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Computed Tomography: The Canonical Imaging Inverse Problem

Computed tomography (CT) is the template for all Fourier-based imaging methods. In X-ray CT, projections through an object are related to the object's Fourier transform via the Fourier slice theorem. This section reviews CT as the canonical inverse problem, establishing the mathematical language that extends to diffraction tomography in subsequent sections.

CT matters for RF imaging for three reasons: (1) it provides the geometric intuition for tomographic reconstruction, (2) the FBP-to-CNN pipeline (FBPConvNet) is the ancestor of the matched-filter-to-U-Net architecture used in RF imaging, and (3) many learned reconstruction methods are first demonstrated on CT before being adapted to RF.

The key difference: CT is a well-conditioned inverse problem (straight-line projections, dense angular sampling), while RF diffraction tomography is ill-conditioned (curved Ewald arcs, limited aperture, near-field effects).

Definition:
The Radon Transform

The Radon transform maps a 2D function $f(x, y)$ to its line integrals. For a projection at angle $\theta$ :

$\mathcal{R}f(\theta, t) = \int_{-\infty}^{\infty} f(t\cos\theta - s\sin\theta,\; t\sin\theta + s\cos\theta)\,ds,$

where $t$ is the signed distance from the origin to the integration line and $s$ parameterizes position along the line.

The collection of projections $\{\mathcal{R}f(\theta, t) : \theta \in [0, \pi)\}$ is the sinogram. Image reconstruction is the inversion of $\mathcal{R}$ .

Theorem: The Fourier Slice Theorem

The 1D Fourier transform of the Radon projection at angle $\theta$ equals a slice of the 2D Fourier transform of $f$ through the origin at the same angle:

$\int_{-\infty}^{\infty} \mathcal{R}f(\theta, t)\, e^{-j2\pi \nu t}\,dt = \tilde{f}(\nu\cos\theta,\,\nu\sin\theta).$

Equivalently, each projection fills a radial line in the 2D Fourier space of $f$ .

A projection collapses the 2D object along one direction. In Fourier space, collapsing a dimension is equivalent to restricting to a line through the origin perpendicular to the collapse direction.

Proof

Write the Radon projection in terms of the 2D inverse FT

Substitute $f(x,y) = \iint \tilde{f}(k_x,k_y)\, e^{j2\pi(k_x x + k_y y)}\,dk_x\,dk_y$ into the Radon integral and evaluate the inner ( $s$ ) integration.

Evaluate the line integral

The $s$ -integration produces $\delta(k_x\sin\theta - k_y\cos\theta)$ , which collapses the 2D Fourier integral to a 1D slice at angle $\theta$ :

$\int \mathcal{R}f(\theta,t)\,e^{-j2\pi\nu t}\,dt = \tilde{f}(\nu\cos\theta,\,\nu\sin\theta). \quad \blacksquare$

Definition:
Filtered Back-Projection for CT

The standard CT filtered back-projection (FBP) formula reconstructs $f(x, y)$ from projections $p(\theta, t) = \mathcal{R}f(\theta, t)$ :

$\hat{f}(x, y) = \int_0^{\pi} \left[\int_{-\infty}^{\infty} |\nu|\,\tilde{p}(\theta, \nu)\, e^{j2\pi\nu t}\,d\nu\right]_{\!t = x\cos\theta + y\sin\theta} d\theta,$

where $|\nu|$ is the ramp filter (Ram-Lak filter) that compensates for the radial sampling density in polar Fourier coordinates.

The inner bracket is the filtered projection; evaluating at $t = x\cos\theta + y\sin\theta$ is the back-projection step.

Radon Transform and FBP Reconstruction

Left: Sinogram of a simple phantom (the Radon transform over all projection angles). Right: FBP reconstruction using the selected filter and number of projections.

Increase $N_v$ to reduce streak artifacts. Compare the Ram-Lak filter (sharpest but noisiest) with the cosine filter (smoother). Observe how each projection adds a radial line of Fourier data.

Parameters

Number of projections

N_v

36

Reconstruction filter

Example: Limited-View CT and Missing Fourier Data

A Shepp-Logan phantom ( $256 \times 256$ ) is imaged with $N_v$ projections uniformly distributed in $[0, \pi)$ .

$N_v$	Fourier coverage	PSNR (FBP)
180	Full (1 deg spacing)	38.2 dB
36	Moderate (5 deg spacing)	28.5 dB
18	Sparse (10 deg spacing)	22.1 dB
9	Very sparse (20 deg spacing)	16.8 dB

Why does halving $N_v$ cause such dramatic degradation?

