Kaihuai Qin

An Adaptable-Multilayer Fractional Fourier Transform Approach for Image Registration

A novel adaptable accurate way for calculating polar FFT and log-polar FFT is developed in this paper, named multilayer fractional Fourier transform (MLFFT). MLFFT is a necessary addition to the pseudo-polar FFT for the following reasons: It has lower interpolation errors in both polar and log-polar Fourier transforms; it reaches better accuracy with the nearly same computing complexity as the pseudo-polar FFT; it provides a mechanism to increase the accuracy by increasing the user-defined computing level. This paper demonstrates both MLFFT itself and its advantages theoretically and experimentally. By emphasizing applications of MLFFT in image registration with rotation and scaling, our experiments suggest two major advantages of MLFFT: 1) scaling up to 5 and arbitrary rotation angles, or scales up to 10 without rotation can be recovered by MLFFT while currently the result recovered by the state-of-the-art algorithms is the maximum scaling of 4; 2) No iteration is needed to obtain large rotation and scaling values of images by MLFFT, hence it is more efficient than the pseudopolar-based FFT methods for image registration.

Efficient wavelet construction with Catmull–Clark subdivision

This paper presents an efficient biorthogonal wavelet construction with the generalized Catmull–Clark subdivision based on the lifting scheme. The subdivision wavelet construction scheme is applicable to all variants of Catmull–Clark subdivision, so it is more universal than the previous wavelet construction for the generalized bicubic B-spline subdivision. Because the analysis and synthesis algorithms of the wavelets are composed of a series of local and in-place lifting operations, they can be performed in linear time. The experiments have demonstrated the stability of the proposed wavelet analysis based on the ordinary Catmull–Clark subdivision. Moreover, the resulting Catmull–Clark subdivision wavelets have better fitting quality than the generalized bicubic B-spline subdivision wavelets at a similar computation cost.

√3-Subdivision-Based Biorthogonal Wavelets

A new efficient biorthogonal wavelet analysis based on the radic3 subdivision is proposed in the paper by using the lifting scheme. Since the radic3 subdivision is of the slowest topological refinement among the traditional triangular subdivisions, the multiresolution analysis based on the radic3 subdivision is more balanced than the existing wavelet analyses on triangular meshes and accordingly offers more levels of detail for processing polygonal models. In order to optimize the multiresolution analysis, the new wavelets, no matter whether they are interior or on boundaries, are orthogonalized with the local scaling functions based on a discrete inner product with subdivision masks. Because the wavelet analysis and synthesis algorithms are actually composed of a series of local lifting operations, they can be performed in linear time. The experiments demonstrate the efficiency and stability of the wavelet analysis for both closed and open triangular meshes with radic3 subdivision connectivity. The radic3-subdivision-based biorthogonal wavelets can be used in many applications such as progressive transmission, shape approximation, and multiresolution editing and rendering of 3D geometric models.

Unlifted loop subdivision wavelets

In this paper, we propose a new wavelet scheme for loop subdivision surfaces. The main idea enabling our wavelet construction is to extend the subdivision rules to be invertible, thus executing each inverse subdivision step in the reverse order makes up the wavelet decomposition rule. As opposed to other existing wavelet schemes for loop surfaces, which require solving a global sparse linear system in the wavelet analysis process, our wavelet scheme provides efficient (linear time and fully in-place) computations for both forward and backward wavelet transforms. This characteristic makes our wavelet scheme extremely suitable for applications in which the speed for wavelet decomposition is critical. We also describe our strategies for optimizing free parameters in the extended subdivision steps, which are important to the performance of the final wavelet transform. Our method has been proven to be effective, as demonstrated by a number of examples.

