Qingde Li

Smooth Piecewise Polynomial Blending Operations for Implicit Shapes

In this paper, we present a new set of blending operations for implicitly defined geometric shapes. The proposed shape operators are piecewise polynomial and blending range controllable, and can be constructed to any required degree of smoothness. The key idea behind these techniques is the introduction of the concept of the smooth absolute functions, which in turn lead to the definition of smooth maximum functions. These novel generalized absolute functions can be constructed recursively or through a recursively defined functions, and can thus be computed cheaply. In addition, the underlying mathematical descriptions of these shape operations are very simple and elegant.

2D piecewise algebraic splines for implicit modeling

2D splines are a powerful tool for shape modeling, either parametrically or implicitly. However, compared with regular grid-based tensor-product splines, most of the high-dimensional spline techniques based on nonregular 2D polygons, such as box spline and simplex spline, are generally very expensive to evaluate. Though they have many desirable mathematical properties and have been proved theoretically to be powerful in graphics modeling, they are not a convenient graphics modeling technique in terms of practical implementation. In shape modeling practice, we still lack a simple and practical procedure in creating a set of bivariate spline basis functions from an arbitrarily specified 2D polygonal mesh. Solving this problem is of particular importance in using 2D algebraic splines for implicit modeling, as in this situation underlying implicit equations need to be solved quickly and accurately. In this article, a new type of bivariate spline function is introduced. This newly proposed type of bivariate spline function can be created from any given set of 2D polygons that partitions the 2D plane with any required degree of smoothness. In addition, the spline basis functions created with the proposed procedure are piecewise polynomials and can be described explicitly in analytical form. As a result, they can be evaluated efficiently and accurately. Furthermore, they have all the good properties of conventional 2D tensor-product-based B-spline basis functions, such as non-negativity, partition of unit, and convex-hull property. Apart from their obvious use in designing freeform parametric geometric shapes, the proposed 2D splines have been shown a powerful tool for implicit shape modeling.

Implicit Fitting Using Radial Basis Functions with Ellipsoid Constraint

Implicit planar curve and surface fitting to a set of scattered points plays an important role in solving a wide variety of problems occurring in computer graphics modelling, computer graphics animation, and computer assisted surgery. The fitted implicit surfaces can be either algebraic or non‐algebraic. The main problem with most algebraic surface fitting algorithms is that the surface fitted to a given data set is often unbounded, multiple sheeted, and disconnected when a high degree polynomial is used, whereas a low degree polynomial is too simple to represent general shapes. Recently, there has been increasing interest in non‐algebraic implicit surface fitting. In these techniques, one popular way of representing an implicit surface has been the use of radial basis functions. This type of implicit surface can represent various shapes to a high level of accuracy. In this paper, we present an implicit surface fitting algorithm using radial basis functions with an ellipsoid constraint. This method does not need to build interior and exterior layers for the given data set or to use information on surface normal but still can fit the data accurately. Furthermore, the fitted shape can still capture the main features of the object when the data sets are extremely sparse. The algorithm involves solving a simple general eigen‐system and a computation of the inverse or psedo‐inverse of a matrix, which is straightforward to implement.

Implicit Reconstruction of Vasculatures Using Bivariate Piecewise Algebraic Splines

Vasculature geometry reconstruction from volumetric medical data is a crucial task in the development of computer guided minimally invasive vascular surgery systems. In this paper, a technique for reconstructing the geometry of vasculatures using bivariate implicit splines is developed. With the proposed technique, an implicit geometry representation of the vascular tree can be accurately constructed based on the voxels extracted directly from the surface of a certain vascular structure in a given volumetric medical dataset. Experimental results show that the geometric representation built using our method can faithfully represent the morphology and topology of vascular structures. In addition, both the qualitative and the quantitative validations have been performed to show that the reconstructed vessel geometry is of high accuracy and smoothness. An virtual angioscopy system has been implemented to indicate one of the strengths of our proposed method.

Skeleton Cuts—An Efficient Segmentation Method for Volume Rendering

Volume rendering has long been used as a key technique for volume data visualization, which works by using a transfer function to map color and opacity to each voxel. Many volume rendering approaches proposed so far for voxels classification have been limited in a single global transfer function, which is in general unable to properly visualize interested structures. In this paper, we propose a localized volume data visualization approach which regards volume visualization as a combination of two mutually related processes: the segmentation of interested structures and the visualization using a locally designed transfer function for each individual structure of interest. As shown in our work, a new interactive segmentation algorithm is advanced via skeletons to properly categorize interested structures. In addition, a localized transfer function is subsequently presented to assign optical parameters via interested information such as intensity, thickness and distance. As can be seen from the experimental results, the proposed techniques allow to appropriately visualize interested structures in highly complex volume medical data sets.

