From approximating to interpolatory non-stationary subdivision schemes with the same generation properties

In this paper we describe a general, computationally feasible strategy to deduce a family of interpolatory non-stationary subdivision schemes from a symmetric non-stationary, non-interpolatory one satisfying quite mild assumptions. To achieve this result we extend our previous work [C.Conti, L.Gemignani, L.Romani, Linear Algebra Appl. 431 (2009), no. 10, 1971-1987] to full generality by removing additional assumptions on the input symbols. For the so obtained interpolatory schemes we prove that they are capable of reproducing the same exponential polynomial space as the one generated by the original approximating scheme. Moreover, we specialize the computational methods for the case of symbols obtained by shifted non-stationary affine combinations of exponential B-splines, that are at the basis of most non-stationary subdivision schemes. In this case we find that the associated family of interpolatory symbols can be determined to satisfy a suitable set of generalized interpolating conditions at the set of the zeros (with reversed signs) of the input symbol. Finally, we discuss some computational examples by showing that the proposed approach can yield novel smooth non-stationary interpolatory subdivision schemes possessing very interesting reproduction properties

A combined approximating and interpolating subdivision scheme with C2 continuity

AbstractIn this paper a combined approximating and interpolating subdivision scheme is presented. The relationship between approximating subdivision and interpolating subdivision is derived by directly performing operations on geometric rules. The behavior of the limit curve produced by our combined subdivision scheme is analyzed by the Laurent polynomial and attains C2 degree of smoothness. Furthermore, a non-uniform combined subdivision with shape control parameters is introduced, which allows a different tension value for every edge of the original control polygon

Six-Point Subdivision Schemes with Cubic Precision

This paper presents 6-point subdivision schemes with cubic precision. We first derive a relation between the 4-point interpolatory subdivision and the quintic B-spline refinement. By using the relation, we further propose the counterparts of cubic and quintic B-spline refinements based on 6-point interpolatory subdivision schemes. It is proved that the new family of 6-point combined subdivision schemes has higher smoothness and better polynomial reproduction property than the B-spline counterparts. It is also showed that, both having cubic precision, the well-known Hormann-Sabin’s family increase the degree of polynomial generation and smoothness in exchange of the increase of the support width, while the new family can keep the support width unchanged and maintain higher degree of polynomial generation and smoothness

Interpolatory Catmull-Clark volumetric subdivision over unstructured hexahedral meshes for modeling and simulation applications

International audienceVolumetric modeling is an important topic for material modeling and isogeometric simulation. In this paper, two kinds of interpolatory Catmull-Clark volumetric subdivision approaches over unstructured hexahedral meshes are proposed based on the limit point formula of Catmull-Clark subdivision volume. The basic idea of the first method is to construct a new control lattice, whose limit volume by the CatmullClark subdivision scheme interpolates vertices of the original hexahedral mesh. The new control lattice is derived by the local push-back operation from one CatmullClark subdivision step with modified geometric rules. This interpolating method is simple and efficient, and several shape parameters are involved in adjusting the shape of the limit volume. The second method is based on progressive-iterative approximation using limit point formula. At each iteration step, we progressively modify vertices of an original hexahedral mesh to generate a new control lattice whose limit volume interpolates all vertices in the original hexahedral mesh. The convergence proof of the iterative process is also given. The interpolatory subdivision volume has C 2-smoothness at the regular region except around extraordinary vertices and edges. Furthermore, the proposed interpolatory volumetric subdivision methods can be used not only for geometry interpolation, but also for material attribute interpolation in the field of volumetric material modeling. The application of the proposed volumetric subdivision approaches on isogeometric analysis is also given with several examples

Subdivision schemes for curve design and image analysis

Subdivision schemes are able to produce functions, which are smooth up to pixel accuracy, in a few steps through an iterative process. They take as input a coarse control polygon and iteratively generate new points using some algebraic or geometric rules. Therefore, they are a powerful tool for creating and displaying functions, in particular in computer graphics, computer-aided design, and signal analysis. A lot of research on univariate subdivision schemes is concerned with the convergence and the smoothness of the limit curve, especially for schemes where the new points are a linear combination of points from the previous iteration. Much less is known for non-linear schemes: in many cases there are only ad hoc proofs or numerical evidence about the regularity of these schemes. For schemes that use a geometric construction, it could be interesting to study the continuity of geometric entities. Dyn and Hormann propose sufficient conditions such that the subdivision process converges and the limit curve is tangent continuous. These conditions can be satisfied by any interpolatory scheme and they depend only on edge lengths and angles. The goal of my work is to generalize these conditions and to find a sufficient constraint, which guarantees that a generic interpolatory subdivision scheme gives limit curves with continuous curvature. To require the continuity of the curvature it seems natural to come up with a condition that depends on the difference of curvatures of neighbouring circles. The proof of the proposed condition is not completed, but we give a numerical evidence of it. A key feature of subdivision schemes is that they can be used in different fields of approximation theory. Due to their well-known relation with multiresolution analysis they can be exploited also in image analysis. In fact, subdivision schemes allow for an efficient computation of the wavelet transform using the filterbank. One current issue in signal processing is the analysis of anisotropic signals. Shearlet transforms allow to do it using the concept of multiple subdivision schemes. One drawback, however, is the big number of filters needed for analysing the signal given. The number of filters is related to the determinant of the expanding matrix considered. Therefore, a part of my work is devoted to find expanding matrices that give a smaller number of filters compared to the shearlet case. We present a family of anisotropic matrices for any dimension d with smaller determinant than shearlets. At the same time, these matrices allow for the definition of a valid directional transform and associated multiple subdivision schemes

Lifting-based subdivision wavelets with geometric constraints.

