Hasan Camci, winter term 22/23
Ultrasonic tomography is a specific application of ultrasonic testing that involves creating cross-sectional images of the inside of an object or material. It is like medical computed tomography (CT) in that it produces images that can visualize the internal structure of the object. It is based on the process, known as image reconstruction, which involves solving an inverse problem to deduce the properties of the object from the measured data.
1.3 Challenge of image reconstruction
1 Introduction
1.1 Ultrasonic testing
Ultrasonic testing is a non-destructive testing (NDT) method that uses ultrasonic waves to inspect the internal structure of an object or material. It is widely used in a variety of applications, including structural health monitoring, nondestructive testing of engineering components, and medical imaging. In ultrasonic testing, an ultrasonic transducer is used to transmit ultrasonic waves into the object or material being tested. Because of the high differences in the impedance and thus the high energy loss in air, a coupling medium, with a lower impedance difference is used in many applications. The waves travel through the object and are reflected to the surface by internal interfaces between different materials or by defects within the material. The reflected waves are then detected by the receiver and used to infer the internal structure of the object. Depending on the accessibility there are different ways to setup the transducer. One way is to locate the receiver directly underneath. Figure 1 illustrates the process of the described testing method [1].
Figure 1: Illustration of ultrasonic testing (Own representation based on [1])
1.2 Ultrasonic tomography
Ultrasonic tomography is a specialized use of ultrasonic testing that generates cross-sectional images of the interior of objects or materials. Like medical computed tomography (CT), ultrasonic tomography provides images that reveal the internal structure of an object. This process involves the reconstruction of an image through the solution of an inverse problem, which utilizes measured data to determine the properties of the object. There are different methods that can be used for image reconstruction, including travel time tomography, attenuation tomography and full waveform inversion. These methods were introduced in order to tackle the challenges of image reconstruction, which are discussed in the following paragraph.
1.3 Challenge of image reconstruction
There are several different aspects that evoke difficulty to recreate images. A few of them are addressed in the following:
- Limited data: Ultrasonic measurements are typically taken at a limited number of locations on the surface of the object, which can make it difficult to accurately reconstruct the entire image.
- Noise and errors in the data: Measurement noise and errors can reduce the accuracy of the reconstructed image.
- Inverse problem: Image reconstruction involves solving an inverse problem, where the goal is to deduce the properties of the object from the measured data. Inverse problems are generally more difficult to solve than forward problems, where the goal is to predict the measured data given the properties of the object.
- Computational complexity: Image reconstruction algorithms can be computationally intensive, which can make them impractical for large or complex objects.
- Limited resolution: The resolution of the reconstructed image is limited by the number and placement of the ultrasonic sensors, as well as the wavelength λ of the ultrasonic waves. For the best resolution, the wavelength should be aimed for. The shorter the wavelength the better the resolution but also the weaker the material penetration.
To address these challenges, researchers have developed a variety of image reconstruction algorithms and techniques such as travel time tomography, attenuation tomography, and full waveform inversion. Each of these methods has different trade-offs in terms of accuracy, computational complexity, and sensitivity to noise and errors in the data [2].
2 Reconstruction Methods
In this sequence the previous mentioned methods for ultrasonic tomography are going to be explained. These are travel time tomography, attenuation tomography and full waveform inversion. For each method, we will discuss the steps involved in implementing it, as well as its assets and drawbacks.
