Supervised Thesis

The need for simulations of soft tissues, like internal organs, arises with the progress of the scientific and medical environments. The goal of my PhD is to develop a novel generic topological and physical model to simulate human organs. Such a model shall be easy to use, perform the simulations in the real time and which accuracy will allow usage for the medical purposes.This thesis explores novel simulation methods and improvement approaches for modeling deformable bodies. The methods aim at fast and robust simulations with physically accurate results. The main interest lies in simulating elastic soft tissues at small and large strains for medical purposes. We show however, that in the existing methods the accuracyto freely simulate deformable bodies and the real-time performance do not go hand in hand. Additionally, to reach the goal of simulating fast, many of the approaches move the necessary calculations to pre-computational part of the simulation, which results in inability to perform topological operations like cutting or refining. The framework used for simulations in this thesis is designed to simulate materials using Mass Spring Systems (MSS) with particular input parameters. Using Mass-Spring System, which is known for its simplicity and ability to perform fast simulations, we present several physically-based improvements to control global features of MSS which play the key role in simulation of real bodies.

The building is a complex system composed by several components. Practically, CAD computer softwares described a building by a set of geometrical shapes. However, there is only few simulating tools that use directly such a description of the building object. In most cases, the simulations represent a building by a graph or an equivalent network, i.e. a topological structure made of vertices and connections between them, representing identifiable portions of the building: premises, walls, junctions, arrays, structures carrier, doublings, ... Identify these entities, build the equivalent graph, extract dimensional characteristics, ... represent as many difficult problems given the wide variety of data handled. Combinatorial maps provide a simple and effective formalism to describe a complex geometry from its topological structure. Such a structure code firstly the connecting links between vertices, curves, surfaces and volumes, then embed it into a geometric support. The creation, modification or deformation of the representation involves a small number of clearly defined operations, which facilitates transcription of the mathematical formalism into computer software. This work should be part of all strategic software tools developed at CSTB (EVE-BIM, ICARE, PHANIE, ACOUBAT, ...) The goal of this thesis is to conceive and develop the numerical tools allowing: (a) the semi-automatic construction of a combinatorial map representing themodel of a building in its immediate surroundings, with the possibility of identifying entire rooms structural elements and components, from CAD data (such as DXF, IFC or equivalent) and GIS. (b) the semi-automatic extraction, from the topological/geometric unified description, of different specific representations for different simulation domains: surface envelope of rooms, adjacency graphs and dimensional properties, estimators depending on the nature of the materials... (c) To simplify and/or to enhance the representation of buildings to facilitate the representation of elements at different levels of detail. This thesis is in an industrial context. Thus there is a strong constraint for software development of the proposed methods. The development will be done in C++ language, related to the 3D geometric modeler Moka and the combinatorial map kernel of CGAL.

A generalized map is a topological model that represents implicitly a set of cells (vertices, edges, faces, volumes, ...) as well as all the incidence and adjacency relations through darts and involutions. Generalized maps are mainly used to model images and 3D objects. There are few tools to analyze and compare generalized maps. Our goal is to define a set of tools for comparing generalized maps. We first define a similarity measure based on the size of the common part between two generalized maps called greatest common submap. We define two types of sub-maps, partial and induced. Induced sub-map must preserved all involutions while partial sub-map allows to remove also some involutions. Partial sub-maps allow to not preserved all the involutions in analogy to the partial subgraph for which the edges can not all be present. Then we define a set of modification operations of darts and sewings and use them to define the generalized map edit distance. This edit distance is equal to the minimum cost generated by all sequences of operations transforming a generalized map into another one. This distance allows the use of labels, with the substitution operation. Labels are placed on the darts and allow to add information to generalized maps. We then show that for certain costs our edit distance can be directly computed from the largest common submap. The computation of the edit distance is an NP-hard problem. We propose a greedy algorithm to compute in polynomial time an approximation of our edit distance. We propose a set of heuristics based on descriptors of the neighborhood darts of the generalized map to guide the greedy algorithm, and we evaluate these heuristics on randomly generated test sets, for which we know a bound of the distance. We propose some possible use of our similarity measures in the field of image and meshes analysis. We compare our edit distance of generalized maps with the edit distance of graphs, often used in structural pattern recognition. We also define a set of heuristics taking into account the labels generalized maps modeling images and meshes. We emphasize the qualitative aspect of our matching, allowing to match areas of the image and mesh points.

