Ioannis P. Ivrissimtzis

Using growing cell structures for surface reconstruction

We study the use of neural network algorithms in surface reconstruction from an unorganized point cloud, and meshing of an implicit surface. We found that for such applications, the most suitable type of neural networks is a modified version of the growing cell structure we propose here. The algorithm works by sampling randomly a target space, usually a point cloud or an implicit surface, and adjusting accordingly the neural network. The adjustment includes the connectivity of the network. Doing several experiments we found that the algorithm gives satisfactory results in some challenging situations involving sharp features and concavities. Another attractive feature of the algorithm is that its speed is virtually independent of the size of the input data, making it particularly suitable for the reconstruction of a surface from a very large point set.

Surface and normal ensembles for surface reconstruction

The majority of the existing techniques for surface reconstruction and the closely related problem of normal reconstruction are deterministic. Their main advantages are the speed and, given a reasonably good initial input, the high quality of the reconstructed surfaces. Nevertheless, their deterministic nature may hinder them from effectively handling incomplete data with noise and outliers. An ensemble is a statistical technique which can improve the performance of deterministic algorithms by putting them into a statistics based probabilistic setting. In this paper, we study the suitability of ensembles in normal and surface reconstruction. We experimented with a widely used normal reconstruction technique [Hoppe H, DeRose T, Duchamp T, McDonald J, Stuetzle W. Surface reconstruction from unorganized points. Computer Graphics 1992;71-8] and Multi-level Partitions of Unity implicits for surface reconstruction [Ohtake Y, Belyaev A, Alexa M, Turk G, Seidel H-P. Multi-level partition of unity implicits. ACM Transactions on Graphics 2003;22(3):463-70], showing that normal and surface ensembles can successfully be combined to handle noisy point sets.

On the support of recursive subdivision

We study the support of subdivision schemes: that is, the region of the subdivision surface that is affected by the displacement of a single control point. Our main results cover the regular case, where the mesh induces a regular Euclidean tesselation of the local parameter space. If n is the ratio of similarity between the tesselations at steps k and k − 1 of the refinement, we show that n determines the extent of this region and largely determines whether its boundary is polygonal or fractal. In particular if n = 2 (or n2 = 2 because we can always take double steps) the support is a convex polygon whose vertices can easily be determined. In other cases, whether the boundary of the support is fractal or not depends on whether there are sufficient points with non-zero coefficients in the edges of the convex hull of the mask. If there are enough points on every such edge, the support is again a convex polygon. If some edges have enough points and others do not, the boundary can consist of a fractal assembly of an unbounded number of line segments.

Mesh Discriminative Features for 3D Steganalysis

We propose a steganalytic algorithm for triangle meshes, based on the supervised training of a classifier by discriminative feature vectors. After a normalization step, the triangle mesh is calibrated by one step of Laplacian smoothing and then a feature vector is computed, encoding geometric information corresponding to vertices, edges and faces. For a given steganographic or watermarking algorithm, we create a training set containing unmarked meshes and meshes marked by that algorithm, and train a classifier using Quadratic Discriminant Analysis. The performance of the proposed method was evaluated on six well-known watermarking/steganographic schemes with satisfactory accuracy rates.

Polygonal mesh watermarking using Laplacian coordinates

We propose a watermarking algorithm for polygonal meshes based on the modification of the Laplacian coordinates. More specifically, we first compute the Laplacian coordinates (x,y,z) of the mesh vertices, then construct the histogram of the lengths of the (x,y,z) vectors, and finally, insert the watermark by altering the shape of that histogram. The watermark extraction is carried out blindly, with no reference to the host model. The proposed method is more robust than several existing high capacity watermarking algorithms. In particular, it is able to resist attacks such as translations, rotations, uniform scaling and vertex reordering, due to the invariance of the histogram of the Laplacian vector lengths under such transformations. Compared to the existing robust watermarking methods, our experiments show that the proposed method can better resist common mesh editing attacks, due to the good behaviour of the Laplacian coordinates under such operations.

On the Geometry of Recursive Subdivision

In this paper we investigate the properties of recursive subdivision from a geometric point of view. We explore the connections between subdivision and such areas of mathematics as spherical trigonometry, inversive geometry, and metric spaces. The methods we develop give new insights to well-known subdivision schemes and they can also be used in the argued construction of new schemes with prescribed properties.

Neural mesh ensembles

This work proposes the use of neural network ensembles to boost the performance of a neural network based surface reconstruction algorithm. Ensemble is a very popular and powerful statistical technique based on the idea of averaging several outputs of a probabilistic algorithm. In the context of surface reconstruction, two main problems arise. The first is finding an efficient way to average meshes with different connectivity, and the second is tuning the parameters for surface reconstruction to maximize the performance of the ensemble. We solve the first problem by voxelizing all the meshes on the same regular grid and taking majority vote on each voxel. We tune the parameters experimentally, borrowing ideas from weak learning methods.

A steganalytic algorithm for 3D polygonal meshes

We propose a steganalytic algorithm for watermarks embedded by Cho et al.s mean-based algorithm [1]. The main observation is that while in a clean model the means of Cho et al.s normalized histogram bins are expected to follow a Gaussian distribution, in a marked model their distribution will be bimodal. The proposed algorithm estimates the number of bins through an exhaustive search and then the presence of a watermark is decided by a tailor made normality test. We also propose a modification of Cho et al.s algorithm which is more resistant to the steganalytic attack and offers an improved robustness/capacity trade-off.

Curvature behaviours at extraordinary points of subdivision surfaces

During the development of subdivision surface methods one of the important questions has been the degree of continuity of the limit surface. In particular whether continuity of curvature can be achieved at the extraordinary points. However, there are several different curvature behaviours, not just two, and this note demonstrates them by examples.

Linear Correlations between Spatial and Normal Noise in Triangle Meshes

We study the relationship between the noise in the vertex coordinates of a triangle mesh and normal noise. First, we compute in closed form the expectation for the angle θ between the new and the old normal when uniform noise is added to a single vertex of a triangle. Next, we propose and experimentally validate an approximation and lower and upper bounds for θ when uniform noise is added to all three vertices of the triangle. In all cases, for small amounts of spatial noise that do not severely distort the mesh, there is a linear correlation between θ and simple functions of the heights of the triangles and thus, θ can be computed efficiently. The addition of uniform spatial noise to a mesh can be seen as a dithered quantization of its vertices. We use the obtained linear correlations between spatial and normal noise to compute the level of dithered quantization of the mesh vertices when a tolerance for the average normal distortion is given.