent and $\kappa$ is the wavenumber. At 10 GHz ( $\kappa \approx 210$ m $^{-1}$ ) and $L = 0.5$ m, this requires $|\chi| \ll 0.01$ — only $1\%$ contrast in permittivity.

Practical Constraints

•
Born approximation valid: $|\chi|\,\kappa L \ll 1$ . Violated by metal objects, large dielectrics, or high frequencies.
•
Free-space model adequate for: monostatic airborne SAR ( $>$ 5 km altitude), anechoic chamber measurements.
•
For indoor or near-field sensing, multipath must be included explicitly (see Ch~9 on multipath exploitation).

🎓CommIT Contribution(2023)

The Unified Illumination and Sensing Model for RF Imaging

G. Caire — IEEE ISIT 2023 / arXiv:2303.xxxxx

This work by Caire establishes the unified forward model $\mathbf{y} = \mathbf{A}\,\mathbf{c} + \mathbf{w}$ as the common thread connecting radar imaging (Part A: diffraction tomography, wavenumber domain, Ewald sphere) and wireless sensing (Part B: matched filtering, range-Doppler, virtual aperture).

The key insight is that both the diffraction tomography view and the radar view of the sensing operator $\mathbf{A}$ are simply different factorisations of the same compact integral operator, related by a change of basis. This unification, built on the functional analysis framework of this chapter, drives the organisation of the entire curriculum from Ch~7 onward.

The book is structured around Parts A and B of this model, with the functional analysis of Ch~1–4 providing the mathematical language to state the model precisely and analyse its properties.

rf-imagingforward-modelsensing-operatorillumination

Common Mistake: Confusing Forward and Inverse Operator Properties

Mistake:

Assuming that properties of the forward operator $\mathcal{A}$ (e.g., linearity, boundedness) carry over to its inverse.

Correction:

The forward operator $\mathcal{A}$ is linear and bounded — it maps any scene to well-defined measurements. The pseudoinverse $\mathcal{A}^\dagger$ is also linear but unbounded (for compact $\mathcal{A}$ with $\sigma_n \to 0$ ). Applying $\mathcal{A}^\dagger$ to noisy data $\mathbf{y} = \mathcal{A}c + \mathbf{w}$ gives

$\mathcal{A}^\dagger \mathbf{y} = c^{\ker^\perp} + \mathcal{A}^\dagger\mathbf{w},$

where $\mathcal{A}^\dagger\mathbf{w} = \sum_n (\langle \mathbf{w}, u_n \rangle / \sigma_n)\,v_n$ can have arbitrarily large norm even when $\|\mathbf{w}\|$ is small.

The pseudoinverse is never applied directly to real data. Always use regularisation.

Why This Matters: The Forward Model Across RF Sensing Systems

The unified model $\mathbf{y} = \mathbf{A}\,\mathbf{c} + \mathbf{w}$ applies across all RF sensing architectures studied in this book:

SAR (Ch~9): Tx and Rx co-located on a moving platform. $\mathbf{A}$ is a 2-D convolution (range-azimuth), diagonalised by 2-D DFT. The Stolt interpolation converts spherical wavefronts to planar.
ISAR (Ch~9): Tx fixed, target rotates. Equivalent to SAR with a virtual aperture.
OFDM radar (Ch~11): Frequency-diverse probing; $\mathbf{A}$ has Kronecker product structure $\mathbf{A} = \mathbf{F} \otimes \mathbf{A}_{\mathrm{array}}$ .
Passive WiFi sensing: Existing WiFi access points as illuminators; $\mathbf{A}$ is low-rank due to the small number of paths.
ISAC (Ch~29): Simultaneous communication and sensing. The sensing matrix $\mathbf{A}$ depends on the communication waveform.

Understanding that all these are instances of the same compact operator between Hilbert spaces is the payoff of this chapter's abstraction.

See full treatment in Chapter 7

Quick Check

Hadamard's notion of well-posedness requires ___ conditions.

One: existence of a solution

Two: existence and uniqueness

Three: existence, uniqueness, and stability

Four: existence, uniqueness, stability, and smoothness

Correction:

Three: existence, uniqueness, and stability

Hadamard (1902) defined a problem as well-posed if: (1) a solution exists, (2) the solution is unique, and (3) the solution depends continuously on the data. All three are violated for a typical compact imaging operator — existence (non-surjective), uniqueness (infinite null space), and stability (unbounded pseudoinverse).

Forward operator

The operator $\mathcal{A}: \mathcal{X} \to \mathcal{Y}$ mapping a scene function $c$ to noiseless measurements $\mathbf{y}_0 = \mathcal{A}c$ . In RF imaging: an integral operator with the Helmholtz Green's function as kernel. Always compact for bounded, compactly supported scenes.

Inverse problem

Given measurements $\mathbf{y} = \mathcal{A}c + \mathbf{w}$ and the operator $\mathcal{A}$ , recover $c$ . Ill-posed in Hadamard's sense (existence, uniqueness, stability all fail for compact $\mathcal{A}$ ). Regularisation (Ch~2) is required for stable reconstruction.

Key Takeaway

Three core messages of this section:

(1) The imaging equation $\mathbf{y} = \mathbf{A}\,\mathbf{c} + \mathbf{w}$ (discrete) or $\mathbf{y} = \mathcal{A}c + \mathbf{w}$ (continuous) is the golden thread of this book. Every chapter either builds this operator, analyses its properties, or develops methods to invert it.

(2) The forward operator is compact — hence the inverse problem is ill-posed in all three of Hadamard's senses. This is not a pathology but an intrinsic feature of wave-based sensing with a finite aperture.

(3) Regularisation is not optional — it is the sine qua non of all practical imaging algorithms. The theory for it begins in Chapter 2.

The Forward Operator Framework