A combinatorial map is a topological model that represents the subdivisions of an nD space into cells and their adjacency relations in n dimensions. This data structure is increasingly used in image processing, but it still lacks tools for analysis. Our goal is to define new tools for combinatorial nD maps. We are particularly interested in the extraction of submaps in a database of maps. We define two combinatorial map signatures: the first one has a quadratic space complexity and may be used to decide of isomorphism with a new map in linear time whereas the second one has a linear space complexity and may be used to decide of isomorphism in quadratic time. They can be used for connected maps, non connected maps, labbeled maps or non labelled maps. These signatures can be used to efficiently search for a map in a database. Moreover, the search time does not depend on the number of maps in the database. We introduce the problem of finding frequent submaps in a database of combinatorial nD maps. We describe two algorithms for solving this problem. The first algorithm extracts the submaps with a breadth-first search approach and the second one uses a depth-first search approach. We compare these two algorithms on synthetic database of maps. Finally, we propose to use the frequent patterns in an image classification application. Each image is described by a map that is transformed into a vector representing the number of occurrences of frequent patterns. From these vectors, we use standard techniques of classification defined on vector spaces. We propose experiments in supervised and unsupervised classification on two image databases.

Some data acquisition devices produce images of several gigabytes. Analyzing such large images raises two main issues. First, the data volume to process forbids a global image analysis, hence a hard partitioning problem. Second, a multi-resolution approach is required to extract global features at low resolution. For instance, regarding histological images, recent improvments in scanners' accuracy allow nowadays to examine cellular structures on the whole slide. However, produced images are up to 18\,GB. Besides, considering a tissue as a particular layout of cells is a global information that is only available at low resolution. Thus, these images combine multi-scale and multi-resolution information. In this work, we define a topological and hierarchical model which is suitable for the segmentation of large images. Our work is based on the models of topological map and combinatorial pyramid. We introduce the tiled map model in order to encode the topology of large partitions and a hierarchical extension, the tiled top-down pyramid, to represent the duality between multi-scale and multi-resolution information. Finally, we propose an application of our model for the segmentation of large images in histology.

A 3D topological map is a model used in image processing which represents the partition of a 3D image into regions. In this work, we introduce some tools that allow to modify a partition presented by a topological map, and we use these tools to propose segmentation algorithms implementing topological criteria. In a first part, we propose three operations. The region merging is defined with a local approach suited for interactive use, and a global approach suited for automatic processing like image segmentation. The region splitting is introduced with a burst into voxel approach, and the split with a guide. Last, a deformation of the partition based on the definition of ML-Simple points: voxels that can be flipped of region without changing the topology of the partition. With these operations, we implement in a second part image segmentation processes using topological maps. First we adapt to our model an existing algorithm using a criterion based on the notion of contrast. Then, we propose methods to compute topological invariants of regions: the Betti numbers. Using these methods we implement a topological criterion that controls the number of tunnels and cavities of the regions. Last, we give an overview of the possibilities of our tools by creating a toolchain to segment brain tumors in medical images.

Virtual architectural (indoor) scenes are often modelled in 3D for various types of simulation systems. For instance, some authors propose methods dedicated to lighting, heat transfer, acoustic or radio wave propagation simulations. These methods rely in most cases on a volumetric representation of the environment, with adjacency and incidence relationships. Unfortunately, many buildings data are only given by 2D plans and the 3D needs varies from one application to another. To solve these problems, we propose a formal representation of consistency constraints dedicated to building interiors and associated with a topological model . We show that such a representation can be used for: (i) reconstructing a 3D model from 2D architectural plans ; (ii) detecting automatically geometrical, topological and semantical inconsistencies ; (iii) designing automatic and semi-automatic operations to correct and enrich a 2D plan. All our constraints are homogeneously defined in 2D and 3D, implemented with generalized maps and used in modeling operations. We explain how this model can be successfully used with various ray-tracing methods.

In this work, we are interested in the hierarchical geometrical modeling with a topological basis. We propose the definition of a generic model in any dimension, and we show a possible application in multi-level segmentation of 3D images. In the first part of this work, we define and study the nD generalized pyramids. This is a generic hierarchical topological model which represents all the cells of a subdivision as well as the adjacency and incidence relations existing between these cells. We propose and compare three possible representations of these pyramids. In order to retrieve the information which corresponds to a cell, we define the notion of generalized orbit. This notion extends the notion of receptive field. We also define a local modification operation of a pyramid level allowing to preserve the model coherence by propagating the modifications at the upper levels. In the second part of this work, we show how to use this model in the case of a multi-level segmentation of 3D images. We define the properties which have to be respected by the pyramid, and we give the algorithms that allow to construct such a pyramid. Then, we show how to use the generalized orbits in order to retrieve the voxels or the inter-voxel elements which compose a region or its boundary. Finally we define an operation allowing to locally modify the segmentation criterion of a region set. This operation is based on the operation defined in the first part in order to preserve the coherence constraints.