Constructive implicit fitting

In this paper, we present a constructive method for fitting both an explicit and an implicit curve or surface to a set of scattered points by using gate functions. With this technique, the data are first partitioned with geometric primitives into small data sets such that each sub-data set can be well fitted by a simple algebraic or non-algebraic shape. These simple shapes are then blended to form an overall fitting for the given data. Compared with some conventional fitting techniques, the proposed method has the following distinct features. First of all, a preset accuracy can always be achieved if the data set is sufficiently finely partitioned. Secondly, a large data set can be dealt with as several small data sets, which can significantly reduce the complexity of the shape to be fitted. Thirdly, the proposed fitting technique can also be used to increase the fitting speed by using simple fitting techniques for each sub-data set. Fourthly, the degree of smoothness of the fitted function can be adjusted to be as smooth as one wishes. Furthermore, the fitting technique provides direct support for parallel computation in curve and surface fitting. When a large data set is partitioned into smaller data sets, these small data sets can then be fitted simultaneously over a parallel system. This will greatly reduce computation time.

Partial shape-preserving splines

A complex geometric shape is often a composition of a set of simple ones, which may differ from each other in terms of their mathematical representations and the ways in which they are constructed. One of the necessary requirements in combining these simple shapes is that their original shapes can be preserved as much as possible. In this paper, a set of partial shape-preserving (PSP) spline basis functions is introduced to smoothly combine a collection of shape primitives with flexible blending range control. These spline basis functions can be considered as a kind of generalization of traditional B-spline basis functions, where the shape primitives used are control points or control polygons. The PSP-spline basis functions have all the advantages of the conventional B-spline technique in the sense that they are nonnegative, piecewise polynomial and of property of partition of unity. However, PSP-spline is a more powerful freeform geometric shape design technique in the sense that it is also a kind of shape-preserving spline. In addition, the PSP-spline technique implicitly integrates the weights of shape control primitives into its basis functions, which allows users to design a required geometric shape based on weighted control primitives. Though its basis functions are simply piecewise polynomial functions, it has the same shape design strengths as the rational piecewise polynomial based spline techniques such as NURBS. In particular, when control shape primitives are specified as a set of control points, PSP-spline behaves like a polygon smoother, with which a shape can be designed to approximate the specified control polygon or control mesh smoothly with any required precision. Consequently, a richer set of geometric shapes can be built using a relatively smaller set of control points.

3D vasculature segmentation using localized hybrid level-set method

BackgroundIntensity inhomogeneity occurs in many medical images, especially in vessel images. Overcoming the difficulty due to image inhomogeneity is crucial for the segmentation of vessel image.MethodsThis paper proposes a localized hybrid level-set method for the segmentation of 3D vessel image. The proposed method integrates both local region information and boundary information for vessel segmentation, which is essential for the accurate extraction of tiny vessel structures. The local intensity information is firstly embedded into a region-based contour model, and then incorporated into the level-set formulation of the geodesic active contour model. Compared with the preset global threshold based method, the use of automatically calculated local thresholds enables the extraction of the local image information, which is essential for the segmentation of vessel images.ResultsExperiments carried out on the segmentation of 3D vessel images demonstrate the strengths of using locally specified dynamic thresholds in our level-set method. Furthermore, both qualitative comparison and quantitative validations have been performed to evaluate the effectiveness of our proposed model.ConclusionsExperimental results and validations demonstrate that our proposed model can achieve more promising segmentation results than the original hybrid method does.

An implicit skeleton-based method for the geometry reconstruction of vasculatures

Due to the high complexity of vascular system network, the geometry reconstruction of vasculatures from raw medical datasets remains a very challenging task. In this paper, we present a novel skeleton-based method for the geometry reconstruction of vascular structures from standard 3D medical datasets. With the proposed techniques, the geometry of vascular structures with high level of smoothness and accuracy can be reconstructed from the raw medical datasets. The experimental results and comparison with other techniques demonstrate that our method can achieve faithful and smooth vascular structures. In addition, quantitative validation has been conducted to evaluate the accuracy and smoothness of the reconstructed vessel geometry based on the proposed method.

A Versatile Optical Model for Hybrid Rendering of Volume Data

In volume rendering, most optical models currently in use are based on the assumptions that a volumetric object is a collection of particles and that the macro behavior of particles, when they interact with light rays, can be predicted based on the behavior of each individual particle. However, such models are not capable of characterizing the collective optical effect of a collection of particles which dominates the appearance of the boundaries of dense objects. In this paper, we propose a generalized optical model that combines particle elements and surface elements together to characterize both the behavior of individual particles and the collective effect of particles. The framework based on a new model provides a more powerful and flexible tool for hybrid rendering of isosurfaces and transparent clouds of particles in a single scene. It also provides a more rational basis for shading, so the problem of normal-based shading in homogeneous regions encountered in conventional volume rendering can be easily avoided. The model can be seen as an extension to the classical model. It can be implemented easily, and most of the advanced numerical estimation methods previously developed specifically for the particle-based optical model, such as preintegration, can be applied to the new model to achieve high-quality rendering results.