The technique of
filtered back projection is one of the most established algorithmic techniques for this problem. It is conceptually simple, tunable and
deterministic. It is also computationally undemanding, with modern scanners requiring only a few milliseconds per image. However, this is not the only technique available: the original EMI scanner solved the tomographic reconstruction problem by
linear algebra, but this approach was limited by its high computational complexity, especially given the computer technology available at the time. More recently, manufacturers have developed
iterative physical model-based
maximum likelihood expectation maximization techniques. These techniques are advantageous because they use an internal model of the scanner's physical properties and of the physical laws of X-ray interactions. Earlier methods, such as filtered back projection, assume a perfect scanner and highly simplified physics, which leads to a number of artifacts, high noise and impaired image resolution. Iterative techniques provide images with improved resolution, reduced noise and fewer artifacts, as well as the ability to greatly reduce the radiation dose in certain circumstances. The disadvantage is a very high computational requirement, but advances in computer technology and
high-performance computing techniques, such as use of highly parallel
GPU algorithms or use of specialized hardware such as
FPGAs or
ASICs, now allow practical use.
Basic principle In this section, the basic principle of tomography in the case that especially uses tomography utilizing the parallel beam irradiation optical system will be explained. Tomography is a technology that uses a tomographic optical system to obtain virtual 'slices' (a tomographic image) of specific cross section of a scanned object, allowing the user to see inside the object without cutting. There are several types of tomographic optical system including the parallel beam irradiation optical system. Parallel beam irradiation optical system may be the easiest and most practical example of a tomographic optical system therefore, in this article, explanation of "How to obtain the Tomographic image" will be based on "the parallel beam irradiation optical system". The resolution in tomography is typically described by the
Crowther criterion. Fig. 3 is intended to illustrate the mathematical model and to illustrate the principle of tomography. In Fig.3, absorption coefficient at a cross-sectional coordinate (x, y) of the subject is modeled as μ(x, y). Consideration based on the above assumptions may clarify the following items. Therefore, in this section, the explanation is advanced according to the order as follows: • (1)Results of measurement, i.e. a series of images obtained by transmitted light are expressed (modeled) as a function p (s,θ) obtained by performing radon transform to μ(x, y), and • (2)μ(x, y) is restored by performing inverse radon transform to measurement results.
(1) The Results of measurement of p(s,θ) in a parallel beam irradiation optical system Consider the mathematical model where the
absorption coefficient of the object at each point (x,y) is represented by the function \mu(x,y) and suppose that the transmission beam penetrates without diffraction, diffusion, or reflection. Also assume the beam is absorbed by the object and its attenuation occurs in accordance with the
Beer-Lambert law. What we want to know then is the values of the function \mu. What we can measure will be the values of the function p(s,\theta). When the
attenuation conforms to the
Beer-Lambert law, the relation between {I}_{0} and I is given by equation () and the
absorbance p_{l} along the light beam path l(t) is given by equation (). Here {I}_{0} is the intensity of the light beam before transmission, while I is the beam intensity after transmission. {{NumBlk|:| \begin{align} I = I_0\exp\left({-\int\mu(x,y)\,dl}\right) = I_0\exp\left({-{\int}_{-\infty}^{\infty}\mu(l(t))\,|\dot{l}(t)|dt}\right) \end{align} {{NumBlk|:| \begin{align} p_{l} = \ln (I/I_0) = -\int\mu(x,y)\,dl = -{\int}_{-\infty}^{\infty}\mu(l(t))\,|\dot{l}(t)|dt \end{align} Here, the direction from the light source toward the screen is defined as the t direction and that perpendicular to the t direction and parallel with the screen is defined as the s direction. (Both the (t,s) and (x,y) coordinate systems are chosen such that they are reflections of each other without mirror-reflective transformation.) By using a parallel beam irradiation optical system, one can experimentally obtain the series of fluoroscopic images (these are one-dimensional images p_{\theta}(s) of a specific cross section of a scanned object) for each angle \theta between the object and the transmitted light beam. In Fig.3, the (x,y) plane rotates counter clockwise. around the point of origin in the plane in such a way "to keep mutual positional relationship between the light source (2) and screen (7) passing through the trajectory (5)." Rotation angle of this case is same as above-mentioned θ. The beam having an angle \theta is the collection of lines {l}_{\theta,s}(t), represented by equation () below. {{NumBlk|:| {l}_{\theta,s}(t) = t \begin{bmatrix} -\sin \theta \\ \cos \theta \\ \end{bmatrix} + \begin{bmatrix} s\cos \theta \\ s\sin \theta \\ \end{bmatrix} The function p_{\theta}(s) is defined by equation (). That p_{\theta}(s) is equal to the line integral of
μ(x,y) along {l}_{[\theta,s]}(t) of (eq. 3) as the same manner of (eq.2). This means that, p(s,\theta) of following (eq. 5) is a resultant of
Radon transformation of μ(x,y). {{NumBlk|:| p_{\theta}(s) = -{\int}_{-\infty}^{\infty}\mu(s\cos \theta -t\sin \theta,s\sin \theta + t\cos \theta )\,dt One can define following function of two variables (). In this article, p(s, \theta) is the collection of
fluoroscopic images. {{NumBlk|:| p(s, \theta) = p_{\theta}(s)
(2)μ(x, y) is restored by performing inverse radon transform to measurement results "What we want to know (μ(x,y))" can be reconstructed from "What we measured ( p(s,θ))" by using
inverse radon transformation . In the above-mentioned descriptions, "What we measured" is p(s,θ) . On the other hand, "What we want to know " is μ(x,y). So, the next will be "How to reconstruct μ(x,y) from p(s,θ)". ==Spiral CT==