2.1 Travel time tomography
Travel time tomography is a method that uses the time it takes for ultrasonic waves to travel through the object to reconstruct the image. It is based on the principle that the speed of sound is slower in materials with higher densities, by evaluating the travel time of the ultrasonic waves, it is possible to infer the density and thus the structure of the object. For the implementation of this method, you first examine the time of flight (ToF) between the transducer pair according to the following equation:
| s_{Travel}=\frac{D}{\Delta t} |
where D is the distance traveled by the ultrasonic wave through the object, s is the speed of sound in the material and is the ToF. This part of the method is called the “Transmission forward model”. It can be described as a system of linear equation as:
| s = A \vec{V} |
where s is the vector of the utilized sound-speed, A is a M x N sensitivity matrix containing the weighted values of the effectiveness of each transducer ray on the pixel matrix and \vec{V} the vector of the discretized sound-speed-spatial-distribution. The main challenge consists of solving the inverse image reconstruction problem, which can be defined as
| \Delta t = A \Delta V +e |
where the additionally unknow is the noise in the measurement and as the velocity profile in different time sections. The challenging part of this inverse problem is that A, with the number of acoustic tomography measurements, is significantly smaller than \Delta{V} with the number of reconstructed image pixels. As a result, the Matrix A is not invertible and has to be dealt as an optimization problem. One way to solve this problem is using the total variation regularization algorithm, which is more detailed explained in [3] The method is easy and fast to implement but is limited in its ability to reconstruct the details of the object [2]. Therefore, different methods have been introduced to tackle this challenge.
2.2 Attenuation tomography
Attenuation tomography is an advanced technique that employs the loss of intensity of ultrasonic waves as they pass through an object to reconstruct an image. It is based on the principle that the intensity of ultrasonic waves decreases as they pass through materials with higher attenuation. By quantifying the intensity of the ultrasonic waves, it is possible to gain information about the internal structure of the object.
To implement attenuation tomography, an ultrasonic pulse is transmitted into the object from a transmitter on the surface. For this example, water is being used as the coupling medium. The pulse travels through the water and object and is captured by a receiver on the surface. The intensity of the pulse is then measured at the receiver, and the attenuation of the pulse is calculated using for example the decay method:
| \int \limits_{ray}^{} a_{0} dl = \frac{1}{f_{c}} \ln\frac{A_{s}}{A_{r}} |
where A_{r} is the received waveform amplitude,A_{s} is the source amplitude measured in a water shot at the same receiver, f_{c} is the central frequency, a_{0} is attenuation parameter to be reconstructed, and l is the length of the ultrasound ray path. This process is repeated for multiple pairs of transmitters and receivers at different locations on the surface of the object. The attenuation measurements can then be used to reconstruct the image of the object using a variety of algorithms. One common algorithm is the bent-ray tomography algorithm to reconstruct the attenuation parameters. For this algorithm, a first time of flight measurements creates a sound-speed distribution, which will be used to trace the bent ray path and attain the attenuation parameters. More comprehensive explaining is given in [4].
2.3 Full waveform inversion
Full waveform inversion is a more advanced procedure that utilizes the entire waveform of ultrasonic waves to reconstruct an image. The shape of the ultrasonic waveform is affected by the properties of the materials through which it passes. By measuring the waveform of the ultrasonic waves, it is possible to obtain the properties of the object. It was introduced to overcome the constraints of limited sensor counts, shape of specimen and internal reflectivity of waves for internal flaws.
To perform full waveform inversion, an initial model of the wave speed of the unflawed structure is used to generate simulated sensor measurements through a process called forward modeling. An advanced gradient-based minimization procedure called L-BFGS is then used to minimize the difference (or "misfit") between the experimental measurements of the flawed specimen and the simulated measurements. This process is iterated to refine the initial model and improve the accuracy of the reconstructed image [6]. Figure 2 describes the adaptive process used for a full waveform inversion.
Figure 2: Process of full waveform inversion (Own representation based on [6])
Full waveform inversion is the most accurate and detailed of the three reconstruction methods described, but it is also the most computationally intensive and may require specialized software to implement. It is sensitive to the specific properties of the materials that make up the object, as well as their overall density and distribution.
3 Outlook
There is ongoing research into improving the accuracy and efficiency of ultrasonic tomography, including the use of deep learning approaches as well as investigations for new transducers.
3.1 Deep Learning Approach
Deep learning is a type of artificial intelligence that involves training a computer to recognize patterns in data using multiple layers of artificial neural networks. It has been shown to be effective in a wide range of applications, and it has the potential to improve the accuracy and speed of ultrasonic tomography.
