Funciones y atribuciones del BCE en una economía dolarizada

The x-ray project a high dimensional object to a lower dimensional detector measurement, this process is mathematically described by the Radon transform, which was introduced in 1917 by Johan Radon. More strictly, the Radon transform is the integral transform which takes a high dimensional function f to a low dimensional function 𝑅(𝑓). The Radon transform is schematically illustrated in Figure 3-4. A function 𝑓(𝑥, 𝑦) which represents the imaged object is defined in x-y coordinate, a line integral 𝐿(𝜃, 𝑡), which represents the an x-ray beam, is performed at an angle 𝜃

the detector function 𝑝𝜃(𝑡) is obtained. We say the detector function 𝑝𝜃(𝑡) is the Radon transform of the function 𝑓(𝑥, 𝑦):

ℛ𝑓(𝜃, 𝑡) = ∫ 𝑓(𝑥, 𝑦)𝑑𝑠 𝐿(𝜃,𝑡)

= 𝑝_𝜃(𝑡). (3-7)

The 𝑥-𝑦 and the 𝜃-𝑡 follows the relationship:

𝑥 cos 𝜃 + 𝑦 sin 𝜃 = 𝑡. (3-8)

Figure 3-4: Schematic illustration of Radon transform.[61]

The detector function 𝑝_𝜃(𝑡) at each imaging/integral angle 𝜃 is called an x-ray projection. If we measure the detector functions 𝑝_𝜃(𝑡) over all imaging angles and combine them together into a single graph, we will obtain the full representation of the Radon transform, also called sinogram. The name “sino-” comes from the fact that a point signal will appears like a sinusoidal

An example of sinogram from a Shepp-Logan phantom is shown in Figure 3-5.

In practice, only limited number of projection images are acquired during one CT scan, which varies from one hundred to three hundred projections, and there is a small gap between two projections. In addition, the detector function 𝑝_𝜃(𝑡) will be discretized to pixels, which have a finite size. Therefore, the continuous Radon transform becomes a discretized format, shown as:

𝑝_𝑖 = 1 ∆𝑡∫ ℛ𝑓(𝜃, 𝑡 + 𝑡 ′_)𝑑𝑡′ ∆𝑡 2⁄ −∆𝑡 2⁄ . (3-9)

Figure 3-5: (a) Shepp-Logan phantom. (b) The sinogram of the Shepp-Logan phantom.

3.2.2 Fourier Slice Theorem

From the Radon transform, we can derive an important theorem, called Fourier slice theorem or projection slice theorem, which is the fundamental of all analytical reconstruction algorithms. The Fourier slice theorem states that the Fourier transform of a detector function 𝑝𝜃(𝑡),

as the line profile of the Fourier transform of the same object function 𝑓(𝑥, 𝑦), as illustrated in Figure 3-6.[61] Fourier slice theorem indicates a connection between the low dimension detector function and the high dimension object function through Fourier transform, thus it provides a way to restore the object function from the measurements.

Figure 3-6: Schematic illustration of the Fourier slice theorem. The radon transform 𝑝𝜃(𝑡) of a function 𝑓(𝑥, 𝑦) is equaled to the line profile of the Fourier transform of the same function at the angle 𝜃.[61]

The Fourier slice theorem can be proved as the following. Consider the Fourier transform of the object function 𝑓(𝑥, 𝑦):

ℱ(𝜁, 𝜂) = ∫ ∫ 𝑓(𝑥, 𝑦)𝑒−𝑖2𝜋(𝜁𝑥 + 𝜂𝑦)𝑑𝑥𝑑𝑦 ∞ −∞ ∞ −∞ , (3-10)

where ℱ(𝜁, 𝜂) is the Fourier transform of the object function 𝑓(𝑥, 𝑦), and 𝜁, 𝜂 is the coordinate of the Fourier space. Remember the Fourier transform of the detector function 𝑝𝜃(𝑡) is:

