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.
Ultrasound imaging is one of the most widely used medical imaging modalities, providing non-invasive, real-time visualization of internal body structures. It plays a crucial role in diagnostics, guiding treatments and monitoring patient health. Its safety, accessibility, and cost-effectiveness make it an essential tool in modern healthcare.
In conventional ultrasound imaging, a series of focused acoustic beams is transmitted into tissue, and the returning echoes are received and processed to form an image [20]. While highly effective for visualizing anatomical structures, this method yields primarily qualitative information about tissue structure, rather than quantitative data about its physical properties. Quantitative ultrasound imaging, in contrast, seeks to reconstruct tissue acoustic parameters from the measured signals. This approach allows for objective tissue characterization and enhances the ability to distinguish between healthy and pathological regions, making it highly relevant for clinical applications.
The pursuit of quantitative imaging has long motivated research in ultrasound, dating back to the development of ultrasound computed tomography in the 1970s [15]. Within this context, diffraction tomography has become a well-established approach [30,32]. Here, a weakly scattering object is illuminated by an incident wave, and information about its internal structure is inferred from measurements of the scattered field. A major advantage of diffraction tomography is its computational efficiency. By employing the first-order Born (or Rytov) approximation, the non-linear inverse scattering problem is simplified to a linear one. The resulting Fourier diffraction theorem [35] establishes a direct link between the Fourier transform of the measured data and the object’s scattering potential, enabling explicit reconstruction via Fourier inversion, commonly referred to as filtered backpropagation [12]. However, classical diffraction tomography relies on idealized assumptions regarding both the incident field and the measurement geometry. Here, the object is illuminated by monochromatic plane waves arriving from a broad range of directions [21,30,35]. While this approach works well for certain imaging scenarios where the object can be surrounded by transmitters and receivers, this setting does not reflect clinical practice in ultrasound imaging, where beams are typically emitted only from one side of the body, and, crucially, these beams are focused. Focusing concentrates acoustic energy at the focal region and thereby enables the visualization of deeper tissue layers with sufficient resolution [26]. This stands in sharp contrast to the plane-wave, full-angle illumination assumed in the classical framework. Recent work [22] 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.
In this work, we go one step further by considering a series of focused beams that are actively scanned across the region of interest. Moreover, recent advances in hardware development, such as multi-aperture [17] and flexible transducer [31], are expanding what is experimentally possible. These devices could, in principle, emit, scan, and record signals from a much broader range of angles and focal positions than conventional probes. To fully exploit these capabilities and to bridge the gap between classical diffraction tomography and practical ultrasound systems, a theoretical framework is required that can accommodate arbitrary scanning geometries. Therefore, we extend the diffraction tomography framework to a general scanning setup, see Figure 1, which combines focused illumination with active scanning. We consider an object in a general Euclidean space R d , d ě 2, probed by a focused beam propagating in a prescribed direction ω P S d´1 :" tx P R d | }x} " 1u. The incident beam is hereby shifted such that the focal point moves on a predefined hyperplane orthogonal to a chosen direction ν P S d´1 . For each position of the focal point, the interaction with the object generates a scattered wave field, which is subsequently measured at every point on the plane R d´1 ˆtLu. We call this setup Raster Scan Diffraction Tomography.
The parameters ω and ν in our formulation can be independently chosen, allowing for a wide range of scanning arrangements from standard linear probes to more exotic configurations. We derive a new Fourier diffraction theorem for such a scan geometry, and analyze which Fourier coefficients of the scattering potential of the object are accessible from the measurements. By systematically characterizing how different choices of beam direction ω and scan normal ν influence the recoverable parts of the object’s Fourier spectrum, we provide new insights into how the scanning geometry governs the object information and which scanning geometries are optimal for practical quantitative reconstructio
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