arXiv subjects

Noemi Naujoks

Publications and source records attributed to Noemi Naujoks.

3 recordsLinked to original sources

arXiv2026

Invertibility of the Fourier Diffraction Relation in Raster Scan Diffraction Tomography

Diffraction tomography aims to recover an object's scattering potential from measured wave fields. In the classical setting, the object is illuminated by plane waves from many directions, and the Fourier diffraction theorem provides a direct relation between the Fourier transform of the object's scattering potential and the Fourier transform of the measurements. In many practical imaging systems, however, focused beams are used instead of plane waves. These beams are then translated across the object to bring different regions of interest into focus. This article discusses what information about the scattering potential can be extracted from such measurements. As in the classical case, the analysis is based on a recently derived Fourier diffraction relation that relates the measurements to the Fourier coefficients of the scattering potential. However, this relation does not immediately provide an explicit reconstruction formula, but instead leads to a linear equation system for the Fourier coefficients. We therefore prove in this work that all Fourier coefficients appearing in these relations are in dimensions higher than two generically uniquely determined. In the two-dimensional case, on the other hand, only a particular subset of the Fourier coverage is uniquely recoverable, while on the remaining region distinct coefficients may produce identical data.

math.NA↗

arXiv2026

Raster Scan Diffraction Tomography

Diffraction tomography is a widely used inverse scattering technique for quantitative imaging of weakly scattering media. In its conventional formulation, diffraction tomography assumes monochromatic plane wave illumination. This assumption, however, represents a simplification that often fails to reflect practical imaging systems such as medical ultrasound, where focused beams are used to scan a region of interest of the human body. Such measurement setups, combining focused illumination with scanning, have not yet been incorporated into the diffraction tomography framework. To bridge this gap, we extend diffraction tomography by modeling incident fields as Herglotz waves, thereby incorporating focused beams into the theory. Within this setting, we derive a new Fourier diffraction relation, which forms the basis for quantitative tomographic reconstruction from scanning data. Using this result, we systematically analyze how different scan geometries influence the reconstruction.

math.NA↗

arXiv2024

Diffraction Tomography for a Generalized Incident Field

Diffraction tomography is an inverse scattering technique used to reconstruct the spatial distribution of the material properties of a weakly scattering object. The object is exposed to radiation, typically light or ultrasound, and the scattered waves induced from different incident field angles are recorded. In conventional diffraction tomography, the incident wave is assumed to be a monochromatic plane wave, an unrealistic simplification in practical imaging scenarios. In this article, we extend conventional diffraction tomography by introducing the concept of customized illumination scenarios, with a pronounced emphasis on imaging with focused beams. We present a new forward model that incorporates a generalized incident field and extends the classical Fourier diffraction theorem to the use of this incident field. This yields a new two-step reconstruction process which we comprehensively evaluate through numerical experiments.

math.NA↗