Topological alterations of 3D digital images under rigid transformations

National audienceRigid transformations in R^n are known to preserve the shape, and are often applied to digital images. However, digitized rigid transformations, defined as digital functions from Z^n to Z^n do not preserve shapes in general\string; indeed, they are almost never bijective and thus alter the topology. In order to understand the causes of such topological alterations, we first study the possible loss of voxel information and modification of voxel adjacencies induced by applications of digitized rigid transformations to 3D digital images. We then show that even very simple structured images such as digital half-spaces may not preserve their topology under these transformations. This signifies that a simple extension of the two-dimensional solution for topology preservation cannot be made in three dimensions

Sufficient conditions for topological invariance of 2D images under rigid transformations

International audienceIn ℝ^2, rigid transformations are topology-preserving operations. However, this property is generally no longer true when considering digital images instead of continuous ones, due to digitization effects. In this article, we investigate this issue by studying discrete rigid transformations (DRTs) on ℤ^2. More precisely, we define conditions under which digital images preserve their topological properties under any arbitrary DRTs. Based on the recently introduced notion of DRT graph and the classical notion of simple point, we first identify a family of local patterns that authorize topological invariance under DRTs. These patterns are then involved in a local analysis process that guarantees topological invariance of whole digital images in linear time

Efficient neighbourhood computing for discrete rigid transformation graph search

International audienceRigid transformations are involved in a wide variety of image processing applications, including image registration. In this context, we recently proposed to deal with the associated optimization problem from a purely discrete point of view, using the notion of discrete rigid transformation (DRT) graph. In particular, a local search scheme within the DRT graph to compute a locally optimal solution without any numerical approximation was formerly proposed. In this article, we extend this study, with the purpose to reduce the algorithmic complexity of the proposed optimization scheme. To this end, we propose a novel algorithmic framework for just-in-time computation of sub-graphs of interest within the DRT graph. Experimental results illustrate the potential usefulness of our approach for image registration

Rigid transformations on 2D digital images : combinatorial and topological analysis

In this thesis, we study rigid transformations in the context of computer imagery. In particular, we develop a fully discrete framework for handling such transformations. Rigid transformations, initially defined in the continuous domain, are involved in a wide range of digital image processing applications. In this context, the induced digital rigid transformations present different geometrical and topological properties with respect to their continuous analogues. In order to overcome the issues raised by these differences, we propose to formulate rigid transformations on digital images in a fully discrete framework. In this framework, Euclidean rigid transformations producing the same digital rigid transformation are put in the same equivalence class. Moreover, the relationship between these classes can be modeled as a graph structure. We prove that this graph has a polynomial space complexity with respect to the size of the considered image, and presents useful structural properties. In particular, it allows us to generate incrementally all digital rigid transformations without numerical approximation. This structure constitutes a theoretical tool to investigate the relationships between geometry and topology in the context of digital images. It is also interesting from the methodological point of view, as we illustrate by its use for assessing the topological behavior of images under rigid transformationsDans cette thèse, nous étudions les transformations rigides dans le contexte de l'imagerie numérique. En particulier, nous développons un cadre purement discret pour traiter ces transformations. Les transformations rigides, initialement définies dans le domaine continu, sont impliquées dans de nombreuses applications de traitement d'images numériques. Dans ce contexte, les transformations rigides digitales induites présentent des propriétés géométriques et topologiques différentes par rapport à leurs analogues continues. Afin de s'affranchir des problèmes inhérents à ces différences, nous proposons de formuler ces transformations rigides dans un cadre purement discret. Dans ce cadre, les transformations rigides sont regroupées en classes correspondant chacune à une transformation digitale donnée. De plus, les relations entre ces classes de transformations peuvent être modélisées par une structure de graphe. Nous prouvons que ce graphe présente une complexité spatiale polynômiale par rapport à la taille de l'image. Il présente également des propriétés structurelles intéressantes. En particulier, il permet de générer de manière progressive toute transformation rigide digitale, et ce sans approximation numérique. Cette structure constitue un outil théorique pour l'étude des relations entre la géométrie et la topologie dans le contexte de l'imagerie numérique. Elle présente aussi un intérêt méthodologique, comme l'illustre son utilisation pour l'évaluation du comportement topologique des images sous des transformations rigide