Qin, Guiming."August 2010."Thesis (M.Phil.)--Chinese University of Hong Kong, 2010.Includes bibliographical references (p. 72-74).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.5Chapter 1.1 --- B splines and B-splines surfaces --- p.5Chapter 1. 2 --- Box spline --- p.6Chapter 1. 3 --- Biorthogonal subdivision wavelets based on the lifting scheme --- p.7Chapter 1.4 --- Geometrically-constrained subdivision wavelets --- p.9Chapter 1.5 --- Contributions --- p.9Chapter 2 --- Explicit symbol formulae for B-splines --- p.11Chapter 2. 1 --- Explicit formula for a general recursion scheme --- p.11Chapter 2. 2 --- Explicit formulae for de Boor algorithms of B-spline curves and their derivatives --- p.14Chapter 2.2.1 --- Explicit computation of de Boor Algorithm for Computing B-Spline Curves --- p.14Chapter 2.2.2 --- Explicit computation of Derivatives of B-Spline Curves --- p.15Chapter 2. 3 --- Explicit power-basis matrix fomula for non-uniform B-spline curves --- p.17Chapter 3 --- Biorthogonal subdivision wavelets with geometric constraints --- p.23Chapter 3. 1 --- Primal subdivision and dual subdivision --- p.23Chapter 3. 2 --- Biorthogonal Loop-subdivision-based wavelets with geometric constraints for triangular meshes --- p.24Chapter 3.2.1 --- Loop subdivision surfaces and exact evaluation --- p.24Chapter 3.2.2 --- Lifting-based Loop subdivision wavelets --- p.24Chapter 3.2.3 --- Biorthogonal Loop-subdivision wavelets with geometric constraints --- p.26Chapter 3. 3 --- Biorthogonal subdivision wavelets with geometric constraints for quadrilateral meshes --- p.35Chapter 3.3.1 --- Catmull-Clark subdivision and Doo-Sabin subdivision surfaces --- p.35Chapter 3.3.1.1 --- Catmull-Clark subdivision --- p.36Chapter 3.3.1.2 --- Doo-Sabin subdivision --- p.37Chapter 3.3.2 --- Biorthogonal subdivision wavelets with geometric constraints for quadrilateral meshes --- p.38Chapter 3.3.2.1 --- Biorthogonal Doo-Sabin subdivision wavelets with geometric constraints --- p.38Chapter 3.3.2.2 --- Biorthogonal Catmull-Clark subdivision wavelets with geometric constraints --- p.44Chapter 4 --- Experiments and results --- p.49Chapter 5 --- Conclusions and future work --- p.60Appendix A --- p.62Appendix B --- p.67Appendix C --- p.69Appendix D --- p.71References --- p.7

A Method of Rendering CSG-Type Solids Using a Hybrid of Conventional Rendering Methods and Ray Tracing Techniques

This thesis describes a fast, efficient and innovative algorithm for producing shaded, still images of complex objects, built using constructive solid geometry ( CSG ) techniques. The algorithm uses a hybrid of conventional rendering methods and ray tracing techniques. A description of existing modelling and rendering methods is given in chapters 1, 2 and 3, with emphasis on the data structures and rendering techniques selected for incorporation in the hybrid method. Chapter 4 gives a general description of the hybrid method. This method processes data in the screen coordinate system and generates images in scan-line order. Scan lines are divided into spans (or segments) using the bounding rectangles of primitives calculated in screen coordinates. Conventional rendering methods and ray tracing techniques are used interchangeably along each scan-line. The method used is detennined by the number of primitives associated with a particular span. Conventional rendering methods are used when only one primitive is associated with a span, ray tracing techniques are used for hidden surface removal when two or more primitives are involved. In the latter case each pixel in the span is evaluated by accessing the polygon that is visible within each primitive associated with the span. The depth values (i. e. z-coordinates derived from the 3-dimensional definition) of the polygons involved are deduced for the pixel's position using linear interpolation. These values are used to determine the visible polygon. The CSG tree is accessed from the bottom upwards via an ordered index that enables the 'visible' primitives on any particular scan-line to be efficiently located. Within each primitive an ordered path through the data structure provides the polygons potentially visible on a particular scan-line. Lists of the active primitives and paths to potentially visible polygons are maintained throughout the rendering step and enable span coherence and scan-line coherence to be fully utilised. The results of tests with a range of typical objects and scenes are provided in chapter 5. These results show that the hybrid algorithm is significantly faster than full ray tracing algorithms

