Andrei Khodakovsky

Progressive geometry compression

We propose a new progressive compression scheme for arbitrary topology, highly detailed and densely sampled meshes arising from geometry scanning. We observe that meshes consist of three distinct components: geometry, parameter, and connectivity information. The latter two do not contribute to the reduction of error in a compression setting. Using semi-regular meshes, parameter and connectivity information can be virtually eliminated. Coupled with semi-regular wavelet transforms, zerotree coding, and subdivision based reconstruction we see improvements in error by a factor four (12dB) compared to other progressive coding schemes.

Wavelet compression of parametrically coherent mesh sequences

We introduce an efficient compression method for animated sequences of irregular meshes of the same connectivity. Our approach is to transform the original input meshes with an anisotropic wavelet transform running on top of a progressive mesh hierarchy, and progressively encode the resulting wavelet details. For temporally coherent mesh sequences we get additional improvement by encoding the differences of the wavelet coefficients. The resulting compression scheme is scalable, efficient, and significantly improves upon the current state of the art for the animated mesh compression.

Near-optimal connectivity encoding of 2-manifold polygon meshes

Encoders for triangle mesh connectivity based on enumeration of vertex valences are among the best reported to date. They are both simple to implement and report the best compressed file sizes for a large corpus of test models. Additionally they have recently been shown to be near-optimal since they realize the Tutte entropy bound for all planar triangulations. In this paper we introduce a connectivity encoding method which extends these ideas to 2-manifold meshes consisting of faces with arbitrary degree. The encoding algorithm exploits duality by applying valence enumeration to both the primal and the dual mesh in a symmetric fashion. It generates two sequences of symbols, vertex valences, and face degrees, and encodes them separately using two context-based arithmetic coders. This allows us to exploit vertex or face regularity if present. When the mesh exhibits perfect face regularity (e.g., a pure triangle or quad mesh) or perfect vertex regularity (valence six or four respectively) the corresponding bit rate vanishes to zero asymptotically. For triangle meshes, our technique is equivalent to earlier valence-driven approaches. We report compression results for a corpus of standard meshes. In all cases we are able to show coding gains over earlier coders, sometimes as large as 50%. Remarkably, we even slightly gain over coders specialized to triangle or quad meshes. A theoretical analysis reveals that our approach is near-optimal as we achieve the Tutte entropy bound for arbitrary planar graphs of two bits per edge in the worst case.

Compression of Normal Meshes

Open image in new window Fig. 1. Partial reconstructions from a progressive encoding of the molecule model. File sizes are given in bytes, errors in multiples of 10−4 and PSNR in dB (model courtesy of The Scripps Research Institute).

Multilevel Solvers for Unstructured Surface Meshes

Parameterization of unstructured surface meshes is of fundamental importance in many applications of digital geometry processing. Such parameterization approaches give rise to large and exceedingly ill-conditioned systems which are difficult or impossible to solve without the use of sophisticated multilevel preconditioning strategies. Since the underlying meshes are very fine to begin with, such multilevel preconditioners require mesh coarsening to build an appropriate hierarchy. In this paper we consider several strategies for the construction of hierarchies using ideas from mesh simplification algorithms used in the computer graphics literature. We introduce two novel hierarchy construction schemes and demonstrate their superior performance when used in conjunction with a multigrid preconditioner.

Fine level feature editing for subdivision surfaces

In many industrial design modeling scenarios the designer wishes to edit small feature lines-such as variable width and height creases--on otherwise smooth surface patches. When the path of such a feature does not align with an iso-parameter line or crosses patch boundaries it becomes increasingly difficult to maintain good editing semantics of the underlying surface. In this paper we describe an algorithm and implementation allowing the interactive creation and manipulation of fine scale feature curves on subdivision surfaces. In particular, our approach addresses the problem of defining the path of such feature curves independent of the location of surface iso-parameter lines and global patch boundaries. The feature lines are modeled as swept displacement curves with variable profiles, providing a rich toolbox of shapes. Furthermore, the hierarchical editing semantics of subdivision surface based representations carry through to our extended setting, ensuring “good” behavior of the feature lines under coarse scale surface edits.

Hybrid meshes: multiresolution using regular and irregular refinement

A hybrid mesh is a multiresolution surface representation that combines advantages from regular and irregular meshes. Irregular operations allow a hybrid mesh to change topology throughout the hierarchy and approximate detailed features at multiple scales. A preponderance of regular refinements allows for efficient data-structures and processing algorithms. We provide a user driven procedure for creating a hybrid mesh from scanned geometry and present a progressive hybrid mesh compression algorithm.

Variational normal meshes

Hierarchical representations of surfaces have many advantages for digital geometry processing applications. Normal meshes are particularly attractive since their level-to-level displacements are in the local normal direction only. Consequently, they only require scalar coefficients to specify. In this article, we propose a novel method to approximate a given mesh with a normal mesh. Instead of building an associated parameterization on the fly, we assume a globally smooth parameterization at the beginning and cast the problem as one of perturbing this parameterization. Controlling the magnitude of this perturbation gives us explicit control over the range between fully constrained (only scalar coefficients) and unconstrained (3-vector coefficients) approximations. With the unconstrained problem giving the lowest approximation error, we can thus characterize the error cost of normal meshes as a function of the number of nonnormal offsets---we find a significant gain for little (error) cost. Because the normal mesh construction creates a geometry driven approximation, we can replace the difficult geometric distance minimization problem with a much simpler least squares problem. This variational approach reduces magnitude and structure (aliasing) of the error further. Our method separates the parameterization construction into an initial setup followed only by subsequent perturbations, giving us an algorithm which is far simpler to implement, more robust, and significantly faster.