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The Bridge Between Scattered Data and the Scene

We have shown that each measurement samples a point in k-space. The Fourier Diffraction Theorem makes the connection precise: the scattered-field data, properly normalized, equals the spatial Fourier transform of the reflectivity at that k-space point. This is the analogue of the Fourier Slice Theorem in X-ray CT, and it is the theoretical foundation for all Fourier-based image reconstruction methods.

Definition:
Spatial Fourier Transform of the Reflectivity

The 3D spatial Fourier transform of the reflectivity function is

$\tilde{c}(\boldsymbol{\kappa}) = \int c(\mathbf{p}) \, e^{-j\boldsymbol{\kappa}^\mathsf{T} \mathbf{p}} \, d\mathbf{p}.$

This maps the spatial distribution of scatterers to the wavenumber domain. The inverse transform recovers the reflectivity:

$c(\mathbf{p}) = \frac{1}{(2\pi)^d} \int \tilde{c}(\boldsymbol{\kappa}) \, e^{j\boldsymbol{\kappa}^\mathsf{T} \mathbf{p}} \, d\boldsymbol{\kappa},$

where $d$ is the spatial dimension (2 or 3).

Theorem: The Fourier Diffraction Theorem

Under the Born approximation, with the far-field/Taylor approximation of Section 6.2, the scattered field for a single Tx-Rx pair at a single frequency satisfies

$x(\mathbf{s}, \mathbf{r}; f) = \alpha(\mathbf{s}, \mathbf{r}, f) \cdot \tilde{c}(\kappa_{\mathbf{s},\mathbf{r}}),$

where $\alpha(\mathbf{s}, \mathbf{r}, f)$ is a known complex scalar (depending on distances, antenna gains, and the propagation phase to/from the target center), and $\tilde{c}(\kappa_{\mathbf{s},\mathbf{r}})$ is the spatial Fourier transform of the reflectivity evaluated at the combined wavenumber $\kappa_{\mathbf{s},\mathbf{r}} = \boldsymbol{\kappa}_s + \boldsymbol{\kappa}_r$ .

In words: Each measurement directly samples the scene's Fourier transform at one point in k-space.

The Born integral with the Taylor approximation reduces to an integral of $c(\tilde{\mathbf{p}}) e^{-j\kappa_{\mathbf{s},\mathbf{r}}^\mathsf{T}\tilde{\mathbf{p}}}$ — which is precisely the definition of the Fourier transform at $\kappa_{\mathbf{s},\mathbf{r}}$ . The prefactor $\alpha$ is known from the system geometry, so dividing by it gives us direct access to $\tilde{c}(\kappa_{\mathbf{s},\mathbf{r}})$ .

Proof

Start from the Taylor-approximated integral

From TTaylor Expansion and Wavenumber Decomposition, the scattered field is

$x(\mathbf{s}, \mathbf{r}; f) = \frac{\sqrt{G^{\text{tx}}G^{\text{rx}}} \, e^{-j\kappa(d(\mathbf{s}, \mathbf{p}_{0}) + d(\mathbf{p}_{0}, \mathbf{r}))}}{d(\mathbf{s}, \mathbf{p}_{0})\,d(\mathbf{p}_{0}, \mathbf{r})} \int_{\tilde{\Omega}} c(\tilde{\mathbf{p}}) \, e^{-j\kappa_{\mathbf{s},\mathbf{r}}^\mathsf{T}\tilde{\mathbf{p}}} \, d\tilde{\mathbf{p}}.$

Identify the Fourier transform

The integral $\int c(\tilde{\mathbf{p}}) e^{-j\kappa_{\mathbf{s},\mathbf{r}}^\mathsf{T}\tilde{\mathbf{p}}} d\tilde{\mathbf{p}}$ is exactly $\tilde{c}(\kappa_{\mathbf{s},\mathbf{r}})$ , the spatial Fourier transform of $c$ evaluated at $\kappa_{\mathbf{s},\mathbf{r}}$ .

Factor out the known coefficient

Defining

$\alpha(\mathbf{s}, \mathbf{r}, f) \triangleq \frac{\sqrt{G^{\text{tx}}G^{\text{rx}}} \, e^{-j\kappa(d(\mathbf{s}, \mathbf{p}_{0}) + d(\mathbf{p}_{0}, \mathbf{r}))}}{d(\mathbf{s}, \mathbf{p}_{0})\,d(\mathbf{p}_{0}, \mathbf{r})},$

we obtain $x(\mathbf{s}, \mathbf{r}; f) = \alpha \cdot \tilde{c}(\kappa_{\mathbf{s},\mathbf{r}})$ . $\blacksquare$

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The Fourier Diffraction Theorem vs. the Fourier Slice Theorem

In X-ray computed tomography (CT), the Fourier Slice Theorem states that a 1D projection of a 2D object at angle $\theta$ corresponds to a line through the origin of the 2D Fourier space, at angle $\theta$ . Rotating the source/detector fills radial lines, leading to filtered backprojection.

