Reconstructing 2D images with natural neighbour interpolation

2012

Radial Basis Functions (RBF) interpolation theory is briefly introduced at the “application level” including some basic principles and computational issues. The RBF interpolation is convenient for un-ordered data sets in n-dimensional space, in general. This approach is convenient especially for a higher dimension N 2 conversion to ordered data set, e.g. using tessellation, is computationally very expensive. The RBF interpolation is not separable and it is based on distance of two points. The RBF interpolation leads to a solution of a Linear System of Equations (LSE) . There are two main groups of interpolating functions: ‘global” and “local”. Application of “local” functions, called Compactly Supporting Functions (CSFBF), can significantly decrease computational cost as they lead to a system of linear equations with a sparse matrix. The RBF interpolation can be used also for image reconstruction, inpainting removal, for solution of Partial Differential Equations (PDE) etc. Key-Word...

Computational Geometry, 2002

We present an algorithm to reconstruct smooth surfaces of arbitrary topology from unorganised sample points and normals. The method uses natural neighbour interpolation, works in any dimension and accommodates nonuniform samples. The reconstructed surface interpolates the data points and is implicitly represented as the zero set of some pseudo-distance function. It can be meshed so as to satisfy a user-defined error bound, which makes the method especially relevant for small point sets. Experimental results are presented for surfaces in R 3 .

'Advanced GIS'. Student Assignment Nr. 2, Lund University, GIS Center, 2010

MSc Student Assignment Nr. 2 'Interpolation' on module ’Geographical Information Systems: Advanced Course (GISA02)’ Lund University, GIS Centre. The work presents a research on interpolation of data in Arc GIS. Local and global interpolations are compared and differences highlighted. 3 interpolation methods are tested: Kriging, Spline and IDW. The IDW, Nearest Neighbor and Spline method are local interpolations: they value distances between the interpolating point and its neighborhood based on the neighbor measured values. Trend Surfaces Interpolation method is a global interpolation method and it is based on general assessing the trends in the surface. Spline gives very smooth surface of the area when interpolating. It minimizes overall roughness and extremities in the relief of the surface. Resulting values of Spline are varying from -2,051.259033 until 6,934.932617. In the practical 9 classes as the most representative are made. The best results are represented by the lowest values of the RMSE (Kriging by Cross-Validation). Method of spatial autocorrelation is tested to measure and to analyze the how much the variables are dependent with each other in a space. It means that the values of the variables are compared with each other and estimated the degree of dependency among the values. Points-based and line-based interpolations are both used for interpolation of the surfaces. Global interpolation methods are more sensible to the outliners, because they are indifferent to the local nuances of the surface, which is not the case with local interpolation methods: the last ones are very sensitive to the neighborhood, can good assess local relief characteristics, and therefore interpolate good a surface for the small outliner-area.

Geographical Analysis, 2010

Our primary interest is in geographical problems and the discussion focuses on examples in which the interpolation estimates are to be made in two dimensions. We believe that the simplest and most sensible method of geographic interpolation consists of the assignment of an average value to the location or locations for which data are required. The set over which the average is taken is obviously important, and, as weighted averages are almost invariably used, the choice of weights is also critical. For spatial variables the relevant set usually consists of values in the vicinity of the locations for which the estimates are desired. Observe here that we implicitly assume that the variable of interest is numerical, and not categorical, so that averages have meaning. Suggestions as to how to proceed when this is not the case may be found in Guptill (1975), Switzer (1975), and Tobler (1979a. We also restrict our attention to arithmetical averages, ignoring geometrical and harmonic averages and medians which may be appropriate in some cases. It should be recognized that no interpolation scheme can overcome the problem of insufficient resolution in the original observations.

Lecture Notes in Computer Science, 2006

Smooth local coordinates have been proposed by Hiyoshi and Sugihara 2000 to improve the classical Sibson's and Laplace coordinates. These smooth local coordinates are computed by integrating geometric quantities over weights in the power diagram. In this paper we describe how to efficiently implement the Voronoi based C 2 local coordinates. The globally C 2 interpolant that Hiyoshi and Sugihara presented in 2004 is then compared to Sibson's and Farin's C 1 interpolants when applied to scattered data interpolation.

Numerical Algorithms, 1993

Based on the work of Robin Sibson concerning Natural Neighbor Interpolant, this paper is devoted to incorporate this concept in Spline theory. To do this, first a new concept, the "Covering Spheres", is presented, which is then linked with Sibson's interpolant. Finally, the interpolant is reformulated to present it as a Bernstein polynomial in local coordinates instead of the usual presentation as rational quartics. As a corollary, the whole idea is presented as modified Vertex Splines.

Brazilian Symposium on Computer Graphics and Image Processing, 2002

Digital image interpolation techniques are frequently used to enlarge pictures, i.e. zooming in, and they are upon increasingly demand for developing product applications using digital still cameras. One common difficulty with conventional interpolation techniques is that of preserving details, i.e. edges, and at the same time smoothing the data for not introducing spurious artifacts. A definitive solution to this is

Citeseer

Subdivision schemes are used to interpolate data samples locally. By using temporary placeholders on a dense grid, we improve one of the best known subdivision scheme (Deslauriers-Dubuc). Interpolated values require 2 steps to stabilize as they are first interpolated on a coarse scale through a tetradic filter and then on a finer scale using a dyadic filter. The interpolants are C 1 and can be made to reproduce polynomials of degree 4 unlike regular subdivision schemes. These generalized interpolatory subdivision schemes have minimal support and no additional memory requirement.

IEEE Transactions on Image Processing, 1998

We discuss possible algorithms for interpolating data given in a set of curves and/or points in the plane. We propose a set of basic assumptions to be satis ed by the interpolation algorithms which lead to a set of models in terms of possibly degenerate elliptic partial di erential equations. The Absolute Minimal Lipschitz Extension model (AMLE) is singled out and studied in more detail. We show experiments suggesting a possible application, the restoration of images with poor dynamic range.

Geographical Analysis, 2005

This paper introduces a procedure for progressively increasing the density of an initial point set that can be used as a basis for interpolating surfaces of variable resolution from sparse samples of data sites. The procedure uses the Simple Recursive Point Voronoi Diagram in which Voronoi concepts are used to tessellate space with respect to a given set of generator points. The construction is repeated every time with a new generator set, which comprises members selected from the previous generator set plus features of the current tessellation. We show how this procedure can be implemented in Arc/Info and present an illustration of its application using a known surface. Initial results suggest that the procedure has considerable potential and we discuss further methods for evaluating and extending it.