A tree-topology preserving pairing for 3D/2D registration

Information Processing in Computer-Assisted Interventions (IPCAI) 2015 Special IssueInternational audiencePurpose: Fusing pre-operative and intra-operative information into a single space aims at taking advantage of two complementary modalities and necessitates a step of registration that must provide good alignment and relevant correspondences. This paper addresses both purposes in the case of 3D/2D vessel tree matching. Method: We propose a registration algorithm endorsing this vascular tree nature by providing a pairing procedure that preserves the tree topology and by integrating this pairing into an iterative algorithm maintaining pairing coherence. In addition, we define two complementary error measures quantifying the resulting alignment error and pairing error. Both are based on manual ground-truth that is independent of the type of transformation to retrieve. Results: Experiments were conducted on a database of 63 clinical cases, evaluating robustness and accuracy of our approach with respect to the iterative closest point algorithm. Conclusion: The proposed method exhibits good results both in term of pairing and alignment as well as low sensitivity to rotations to be compensated (up to 30 degrees)

A Simple Regularizer for B-spline Nonrigid Image Registration That Encourages Local Invertibility

Nonrigid image registration is an important task for many medical imaging applications. In particular, for radiation oncology it is desirable to track respiratory motion for thoracic cancer treatment. B-splines are convenient for modeling nonrigid deformations, but ensuring invertibility can be a challenge. This paper describes sufficient conditions for local invertibility of deformations based on B-spline bases. These sufficient conditions can be used with constrained optimization to enforce local invertibility. We also incorporate these conditions into nonrigid image registration methods based on a simple penalty approach that encourages diffeomorphic deformations. Traditional Jacobian penalty methods penalize negative Jacobian determinant values only at grid points. In contrast, our new method enforces a sufficient condition for invertibility directly on the deformation coefficients to encourage invertibility globally over a 3-D continuous domain. The proposed penalty approach requires substantially less compute time than Jacobian penalties per iteration.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85951/1/Fessler21.pd

2D and 3D surface image processing algorithms and their applications

This doctoral dissertation work aims to develop algorithms for 2D image segmentation application of solar filament disappearance detection, 3D mesh simplification, and 3D image warping in pre-surgery simulation. Filament area detection in solar images is an image segmentation problem. A thresholding and region growing combined method is proposed and applied in this application. Based on the filament area detection results, filament disappearances are reported in real time. The solar images in 1999 are processed with this proposed system and three statistical results of filaments are presented. 3D images can be obtained by passive and active range sensing. An image registration process finds the transformation between each pair of range views. To model an object, a common reference frame in which all views can be transformed must be defined. After the registration, the range views should be integrated into a non-redundant model. Optimization is necessary to obtain a complete 3D model. One single surface representation can better fit to the data. It may be further simplified for rendering, storing and transmitting efficiently, or the representation can be converted to some other formats. This work proposes an efficient algorithm for solving the mesh simplification problem, approximating an arbitrary mesh by a simplified mesh. The algorithm uses Root Mean Square distance error metric to decide the facet curvature. Two vertices of one edge and the surrounding vertices decide the average plane. The simplification results are excellent and the computation speed is fast. The algorithm is compared with six other major simplification algorithms. Image morphing is used for all methods that gradually and continuously deform a source image into a target image, while producing the in-between models. Image warping is a continuous deformation of a: graphical object. A morphing process is usually composed of warping and interpolation. This work develops a direct-manipulation-of-free-form-deformation-based method and application for pre-surgical planning. The developed user interface provides a friendly interactive tool in the plastic surgery. Nose augmentation surgery is presented as an example. Displacement vector and lattices resulting in different resolution are used to obtain various deformation results. During the deformation, the volume change of the model is also considered based on a simplified skin-muscle model