In diffraction tomography, the Fourier Diffraction Theorem is the wave analogue: each measurement corresponds to a point on an Ewald sphere arc (not a line). The key differences:

CT uses straight rays (no diffraction) → radial lines in k-space.
Diffraction tomography uses scattered waves → arcs on the Ewald sphere.
CT: varying angle fills k-space along radial spokes.
DT: varying Tx/Rx angles fills k-space along arc segments.

For small objects (Born approximation valid) at high frequencies (Ewald sphere radius $\gg$ object size), the arcs flatten and the Fourier Diffraction Theorem approaches the Fourier Slice Theorem.

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X-ray CT vs. Diffraction Tomography

Property	X-ray CT	RF Diffraction Tomography
Wave model	Straight rays (no diffraction)	Scattered waves (Born approximation)
k-Space sampling geometry	Radial lines through origin	Arcs on the Ewald sphere
Key theorem	Fourier Slice Theorem	Fourier Diffraction Theorem
Reconstruction	Filtered backprojection (FBP)	Fourier interpolation + inverse FFT
Wavelength vs object size	$\lambda \ll$ object (geometric optics limit)	$\lambda \sim$ object features (diffraction regime)
Primary application	Medical imaging	Ultrasound, microwave, RF imaging

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Example: Fourier Diffraction Theorem for a Point Scatterer

A single point scatterer at position $\mathbf{p}_{0}$ with reflectivity $c_0$ . Verify the Fourier Diffraction Theorem: show that the scattered data measured by all Tx-Rx-frequency triples is consistent with a constant Fourier transform $\tilde{c}(\boldsymbol{\kappa}) = c_0$ for all $\boldsymbol{\kappa}$ .

Solution

Reflectivity of a point scatterer

$c(\mathbf{p}) = c_0 \, \delta(\mathbf{p} - \mathbf{p}_{0})$ . Its Fourier transform is $\tilde{c}(\boldsymbol{\kappa}) = c_0 \, e^{-j\boldsymbol{\kappa}^\mathsf{T}\mathbf{p}_{0}}$ . If we set the coordinate origin at $\mathbf{p}_{0}$ (i.e., $\mathbf{p}_{0} = \mathbf{0}$ ), then $\tilde{c}(\boldsymbol{\kappa}) = c_0$ for all $\boldsymbol{\kappa}$ .

Scattered data

By the FDT, $x(\mathbf{s}_{i}, \mathbf{r}_{j}; f_k) = \alpha_{i,j,k} \cdot c_0$ — each measurement is proportional to $c_0$ with a known factor $\alpha$ . All k-space samples have the same value $c_0$ .

Physical interpretation

A point scatterer has a "flat" spatial spectrum — it contributes equally at all spatial frequencies. This is the 3D analogue of a Dirac delta having a flat Fourier transform. Imaging a point scatterer is essentially sampling a constant function on an irregular k-space grid — the constant value gives the reflectivity, and the pattern of the sampling points determines the point-spread function.

Common Mistake: The k-Space Sampling Is (Almost) Never Uniform

Mistake:

Assuming that the sampled k-space points $\{\boldsymbol{\kappa}_{i,j,k}\}$ lie on a regular grid, and applying a standard FFT for reconstruction.

Correction:

The Ewald sphere construction produces irregular, non-uniform sampling in k-space. Direct application of the FFT is not possible. Instead, one must either: (1) Interpolate the irregular samples onto a regular grid and then apply FFT (NUFFT approach), or (2) Solve the linear system $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$ directly using iterative methods (conjugate gradient, ADMM, etc.), which implicitly handles the non-uniformity. Approach (2) is generally preferred in RF imaging because it allows incorporating regularization.

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Fourier Diffraction Theorem

The theorem stating that, under the Born approximation, the scattered field from a single Tx-Rx pair at a single frequency is proportional to the spatial Fourier transform of the reflectivity evaluated at the combined wavenumber $\kappa_{\mathbf{s},\mathbf{r}}$ . It is the wave-scattering analogue of the Fourier Slice Theorem used in X-ray CT.

Quick Check

According to the Fourier Diffraction Theorem, if we could measure the scattered field for ALL possible Tx directions, ALL possible Rx directions, and ALL frequencies up to some maximum, we would sample k-space:

Only along radial lines through the origin

Uniformly on a regular grid

Everywhere within a ball of radius $2\kappa_{\max}$

Only on the surface of a sphere of radius $2\kappa_{\max}$