Nondyadic and nonlinear multiresolution image approximations

This thesis focuses on the development of novel multiresolution image approximations. Specifically, we present two kinds of generalization of multiresolution techniques: image reduction for arbitrary scales, and nonlinear approximations using other metrics than the standard Euclidean one. Traditional multiresolution decompositions are restricted to dyadic scales. As first contribution of this thesis, we develop a method that goes beyond this restriction and that is well suited to arbitrary scale-change computations. The key component is a new and numerically exact algorithm for computing inner products between a continuously defined signal and B-splines of any order and of arbitrary sizes. The technique can also be applied for non-uniform to uniform grid conversion, which is another approximation problem where our method excels. Main applications are resampling and signal reconstruction. Although simple to implement, least-squares approximations lead to artifacts that could be reduced if nonlinear methods would be used instead. The second contribution of the thesis is the development of nonlinear spline pyramids that are optimal for lp-norms. First, we introduce a Banach-space formulation of the problem and show that the solution is well defined. Second, we compute the lp-approximation thanks to an iterative optimization algorithm based on digital filtering. We conclude that l1-approximations reduce the artifacts that are inherent to least-squares methods; in particular, edge blurring and ringing. In addition, we observe that the error of l1-approximations is sparser. Finally, we derive an exact formula for the asymptotic Lp-error; this result justifies using the least-squares approximation as initial solution for the iterative optimization algorithm when the degree of the spline is even; otherwise, one has to include an appropriate correction term. The theoretical background of the thesis includes the modelisation of images in a continuous/discrete formalism and takes advantage of the approximation theory of linear shift-invariant operators. We have chosen B-splines as basis functions because of their nice properties. We also propose a new graphical formalism that links B-splines, finite differences, differential operators, and arbitrary scale changes

IGA-based Multi-Index Stochastic Collocation for random PDEs on arbitrary domains

This paper proposes an extension of the Multi-Index Stochastic Collocation (MISC) method for forward uncertainty quantification (UQ) problems in computational domains of shape other than a square or cube, by exploiting isogeometric analysis (IGA) techniques. Introducing IGA solvers to the MISC algorithm is very natural since they are tensor-based PDE solvers, which are precisely what is required by the MISC machinery. Moreover, the combination-technique formulation of MISC allows the straight-forward reuse of existing implementations of IGA solvers. We present numerical results to showcase the effectiveness of the proposed approach.Comment: version 3, version after revisio

PAN AIR: A computer program for predicting subsonic or supersonic linear potential flows about arbitrary configurations using a higher order panel method. Volume 4: Maintenance document (version 1.1)

The Maintenance Document is a guide to the PAN AIR software system, a system which computes the subsonic or supersonic linear potential flow about a body of nearly arbitrary shape, using a higher order panel method. The document describes the over-all system and each program module of the system. Sufficient detail is given for program maintenance, updating and modification. It is assumed that the reader is familiar with programming and CDC (Control Data Corporation) computer systems. The PAN AIR system was written in FORTRAN 4 language except for a few COMPASS language subroutines which exist in the PAN AIR library. Structured programming techniques were used to provide code documentation and maintainability. The operating systems accommodated are NOS 1.2, NOS/BE and SCOPE 2.1.3 on the CDC 6600, 7600 and Cyber 175 computing systems. The system is comprised of a data management system, a program library, an execution control module and nine separate FORTRAN technical modules. Each module calculates part of the posed PAN AIR problem. The data base manager is used to communicate between modules and within modules. The technical modules must be run in a prescribed fashion for each PAN AIR problem. In order to ease the problem of supplying the many JCL cards required to execute the modules, a separate module called MEC (Module Execution Control) was created to automatically supply most of the JCL cards. In addition to the MEC generated JCL, there is an additional set of user supplied JCL cards to initiate the JCL sequence stored on the system

PAN AIR: A computer program for predicting subsonic or supersonic linear potential flows about arbitrary configurations using a higher order panel method. Volume 4: Maintenance document (version 3.0)

