Research on reconstruction of diffraction tomography based on prior information

Under coherent light illumination, several approaches need either angle scanning or diffuser rotating to reconstruct the image through opaque scattering media. We propose a linear model to restore the hidden object through the actual power spectrum with disturbance of the scattering layer. The experimental results confirm that, the algorithm quickly converge to the only correct reconstruction solution with the accuracy power spectrum pattern of Fourier transform, and the method can reconstruct the high accuracy image of the object hidden by the scattering media with one-shot power spectrum.


Introduction
Computer tomography (CT) is a revolutionary technology in the field of medical diagnosis, which reconstructs cross-sectional images by measuring the projections of human internal tissues under X-ray [1].The success of this medical technology has quickly promoted it to many other non-medical imaging fields, such as geological resource exploration, industrial non-destructive detection, celestial brightness distribution, electron microscopy and so on [2].In order to identify submarine objects, the current popular method is to rely on high-resolution sonar imaging systems, using acoustic lenses to form very narrow beams during pulse transmission and echo reception, and distinguishing different location features of targets to form images [3].However, due to high requirements on working frequency of the narrow beam, this method faces the limitation of the short range of sonar (2~5m).Therefore, the technology of CT is introduced into the field of ocean imaging to form the acoustic diffraction tomography technology, also known as echo tomography or circular synthetic aperture imaging technology [4].
Circumferential synthetic aperture imaging is an imaging technique that reconstructs the geometry and internal structural characteristics of an object by measuring one-dimensional projection data from different sides [5].The circumferential synthetic aperture sonar makes a circular or partial circular motion around the center of the measured target or the area under test.During this movement, the sonar beam always irradiates the same area, and at the same time obtains the echo data of the observation water environment or underwater suspicious targets in different sides or attitudes, divides the detection area into a series of faults along the height direction, focuses on each fault separately and obtains a twodimensional tomographic image sequence, and finally reconstructs the three-dimensional image of the detection area.
However, due to factors such as wind and waves and ship operability, the sonar carrier will deviate from the ideal motion track, and the circumferential synthetic aperture sonar often cannot be observed in all directions [6], resulting in increased difficulty in imaging.In this paper, a diffraction tomography method based on prior information is proposed.

Method
For the monostatic sonar system (source and receiving device are juxtaposed, vertical transmission and reception), it can receive and send acoustic signals.as shown in Figure 1.
where −∞ <  < ∞，0 ≤  < , and ℛ is Radon transform (projection operator).The Fourier slice theorem can be used to establish the relationship between the above-mentioned projection and the two-dimensional Fourier transform of the target cross-section [7][8][9].The Fourier slice theorem plays a crucial role in various imaging applications, including medical imaging, where it enables the reconstruction of 2D or 3D images from a series of projection images acquired from multiple angles.It allows us to recover detailed structural information from limited projections, thus aiding in the diagnosis and analysis of various conditions.In the Fourier slice theorem, we consider a specific line profile in the 2D projection image.This line profile is essentially a set of intensity values taken along a straight line passing through the center of the image.The Fourier slice theorem states that the Fourier transform of this line profile is equivalent to a slice or a cross-section of the 2D Fourier transform of the original object.This relationship can help accurately reconstruct the cross-sectional image.However, the technology of circular synthetic aperture imaging requires multiple angle scanning and projection in the range of [0, ] to complete a clear reconstruction of target image, which puts forward higher requirements for the actual data acquisition process.Because of the complex environment of the seafloor and the difficulty of collecting echo signals, the projection data used for reconstruction might be missing, and the reconstruction resolution will be too low.These situations will cause the reconstruction fail and the target features cannot be recognized.To this end, we proceed from the actual situation of the detection process and based on the known prior information of the specific target, we can only use a few sets of projection data to obtain the key parameters of the target, thereby achieving the purpose of target recognition and detection.Combining the method of circular synthetic aperture imaging and applying the symmetry of a special object, the following physical model diagram is established in Figure 2: Schematic diagram of data collection system.Using the symmetry of the target can greatly reduce the amount of projection data needed to collect.Due to geometric symmetry of the cylinder, it can be predicted that the projection data distribution characteristics of each angle are similar.Therefore, in theory, the projection data can be approximately extended to each measurement in the range (0 to π) through one set of data, and then the method of filtered back projection (FBP) can be used for reconstruction to obtain the geometric characteristics (diameter in this paper) of the cylinder [10][11].The physical process is shown in Figure 3: Filtered back projection is a reconstruction technique commonly used in computed tomography (CT) imaging to reconstruct an image from a set of projection data.It involves two main steps: filtering and back projection.Filtering: In the first step, the raw projection data is filtered using a specific filter function.The purpose of filtering is to enhance image quality by selectively attenuating certain frequencies or spatial components in the projection data.This helps to reduce noise and artifacts in the reconstructed image.Back projection: Once the projection data is filtered, the back projection step takes place.During back projection, each filtered projection is assigned to its corresponding location in the reconstruction grid.By back projecting the filtered data, information from all angles is spread throughout the image space, contributing to the final reconstructed image.By combining the filtering and back projection steps, filtered back projection enables the reconstruction of high-resolution images from a series of projection data obtained from different angles.
That is: Project a special object (, ) (e.g.cylinder) at a specific angle ; Use the detector to collect the receiving data (,   ); Expand the data to 2π, and then arrange these 360 sets of data in order to obtain the signal (, ) ; Perform one-dimensional Fourier transform ℱ 1 on the signal to obtain the Fourier spectrum (, ) of the echo data; Apply the Fourier slice theorem (or Fourier diffraction projection theorem) to know that the spectrum is consistent with the two-dimensional Fourier transform spectrum ℱ  (, ); Then convert the polar coordinates to rectangular coordinates (bilinear interpolation) to obtain the standard Fourier spectrum (  ,   ) that can be used for Fourier reconstruction.Subsequently, in theory, the two-dimensional inverse Fourier transform ℱ 2 −1 can be directly applied to obtain the reconstructed object function (, ); and then the geometric feature (e.g.diameter) of the cylinder can be directly obtained.