One area where deep learning has shown promise is in the image reconstruction process. Traditional image reconstruction methods rely on mathematical models to relate the measured ultrasonic data to the properties of the object being imaged. These models can be complex and computationally intensive, and they may not always accurately capture the underlying physics of the problem. Deep learning approaches, on the other hand, can learn to recognize patterns in the data directly from the data itself, without the need for explicit mathematical models. This leads to more accurate and efficient image reconstruction [7], [8].
3.2 Hardware Development
There are also ongoing efforts to develop new ultrasonic sensors that can improve the resolution, geometrical constraints, and sensitivity of ultrasonic tomography.
For example, researchers are exploring the use of metamaterials, which are artificial materials with unique electromagnetic properties, to design transducers that can focus and steer ultrasonic waves in specific directions. Other researchers are working on developing miniature ultrasonic sensors that can be used to create high-resolution images of small objects or tissues [9]. Another promising research approach is the development of flexible ultrasonic transducers. With the flexibility, the hardware can overcome the constraints of not being capable of conforming to complex shapes and therefore not maximizing the transferred wave energy. By making the transducer flexible high-quality images can be achieved. The only considerable challenge is the manufacturing effort [10].
Overall, the outlook for ultrasonic tomography is promising, with ongoing research and development aimed at improving the accuracy, efficiency, and capabilities of this important imaging technique.
4 References
[1] Tant, K. M. M.; Galetti, E.; Mulholland, A. J.; Curtis, A.; Gachagan, A. (Hrsg.): A transdimensional Bayesian approach to ultrasonic travel-time tomography for non-destructive testing. In: Inverse Problems 34 (9), (2018), S. 95002. DOI: 10.1088/1361-6420/aaca8f.
[2] Koulountzios, P., Rymarczyk, T. & Soleimani, M. A Quantitative Ultrasonic Travel-Time Tomography to Investigate Liquid Elaborations in Industrial Processes. Sensors, 23 (19), (2019), S. 5117, URL: http://dx.doi.org/10.3390/s19235117.
[3] Li, F.; Abascal, J.F.; Desco, M.; Soleimani, M. Total Variation Regularization with Split Bregman-Based Method in Magnetic Induction Tomography Using Experimental Data. IEEE Sens. J. (2017), 17, 976–985.
[4] A. Hormati, I. Jovanović, O. Roy, M. Vetterli, "Robust ultrasound travel-time tomography using the bent ray model," Proc. SPIE 7629, Medical Imaging 2010: Ultrasonic Imaging, Tomography, and Therapy, 76290I, (2010); https://doi.org/10.1117/12.844693
[5] Li, C.; Duric, N.; Huang, L. (Hrsg.): Comparison of ultrasound attenuation tomography methods for breast imaging. Proceedings of SPIE - The International Society for Optical Engineering. 10.1117/12.771433. (2008).
[6] Seidl, R.: Full Waveform Inversion for Ultrasonic Nondestructive Testing, Dissertation, Technische Universität München, (2018).
[7] Ahishakiye, Emmanuel; van Gijzen, Martin Bastiaan; Tumwiine, Julius; Wario, Ruth; Obungoloch, Johnes: A survey on deep learning in medical image reconstruction. In: Intelligent Medicine 1 (3), (2021), S. 118–127. DOI: 10.1016/j.imed.2021.03.003.
[8] Dürrmeier, F.: Anwendung von Deep Learning Ansätzen in der Luftultraschallprüfung. Semesterarbeit. Technische Universität München, (2022).
[9] M. R. K. Bhukya, A. K. Singh, and S. K. Singh, "Recent advances in ultrasonic tomography: A review," Measurement, vol. 158, (2019) , pp. 207-225.
[10] Ren, D.; Yin, Y.; Li, C.; Chen, R.; Shi, J. Recent Advances in Flexible Ultrasonic Transducers: From Materials Optimization to Imaging Applications. Micromachines, (2023), 14, 126. https://doi.org/10.3390/mi14010126.
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