1. X-ray Imaging (Radiography)

X-rays are electromagnetic waves (light) with short wavelengths. One of their major characteristics is their high penetration power through materials, as shown in Figure 1. X-ray imaging (radiography) is a method of capturing the "shadow" of an object by utilizing this high penetration power.

Figure 1: X-rays have high transmittance through various materials

When the intensity of X-rays irradiated onto a subject is $I_0$, the transmitted intensity $I$ is given by:

Here, $\mu$ represents the linear attenuation coefficient, and $t$ is the thickness of the subject. Generally, the denser (heavier) the material, the larger $\mu$ becomes, making it harder for X-rays to pass through.

Consequently, in a radiograph, denser parts of the subject appear darker, while lighter parts appear brighter. (Note: Since X-ray film is developed with inverted brightness, denser areas are displayed as brighter on traditional film.)

2. X-ray CT

Because X-ray absorption depends on two variables—the thickness ($t$) and the linear attenuation coefficient ($\mu$, related to density)—it is impossible to distinguish between objects that have different thickness and attenuation coefficients but the same product ($\mu t$), as shown in Figure 2.

Figure 2: Radiography cannot distinguish objects if the product $\mu t$ is the same

Furthermore, since absorption is an integrated value along the path of X-ray transmission, there is an inherent problem where the front-to-back relationship of the object cannot be detected (lack of depth resolution), as shown in Figure 3.

Figure 3: Depth direction cannot be distinguished

X-ray CT was invented to solve this problem. Below is an explanation of its principle using a highly simplified model.

Consider a sample composed of $3 \times 3$ cells, as shown in Figure 4. Let the linear attenuation coefficient of each cell be $\mu_{i,j}$, and assume the thickness $t$ is constant for all cells. The goal of CT is to determine the distribution of these linear attenuation coefficients.

When X-rays of intensity $I_0$ are incident from the left, the intensity of X-rays transmitted through cell (1,1) is:

For the next cell (1,2), $I_{11}$ becomes the incident intensity, so the intensity $I_{12}$ is:

Similarly, the intensity $I_{13}$ emerging from the last cell is:

Substituting $I_0$ into the equation gives:

Similarly, for X-rays emerging from cells (2,3) and (3,3):

From this, it is clear that we only need to consider the summation of the linear attenuation coefficients for each cell.

Figure 4: Principle of X-ray CT (1)

When X-rays are irradiated from the left, top, right, and bottom directions, the sums of the linear attenuation coefficients are as shown in Figure 5.

Figure 5: Calculating the sum of linear attenuation coefficients for each direction

When these sums are rearranged with the angle on the vertical axis and the sums on the horizontal axis, we get Figure 6. This representation is called a Sinogram. Since this data is obtained through measurement, the distribution of linear attenuation coefficients is calculated from this sinogram.

Figure 6: Table summarizing Figure 5: Sinogram

Specifically, the calculation is performed by following the reverse procedure of Figure 5 (Figure 7):

1. Initialize all cells to zero.

2. For data obtained from the left, add the values to each cell from the right.

3. Similarly, for data from the top, add the values from the bottom.

4. Repeat for data obtained from the right and bottom.

5. Divide the values of all cells by the number of projection angles (in this case, 4).

Figure 7: CT (Back-projection) Calculation

Because the data is "projected back" and summed, this method is called "back-projection." Comparing the result of this back-projection with the original distribution shows that the trends are somewhat similar (Figure 8).

Figure 8: Comparison between the original distribution and the back-projection result (reconstruction)

The primary reason they do not match perfectly is that more additions occur in the central region. To solve this problem, back-projection is generally performed after applying a high-pass filter (convolution). This method is known as Filtered Back-projection (FBP). Various types of filters exist, with the most famous being the Shepp-Logan filter.

A real CT measurement follows these steps:

1. Rotate the sample and acquire multiple projection images.

2. Calculate the sinogram for each position.

3. Apply filter convolution.

4. Perform back-projection.

Figure 10 shows cross-sectional images reconstructed without a filter, with the Shepp-Logan (SL) filter, and with a noise-reduction filter (which suppresses high-frequency components).

Figure 10: Reconstruction examples using various filter functions (Electrolytic Capacitor)