1. Introduction

Since a long time, the interpolation is used to model and visualize data. Indeed, a way to analyze experimental data is to represent them on a 2D or 3D graph; it fills this gap by adding intermediate points that could simulate a more detailed experiment. Our proposed approach is to achieve a 3D fractal interpolation, more adapted to many natural phenomena. We use the fractal interpolation to perform the image restoration. The principle of this technique is explained hereafter [1–18].

In this research, we are interested in the 3D geometrical models of human organs [2–11]; the three-dimensional reconstruction of the vascular networks presents a major medical interest for the diagnostic and prognostic monitoring of several diseases such as atheromatous disease. However, the disadvantage of threedimensional reconstruction approaches is too costly in terms of storage capacity and transmission time. The objective of this research is to create a technique that allows generating with very little information at the beginning, a large amount of data with few errors. To accomplish this goal, we propose an original method based on mathematical morphology in order to provide a 3D reconstruction taking into account all the data of the problem [11–18].

A lot of available works focuses on 3D reconstruction by the 3D fractal interpolation. These approaches were used to generate a 3D model with a high degree of realism.

Guedri et al. [11] have proposed a method for the 3D model blood vessel reconstruction by using the 3D iterated function systems (IFS) and 3D fractal interpolation, and the approach is within the framework of 2D/3D techniques. Sun [12, 13] has presented the principles of the multifractal interpolation surface, and he described the methods of dividing local interpolation neighborhoods and determining multiple vertical compression ratios. Then, he gave a practical MATLAB program for the multifractal interpolation surface, and he explained the main parameters in his proposed program. Chen and Bi [14] have created fascinating fractal scenes by using the 3D IFS, and they proposed an efficient coloring, lighting, and mist effect scheme. Guérin et al. [15] have used a fractal model called projected iterated function system (IFS) model that allows the extension of the iteration space to a barycentric space Rn2 by enriching the classical IFS model with a set of control points for approximating smooth or rough surfaces defined in R3 . He et al. [16] have proposed a probabilitybased method to speed up the fractal interpolation execution to three-dimensional terrain reconstruction by a few sparse points in the digital elevation model (DEM). Huang and Chen [17] have adopted two sets of real-world 3D terrain profile data to precede data reducing and then reconstruct them through 3D fractal reconstruction. Xiong [18] has analyzed the reasons that fractal can be used in 3D terrain surface reconstruction and introduced a fractional Brownian motion. Then, they presented an interpolating algorithm to reconstruct 3D terrain surface.

The chapter is organized as follows. In the first step, we give the image source and introduce its features. Then, we present preprocessing of the 2D image datasets and the 3D reconstruction model. Then, a linear simplification method is provided, the Douglas-Peucker method, in order to detect the control points and reduce the data. Thereafter, we develop a reconstruction algorithm which adds new data points by 3D fractal interpolation; this algorithm generates more data points to restore the 3D original model. To evaluate the results of the fractal interpolation method, some blood vessel samples extracted from a real retinal image are used to make the fractal interpolation.
