3.3. 3D interpolation fractal

The control point's density is large in the regions having a large curvature. The original shape of the blood vessel is preserved. These control points are used to reconstruct the 3D blood vessel via fractal interpolation. The affine transformations Wn are performed out. The execution time, the number of interpolated points, and the error are calculated versus the iteration number.

In addition, to analyze the performance of our proposed interpolation algorithm, we calculate the error rate between the original image and the interpolated image in the equation below:

$$Er(\%) = \frac{NE \, P}{TP} \times 100\,\tag{7}$$

where NEP is the number of erroneous pixels and TP is the total pixel. The number of erroneous pixels is the pixel numbers that do not appear in the interpolated image and the total pixels are the numbers of pixels in the original image.

Table 3 shows the evolution of the iteration number, as well as the error between original and interpolated image and the number of iterations for two test images used; for a small iteration number (N = 50), the error is about 7%. Moreover, For N= 400 iterations, the error decreases to 2.3% for the first image 12 (im0139.vk. jpg) and 2% for the second image used (02\_h.tif) For N = 1000, the minimum error rate (0.3%) is obtained from the two test images.

The algorithm described in this document takes a lower time of 2 s to find the curve for a blood vessel and the 3D reconstruction. Moreover, the remnants of execution times are devoted to 3D fractal interpolation. The execution times were between 4 and 14 s. For a small iteration number (N = 50), the execution times is about 4 s. By looking at Table 3, we can see that the value of the execution time increases in the last three tests; this is justified by the increased number of interpolated point, for example, for N = 1000 iterations, the execution time is about 14 s.

Increasing the number of iterations reduces the error but dramatically increases the number of interpolated points and execution times. A trade-off should be looked from these two parameters.


Table 3.

Performance evaluation for the image "im0139.vk.jpg" and image "02\_h.tif".

Figure 11. (a) 3D models reconstructed. (b) The original models.

Reconstruction of Three-Dimensional Blood Vessel Model Using Fractal Interpolation DOI: http://dx.doi.org/10.5772/intechopen.82247

Figure 11a shows the test experimental results for the reconstructed blood vessel using 3D fractal interpolation with ε = 0.5 and N = 1000 iterations from image im0139.ah.ppm with different arc lengths (from 50 to 385 pixels); Figure 11b depicts the original 3D blood vessels.

The obtained results are qualitatively similar and quantitatively more than the original data.