Medical imaging analysis with artificial neural networks

Given that neural networks have been widely reported in the research community of medical imaging, we provide a focused literature survey on recent neural network developments in computer-aided diagnosis, medical image segmentation and edge detection towards visual content analysis, and medical image registration for its pre-processing and post-processing, with the aims of increasing awareness of how neural networks can be applied to these areas and to provide a foundation for further research and practical development. Representative techniques and algorithms are explained in detail to provide inspiring examples illustrating: (i) how a known neural network with fixed structure and training procedure could be applied to resolve a medical imaging problem; (ii) how medical images could be analysed, processed, and characterised by neural networks; and (iii) how neural networks could be expanded further to resolve problems relevant to medical imaging. In the concluding section, a highlight of comparisons among many neural network applications is included to provide a global view on computational intelligence with neural networks in medical imaging

Shape analysis and description based on the isometric invariances of topological skeletonization

ilustracionesIn this dissertation, we explore the problem of how to describe the shape of an object in 2D and 3D with a set of features that are invariant to isometric transformations. We focus to based our approach on the well-known Medial Axis Transform and its topological properties. We aim to study two problems. The first is how to find a shape representation of a segmented object that exhibits rotation, translation, and reflection invariance. The second problem is how to build a machine learning pipeline that uses the isometric invariance of the shape representation to do both classification and retrieval. Our proposed solution demonstrates competitive results compared to state-of-the-art approaches. We based our shape representation on the medial axis transform (MAT), sometimes called the topological skeleton. Accepted and well-studied properties of the medial axis include: homotopy preservation, rotation invariance, mediality, one pixel thickness, and the ability to fully reconstruct the object. These properties make the MAT a suitable input to create shape features; however, several problems arise because not all skeletonization methods satisfy all the above-mentioned properties at the same time. In general, skeletons based on thinning approaches preserve topology but are noise sensitive and do not allow a proper reconstruction. They are also not invariant to rotations. Voronoi skeletons also preserve topology and are rotation invariant, but do not have information about the thickness of the object, making reconstruction impossible. The Voronoi skeleton is an approximation of the real skeleton. The denser the sampling of the boundary, the better the approximation; however, a denser sampling makes the Voronoi diagram more computationally expensive. In contrast, distance transform methods allow the reconstruction of the original object by providing the distance from every pixel in the skeleton to the boundary. Moreover, they exhibit an acceptable degree of the properties listed above, but noise sensitivity remains an issue. Therefore, we selected distance transform medial axis methods as our skeletonization strategy, and focused on creating a new noise-free approach to solve the contour noise problem. To effectively classify an object, or perform any other task with features based on its shape, the descriptor needs to be a normalized, compact form: $\Phi$ should map every shape $\Omega$ to the same vector space $\mathrm{R}^{n}$. This is not possible with skeletonization methods because the skeletons of different objects have different numbers of branches and different numbers of points, even when they belong to the same category. Consequently, we developed a strategy to extract features from the skeleton through the map $\Phi$, which we used as an input to a machine learning approach. After developing our method for robust skeletonization, the next step is to use such skeleton into the machine learning pipeline to classify object into previously defined categories. We developed a set of skeletal features that were used as input data to the machine learning architectures. We ran experiments on MPEG7 and ModelNet40 dataset to test our approach in both 2D and 3D. Our experiments show results comparable with the state-of-the-art in shape classification and retrieval. Our experiments also show that our pipeline and our skeletal features exhibit some degree of invariance to isometric transformations. In this study, we