## 2 New Tomographic Techniques
### Summary  
In this subsection, as the summary of this section, let us compare the tomographic techniques introduced in this section. Before the summary, let us note that there is another tomographic technique called magnetic resonance electrical impedance tomography (MREIT, [11]). This technique is a combination of MRI and EIT, which is not a tomographic technique without X-ray nor MRI, hence omitted to be introduced in this section, for the reasons mentioned in the introduction of this paper.  
In acoustic tomography, the device being not so expensive, a number of observation data are required for the reconstruction of the velocity distribution \(C(x)\). We cannot say that the reconstruction of the acoustic tomography is very sharp, for the time being. If a rough sketch of the velocity distribution \(C(x)\) is sufficient or if the object is very big, like an ocean tomography, then this method can play an important role (cf. [3]).  
In EIT, mathematical treatment of the problem being same as acoustic tomography, the device is more expensive and the reconstruction is sharper, but more development is still required to function as a sharp medical imaging technique like CT and MRI.  
Optical tomography and PAT being new and interesting non-invasive inspection techniques, they should also be more improved as sharp medical imaging techniques.  
In many cases, these new techniques are applied to obtain some rough sketches of the object for examination, not for an inspection. Note that by 'examination', especially in medical application, we mean the cheaper-running and rougher-sketch-obtaining inspection in order to determine whether a more detailed inspection is necessary or not.  
We would like a new technique which can be an alternative to X-ray CT or MRI for non-destructive or non-invasive inspection of the object, not for rough examination. In this view, ultrasonic tomography seems to be possible to be the sharpest imaging technique by virtue of Property 1 and Problem 1, where we can observe as sharp data as the X-ray tomography except that the orbit of the ultrasound is not a line. However, ultrasonic CT cannot be applied for the non-invasive inspection for the human brain, because of Property 1. The ultrasound travels along the skull, where the ultrasonic speed is very fast, and would not reach the interior brain, in view of which, optical tomography is expected to be an alternative to brain CT or brain MRI. While PAT, by its physical nature, is expected to be an alternative of positron emission tomography (PET), where the X-rays are discharged at the expected interior point in the human body. We note that the required device is the cheapest in ultrasonic CT, among these three new tomographic techniques.  
In spite of the above demerit, the ultrasonic tomography is possible to be the sharpest imaging technique in the new tomographic techniques introduced in this section, by virtue of Property 1 and Problem 1. This technique can be an alternative of CT and MRI except for the non-invasive inspection of the brain. One of the best merits of the ultrasonic tomography is that the device and its running cost is much cheaper than CT and MRI and that no protective facilities are required. It may enable small clinics and hospitals to have their own CT devices, which can contribute to the development of clinical medicine all over the world. We also note that the cost of the ultrasonic tomography can be the cheapest in the new tomographic techniques introduced in this section. By the inexpensiveness of its cost and by the ease of its application, the ultrasonic tomography can be an effective technique to non-destructively inspect the concrete structures, where it is not suitable to spend too much cost since the concrete itself is very cheap.  
In view of the above argument, we shall focus on the ultrasonic tomography and study it in the next section.