The Maintenance Document Version 3.0 is a guide to the PAN AIR software system, a system which computes the subsonic or supersonic linear potential flow about a body of nearly arbitrary shape, using a higher order panel method. The document describes the overall system and each program module of the system. Sufficient detail is given for program maintenance, updating, and modification. It is assumed that the reader is familiar with programming and CRAY computer systems. The PAN AIR system was written in FORTRAN 4 language except for a few CAL language subroutines which exist in the PAN AIR library. Structured programming techniques were used to provide code documentation and maintainability. The operating systems accommodated are COS 1.11, COS 1.12, COS 1.13, and COS 1.14 on the CRAY 1S, 1M, and X-MP computing systems. The system is comprised of a data base management system, a program library, an execution control module, and nine separate FORTRAN technical modules. Each module calculates part of the posed PAN AIR problem. The data base manager is used to communicate between modules and within modules. The technical modules must be run in a prescribed fashion for each PAN AIR problem. In order to ease the problem of supplying the many JCL cards required to execute the modules, a set of CRAY procedures (PAPROCS) was created to automatically supply most of the JCL cards. Most of this document has not changed for Version 3.0. It now, however, strictly applies only to PAN AIR version 3.0. The major changes are: (1) additional sections covering the new FDP module (which calculates streamlines and offbody points); (2) a complete rewrite of the section on the MAG module; and (3) strict applicability to CRAY computing systems

Conversion of trimmed NURBS surfaces to Catmull-Clark subdivision surfaces

This paper introduces a novel method to convert trimmed NURBS surfaces to untrimmed subdivision surfaces with Bézier edge conditions. We take a NURBS surface and its trimming curves as input, from this we automatically compute a base mesh, the limit surface of which fits the trimmed NURBS surface to a specified tolerance. We first construct the topology of the base mesh by performing a cross-field based decomposition in parameter space. The number and positions of extraordinary vertices required to represent the trimmed shape can be automatically identified by smoothing a cross field bounded by the parametric trimming curves. After the topology construction, the control point positions in the base mesh are calculated based on the limit stencils of the subdivision scheme and constraints to achieve tangential continuity across the boundary. Our method provides the user with either an editable base mesh or a fine mesh whose limit surface approximates the input within a certain tolerance. By integrating the trimming curve as part of the desired limit surface boundary, our conversion can produce gap-free models. Moreover, since we use tangential continuity across the boundary between adjacent surfaces as constraints, the converted surfaces join with G1 continuity. © 2014 The Authors.EPSRC, Chinese Government (PhD studentship) and Cambridge Trust

Automatic, computer aided geometric design of free-knot, regression splines

A new algorithm for Computer Aided Geometric Design of least squares (LS) splines with variable knots, named GeDS, is presented. It is based on interpreting functional spline regression as a parametric B-spline curve, and on using the shape preserving property of its control polygon. The GeDS algorithm includes two major stages. For the first stage, an automatic adaptive, knot location algorithm is developed. By adding knots, one at a time, it sequentially "breaks" a straight line segment into pieces in order to construct a linear LS B-spline fit, which captures the "shape" of the data. A stopping rule is applied which avoids both over and under fitting and selects the number of knots for the second stage of GeDS, in which smoother, higher order (quadratic, cubic, etc.) fits are generated. The knots appropriate for the second stage are determined, according to a new knot location method, called the averaging method. It approximately preserves the linear precision property of B-spline curves and allows the attachment of smooth higher order LS B-spline fits to a control polygon, so that the shape of the linear polygon of stage one is followed. The GeDS method produces simultaneously linear, quadratic, cubic (and possibly higher order) spline fits with one and the same number of B-spline regression functions. The GeDS algorithm is very fast, since no deterministic or stochastic knot insertion/deletion and relocation search strategies are involved, neither in the first nor the second stage. Extensive numerical examples are provided, illustrating the performance of GeDS and the quality of the resulting LS spline fits. The GeDS procedure is compared with other existing variable knot spline methods and smoothing techniques, such as SARS, HAS, MDL, AGS methods and is shown to produce models with fewer parameters but with similar goodness of fit characteristics, and visual quality