Pascal Schreck

Sketch-based pruning of a solution space within a formal geometric constraint solver

In CAD systems, formal geometric solvers enable the designer to draw a sketch and to provide constraints that are compiled into a construction plan by symbolic geometric reasoning. Then the plan is interpreted in order to generate the required figure. In case there are multiple solutions, they allow to scan the entire solution space. But when the number of solutions becomes too high, it is very time-consuming to examine each of them to determine which one is the closest to the user’s will. In this paper, we introduce a sketch-based heuristic that enables to easily eliminate most of the solutions and to keep, among a solution space represented by a tree, only one branch, or at the worst a small subtree of solutions, that has the best likeness with the original sketch.

Multi-criteria trajectory planning for hepatic radiofrequency ablation

In this paper, we propose a method based on multiple criteria to assist physicians in planning percutaneous RFA on liver. We explain how we extracted information from literature and interviews with radiologists, and formalized them into geometric constraints. We expose then our method to compute the most suitable needle insertion in two steps: computation of authorized insertion zones and multi-criteria optimization of the trajectory within this zones. We focus on the combination of the criteria to optimize and on the optimization step.

Geometric construction by assembling solved subfigures

Abstract Among the expected contributions of Artificial Intelligence to Computer-Aided Design is the possibility of constructing a geometric object, the description of which is given by a system of topological and dimensional constraints. This paper presents the theoretical foundations of an original approach to formal geometric construction of rigid bodies in the Euclidian plane, based on invariance under displacements and relaxation of positional constraints. This general idea allows to explain in greater detail several methods proposed in the literature. One of the advantages of this approach is its ability to efficiently generalize and join together different methods for local solving. The paper also describes the main features of a powerful and extensible operational prototype based on these ideas, which can be viewed as a simple multi-agent system with a blackboard. Finally, some significant examples solved by this prototype are presented.

Trajectory optimization for the planning of percutaneous radiofrequency ablation of hepatic tumors

Radiofrequency ablation is increasingly used in the treatment of hepatic tumors. Planning the percutaneous intervention is essential and particularly difficult. In this paper, we focus on an automated computation of optimal needle insertion in computer-assisted surgery with 3D visualization. First, we review our method which delineates on the skin of a virtual patient the candidate zones for needle insertion, i.e., those which allow safe access to the tumor. In each case, we look for the trajectory that minimizes the volume of burnt tissue. Secondly, we introduce a quasi-exhaustive method that allies sampling and certified minimization to form a strong argument for the accuracy of our results. We also compare results of applying both methods on 7 representative reconstructed patient cases.

Formal resolution of geometrical constraint systems by assembling

Handling geometric objects described declaratively by a system of geometric constraints is an important issue in CAD. But until now, this requires the effective geometric construction of the objects. This paper presents an original approach to formal geometric constructions in the Euclidian plane, based on invariance under displacements and relaxation of positional constraints. This approach allows to efficiently generalize and join different methods for local solving. The paper also describes the main features of a powerful and extensible operational prototype based on these ideas, which can be viewed as a simple multi-agent system with a blackboard.

Optimal trajectories computation within regions of interest for hepatic RFA planning

Percutaneous radiofrequency ablation has become a frequently used technique for the treatment of liver cancers, but still remains very difficult to plan. In this paper, we propose a robust method to delineate on the skin of a 3D reconstructed patient the zones that are candidate for an insertion, because they allow a safe access to the tumor without meeting any organ, and to compute automatically within these zones an optimal trajectory minimizing the volume of necrosis covering the tumor.

Geometric constraints solving: some tracks

This paper presents some important issues and potential research tracks for Geometric Constraint Solving: the use of the simplicial Bernstein base to reduce the wrapping effect in interval methods, the computation of the dimension of the solution set with methods used to measure the dimension of fractals, the pitfalls of graph based decomposition methods, the alternative provided by linear algebra, the witness configuration method, the use of randomized provers to detect dependences between constraints, the study of incidence constraints, the search for intrinsic (coordinate-free) formulations and the need for formal specifications.

Formalization of wu's simple method in coq

We present in this paper the integration within the Coq proof assistant, of a method for automatic theorem proving in geometry. We use an approach based on the validation of a certificate. The certificate is generated by an implementation in Ocaml of a simple version of Wus method.

Precise determination of regions of interest for hepatic RFA planning

Percutaneous radiofrequency ablation is one of the most promising alternatives to open surgery for the treatment of liver cancer. This operation is a minimally invasive procedure that consists in inserting a needle in targeted tissues that are destroyed by heat. The success of such an operation mainly depends on the accuracy of the needle insertion, making it possible to destroy the whole tumor, while avoiding damages on other organs and minimizing risks of a local recurrence. We are developing a software that applies planning rules on patient-specific 3D reconstructions, in order to suggest relevant options for the choice of a path to the tumor, and that displays various information allowing to adjust the final choice. In this context we propose a method to compute automatically, quickly, and accurately, the possible insertion areas on the skin. Within these areas, an insertion of the probe targeting the tumor respects the numerous strong (boolean) constraints required for a radiofrequency ablation. Besides, these insertion zones define the research domain of the optimization process, taking into account soft constraints to refine the solutions. They are also displayed on the skin of the virtual patient to inform the physician about the different possibilities specific to each case, allowing him at the end of the automatic process, to modify interactively the proposed strategy, with a real-time update of the related information. We discuss in this paper about the importance of a precise delineation of these areas.

INCIDENCE CONSTRAINTS: A COMBINATORIAL APPROACH

The simplest geometric constraints are incidences between points and lines in the projective plane. This problem is universal, in the sense that all algebraic systems reduce to such geometric constraints. Detecting incidence dependences between these geometric constraints is NP-complete. New methods to prove incidence theorems are proposed, which use strictly no computer algebra but only combinatorial arguments.