Topics (~chapters):

(1) A transmission system for triangular mesh.

We describe a conceptual transmission system for texutred meshes and identified  the problem to address in order to obtain an progressive and  bandwidth-adaptive transmission.

(2) A journey in piecewise-linear function approximation.

We review a set of optimal and greedy simplification strategies. We use the simple polyline model and evaluate the computational complexity of each approach. Then,
we compare each method is term of approximation quality across rate (measured as the number of polyline knots).

(3) Computational analysis of 4-8 meshes. 

We introduce a type semi-regular mesh whose subdivision connectivity is obtained with the 4-8 scheme. We analyse a set of simplification operations: vertex insertion and decimation. Then, we explain several properties of the mesh construction. These propertis are useful for geometry simplification algorithms. 

(4) Quadtree data structure for efficient storage and access of 4-8 meshes 

We introduce a quadtree data structure to store efficient semi-regular 4-8 meshes. To quadtree store the 4-8 mesh without redundancy and provide an efficient constant-time access mechanism to query and process the dataset.

(5) Progressive meshes in an operational rate-distortion sense. 

Given a semi-regular mesh whose subdivision connectivity is obtained with the 4-8 scheme, we use a tree-driven, fine to coarse approach to simplify the mesh using vertex decimation. Our method uses global error and a generalized vertex decimation technique borrowed from optimal tree pruning algorithms used in compression. The output representation is adaptive and progressive. 

(6) Joint Mesh-Texture optimization.

Given a bit budget to spend on the representation of a textured polygonal mesh, how can the texture and the geometry information be balanced in order to maximize the rendered quality? We propose a greedy, yet efficient optimization method to address this problem.