Loïc Mazo

About Multigrid Convergence of Some Length Estimators

An interesting property for curve length digital estimators is the convergence toward the continuous length and the associate convergence speed when the digitization step h tends to 0. On the one hand, it has been proved that the local estimators do not verify this convergence. On the other hand, DSS and MLP based estimators have been proved to converge but only under some convexity and smoothness or polygonal assumptions. In this frame, a new estimator class, the so called semi-local estimators, has been introduced by Daurat et al. in [4]. For this class, the pattern size depends on the resolution but not on the digitized function. The semi-local estimator convergence has been proved for functions of class \(\mathcal{C}^2\) with an optimal convergence speed that is a \(\mathcal{O}(h^{\frac 1 2})\) without convexity assumption (here, optimal means with the best estimation parameter setting). A semi-local estimator subclass, that we call sparse estimators, is exhibited here. The sparse estimators are proved to have the same convergence speed as the semi-local estimators under the weaker assumptions. Besides, if the continuous function that is digitized is concave, the sparse estimators are proved to have an optimal convergence speed in h. Furthermore, assuming a sequence of functions \(G_h\colon h\mspace{1.0mu}\mathbb{Z} \to h\mspace{1.0mu}\mathbb{Z}\) discretizing a given Euclidean function as h tends to 0, sparse length estimation computational complexity in the optimal setting is a \(\mathcal{O}(h^{-\frac{1}{2}})\).

Topological Properties of Thinning in 2-D Pseudomanifolds

Preserving topological properties of objects during thinning procedures is an important issue in the field of image analysis. In the case of 2-D digital images (i.e. images defined on ℤ2) such procedures are usually based on the notion of simple point. In contrast to the situation in ℤn, n≥3, it was proved in the 80s that the exclusive use of simple points in ℤ2 was indeed sufficient to develop thinning procedures providing an output that is minimal with respect to the topological characteristics of the object. Based on the recently introduced notion of minimal simple set (generalising the notion of simple point), we establish new properties related to topology-preserving thinning in 2-D spaces which extend, in particular, this classical result to cubical complexes in 2-D pseudomanifolds.

Curve Digitization Variability

This paper presents a study on the set of digitizations generated by the action of a group of transformations on a continuous curve before the digitization step. An upper bound for the cardinal of this digitization set under the translation group action is exhibited. Then this bound is tested on several functions. Finally, a representation of this digitization set is proposed and an illustration of its potential use is given on a length estimator.

A framework for label images

Label images need a specific topological model to take into account not only the topologies of the regions but also the topology of the partition. We propose a framework for label images in which all the regions of the initial partition and of any coarser partition of the space can be explicitly represented. Some properties of the model are given and a local transformation that preserves the weak homotopy types of all the regions of all the partitions is defined.

Object digitization up to a translation

Abstract This paper presents a study on the set of the digitizations generated by all the translations of a planar body on a square grid. First the translation vector set is reduced to a bounded subset, then the dual introduced in [1] linking the translation vector to the corresponding digitization is proved to be piecewise constant. Finally, a new algorithm is proposed to compute the digitization set using the dual.

On 2-dimensional Simple Sets in n-dimensional Cubic Grids

Preserving topological properties of objects during reduction procedures is an important issue in the field of discrete image analysis. Such procedures are generally based on the notion of simple point, the exclusive use of which may result in the appearance of “topological artifacts.” This limitation leads to consider a more general category of objects, the simple sets, which also enable topology-preserving image reduction. A study of two-dimensional simple sets in two-dimensional spaces has been proposed recently. This article is devoted to the study of two-dimensional simple sets in spaces of higher dimension (i.e., n-dimensional spaces, n≥3). In particular, several properties of minimal simple sets (i.e., which do not strictly include any other simple sets) are proposed, leading to a characterisation theorem. It is also proved that the removal of a two-dimensional simple set from an object can be performed by only considering the minimal ones, thus authorising the development of efficient thinning algorithms.

Topology-preserving thinning in 2-D pseudomanifolds

Preserving topological properties of objects during thinning procedures is an important issue in the field of image analysis. In the case of 2-D digital images (i.e. images defined on Z2) such procedures are usually based on the notion of simple point. By opposition to the case of spaces of higher dimensions (i.e. Zn, n ≥ 3), it was proved in the 80s that the exclusive use of simple points in Z2 was indeed sufficient to develop thinning procedures providing an output that is minimal with respect to the topological characteristics of the object. Based on the recently introduced notion of minimal simple set (generalising the notion of simple point), we establish new properties related to topology-preserving thinning in 2-D spaces which extend, in particular, this classical result to more general spaces (the 2-D pseudomanifolds) and objects (the 2-D cubical complexes).

Study on the digitization dual combinatorics and convex case

The action of a translation on a continuous object before its digitization generates several digitizations. The dual, introduced by the authors in a previous paper, stands for these digitizations in function of the translation parameters. This paper focuses on the combinatorics of the dual by making a link between the digitization number and the boundary curve, especially through its dual representation. The convex case is then studied and a few significant examples are exhibited.

Joint 3D alignment-reconstruction multi-scale approach for cryo electron tomography

3D volume reconstruction in cryo-electron tomography is possible by using Transmission Electron Microscope (TEM) images from different tilt angles. The misalignment of these images is one of the limits to the quality of the reconstructed object. There are many alignment techniques to deal with this problem. Their common feature is to correct the 2D geometric transformation in the projection images. Nevertheless, 3D geometric transformation can occur in the TEM acquisition including tilt angular uncertainty. In this paper, we proposed a new multi-scale approach based on a Conjugate Gradient optimization of a cost function between the 3D reconstructed and the projection images with the purpose to find all the 3D parameters of geometric transformation. Tests on synthetic and real data prove the accuracy of our geometric transformation estimation.

Angular Uncertainty Refinement and Image Reconstruction Improvement in Cryo-electron Tomography

In the field of cryo-electron tomography (cryo-ET), numerous approaches have been proposed to tackle the difficulties of the three-dimensional reconstruction problem. And that, in order to cope with (1) t e missing and noisy data from the collected projections, (2) errors in projection images due to acquisition problems, (3) the capacity of processing large data sets and parameterizing the contrast function of the electron microscopy. In this paper, we present a novel approach for dealing with angular uncertainty in cryo-ET. To accomplish this task we propose a cost function and with the use of the nonlinear version of the optimization algorithm called Conjugate Gradient, we minimize it. We test the efficiency of our algorithm with both simulated and real data.