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Image Compression by Linear Splines over Adaptive Triangulations
"... This paper proposes a new method for image compression. The method is based on the approximation of an image, regarded as a function, by a linear spline over an adapted triangulation, D(Y ), which is the Delaunay triangulation of a small set Y of significant pixels. The linear spline minimizes the d ..."
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Cited by 38 (9 self)
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This paper proposes a new method for image compression. The method is based on the approximation of an image, regarded as a function, by a linear spline over an adapted triangulation, D(Y ), which is the Delaunay triangulation of a small set Y of significant pixels. The linear spline minimizes the distance to the image, measured by the mean square error, among all linear splines over D(Y ). The significant pixels in Y are selected by an adaptive thinning algorithm, which recursively removes less significant pixels in a greedy way, using a sophisticated criterion for measuring the significance of a pixel. The proposed compression method combines the approximation scheme with a customized scattered data coding scheme. We demonstrate that our compression method outperforms JPEG2000 on two geometric images and performs competitively with JPEG2000 on three popular test cases of real images.
LIGHT DETECTION AND RANGING (LIDAR) DATA COMPRESSION
, 2005
"... Light Detection and Ranging (LIDAR) data compression has been an active research field for last few years because of its large storage size. When LIDAR has large number of data points, the surface generation represented by interpolation methods may be inefficient in both storage and computational re ..."
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Cited by 1 (0 self)
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Light Detection and Ranging (LIDAR) data compression has been an active research field for last few years because of its large storage size. When LIDAR has large number of data points, the surface generation represented by interpolation methods may be inefficient in both storage and computational requirements. This paper presents a newly developed compression scheme for the LIDAR data based on second generation wavelet. A new interpolation wavelet filter has been applied in two steps, namely splitting and elevation. In the splitting step, a triangle has been divided into several subtriangles and the elevation step has been used to ‘modify’ the point values (point coordinates for geometry) after the splitting. Then, this data set is compressed at the desired locations by using second generation wavelets. The quality of geographical surface representation after using proposed technique is compared with the original LIDAR data. The results show that this method can be used for significant reduction of data set.
On the relation between piecewise polynomial and rational approximation
 in Lp(R 2 ), Constr. Approx
"... Abstract: In the univariate case there are certain equivalences between the nonlinear approximation methods that use piecewise polynomial and those that use rational functions. It is known that for certain parameters the respective approximation spaces are identical and may be described as Besov spa ..."
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Abstract: In the univariate case there are certain equivalences between the nonlinear approximation methods that use piecewise polynomial and those that use rational functions. It is known that for certain parameters the respective approximation spaces are identical and may be described as Besov spaces. The characterization of the approximation spaces of the multivariate nonlinear approximation by piecewise polynomials and by rational functions is not known. In this work we compare between the two methods in the bivariate case. We show some relations between the approximation spaces of piecewise polynomials defined on n triangles and those of bivariate rational functions of total degree n which are described by n parameters. Thus we compare two classes of approximants with the same number Cn of parameters. We consider this the proper comparison between the two methods.
Corresponding Author (especially for galley proofs):
, 2005
"... Number of figures: 6 Figures and their captions: Figure 1[on page 8] Removal of the vertex y ∈ D(Y), and the Delaunay triangulation of its cell C(y). The five triangles of the cell C(y) in (a) are replaced by the three triangles in (b). Figure 2[on page 16] Geometric test image Chessboard. ..."
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Number of figures: 6 Figures and their captions: Figure 1[on page 8] Removal of the vertex y ∈ D(Y), and the Delaunay triangulation of its cell C(y). The five triangles of the cell C(y) in (a) are replaced by the three triangles in (b). Figure 2[on page 16] Geometric test image Chessboard.