sought to design an isometric invariant shape descriptor through robust skeletonization enforced by a feature extraction pipeline that exploits such invariance through a machine learning methodology. We conducted a set of classification and retrieval experiments over well-known benchmarks to validate our proposed method. (Tomado de la fuente)En esta disertación se explora el problema de cómo describir la forma de un objeto en 2D y 3D con un conjunto de características que sean invariantes a transformaciones isométricas. La metodología propuesta en este documento se enfoca en la Transformada del Eje Medio (Medial Axis Transform) y sus propiedades topológicas. Nuestro objetivo es estudiar dos problemas. El primero es encontrar una representación matemática de la forma de un objeto que exhiba invarianza a las operaciones de rotación, translación y reflexión. El segundo problema es como construir un modelo de machine learning que use esas invarianzas para las tareas de clasificación y consulta de objetos a través de su forma. El método propuesto en esta tesis muestra resultados competitivos en comparación con otros métodos del estado del arte. En este trabajo basamos nuestra representación de forma en la transformada del eje medio, a veces llamada esqueleto topológico. Algunas propiedades conocidas y bien estudiadas de la transformada del eje medio son: conservación de la homotopía, invarianza a la rotación, su grosor consiste en un solo pixel (1D), y la habilidad para reconstruir el objeto original a través de ella. Estas propiedades hacen de la transformada del eje medio un punto de partida adecuado para crear características de forma. Sin embargo, en este punto surgen varios problemas dado que no todos los métodos de esqueletización satisfacen, al mismo tiempo, todas las propiedades mencionadas anteriormente. En general, los esqueletos basados en enfoques de erosión morfológica conservan la topología del objeto, pero son sensibles al ruido y no permiten una reconstrucción adecuada. Además, no son invariantes a las rotaciones. Otro método de esqueletización son los esqueletos de Voronoi. Los esqueletos de Voronoi también conservan la topología y son invariantes a la rotación, pero no tienen información sobre el grosor del objeto, lo que hace imposible su reconstrucción. Cuanto más denso sea el muestreo del contorno del objeto, mejor será la aproximación. Sin embargo, un muestreo más denso hace que el diagrama de Voronoi sea más costoso computacionalmente. Por el contrario, los métodos basados en la transformada de la distancia permiten la reconstrucción del objeto original, ya que proporcionan la distancia desde cada píxel del esqueleto hasta su punto más cercano en el contorno. Además, exhiben un grado aceptable de las propiedades enumeradas anteriormente, aunque la sensibilidad al ruido sigue siendo un problema. Por lo tanto, en este documento seleccionamos los métodos basados en la transformada de la distancia como nuestra estrategia de esqueletización, y nos enfocamos en crear un nuevo enfoque que resuelva el problema del ruido en el contorno. Para clasificar eficazmente un objeto o realizar cualquier otra tarea con características basadas en su forma, el descriptor debe ser compacto y estar normalizado: $\Phi$ debe relacionar cada forma $\Omega$ al mismo espacio vectorial $\mathrm{R}^{n}$. Esto no es posible con los métodos de esqueletización en el estado del arte, porque los esqueletos de diferentes objetos tienen diferentes números de ramas y diferentes números de puntos incluso cuando pertenecen a la misma categoría. Consecuentemente, en nuestra propuesta desarrollamos una estrategia para extraer características del esqueleto a través de la función $\Phi$, que usamos como entrada para un enfoque de aprendizaje automático. % TODO completar con resultados. Después de desarrollar nuestro método de esqueletización robusta, el siguiente paso es usar dicho esqueleto en un modelo de aprendizaje de máquina para clasificar el objeto en categorías previamente definidas. Para ello se desarrolló un conjunto de características basadas en el eje medio que se utilizaron como datos de entrada para la arquitectura de aprendizaje automático. Realizamos experimentos en los conjuntos de datos: MPEG7 y ModelNet40 para probar nuestro enfoque tanto en 2D como en 3D. Nuestros experimentos muestran resultados comparables con el estado del arte en clasificación y consulta de formas (retrieval). Nuestros experimentos también muestran que el modelo desarrollado junto con nuestras características basadas en el eje medio son invariantes a las transformaciones isométricas. (Tomado de la fuente)Beca para Doctorados Nacionales de Colciencias, convocatoria 725 de 2015DoctoradoDoctor en IngenieríaVisión por computadora y aprendizaje automátic