Chapter Summary

Chapter 15 Summary: Diffraction Tomography

Key Points

• 1.

  CT as the canonical inverse problem. The Radon transform and Fourier slice theorem provide the mathematical template for tomographic imaging. Each projection fills a radial line in Fourier space; limited views create spectral gaps and streak artifacts. FBP inverts the Radon transform using a ramp filter and back-projection.

• 2.

  The Fourier diffraction theorem. Under the Born approximation, scattered far-field data map to the object spectrum on Ewald circles (not straight lines as in CT). The Ewald vector $\mathbf{K} = k_0(\hat{\mathbf{k}}_s - \hat{\mathbf{k}}_i)$ links each measurement to a Fourier-space point. In the high-frequency limit, the FDT reduces to the Fourier slice theorem.

• 3.

  Direct inversion. Gridding + inverse FFT provides fast reconstruction for well-sampled configurations. The density compensation function corrects for non-uniform k-space sampling. NUFFT provides guaranteed error bounds. Iterative methods (SIRT, ART, CGLS) handle incomplete angular coverage with implicit regularization.

• 4.

  Multi-frequency coverage. Each frequency contributes an Ewald sphere of different radius. Bandwidth controls range resolution ($\delta_r = c/2W$); angular aperture controls cross-range resolution ($\delta_{\text{cr}} = \lambda/(4\sin(\Delta\theta/2))$). Both diversities are needed for high-quality reconstruction.

• 5.

  Near-field diffraction tomography. Most RF imaging systems operate in the near field where spherical wavefront modeling is essential. The NF-FF transformation converts near-field data to far-field equivalent. XL-MIMO arrays have far-field distances of hundreds of meters, making near-field imaging the default regime. Evanescent waves enable sub-wavelength resolution at very short range.