## NNr a Ri o e j t Si2

S(t, &o) = m^st)) Co g —R\—exp )sm(mt - Si)

where Si = 2Ri/c and S(x)isthe 1D echogram generated by a set of Ne scatterers located in (Ri, &a < &i < &b). The overall distribution backscattering cross-section ai(Ri, ei ± S&) is given by Eq. (1.11).

### 1.5.3 2D Echogram Generation

The procedure to obtain the 2D simulated IVUS is the following: A rotatory transducer with angular velocity m (Fig. 1.23(a)) is located at the center of the simulated arterial configuration given by Eq. (1.11). The transducer emits an ultrasound pulse radially focused at frequency fo along angular direction 01 (Fig. 1.23(a)). The pulse progressively penetrates each one of the layers of the simulated arterial structure according to Eq. (1.15). Each one of the layers generates a profile of amplitude or echoes in time, which can be transformed into a profile of amplitude as a function of the penetration depth (Fig. 1.23(b)). Therefore, the depth can be calculated using Eq. (1.1). As the penetration depth is coincident with the axial beam direction, the radial coordinate R is thus determined. This procedure is repeated n times for angles, (01,..., 0n) and the 2D image is generated. The generated echo profiles are transformed to a polar

Figure 1.23: The transducer emits from the artery center (a), echo profile transformed into penetration depth (b), the echo profiles are transformed to a polar image (c), and empty pixels filled and the final IVUS image is smoothed (d).

image, and the intermediate beams are computed (Fig. 1.23(c)). The image is transformed to Cartesian form and the empty pixels are filled (Fig. 1.23(d)).

Using the ultrasound reflected signal S(t, O) for a finite set of N reflecting scatterers with coordinates (R, O, Z) and spatial distribution of the differential backscattering cross-section, a (R, O, Z), the 2D echo signal S(t, O) can be written as:

i=1 j=1 1 Ril where S(t, O) is the temporally generated signal by a set NR of scatterers, which are localized in angular position 0, 0 e [0a, 0b], N0i is the total scatterers number in the angular position 0a < © < Qb for a radial position Ri. We consider two forms of ©:

• with no uniform distributed scatterers:

• with uniform distributed scatterers:

1.5.4 Final Image Processing

The actual image obtained with only the original beams is very poor; we must explore several smoothing procedures to improve the image appearance. The procedures to obtain the final simulated image are as follows:

1. The echoes are obtained by the pivoting transducer (Fig. 1.23(a)).

2. Each echo profile is ordered according to the angular position (Fig. 1.23(b)).

3. The original image is transformed to a polar form (Fig. 1.23(c)).

4. Secondary beams are computed between two original neighboring beams (Fig. 1.23(c)).

5. The image is smoothed by a 2 x 2 median filter.

6. The image is again transformed to Cartesian form. As a result of this transformation, a significant number of pixels will be empty (Fig. 1.23(d)).

7. The empty pixels are filled in a recursive way form, using an average of the eight nearest neighbors (Fig. 1.23(d)).

8. An image reference reticle is added and a Gaussian filter is applied.

Figure 1.24 shows the scatterers distribution for a concentric arterial structure and an axial ultrasound beam position (a), and its corresponding echo profiles (b). Each axial echo is positioned by an angular position (c). In this way, the 2D echogram is constructed (d). The procedure of image smoothing is described in Section 1.5.4.

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