General matrix representations for B-splines

The concept of basis matrix of B-splines is presented. A general matrix representation, which results in an explicitly recursive matrix formula, for nonuniform B-spline curves of an arbitrary degree is also presented by means of Toeplitz matrix. New recursive matrix representations for uniform B-spline curves and Bezier curves of an arbitrary degree are obtained as special cases of that for nonuniform B-spline curves. The recursive formula for basis matrix can be substituted for de Boor-Coxs one for B-splines, and it has better time complexity than de Boor-Coxs formula when used for conversion and computation of B-spline curves and surfaces between different CAD systems. Finally, some applications of the matrix representations are presented.

Interactive Approximate Rendering of Reflections, Refractions, and Caustics

Reflections, refractions, and caustics are very important for rendering global illumination images. Although many methods can be applied to generate these effects, the rendering performance is not satisfactory for interactive applications. In this paper, complex ray-object intersections are simplified so that the intersections can be computed on a GPU, and an iterative computing scheme based on the depth buffers is used for correcting the approximate results caused by the simplification. As a result, reflections and refractions of environment maps and nearby geometry can be rendered on a GPU interactively without preprocessing. We can even achieve interactive recursive reflections and refractions by using an object-impostor technique. Moreover, caustic effects caused by reflections and refractions can be rendered by placing the eye at the light. Rendered results prove that our method is sufficiently efficient to render plausible images interactively for many interactive applications

A genetic algorithm for the minimum weight triangulation

In this paper, a new method for the minimum weight triangulation of points on a plane, called genetic minimum weight triangulation (GMWT), is presented based on the rationale of genetic algorithms. Polygon crossover and its algorithm for triangulations are proposed. New adaptive genetic operators, or adaptive crossover and mutation operators, are introduced. It is shown that the new method for the minimum weight triangulation can obtain more optimal results of triangulations than the greedy algorithm.

A new method for speeding up ray tracing NURBS surfaces

In this paper, a new method for computing the intersection of a ray and a NURBS surface is presented. The method is based on Newton iteration for a system of nonlinear equations, but its convergence is accelerated by using the polynomial extrapolation. Additionally, a ray is defined to be the intersection of two planes, which are usually non-orthogonal, so that theoperations are significantly reduced. A trapezoid prism is adopted for the bounding box enclosing the patch as tightly as possible. It can be used both for getting a good start point for the Newton iteration and for efficiently detecting whether a ray intersects a patch. The method proposed for speeding up ray tracing NURBS surfaces can reduce both the iteration steps and arithmetic operations, thus much CPU time is saved. The new method is faster than the previous methods for which published performance data allow reliable comparison. The acceleration techniques with extrapolation presented for solution to a system of nonlinear equations can also be used to speed up tracing the intersection of two parametric surfaces as well as others that need Newton iteration.

Estimating subdivision depth of Catmull-Clark surfaces

In this paper, both general and exponential bounds of the distance between a uniform Catmull-Clark surface and its control polyhedron are derived. The exponential bound is independent of the process of subdivision and can be evaluated without recursive subdivision. Based on the exponential bound, we can predict the depth of subdivision within a user-specified error tolerance. This is quite useful and important for pre-computing the subdivision depth of subdivision surfaces in many engineering applications such as surface/surface intersection, mesh generation, numerical control machining and surface rendering.

Biorthogonal wavelets based on gradual subdivision of quadrilateral meshes

This paper introduces a new biorthogonal wavelet based on a variant of 2 subdivision by using the lifting scheme. The greatest advantage of this wavelet is its very slow gradual refinement for quadrilateral meshes, which offers the biggest number of resolution levels to control a quadrilateral mesh. Moreover, the resulting wavelet transforms have a linear computational complexity, as they are composed of local and in-place lifting operations only. Feature lines can also be effectively integrated into the wavelet transforms as self-governed boundary curves. The introduced wavelet analysis can be used in a variety of applications such as progressive transmission, data compression, shape approximation and multiresolution rendering. The experiments have shown sufficient stability as well as better performance of the introduced wavelet analysis, as compared to the existing wavelet analyses for quadrilateral meshes of arbitrary topology.