Browsing by Author "Bartels, Richard"

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    √3 Multiresolution by Local Least Squares: The Diagrammatic Approach
    (2015-10-19) Bartels, Richard; Mahdavi-Amiri, Ali; Samavati, Faramarz
    In [2, 3, 20, 21] the authors explored a construction to produce multiresolutions from given subdivisions. Certain assumptions carried through that work, two of which we wish to challenge: (1) that multiresolutions for irregular meshes have to be constructed on the fly rather than being prepared beforehand and (2) that the connectivity graph of the coarse mesh would have to be a subgraph of the connectivity graph of the fine mesh. Kobbelt's √3 subdivision [11] lets us engage both of these assumptions. With respect to (2), the √3, post-subdivision connectivity graph shares no interior edges with the pre-subdivision connectivity graph. With respect to (1), we observe that subdivision does not produce an arbitrary connectivity graph. Rather, there are local regularities that subdivision imposes on the fine mesh that are exploitable to establish, in advance, the decomposition and reconstruction filters of a multiresolution for an irregular coarse mesh.
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    Diagrammatic Tools for Generating Biothogonal Multiresolutions
    (2003-10-22) Samavati, Faramarz; Bartels, Richard
    In a previous work we introduced a construction designed to produce biorthogonal multiresolutions from given subdivisions. This construction was formulated in matrix terms, which is appropriate for curves and tensor-product surfaces. For mesh surfaces of non-tensor connectivity, however, matrix notation is inconvenient. This work introduces diagrams and diagram interactions to replace matricies and matrix multiplication. The diagrams we use are patterns of value-labeled nodes, one type of diagram corresponding to each row or column of one of the matricies of a biorthogonal system. All types of diagrams used in the construction are defined on a common mesh of the multiresolution.
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    Multiresolutions Numerically from Subdivisions
    (2010-12-02T23:05:36Z) Samavati, Faramarz; Bartels, Richard
    In previous work we introduced a construction to produce multiresolutions from given subdivisions. A portion of that construction required solving bilinear equations using a symbolic algebra system. Here we replace the bilinear equations with a pair of linear equation systems, resulting in a completely numerical construction. Diagrammatic tools provide assistance in carrying this out. The construction is shown for an example of univariate subdivision. The results for a bivariate subdivision are given to illustrate the construction's ability to handle multivariate meshes, as well as special points, without requiring any modi cation of approach. The construction usually results in analysis and reconstruction lters that are nite, since it seeks each lter locally for the neighborhood of the mesh to which it applies. The use of a set of lters constructed in this way is compared with lters based on spline wavelets for image compression to show that the construction can yield satisfactory results.
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    REVERSING SUBDIVISION USING LOCAL LINEAR CONDITIONS: GENERATING MULTIRESOLUTIONS ON REGULAR TRIANGULAR MESHES
    (2002-12-02) Samavati, Faramarz; Bartels, Richard
    In a previous work [1] we investigated how to reverse subdivision rules using local linear conditions based upon least squares approximation. We outlined a general approach for reversing subdivisions and showed how to use the approach to construct multiresolutions with finite decomposition and reconstruction filters. These multiresolutions correspond to biorthogonal wavelet systems that use inner products implicitly defined by the construction. We gave evidence through a number of example subdivision rules that the approach works for curves and tensor-product surfaces. In [14] some of this material was put to work on non-tensor-product surface meshes of arbitrary connectivity. The price to be paid for such connectivity is a limitation on the flexibility one has in formulating the linear conditions for reversal and the complexity in assessing the face topology of the mesh. The full sweep of the general approach is lost in the irregularity of the application. In this work we take regular, triangular meshes and use one interpolating and two noninterpolating subdivisions: the Butterfly subdivision [6], Loop's subdivision [12], and a quasi-interpolation based subdivision [11], as examples. We visit the general approach for curves once again and, using these example subdivisions, show that the approach can be applied with success to produce finite filter multiresolutions in the triangular mesh case as well. In the process, we introduce graphical insights that provide a mask-based development in place of our previous matrix-based development, suggesting that our construction is not limited to triangle mesh geometry. To overcome a limitation we encountered in symbolic algebra systems, we invoke the lifting process [19] in a nonstandard way.