Investigation of Different Video Compression Schemes Using Neural Networks

Image/Video compression has great significance in the communication of motion pictures and still images. The need for compression has resulted in the development of various techniques including transform coding, vector quantization and neural networks. this thesis neural network based methods are investigated to achieve good compression ratios while maintaining the image quality. Parts of this investigation include motion detection, and weight retraining. An adaptive technique is employed to improve the video frame quality for a given compression ratio by frequently updating the weights obtained from training. More specifically, weight retraining is performed only when the error exceeds a given threshold value. Image quality is measured objectively, using the peak signal-to-noise ratio versus performance measure. Results show the improved performance of the proposed architecture compared to existing approaches. The proposed method is implemented in MATLAB and the results obtained such as compression ratio versus signalto- noise ratio are presented

Numerical simulation of three-dimensional free surface film flow over or around obstacles on an inclined plane

Within the bearing chamber of a gas turbine aero-engine, lubrication of the shaft and other bearings is achieved by an oil film which may become significantly disturbed by interacting with a range of chamber geometries which protrude from the chamber wall. Minimizing these disturbances and preventing possible dry areas is crucial in optimizing a bearing chambers design. In addition, multiple obstructions may be located close to one another, resulting in a more complex disturbed film profile than by individual obstacles. Prediction of the disturbance of the film is an important aspect of bearing chamber design. For analysis of the film profile over or around a local obstacle, typical bearing chamber flows can be approximated as an incompressible thin film flow down an inclined wall driven by gravity. The Reynolds number of thin film flows is often small, and for the bulk of this thesis a Stokes flow assumption is implemented. In addition, thin films are often dominated by surface tension effects, which for accurate modelling require an accurate representation of the free surface profile. Numerical techniques such as the volume of fluid method fail to track the surface profile specifically, and inaccuracies will occur in applying surface tension in this approach. A numerical scheme based on the boundary element method tracks the free surface explicitly, alleviating this potential error source and is applied throughout this thesis. The evaluation of free surface quantities, such as unit normal and curvature is achieved by using a Hermitian radial basis function interpolation. This hermite interpolation can also be used to incorporate the far field boundary conditions and to enable contact line conditions to be satisfied for cases where the obstacle penetrates the free surface. Initial results consider a film flowing over an arbitrary hemispherical obstacle, fully submerged by the fluid for a range of flow configurations. Comparison is made with previously published papers that assume the obstacle is small and / or the free surface deflection and disturbance velocity is small. Free surface profiles for thin film flows over hemispherical obstacles that approach the film surface are also produced, and the effects of near point singularities considered. All free surface profiles indicate an upstream peak, followed by a trough downstream of the obstacle with the peak decaying in a “horseshoe” shaped surface deformation. Flow profiles are governed by the plane inclination, the Bond number and the obstacle geometry; effects of these key physical parameters on flow solutions are provided. The disturbed film profiles over multiple obstacles will differ from the use of a single obstacle analysis as their proximity decreases. An understanding of the local interaction of individual obstacles is an important aspect of bearing chamber design. In this thesis the single obstacle analysis is extended to the case of flow over multiple hemispheres. For obstacles that are separated by a sufficiently large distance the flow profiles are identical to those for a single obstacle. However, for flow over multiple obstacles with small separation, variations from single obstacle solutions maybe significant. For flow over two obstacles placed in-line with the incident flow, variations with flow parameters are provided. To identify the flexibility of this approach, flows over three obstacles are modelled. The calculation of flows around obstacles provides a greater challenge. Notably, a static contact line must be included such that the angle between the free surface and the obstacle is introduced as an extra flow parameter that will depend both on the fluid and the obstacle surface characteristics. The numerical models used for flow over hemispheres can be developed to consider film flow around circular cylinders. Numerical simulations are used to investigate flow parameters and boundary conditions. Solutions are obtained where steady flow profiles can be found both over and around a cylindrical obstacle raising the awareness of possible multiple solutions. Flow around multiple obstacles is also analyzed, with profiles produced for flow around two cylinders placed in various locations relative to one another. As for flow over two hemispheres, for sufficiently large separations the flow profiles are identical to a single obstacle analysis. For flow around two obstacles spaced in the direction of the flow, effects of altering the four governing parameters; plane inclination angle, Bond number, obstacle size, and static contact angle are examined. The analysis of flow around three cylinders in two configurations is finally considered. In addition, for two obstacles spaced in-line with the incident flow, the numerical approaches for flow over and flow around are combined to predict situations where flow passes over an upstream cylinder, and then around an identical downstream cylinder. The final section of this thesis removes the basic assumption of Stokes flow, through solving the full Navier-Stokes equations at low Reynolds number and so incorporating the need to solve nonlinear equations through the solution domain. An efficient numerical algorithm for including the inertia effects is developed and compared to more conventional methods, such as the dual reciprocity method and particular integral techniques for the case of a three-dimensional lid driven cavity. This approach is extended to enable calculation of low Reynolds number film profiles for both flow over and around a cylinder. Results are compared to the analysis from previous Stokes flow solutions for modest increases in the Reynolds number