Browsing by Author "Jansen, Maarten"

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    Multiscale Approximation of Piecewise Smooth Two-Dimensional Function using Normal Triangulated Meshes
    (2005-07-01) Jansen, Maarten; Baraniuk, Richard G.; Lavu, Sridhar; Digital Signal Processing (http://dsp.rice.edu/)
    Multiresolution triangulation meshes are widely used in computer graphics for representing three-dimensional(3-d) shapes. We propose to use these tools to represent 2-d piecewise smooth functions such as grayscale images,because triangles have potential to more efficiently approximate the discontinuities between the smooth pieces than other standard tools like wavelets. We show that normal mesh subdivision is an efficient triangulation, thanks to its local adaptivity to the discontinuities. Indeed, we prove that, within a certain function class, the normal mesh representation has an optimal asymptotic error decay rate as the number of terms in the representation grows. This function class is the so-called horizon class comprising constant regions separated by smooth discontinuities,where the line of discontinuity is C2 continuous. This optimal decay rate is possible because normal meshes automatically generate a polyline (piecewise linear) approximation of each discontinuity, unlike the blocky piecewise constant approximation of tensor product wavelets. In this way, the proposed nonlinear multiscale normal mesh decomposition is an anisotropic representation of the 2-d function. The same idea of anisotropic representations lies at the basis of decompositions such as wedgelet and curvelet transforms, but the proposed normal mesh approach has a unique construction.
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    Multiscale Image Processing Using Normal Triangulated Meshes
    (2001-10-01) Jansen, Maarten; Choi, Hyeokho; Lavu, Sridhar; Baraniuk, Richard G.; Digital Signal Processing (http://dsp.rice.edu/)
    Multiresolution triangulation meshes are widely used in computer graphics for 3-d modelling of shapes. We propose an image representation and processing framework using a multiscale triangulation of the grayscale function. Triangles have the potential of approximating edges better than the blocky structures of tensor-product wavelets. Among the many possible triangulation schemes, normal meshes are natural for efficiently representing singularities in image data thanks to their adaptivity to the smoothness of the modeled image. Our non-linear, multiscale image decomposition algorithm, based on this subdivision scheme, takes edges into account in a way that is closely related to wedgelets and curvelets. The highly adaptive property of the normal meshes construction provides a very efficient representation of images, which potentially outperforms standard wavelet transforms. We demonstrate the approximation performance of the normal mesh representation through mathematical analyses for simple functions and simulations for real images.