EXPERIMENTS
In actual experiments, we applied the method named filtered back projection (FBP) to reconstruct the images of special object (circular symmetry).We designed a numerical simulation experiment in which a cylinder with a diameter of 20 cm was used as the target.After analysing the projection data in each direction of the cylinder, we can see that the symmetry results in the extremely similar distribution of the projection data at each angle (as shown in Figure 4 (a)).Therefore, we use the shape feature (symmetry) of the target as a priori information.Base on this, we can only collect one set of projection data under a certain angle, and expand it to other angles to meet the requirement of the data volume in FBP method (The process of FBP is shown in Figure 4 (b)).The numerical simulation results of our method are shown in Figure 4 (c), which can prove that the projected data after extension can accurately reconstruct the important characteristic parameters (e.g.diameter) of the target.

Conclusion
Based on the traditional technology of circular synthetic aperture, we have successfully realized the diffraction tomography with fewer number of measurements by using the prior information of special ICFOST-2023 Journal of Physics: Conference Series 2718 (2024) 012053 targets.By using the symmetry of spherical objects, the key features (diameter parameters) of the object can be obtained successfully with only 11 sets of projection data, which reduces the dependence of 360 degree measurement in the traditional method.This study will shorten the time of data acquisition in the circular synthetic aperture technology and further broaden the practicability of the technology.

Figure 1 .
Figure 1.Schematic diagram of the monostatic sonar system [4]. 0 is the distance of the origin from the transducer.  is the perpendicular distance of the line from the origin O.In Figure 1, a Cartesian rectangular coordinate is established, which perpendiculars to the plane.The object function (, ) is reconstructed from the three-dimensional reflection projection function (, , )(acoustic reflection tomography) of the object: (, ) = ∫ (, , ).After derivation, it can be seen that the total projection (, ) is the Radon transform of the object function (, ):

Figure 3 .
Figure 3. Schematic diagram of reconstruction method based on the prior information.Filtered back projection is a reconstruction technique commonly used in computed tomography (CT) imaging to reconstruct an image from a set of projection data.It involves two main steps: filtering and back projection.Filtering: In the first step, the raw projection data is filtered using a specific filter function.The purpose of filtering is to enhance image quality by selectively attenuating certain frequencies or spatial components in the projection data.This helps to reduce noise and artifacts in the reconstructed image.Back projection: Once the projection data is filtered, the back projection step takes place.During back projection, each filtered projection is assigned to its corresponding location in the reconstruction grid.By back projecting the filtered data, information from all angles is spread throughout the image space, contributing to the final reconstructed image.By combining the filtering and back projection steps, filtered back projection enables the reconstruction of high-resolution images from a series of projection data obtained from different angles.That is: Project a special object (, ) (e.g.cylinder) at a specific angle ; Use the detector to collect the receiving data (,   ); Expand the data to 2π, and then arrange these 360 sets of data in order to obtain the signal (, ) ; Perform one-dimensional Fourier transform ℱ 1 on the signal to obtain the Fourier spectrum (, ) of the echo data; Apply the Fourier slice theorem (or Fourier diffraction projection theorem) to know that the spectrum is consistent with the two-dimensional Fourier transform spectrum ℱ  (, ); Then convert the polar coordinates to rectangular coordinates (bilinear interpolation)

Figure 4 .
Figure 4.The numerical simulation results of our method.(a) circular symmetry.(b) The reconstruction process.(c) FBP results.