Ρ(𝜔) = ∫ 𝑝𝜃(𝑡)𝑒−𝑖2𝜋𝜔𝑡𝑑𝑡 ∞

3-8 into Equation 3-11, we obtain: Ρ(𝜔) = ∫ ∫ 𝑓(𝑥, 𝑦)𝑒−𝑖2𝜋𝜔(𝑥 cos 𝜃+𝑦 sin 𝜃)𝑑𝑥𝑑𝑦 ∞ −∞ ∞ −∞ = ℱ(𝜔 cos 𝜃 , 𝜔 sin 𝜃). (3-12)

Since ℱ(𝜔 cos 𝜃 , 𝜔 sin 𝜃) is the line profile of the ℱ(𝜁, 𝜂) at the angle 𝜃, the Fourier slice theorem is proved.

3.2.3 Reconstruction Based on Fourier Slice Theorem

Based on the Fourier slice theorem, one can Fourier transform each projection in the sinogram, map the results to the Fourier space in the polar coordinate, then interpolate the Fourier space to find the values of Fourier space in Cartesian coordinate, and finally perform an inverse Fourier transform to obtain the original image. However, this approach has two main drawbacks: first, the interpolation that transforms the polar coordinate to the Cartesian coordinate is prone to errors. Second, the Fourier space is well-sampled in the low-frequency regions but not in the high- frequency region. This would result in a loss of high-frequency information in the original image and introduce image blur. To fix this problem, several Fourier projectors have been investigated. Fessler proposed a non-uniform fast Fourier transform, in which the inverse Fourier transform in the last step is performed on a non-uniform grid.[62]–[64] An iterative reconstruction algorithm is also developed based on the pseudo-polar fast Fourier transform, as shown in Figure 3-7.[65] These algorithms produce better image quality than the original Fourier-based method, however, they are computationally more demanding or hard to adapt to complex imaging geometries such as the helical CT scan.

Figure 3-7: Schematic illustration of (a) Pseodopolar fast Fourier transform, (b) iterative reconstruction with constraints based on the Fourier slice theorem.[65]

3.2.4 Filtered Back Projection

Filtered back projection (FBP) is another image reconstruction method that is also based on the Fourier slice theorem but avoids the drawbacks of direct interpolation on the Fourier sampling space. There are two steps in FBP method: filtering and back projection (BP). The

efficient and computationally faster. The filtered back projection algorithm can be derived using the Fourier slice theorem. Mathematically, BP is a summation of detector functions over all imaging angles, and it can be expressed as:

𝑓_𝐵𝑃(𝑥, 𝑦) = ∫ 𝑝_𝜃(𝑡)𝑑𝜃 2𝜋 0 = ∫ 𝑝_𝜃(𝑥 cos 𝜃 + 𝑦 sin 𝜃)𝑑𝜃 2𝜋 0 . (3-13)

where 𝑓_𝐵𝑃(𝑥, 𝑦) is the image obtained by BP the detector function 𝑝_𝜃(𝑡). We can then substitute the detector function 𝑝_𝜃(𝑡) with its Fourier transform 𝑝_𝜃(𝑡) = ℱ−1_{(Ρ(𝜔)) and obtained:}

𝑓_𝐵𝑃(𝑥, 𝑦) = ∫ 𝑝𝜃(𝑡)𝑑𝜃 2𝜋 0 = ∫ ℱ−1_{(Ρ(𝜔))𝑑𝜃} 2𝜋 0 = ∫ ℱ−1(ℱ(𝜔 cos 𝜃 , 𝜔 sin 𝜃))𝑑𝜃 2𝜋 0 = ∫ ∫ ℱ(𝜔 cos 𝜃 , 𝜔 sin 𝜃)𝑒𝑖2𝜋𝜔𝑡_{𝑑𝜔𝑑𝜃} ∞ −∞ 2𝜋 0 = ∫ ∫ ℱ(𝜁, 𝜂)𝑒𝑖2𝜋(𝜁𝑥 + 𝜂𝑦)𝑑𝜔𝑑𝜃 ∞ −∞ , 2𝜋 0 (3-14)

where we also used the Fourier slice theorem: Ρ(𝜔) = ℱ(𝜔 cos 𝜃 , 𝜔 sin 𝜃) and the coordinate transformation 𝑡 = 𝑥 cos 𝜃 + 𝑦 sin 𝜃. Note that the Fourier transform of the original object function is: 𝑓(𝑥, 𝑦) = ∫ ∫ ℱ(𝜁, 𝜂)𝑒𝑖2𝜋(𝜁𝑥 + 𝜂𝑦)_{𝑑𝑥𝑑𝑦} ∞ −∞ ∞ −∞ . (3-15)

