Yan Gérard

Shape-from-Template

We study a problem that we call Shape-from-Template, which is the problem of reconstructing the shape of a deformable surface from a single image and a 3D template. Current methods in the literature address the case of isometric deformations, and relax the isometry constraint to the convex inextensibility constraint, solved using the so-called maximum depth heuristic. We call these methods zeroth-order since they use image point locations (the zeroth-order differential structure) to solve the shape inference problem from a perspective image. We propose a novel class of methods that we call first-order. The key idea is to use both image point locations and their first-order differential structure. The latter can be easily extracted from a warp between the template and the input image. We give a unified problem formulation as a system of PDEs for isometric and conformal surfaces that we solve analytically. This has important consequences. First, it gives the first analytical algorithms to solve this type of reconstruction problems. Second, it gives the first algorithms to solve for the exact constraints. Third, it allows us to study the well-posedness of this type of reconstruction: we establish that isometric surfaces can be reconstructed unambiguously and that conformal surfaces can be reconstructed up to a few discrete ambiguities and a global scale. In the latter case, the candidate solution surfaces are obtained analytically. Experimental results on simulated and real data show that our isometric methods generally perform as well as or outperform state of the art approaches in terms of reconstruction accuracy, while our conformal methods largely outperform all isometric methods for extensible deformations.

An elementary digital plane recognition algorithm

A naive digital plane is a subset of points (x, y, z) ∈ Z3 verifying h ≤ ax + by + cz < h + max{|a|, |b|, |c|}, where (a, b, c, h) ∈ Z4. Given a finite unstructured subset of Z3, the problem of the digital plane recognition is to determine whether there exists a naive digital plane containing it. This question is rather classical in the field of digital geometry (also called discrete geometry). We suggest in this paper a new algorithm to solve it. Its asymptotic complexity is bounded by O(n7) but its behavior seems to be linear in practice. It uses an original strategy of optimization in a set of triangular facets (triangles). The code is short and elementary (less than 300 lines) and available on http://www.loria.fr/~debled/plane.

Estimation of the derivatives of a digital function with a convergent bounded error

We provide a new method to estimate the derivatives of a digital function by linear programming or other geometrical algorithms. Knowing the digitization of a real continuous function f with a resolution h, this approach provides an approximation of the kth derivative f(k)(x) with a maximal error in O(h1/1+k) where the constant depends on an upper bound of the absolute value of the (k + 1)th derivative of f in a neighborhood of x. This convergence rate 1/k+1 should be compared to the two other methods already providing such uniform convergence results, namely 1/3 from Lachaud et. al (only for the first order derivative) and (2/3)k from Malgouyres et al..

The chords' problem

The chords’ problem is a variant of an old problem of computational geometry: given a set of points of Rn, one can easily build the multiset of the distances between the points of the set but the converse construction is known, for a longtime, as to be difficult. The problem that we are going to investigate is also a converse construction with the difference that it is not one of the distances’ multisets but one of the chords’ multisets. In dimension 1, the old distances’ problem and the chords’ problem coincide with each other whereas in other dimensions, the chords’ multisets contain more information on the sets than their distances’ multisets. This paper provides, in dimension 1, two different algorithms to reconstruct the set of points according to their chords’ multiset. The first one is given for its effectiveness in spite of an uncertain complexity whereas the second one is the first polynomial algorithm solving the chords’ problem. At least, we will explain how to transform a chords’ problem in dimension n into an equivalent chords’ problem in dimension 1.

Local Configurations of Digital Hyperplanes

The aim of this article is to provide some arithmetical tools in order to study the local properties of digital hyperplanes. With the help of the new general notion of configuration, we investigate the arrangement of the different combinatorial structures contained in a digital hyperplane. The regularity of this deployment is controlled by two arithmetical functions that we call code(I) and boundary(I). By using these two simple tools, we prove that the local configurations in a functional digital hyperplane only depends on its normal vector and that their number is less than the size of the chosen neighborhood.

Introduction to digital level layers

We introduce the notion of Digital Level Layer, namely the subsets of Zd characterized by double-inequalities h1 ??? f(x) ???? h2. The purpose of the paper is first to investigate some theoretical properties of this class of digital primitives according to topological and morphological criteria. The second task is to show that even if we consider functions f of high degree, the computations on Digital Level Layers, for instance the computation of a DLL containing an input set of points, remain linear. It makes this notion suitable for applications, for instance to provide analytical characterizations of digital shapes.

Recognition of digital hyperplanes and level layers with forbidden points

We consider a new problem of recognition of digital primitives - digital hyperplanes or level layers - arising in a new practical application of surface segmentation. Such problems are usually driven by a maximal thickness criterion which is not satisfactory for applications as soon as the dimension of the primitives becomes greater than 1. It is a good reason to introduce a more flexible approach where the set to recognize (whose points are called inliers) is given along with two other sets of outliers that should each remain on his own side of the primitive. We reduce this problem of recognition with outliers to the separation of three point clouds of Rd by two parallel hyperplanes and we provide a geometrical algorithm derived from the well-known GJK algorithm to solve the problem.

Gift-wrapping based preimage computation algorithm

The aim of the paper is to define an algorithm for computing preimages - roughly the sets of naive digital planes containing a finite subset S of Z3. The method is based on theoretical results: the preimage is a polytope that vertices can be decomposed in three subsets, the upper vertices, the lower vertices and the intermediary ones (equatorial). We provide a geometrical understanding (as facets on S or S ⊖ S) of each kind of vertices. These properties are used to compute the preimage by gift-wrapping some regions of the convex hull of S or of S⊖S∪{(0, 0, 1)}.

An Elementary Algorithm for Digital Arc Segmentation

Abstract This paper is concerned with the digital circle recognition problem and more precisely with the circular separating algorithm . It tries to go further in implementation details, giving pseudo-code algorithms for the main points, and avoids using the sophisticated machinery coming either from Computational Geometry or from Linear Programming found in previous papers on this subject. After recalling the geometrical meaning of the separating circle problem, we present an incremental algorithm to segment a discrete curve into digital arc.

Digital Level Layers for Digital Curve Decomposition and Vectorization

The purpose of this paper is to present Digital Level Layers and show the motivations for working with such analytical primitives in the framework of Digital Geometry. We rst compare their properties to morphological and topological counterparts, and then we explain how to recognize them and use them to decompose or vectorize digital curves and contours.