Solution

Fourier coverage analysis

Each projection fills a radial line in 2D Fourier space. With $N_v$ uniformly spaced projections, the angular gap between adjacent lines is $\pi / N_v$ . Halving $N_v$ doubles the angular gap, leaving wedge-shaped spectral regions unsampled.

Artifact mechanism

The missing Fourier wedges produce streak artifacts aligned with the gap directions. The streaks are the point spread function (PSF) sidelobes caused by incomplete angular sampling. Each halving of $N_v$ removes approximately half the Fourier data, causing a roughly 6 dB PSNR loss.

Connection to RF imaging

In RF diffraction tomography, the situation is worse: the Fourier data lies on curved arcs (Ewald circles) rather than straight lines, and practical systems have far fewer views ( $N_v \sim 4$ -- $36$ ). This motivates the sparse reconstruction methods of Chapter 13.

Historical Note: The Invention of CT

1917--1979

Godfrey Hounsfield built the first clinical CT scanner in 1971, sharing the 1979 Nobel Prize in Physiology or Medicine with Allan Cormack, who had independently developed the mathematical theory. The Fourier slice theorem had been known since Johann Radon's 1917 paper, but it took 54 years for engineering to catch up with the mathematics. Today, CT reconstruction algorithms process hundreds of projections in milliseconds -- the same algorithmic ideas (FBP, gridding, iterative refinement) underpin modern RF diffraction tomography.

Common Mistake: CT Assumes Ray Optics -- Diffraction Invalidates the Model

Mistake:

Applying the Fourier slice theorem directly to RF measurements assumes straight-line projections (no diffraction, no refraction). This is valid when the wavelength $\lambda$ is much smaller than the object features ( $\lambda \ll a$ ) and the refractive index contrast is small.

In RF imaging, $\lambda$ is often comparable to the object size (e.g., $\lambda = 3$ cm at 10 GHz, imaging a 30 cm object). The Fourier slice theorem fails, and the straight lines in Fourier space become curved arcs.

Correction:

Use the Fourier diffraction theorem (Section s02), which accounts for wave diffraction. Each measurement maps to a point on an Ewald circle in Fourier space, not a radial line.

From FBP to FBPConvNet: Why CT Matters for RF Imaging

The FBP algorithm is fast but produces artifacts with limited data. FBPConvNet (Jin et al., 2017) replaces the post-processing step with a U-Net trained on paired (FBP artifact, ground truth) images. This FBP-to-CNN pipeline is the direct ancestor of the matched-filter-to-U-Net architecture used in RF imaging (Chapter 12, Chapter 13).

The pattern is general: start with a fast, physics-based initialization (FBP or matched filter), then refine with a learned post-processor. Understanding FBP deeply is essential for designing the physics-based backbone of learned RF imaging methods.

Quick Check

In the Fourier slice theorem, each CT projection at angle $\theta$ provides samples of the 2D Fourier transform along:

A circle of radius $k_0$ centered at the origin

A radial line through the origin at angle $\theta$

A horizontal line at height $\theta$

A spiral path determined by the projection angle

Correction:

A radial line through the origin at angle

\theta

Each projection fills a radial line in Fourier space -- this is the Fourier slice theorem.

Radon Transform

A linear integral transform that maps a 2D function to the set of its line integrals over all orientations and offsets. Inverting the Radon transform is the mathematical core of computed tomography.

Sinogram

The 2D representation of all Radon projections of an object, with projection angle $\theta$ on one axis and offset $t$ on the other. A point scatterer traces a sinusoidal curve in the sinogram, hence the name.

Related: Radon Transform

Filtered Back-Projection (FBP)

The standard analytical algorithm for inverting the Radon transform. Each projection is filtered with a ramp (or modified ramp) filter in frequency domain, then back-projected (smeared) across the image at its original angle. Complexity: $O(N^2 N_v)$ for an $N \times N$ image with $N_v$ projections.

Related: Radon Transform, Sinogram

Key Takeaway

CT is the canonical template for tomographic imaging. The Fourier slice theorem says each projection fills a radial line in Fourier space; FBP inverts this relationship. Limited-view imaging creates spectral gaps and streak artifacts. CT assumes ray optics (straight-line propagation); when $\lambda$ is comparable to the object size, diffraction bends the rays into arcs, and the Fourier diffraction theorem replaces the Fourier slice theorem.

Computed Tomography as the Canonical Inverse Problem