Summary form only given. We propose a new method for 3D shape metamorphosis, where the in-between objects are constructed by using T-spline scalar functions. The use of T-spline level sets offers several advantages: First, it is convenient to handle complex topology changes without the need of model parameterization. Second, the constructed objects are smooth (C2 in our case). Third, high quality meshes can be easily obtained by using the marching triangulation method. Fourth, the distribution of the degrees of freedom can be adapted to the geometry of the object. Given one source object and one target object, we firstly find a global coordinate transformation to approximately align the two objects. The T-spline control grid is adoptively generated according to the geometry of the aligned objects, and the initial T-spline level set is found by approximating the signed distance function of the source object. Then we use an evolution process, which is governed by a combination of the signed distance function of the target object and a curvature-dependent speed function, to deform the T-spline level set until it converges to the target shape. Additional intermediate objects are inserted at the beginning/end of the sequence of generated T-spline level sets, by gradually projecting the source/target object to the initial/final T-spline level set. A fully automatic algorithm is developed for the above procedures. Experimental results are presented to demonstrate the effectiveness of our method.

We propose a new method for constructing a piecewise smooth mesh from a set of unorganized data points, which may be non-uniformly sampled, noisy, and even containing holes. The method is based on the construction of an implicit representation of the surface, by using smooth (C2in our case) T-spline scalar functions. We first generate the T- spline control grid, and use an evolution process such that the resulting T-spline level sets capture the topology and outline of the object to be reconstructed. The initial mesh with high quality is obtained from the implicit T-spline function through the marching triangulation method. Then we project each data point to the initial mesh, and get a scalar displacement field. Detailed features will be captured by the displaced mesh. We also propose an additional evolution process, which combines data-driven velocities and feature- preserving bilateral filters, in order to reproduce sharp features.

We present a complete framework for computing a subdivision surface to approximate unorganized point sample data, which is a separable nonlinear least squares problem. We study the convergence and stability of three geometrically-motivated optimization schemes and reveal their intrinsic relations with standard methods for constrained nonlinear optimization. A commonly-used method in graphics, called point distance minimization, is shown to use a variant of the gradient descent step and thus has only linear convergence. The second method, called tangent distance minimization, which is well-known in computer vision, is shown to use the Gauss-Newton step, and thus demonstrates near quadratic convergence for zero residual problems but may not converge otherwise. Finally, we show that an optimization scheme called squared distance minimization, recently proposed by Pottmann et al., can be derived from the Newton method. Hence, with proper regularization, tangent distance minimization and squared distance minimization are more efficient than point distance minimization. We also investigate the effects of two step size control methods -- Levenberg-Marquardt regularization and the Armijo rule -- on the convergence stability and efficiency of the above optimization schemes.

We study the evolution of T-spline level sets (i.e, implicitly defined T-spline curves and surfaces). The use of T-splines leads to a sparse representation of the geometry and allows for an adaptation to the given data, which can be unorganized points or images. The evolution process is governed by a combination of prescribed, data-driven normal velocities, and additional distance field constraints. By incorporating the distance field constraints we are able to avoid additional branches and singularities of the T-spline level sets without having to use re-initialization steps. Experimental examples are presented to demonstrate the effectiveness of our approach.

We present a novel algorithm based on least-squares minimization to approximate point cloud data in 2D plane with a smooth B-spline curve. The point cloud data may represent an open curve with self intersection and sharp corner. Unlike other existing methods, such as the moving least-squares method and the principle curve method, our algorithm does not need a thinning process. The idea of our algorithm is intuitive and simple - we make a B-spline curve grow along the tangential directions at its two end-points following local geometry of point clouds. Our algorithm generates appropriate control points of the fitting B-spline curve in the least squares sense. Although presented for the 2D case, our method can be extended in a straightforward manner to fitting data points by a B-spline curve in higher dimensions

We study the reconstruction of smooth surfaces from point clouds. We use a new squared distance error term in optimization to fit a subdivision surface to a set of unorganized points, which defines a closed target surface of arbitrary topology. The resulting method is based on the framework of squared distance minimization (SDM) proposed by Pottmann et al. Specifically, with an initial subdivision surface having a coarse control mesh as input, we adjust the control points by optimizing an objective function through iterative minimization of a quadratic approximant of the squared distance function of the target shape. Our experiments show that the new method (SDM) converges much faster than the commonly used optimization method using the point distance error function, which is known to have only linear convergence. This observation is further supported by our recent result that SDM can be derived from the Newton method with necessary modifications to make the Hessian positive definite and the fact that the Newton method has quadratic convergence.

We present a new method for converting a photo or image to a synthesized painting following the painting style of an example painting. Treating painting styles of brush strokes as sample textures, we reduce the problem of learning an example painting to a texture synthesis problem. The proposed method uses a hierarchical patch-based approach to the synthesis of directional textures. The key features of our method are: 1) Painting styles are represented as one or more blocks of sample textures selected by the user from the example painting; 2) image segmentation and brush stroke directions defined by the medial axis are used to better represent and communicate shapes and objects present in the synthesized painting; 3) image masks and a hierarchy of texture patches are used to efficiently synthesize high-quality directional textures. The synthesis process is further accelerated through texture direction quantization and the use of Gaussian pyramids. Our method has the following advantages: First, the synthesized stroke textures can follow a direction field determined by the shapes of regions to be painted. Second, the method is very efficient; the generation time of a synthesized painting ranges from a few seconds to about one minute, rather than hours, as required by other existing methods, on a commodity PC. Furthermore, the technique presented here provides a new and efficient solution to the problem of synthesizing a 2D directional texture. We use a number of test examples to demonstrate the efficiency of the proposed method and the high quality of results produced by the method.