Christian Brechbühler

Parametrization of closed surfaces for 3-D shape description

This paper presents procedures for the explicit parametric representation and global description of surfaces of simply connected 3-D objects. The novel techniques overcome severe limitations of earlier methods (restriction to star-shaped objects (D. H. Ballard and Ch. M. Brown, Computer Vision, Prentice-Hall, Englewood Cliffs, NJ, 1981), constraints on positioning and shape of cross-sections (F. Solina and R. Bajcsy, IEEE Trans. Pattern Anal. Much. Intell. 12(2), 1990, 131-147; L. H. Staib and J. S. Duncan, in Visualization in Biomedical Computing 1992 (R. A. Robb, Ed.), Vol. Proc. SPIE 108, pp. 90-104, 1992), and nonhomogeneous distribution of parameter space). We parametrize the surface by defining a continuous, one-to-one mapping from the surface of the original object to the surface of a unit sphere. The parametrization is formulated as a constrained optimization problem. Practicable starting values are obtained by an initial mapping based on a heat conduction model. The parametrization enables us to expand the object surface into a series of spherical harmonic functions, extending to 3-D the concept of elliptical Fourier descriptors for 2-D closed curves (E. Persoon and K. S. Fu, IEEE Trans. Syst. Man Cybernetics 7(3), 1977, 388-397; F. P. Kuhl and Ch. R. Giardina, Comput. Graphics Image Process. 18(3), 1982, 236-258). Invariant, object-centered descriptors are obtained by rotating the parameter net and the object into standard positions. The new methods are illustrated with 3-D test objects. Potential applications are recognition, classification, and comparison of convoluted surfaces or parts of surfaces of 3-D shapes.

Parametric estimate of intensity inhomogeneities applied to MRI

Presents a new approach to the correction of intensity inhomogeneities in magnetic resonance imaging (MRI) that significantly improves intensity-based tissue segmentation. The distortion of the image brightness values by a low-frequency bias field impedes visual inspection and segmentation. The new correction method called parametric bias field correction (PABIC) is based on a simplified model of the imaging process, a parametric model of tissue class statistics, and a polynomial model of the inhomogeneity field. The authors assume that the image is composed of pixels assigned to a small number of categories with a priori known statistics. Further they assume that the image is corrupted by noise and a low-frequency inhomogeneity field. The estimation of the parametric bias field is formulated as a nonlinear energy minimization problem using an evolution strategy (ES). The resulting bias field is independent of the image region configurations and thus overcomes limitations of methods based on homomorphic filtering. Furthermore, PABIC can correct bias distortions much larger than the image contrast. Input parameters are the intensity statistics of the classes and the degree of the polynomial function. The polynomial approach combines bias correction with histogram adjustment, making it well suited for normalizing the intensity histogram of datasets from serial studies. The authors present simulations and a quantitative validation with phantom and test images. A large number of MR image data acquired with breast, surface, and head coils, both in two dimensions and three dimensions, have been processed and demonstrate the versatility and robustness of this new bias correction scheme.

Segmentation of 2-D and 3-D objects from MRI volume data using constrained elastic deformations of flexible Fourier contour and surface models

This paper describes a new model-based segmentation technique combining desirable properties of physical models (snakes), shape representation by Fourier parametrization, and modelling of natural shape variability. Flexible parametric shape models are represented by a parameter vector describing the mean contour and by a set of eigenmodes of the parameters characterizing the shape variation. Usually the segmentation process is divided into an initial placement of the mean model and an elastic deformation restricted to the model variability. This, however leads to a separation of biological variation due to a global similarity transform from small-scale shape changes originating from elastic deformations of the normalized model contours only. The performance can be considerably improved by building shape models normalized with respect to a small set of stable landmarks (AC-PC in our application) and by explaining the remaining variability among a series of images with the model flexibility. This way the image interpretation is solved by a new coarse-to-fine segmentation procedure based on the set of deformation eigenmodes, making a separate initialization step unnecessary. Although straightforward, the extension to 3-D is severely impeded by difficulties arising during the generation of a proper surface parametrization for arbitrary objects with spherical topology. We apply a newly developed surface parametrization which achieves a uniform mapping between object surface and parameter space. The 3-D procedure is demonstrated by segmenting deep structures of the human brain from MR volume data.

