Direct volume rendering renders volume datasets without intermediate surface representation. However, due to the large data size, high quality volume rendering is a challenging research problem. Researchers have proposed various volumetric lighting models that consider no scattering, single scattering and multiple scattering. In practice, nearly all numerical solutions of graphics and visualization problems assume a discretization of the simulation domain into a grid of cells or points. In most cases, the discretization is regular and generates a lattice. This work presents several new techniques and algorithms for rendering lattice volumes. The 3D Cartesian Cubic (CC) lattice is the most frequently used lattice for volumes. Because of its simplicity, it is natively supported by many kinds of hardware, especially on the graphics processing unit (GPU). Three algorithms are presented for rendering 3D CC lattices without scattering: an object order cell projection algorithm based on min-max octree data structure that renders large datasets beyond GPU memory capacity, a hybrid method that improves the rendering speed by CPU/GPU parallelism, and a method of ray tracing height field on the GPU. For rendering the CC lattice with single scattering, a half angle splatting algorithm has achieved real time rendering of smoke. To overcome the popping artifact of half angle based method, a lighting volume based method is proposed and further accelerated with the GPU. A novel volumetric global illumination framework based on the Face-Centered Cubic (FCC) lattice is proposed to compute multiple scattering for volumetric global illumination. An FCC lattice has better sampling efficiency than the CC lattice. Furthermore, it has the maximal possible kissing number, which provides optimal 3D angular discretization. The proposed algorithms greatly simplify the computation of multiple scattering and minimize illumination information storage due to angular discretization. Two distinct applications of lattice computation and rendering have been developed, dispersion visualization for the flow simulation of lattice Boltzmann Method (LBM) and computer aided polyp detection for virtual colonoscopy. The LBM simulates the flow field by the micro-scale Boltzmann kinetics of fluid elements on a lattice, and the simulation results are rendered on GPUs and GPU clusters. The second application focuses on a new pipeline for computer aided polyp detection for virtual colonoscopy. The new pipeline has a volume rendering stage to generate the electronic biopsy image of a conformal flattened colon. Then, the polyps are detected on the single 2D biopsy image with texture analysis, which is significantly faster than traditional shape analysis methods.