Jianmin Zheng

T-spline simplification and local refinement

A typical NURBS surface model has a large percentage of superfluous control points that significantly interfere with the design process. This paper presents an algorithm for eliminating such superfluous control points, producing a T-spline. The algorithm can remove substantially more control points than competing methods such as B-spline wavelet decomposition. The paper also presents a new T-spline local refinement algorithm and answers two fundamental open questions on T-spline theory.

Non-uniform recursive subdivision surfaces

Doo-Sabin and Catmull-Clark subdivision surfaces are based on the notion of repeated knot insertion of uniform tensor product B-spline surfaces. This paper develops rules for non-uniform Doo-Sabin and Catmull-Clark surfaces that generalize non-uniform tensor product Bspline surfaces to arbitrary topologies. This added flexibility allows, among other things, the natural introduction of features such as cusps, creases, and darts, while elsewhere maintaining the same order of continuity as their uniform counterparts.

On linear independence of T-spline blending functions

This paper shows that, for any given T-spline, the linear independence of its blending functions can be determined by computing the nullity of the T-spline-to-NURBS transform matrix. The paper analyzes the class of T-splines for which no perpendicular T-node extensions intersect, and shows that the blending functions for any such T-spline are linearly independent.

User-Friendly Interactive Image Segmentation Through Unified Combinatorial User Inputs

One weakness in the existing interactive image segmentation algorithms is the lack of more intelligent ways to understand the intention of user inputs. In this paper, we advocate the use of multiple intuitive user inputs to better reflect a users intention. In particular, we propose a constrained random walks algorithm that facilitates the use of three types of user inputs: 1) foreground and background seed input, 2) soft constraint input, and 3) hard constraint input, as well as their combinations. The foreground and background seed input allows a user to draw strokes to specify foreground and background seeds. The soft constraint input allows a user to draw strokes to indicate the region that the boundary should pass through. The hard constraint input allows a user to specify the pixels that the boundary must align with. Our proposed method supports all three types of user inputs in one coherent computational framework consisting of a constrained random walks and a local editing algorithm, which allows more precise contour refinement. Experimental results on two benchmark data sets show that the proposed framework is highly effective and can quickly and accurately segment a wide variety of natural images with ease.

Robust Interactive Image Segmentation Using Convex Active Contours

The state-of-the-art interactive image segmentation algorithms are sensitive to the user inputs and often unable to produce an accurate boundary with a small amount of user interaction. They frequently rely on laborious user editing to refine the segmentation boundary. In this paper, we propose a robust and accurate interactive method based on the recently developed continuous-domain convex active contour model. The proposed method exhibits many desirable properties of an effective interactive image segmentation algorithm, including robustness to user inputs and different initializations, the ability to produce a smooth and accurate boundary contour, and the ability to handle topology changes. Experimental results on a benchmark data set show that the proposed tool is highly effective and outperforms the state-of-the-art interactive image segmentation algorithms.

A diffusion approach to seeded image segmentation

Seeded image segmentation is a popular type of supervised image segmentation in computer vision and image processing. Previous methods of seeded image segmentation treat the image as a weighted graph and minimize an energy function on the graph to produce a segmentation. In this paper, we propose to conduct the seeded image segmentation according to the result of a heat diffusion process in which the seeded pixels are considered to be the heat sources and the heat diffuses on the image starting from the sources. After the diffusion reaches a stable state, the image is segmented based on the pixel temperatures. It is also shown that our proposed framework includes the RandomWalk algorithm for image segmentation as a special case which diffuses only along the two coordinate axes. To better control diffusion, we propose to incorporate the attributes (such as the geometric structure) of the image into the diffusion process, yielding an anisotropic diffusion method for image segmentation. The experiments show that the proposed anisotropic diffusion method usually produces better segmentation results. In particular, when the method is tested using the groundtruth dataset of Microsoft Research Cambridge (MSRC), an error rate of 4.42% can be achieved, which is lower than the reported error rates of other state-of-the-art algorithms.

Variational mesh decomposition

The problem of decomposing a 3D mesh into meaningful segments (or parts) is of great practical importance in computer graphics. This article presents a variational mesh decomposition algorithm that can efficiently partition a mesh into a prescribed number of segments. The algorithm extends the Mumford-Shah model to 3D meshes that contains a data term measuring the variation within a segment using eigenvectors of a dual Laplacian matrix whose weights are related to the dihedral angle between adjacent triangles and a regularization term measuring the length of the boundary between segments. Such a formulation simultaneously handles segmentation and boundary smoothing, which are usually two separate processes in most previous work. The efficiency is achieved by solving the Mumford-Shah model through a saddle-point problem that is solved by a fast primal-dual method. A preprocess step is also proposed to determine the number of segments that the mesh should be decomposed into. By incorporating this preprocessing step, the proposed algorithm can automatically segment a mesh into meaningful parts. Furthermore, user interaction is allowed by incorporating the users inputs into the variational model to reflect the users special intention. Experimental results show that the proposed algorithm outperforms competitive segmentation methods when evaluated on the Princeton Segmentation Benchmark.

The mu-basis of a rational ruled surface

Abstract The mu-basis of a planar rational curve is a polynomial ideal basis comprised of two polynomials that greatly facilitates computing the implicit equation of the curve. This paper defines a mu-basis for a rational ruled surface, and presents a simple algorithm for computing the mu-basis. The mu-basis consists of two polynomials p(x,y,z,s) and q(x,y,z,s) that are linear in x,y,z and degree μ and m−μ in s respectively, where m is the degree of the implicit equation. The implicit equation of the surface is then obtained by merely taking the resultant of p and q with respect to s . This implicitization algorithm is faster and/or more robust than previous methods.

Target curvature driven fairing algorithm for planar cubic B-spline curves

This paper presents a new algorithm for fairing planar cubic B-spline curves. Target curvature plots prescribed by designers according to design intent are used to identify bad points and bad curve segments. The corresponding control points are then modified using local constrained optimization. The objective function is a weighed combination of two components, which are associated with the fairness in the sense of energy minimization and the coherence to the original design, respectively. Hence, designers have more control over the fairing process by using appropriate weights for the two components. Several numerical examples are provided to demonstrate the effectiveness of the algorithm.

Adaptive T-spline surface fitting to z-map models

Surface fitting refers to the process of constructing a smooth representation for an object surface from a fairly large number of measured 3D data points. This paper presents an automatic algorithm to construct smooth parametric surfaces using T-splines from z-map data. The algorithm begins with a rough surface approximation and then progressively refines it in the regions where the approximation accuracy does not meet the requirement. The topology of the resulting T-spline surface is determined adaptively based on the local geometric character of the input data and the geometry of the control points is obtained by a least squares procedure. The advantage of the approach is that the resulting surface is C2 continuous and the refinement is essentially local, resulting in a small number of control points for the surface.