Progressive Coding of 3-D Objects Based on Overcomplete Decompositions

This paper presents a progressive coding scheme for 3-D objects, based on overcomplete signal expansions on the 2-D sphere. Due to increased freedom in the basis construction, redundant expansions have shown interesting approximation properties in the decomposition of signals with multidimensional singularities organized along embedded submanifolds. We propose to map simple 3-D models on 2-D spheres and then to decompose the signal over a redundant dictionary of oriented and anisotropic atoms living on the sphere. The signal expansion is computed iteratively with a matching pursuit algorithm, which greedily selects the most prominent components of the 3-D model. The decomposition therefore inherently represents a progressive stream of atoms, which is advantageously used in the design of scalable representations. An encoder is proposed that compresses the stream of atoms by adaptive coefficient quantization and entropy coding of atom indexes. Experimental results show that the novel coding strategy outperforms state-of-the-art progressive coders in terms of distortion, mostly at low bit rates. Furthermore, since the dictionary is built on structured atoms, the proposed representation simultaneously offers an increased flexibility for easy stream manipulations. We finally illustrate that advantage in the design of a view-dependent transmission scheme.

Progressive coding of 3D objects based on overcomplete decompositions

Pascal Frossard, Ivana Tosic, Pierre Vandergheynst

This paper presents a progressive coding scheme for 3D objects, based on an overcomplete decomposition of the 3D model on a sphere. Due to increased freedom in the bases construction, redundant expansions have shown interesting approximation properties in the decomposition of signals with multidimensional singularities organized along embedded submanifolds. We propose to map simple 3D models on 2D spheres and then to decompose the signal over a redundant dictionary of oriented and anisotropic atoms that live on the sphere. The signal expansion is computed iteratively with a Matching Pursuit algorithm, which greedily selects the most prominent components of the 3D model. The decomposition therefore inherently represents a progressive stream of atoms, which is advantageously used in the design of scalable representations. An encoder is proposed that compresses the stream of atoms by adaptive coefficient quantization, and entropy coding of atom indexes. Experimental results show that the novel coding strategy outperforms state-of-the-art progressive coders in terms of distortion, mostly at low bit rate. Furthermore, since the dictionary is built on structured atoms, the representation simultaneously offers an increased flexibility. This enables easy stream manipulations, and we finally illustrate this advantage in the design of a view-dependent transmission scheme.

2005

Tree-Based Pursuit: Algorithm and Properties

Pascal Frossard, Philippe Jost, Pierre Vandergheynst

This paper proposes a tree-based pursuit algorithm that efficiently trades off complexity and approximation performance for overcomplete signal expansions. Finding the sparsest representation of a signal using a redundant dictionary is, in general, a NP-Hard problem. Even sub-optimal algorithms such as Matching Pursuit remain highly complex. We propose a structuring strategy that can be applied to any redundant set of functions, and which basically groups similar atoms together. A measure of similarity based on coherence allows for representing a highly redundant sub-dictionary of atoms by a unique element, called molecule. When the clustering is applied recursively on atoms and then on molecules, it naturally leads to the creation of a tree structure. We then present a new pursuit algorithm that uses the structure created by clustering as a decision tree. This tree-based algorithm offers important complexity reduction with respect to Matching Pursuit, as it prunes important parts of the dictionary when traversing the tree. Recent results on incoherent dictionaries are extended to molecules, while the true highly redundant nature of the dictionary stays hidden by the tree structure. We then derive recovery conditions on the structured dictionary, under which tree-based pursuit is guaranteed to converge. Experimental results finally show that the gain in complexity offered by tree-based pursuit does in general not have a high penalty on the approximation performance. They show that the dimensionality of the problem is reduced thanks to the tree construction, without significant loss of information at hand.

2005

Progressive low bit rate coding of simple 3D objects with Matching Pursuit

Pascal Frossard, Ivana Tosic, Pierre Vandergheynst

This paper presents a low rate progressive 3D mesh compression scheme for simple genus-zero 3D objects. The proposed scheme is based on signal representation using redundant expansions. 3D model representation is constructed using a Matching Pursuit algorithm, with an over-complete dictionary of atoms defined on a sphere. A specific dictionary construction is proposed, that is adapted to the characteristics of 3D models. The dictionary is built on both low-frequency atoms, and anisotropic refinement to capture singularities of the signal, living on a 2-D sphere. The novel coding method is shown to favorably compare to state-of-the-art compression schemes at low bit rate, while providing a flexible and progressive representation. It therefore represents an very interesting alternative for simple 3D model representations, especially in view-dependent or scalable applications.

2005