Equation 3-15 is surprisingly similar to this expression, except that Equation 3-14 is written in polar coordinate. We then transform the Cartesian coordinate in Equation 3-15 to the polar coordinate and obtain:

𝑓(𝑥, 𝑦) = ∫ ∫ ℱ(𝜁, 𝜂)𝑒𝑖2𝜋(𝜁𝑥 + 𝜂𝑦)𝜔𝑑𝜔𝑑𝜃 ∞

= ∫ (∫ ℱ(𝜁, 𝜂)𝑒𝑖2𝜋(𝜁𝑥 + 𝜂𝑦)_𝜔𝑑𝜔 ∞ −∞ ) 2𝜋 0 𝑑𝜃 = ∫ ℱ−1(Ρ(𝜔)̃ )𝑑𝜃 2𝜋 0 = 𝑓_𝐹𝐵𝑃(𝑥, 𝑦),

where Ρ(𝜔)̃ is the filtered Ρ(𝜔) on the Fourier space:

Ρ(𝜔)̃ = Ρ(𝜔) ∙ |𝜔|. (3-17)

Equation 3-16 and 3-17 suggests that the image obtained from BP is a blurred version of the original object function 𝑓(𝑥, 𝑦). By filtering the detector function 𝑝_𝜃(𝑡) on the Fourier domain and then BP the result, an exact solution of the original object function will be obtained, and this is the exactly how FBP reconstructs images. Equation 3-16 also suggests that a filtering can be done after the BP, however, it is rarely implemented for computational speed reason.

The filter design is critical in FBP, as it strongly affects the reconstruction image quality. A high-pass ramp filter with a cut-off frequency is often used in FBP, which is defined as:

ℎ(𝜔) = {|𝜔|, |𝜔| < 𝜔0 0, |𝜔| ≥ 𝜔0,

(3-18)

where 𝜔₀ is the cut-off frequency that controls how much high-frequency information is passed to the final reconstruction images. Generally, a larger cut-off frequency would result a better spatial resolution, but at the same time the high-frequency noise also increases. In contrast, a small cut- off frequency would reduce the noise in the reconstruction image, however, it reduces the image resolution as well and the reconstruction will look blurry. Examples of filters used in FBP are BlackMan filter, Butterworth filter, Cosine filter, Hamming filter, Shepp-Logan filter and RamLak filter. Each filter has different shape and different cut-off frequency, which changes the image characteristic at different frequencies. The filter performance is application dependent (noise level

than the other. With that being said, despite different designs, all filters have to trade off one or the other between the image resolution and the image noise. A collection of FBP filters are shown in Figure 3-8.[66]

Figure 3-8: A collection of filters used in FBP. [66]

For many years, FBP is the predominant image reconstruction algorithm, it generates sharp images within a reasonable time. Nowadays, it is still widely used in many CT and DTS systems. Thanks to the fast Fourier transform, FBP algorithm is computationally efficient, and doctors can see the reconstruction images in a short time. However, as an analytical method, FBP requires a sufficient sampling size, which should be above the Nyquist rate and it assumes no noise in the raw data. These two conditions ensure the accuracy of the FBP reconstruction, however, they could be hard to achieve in certain clinical applications. For example, the number of projection images

in the raw data especially with low dose imaging. Currently, with the increasing concerns of radiation-induced cancer, tremendous effects are made on developing low dose clinical protocols and imaging techniques, where the image noise can no longer be ignored and/or the projection data becomes insufficient. This facilitates the development of a new reconstruction method which produces better images with noisy and/or insufficient data.

3.3 Algebraic Reconstruction Methods