A Study Of Lattice Boltzmann Models And Their Applications In Image Processing

Lattice Boltzmann model (LBM) is a newly developed numerical tool for resolving partial differential equation (PDE). The most important difference between the LBM and the traditional numerical methods is that LBM is based on the microscopic description of the physical systems while the traditional numerical method is based on the discretization of the PDE which is a macroscopic description of the physical systems. By simulating the microscopic behavior of the physical systems, the global equation of LBM can match a certain PDE. Although, until nowadays LBM is rarely applied in image processing, it has a broad application prospect in image processing for several reasons. First, as the origin of LBM, Cellular Automata (CA) has been widely used in image processing since 1960s. Second, LBM inherits the advantages of the CA, for example, the simplicity of the algorithm, the high degree of the parallel computing and better numerical stability. Third, LBM is a numerical tool for resolving the PDE, and according to the methods and models based on PDE, LBM can be used in image processing. Last, the evolution equation of the LBM is much easer to design than CA rules because it is designed according to the microscopic behavior of the physical systems, and the parameters of LBM can be determined by the PDE which wants to be resolved.In this dissertation, in order to obtain a simple, fast and stable PDE-based method for image processing, we started from simulating the physical system, and then established three lattice Boltzmann models. According to the methods and models based on PDE, these lattice Boltzmann models were applied to image processing.The achievements of the dissertation are enumerated as follows.The design of the LBM evolution equation and its effect to the stability of the lattice Boltzmann model was discussed. A stable lattice Boltzmann isotropic diffusion model (LBIDM) was presented. By adding semi-permeable membranes into LBIDM, the lattice Boltzmann anisotropic diffusion model (LBADM) and the lattice Boltzmann convection diffusion model (LBCDM) were proposed. The mathematical derivation showed that these three Lattice Boltzmann models can be used for resolving the isotropic diffusion equation, the anisotropic diffusion equation and the convection diffusion equation respectively.The backward diffusion process is the main reason for the instability and the staircasing effect of the Perona-Malik's method. To overcome the two drawbacks, a new algorithm was proposed based on the LBADM. In macroscopic sense, the particles always move from the cells with high particle density to those with low particle density. Therefore, the force represented by the forward diffusion process dominates in the LBADM. This means the Lattice Boltzmann model LBADM is stable. The experiment and discussion showed that our algorithm can overcome the staircasing effect of the Perona-Malik's algorithm well, and had higher peak signal-to-noise ratio compared to the Perona-Malik's algorithm. In addition to this, the stability analysis of the LBADM was given.To promote the accuracy of the edge detection and the capacity for noise immunity, an edge detector based on the LBIDM was proposed, called LBMED (Lattice Boltzmann Model based Edge Detector). LBMED approximates the behavior of LoG operator, and detects the image edges based on the second derivative. This means LBMED can detect the edge accurately. In order to overcome the second derivative method's drawback, its high sensitivity to noise, a parameter called "derivative scale" was introduced. A small derivative scale can obtain more edge information, but will be more sensitivity to the noise. A large derivative scale can reduce weak edges and noise, but will lead to a rough edge contour. By comparing our algorithm with the traditional method and the existent CA method, the qualitative evaluation was implemented. In terms of the Pratt index, our method was evaluated quantitatively. The experiment and discussion showed that our algorithm can detect the image edge accurately and eliminate the noise effectively.To obtain a continuous and closed segment contour when applied the anisotropic diffusion equation in image segmentation, we proposed an algorithm based on LBADM. In the algorithm, the anisotropic diffusion equation was regarded as the diffusion process of a thermal field. Thus, the evolution of the segment contour could be simulated implicitly by the thermal field's diffusion process. The results of the experiments showed that the algorithm based on the LBADM can segment the CT, MRI and DSA images effectively. The segment contour was continuous and closed. Compared to the Level Set method, the segment speed was faster. To speed up the Level Set method in image segmentation, an algorithm based on the LBCDM was also proposed. By replacing the curvature term by the diffusion term, we reduced the computational complexity dramatically. The results of the experiments showed that the algorithm based on the LBCDM can segment the CT, MRI and DSA images effectively. The segment contour was continuous and closed. Compared to the Level Set method, the segment speed was faster. In addition, the stability analysis of the LBCDM was given.Nowadays the PDE-based methods for image processing have been widely applied to many fields, for example, physics, astronomy and electronics etc. However, the PED-based image processing algorithm is always very complex and time consuming. For simplicity of implementation, fast computation and good stability, applying LBM to the image processing will lead to better applied perspective of the PDE theory in image processing.