Symbolic Description of 3-D Structures Applied to Cerebral Vessel Tree Obtained from MR Angiography Volume Data

The present paper focuses on the conversion of multidimensional image structures to an object-centered, abstract description encoding shape features and structure relationships. We describe a prototype system that extracts three-dimensional (3-D) curvilinear structures from volume image data and transforms them into a symbolic description which represents topological and geometrical features of tree-like, filamentous objects.

Segmentation of 3D Objects from MRI Volume Data Using Constrained Elastic Deformations of Flexible Fourier Surface Models

This paper describes a new model-based segmentation technique combining desirable properties of physical models (snakes, [2]), shape representation by Fourier parametrization (Fourier snakes, [12]), and modelling of natural shape variability (eigenmodes, [7, 10]). Flexible shape models are represented by a parameter vector describing the mean contour and by a set of eigenmodes of the parameters characterizing the shape variation with respect to a small set of stable landmarks (AC-PC in our application) and explaining the remaining variability among a series of images with the model flexibility. Although straightforward, the extension to 3-D is severely impeded by finding a proper surface parametrization for arbitrary objects with spherical topology. We apply a newly developed surface parametrization [16, 17] which achieves a uniform mapping between object surface and parameter space. The 3D model building and Fourier-snake procedure are demonstrated by segmenting deep structures of the human brain from MR volume data.

Compensation of Spatial Inhomogeneity in MRI Based on a Parametric Bias Estimate

A novel bias correction technique is proposed based on the estimation of the parameters of a polynomial bias field directly from image data. The procedure overcomes difficulties known from homomorphic filtering or from techniques assuming an initial presegmented image. The only parameters are a set of expected class means and the standard deviation. Applications to various MR images illustrate the performance.

Imposing Hard Constraints on Soft Snakes

An approach is presented for imposing generic hard constraints on deformable models at a low computational cost, while preserving the good convergence properties of snake-like models. We believe this capability to be essential not only for the accurate modeling of individual objects that obey known geometric and semantic constraints but also for the consistent modeling of sets of objects.

Imposing Hard Constraints on Deformable Models through Optimization in Orthogonal Subspaces

An approach is presented for imposing generic hard constraints on deformable models at a low computational cost, while preserving the good convergence properties of snake-like models. We believe this capability to be essential not only for the accurate modeling of individual objects that obey known geometric and semantic constraints but also for the consistent modeling of sets of objects. Many of the approaches to this problem that have appeared in the vision literature rely on adding penalty terms to the objective functions. They rapidly become intractable when the number of constraints increases. Applied mathematicians have developed powerful constrainted optimization algorithms that, in theory, can address this problem. However, these algorithms typically do not take advantage of the specific properties of snakes. We have therefore designed a new algorithm that is closely related to Lagrangian methods but is tailored to accommodate the particular brand of deformable models used in the image understanding community. We demonstrate the validity of our approach first in two dimensions using synthetic images and then in three dimensions using real aerial images to simultaneously model terrain, roads, and ridgelines under consistency constraints.

Analysis of MR Angiography Volume Data Leading to the Structural Description of the Cerebral Vessel Tree

The performance of computer assisted systems for presentation, manipulation and quantitation of objects obtained from multidimensional image data depends critically on the ability to segment and describe structures in images. We describe the development of a prototype system that extracts three-dimensional (3-D) curvilinear structures from volume image data and converts them into a symbolic description which is more appropriate to assess features of tree-like, filamentous objects. The initial segmentation is performed by 3-D line filtering and/or 3-D hysteresis thresholding. A skeletal structure is derived by 3-D binary thinning, approximating the center-line by pseudo-parallel erosion while fully preserving the 3-D topology. The final graph data-structure encodes the spatial course of line sections, the estimate of the local diameter, and the topology at important key locations like branchings and end-points. The system is applied to analyze the cerebral vascular system resulting from magnetic resonance angiography (MRA).

3D Verdünnung zur symbolischen Beschreibung von verästelten, räumlichen Strukturen

Die Analyse von 3-D Strukturen erfordert Segmentierung und objektzentrierte Beschreibung. Wahrend bei der Segmentierung von 3-D Bilddaten Erfolge erzielt wurden, stehen Beschreibungsmethoden, die Zugriff auf geometrische Formkriterien erlauben, immer noch weitgehend aus.