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 gives a direct relation between the Fourier transform of the scattering potential of the object and the Fourier transformed 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. The Fourier diffraction relation adapted to this setting differs in one crucial point from the plane-wave case: while certain Fourier coefficients of the measurements still directly correspond to individual Fourier coefficients of the scattering potential, others are given by linear combinations of two Fourier coefficients of the scattering potential. This article investigates which Fourier coefficients of the scattering potential can be uniquely recovered from these relations. We show that in dimensions higher than two, all coefficients appearing in the equations are typically uniquely determined. In two dimensions, however, only part of the Fourier coverage is uniquely recoverable, while on the remaining subset, distinct coefficients can produce identical data.
Diffraction tomography is a widely used inverse scattering technique that aims to reconstruct the spatial distribution of an object's scattering potential from measurements of scattered wave fields. A prominent application is ultrasound tomography [1,4,12], where both amplitude and phase information of the scattered fields are available.
Diffraction tomography relies on the Born approximation, under which the dependence of the scattered field on the scattering potential becomes linear and the corresponding Fourier diffraction theorem [14] provides an explicit representation of the scattering potential in terms of the measurements in Fourier space. More precisely, it states that the Fourier-transformed measurements coincide with the Fourier transform of the scattering potential along a semicircle in two dimensions or a hemisphere in higher dimensions. Incorporating a collection of measurements from multiple views then generates data on a union of distinct hemispheres in Fourier space, enabling reconstruction of the scattering potential through filtered backpropagation [3,7,9,14].
However, the classical theory relies on idealized assumptions about both the incident field and the measurement geometry. In particular, the object is assumed to be illuminated by monochromatic plane waves arriving from a broad range of directions [7,10,14]. In practice, though, many imaging systems, such as medical ultrasound [6], use focused beams that concentrate the energy at a focal region to improve the spatial resolution. Rather than illuminating the entire object from different directions, these beams are translated to bring different regions into focus. This scanning setup differs fundamentally from the plane-wave, full-angle illumination assumed in the conventional framework, highlighting a gap between theory and practical setups.
To address this discrepancy, recent work [8] extended the classical diffraction tomography framework to account for focused beams by modeling them as superpositions of plane waves while still assuming full-angle data. Building on this, [5] further extended diffraction tomography to a raster scan geometry in which a focused beam is emitted from only one side and then translated across the object along a fixed hyperplane. Within this framework, a new Fourier diffraction relation was derived, expressing the Fourier-transformed measurements in terms of the Fourier coefficients of the scattering potential of the object. Here, depending on the scan configuration, some measured data correspond directly to individual coefficients, while others are linear combinations of two coefficients.
To obtain the involved Fourier data of the scattering potential, we therefore still have to solve a linear equation system. But before doing so, we want to check in this article for every scan setup which Fourier coefficients are in fact uniquely determined by these equations.
While the physically relevant case is cleary the three-dimensional setting, we would like to discuss the problem in arbitrary dimensions, with particular attention to the theoretically interesting two-dimensional case where the dimensions of the gathered data and the Fourier data of the scattering potential are equal.
Our results show that in dimensions higher than two, all Fourier coefficients appearing in the equations are, at least in the generic case, uniquely recoverable from the measured data. In two dimensions, however, only a subset of the coefficients is uniquely determined, while on the remaining region distinct values for the Fourier coefficients produce identical measurements.
Outline. We begin in Section 2 with the formulation of the inverse problem and recall the Fourier diffraction relation obtained in [5]. In Section 3, we then investigate which pairs of Fourier coefficients of the scattering potential are interconnected by these relations. This structural analysis already reveals fundamental differences between the two-dimensional, the three-dimensional, and the higher-dimensional cases. Accordingly, we address these situations separately in Section 4, Section 5, and Section 6, where we start with the problem in more than three dimensions as the surplus of measurement dimensions simplifies the analysis, and we end with the critical two-dimensional case.
We begin by briefly outlining the experimental setup underlying to our analysis, see Figure 1. The aim is to image an unknown object, represented by its complex-valued scattering potential f ∈ L 1 (R d ) in the arbitrary spatial dimension d ∈ N \ {1}, where we are, of course, mainly interested in the physically relevant cases d ∈ {2, 3}. We assume that the object is embedded in a homogeneous background medium. More precisely, we assume that supp(f ) ⊆ B d r , where we denote by
the open ball in R d of radius r > 0 centered at the origin.
As incident wave, we consider a focused acoustic beam which propagates in the direction ω ∈ S d-1 := ∂B d 1 and which is then successivel
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