Reversing subdivision rules: local linear conditions and observations on inner products

AbstractIn a previous work (Samavati and Bartels, Comput. Graphics Forum 18 (1998) 97–119) we investigated how to reverse subdivision rules using global least-squares fitting. This led to multiresolution structures that could be viewed as semiorthogonal wavelet systems whose inner product was that for finite-dimensional Cartesian vector space. We produced simple and sparse reconstruction filters, but had to appeal to matrix factorization to obtain an efficient, exact decomposition. We also made some observations on how the inner product that defines the semiorthogonality influences the sparsity of the reconstruction filters. In this work we carry the investigation further by studying biorthogonal systems based upon subdivision rules and local least-squares fitting problems that reverse the subdivision. We are able to produce multiresolution structures for some common univariate subdivision rules that have both sparse reconstruction and decomposition filters. Three will be presented here – for quadratic and cubic B-spline subdivision and for the four-point interpolatory subdivision of Dyn et al. We observe that each biorthogonal system we produce can be interpreted as a semiorthogonal system with an inner product induced on the multiresolution that is quite different from that normally used. Some examples of the use of this approach on images, curves, and surfaces are given

Systematic review using a spiral approach with machine learning

Abstract With the accelerating growth of the academic corpus, doubling every 9 years, machine learning is a promising avenue to make systematic review manageable. Though several notable advancements have already been made, the incorporation of machine learning is less than optimal, still relying on a sequential, staged process designed to accommodate a purely human approach, exemplified by PRISMA. Here, we test a spiral, alternating or oscillating approach, where full-text screening is done intermittently with title/abstract screening, which we examine in three datasets by simulation under 360 conditions comprised of different algorithmic classifiers, feature extractions, prioritization rules, data types, and information provided (e.g., title/abstract, full-text included). Overwhelmingly, the results favored a spiral processing approach with logistic regression, TF-IDF for vectorization, and maximum probability for prioritization. Results demonstrate up to a 90% improvement over traditional machine learning methodologies, especially for databases with fewer eligible articles. With these advancements, the screening component of most systematic reviews should remain functionally achievable for another one to two decades

L-systems in Geometric Modeling

We show that parametric context-sensitive L-systems with affine geometry interpretation provide a succinct description of some of the most fundamental algorithms of geometric modeling of curves. Examples include the Lane-Riesenfeld algorithm for generating B-splines, the de Casteljau algorithm for generating Bezier curves, and their extensions to rational curves. Our results generalize the previously reported geometric-modeling applications of L-systems, which were limited to subdivision curves.Comment: In Proceedings DCFS 2010, arXiv:1008.127

REVERSING SUBDIVISION USING LOCAL LINEAR CONDITIONS: GENERATING MULTIRESOLUTIONS ON REGULAR TRIANGULAR MESHES

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.We are currently acquiring citations for the work deposited into this collection. We recognize the distribution rights of this item may have been assigned to another entity, other than the author(s) of the work.If you can provide the citation for this work or you think you own the distribution rights to this work please contact the Institutional Repository Administrator at [email protected]

Local filters of b-spline wavelets

Haar wavelets have been widely used in Biometrics. One advantage of Haar wavelets is the simplicity and the locality of their decomposition and reconstruction filters. However, Haar wavelets are not satisfactory for some applications due to their non-continuous behaviour. Having a particular level of smoothness is important for many applications. B-spline wavelets are capable of being applied to signals and functions of any smoothness. However, the conventional B-spline wavelets results ”non-local ” decomposition filters and consequently, they are not efficient as are the Haar wavelets. We present our recently developed local filters of Bspline wavelets. Here, we focus on quadratic case that guarantees once-differentiable smoothness. Practical issues for the efficient implementation are discussed. We show that how the resulting filters can be applied to curves, images and surfaces

Diagrammatic Tools for Generating Biothogonal Multiresolutions

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.We are currently acquiring citations for the work deposited into this collection. We recognize the distribution rights of this item may have been assigned to another entity, other than the author(s) of the work.If you can provide the citation for this work or you think you own the distribution rights to this work please contact the Institutional Repository Administrator at [email protected]

Multiresolutions Numerically from Subdivisions

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.N

Shape Defined Panoramas

Panoramic projections are often defined by the geometric surfaces used to derive the projections’ equations (e.g., spherical and cylindrical panoramas). The parameterization of these surfaces greatly affects the resulting projection equations and image properties. Problematically, unusual parameterization can reproduce panoramas associated with other shapes. In this paper, we ensure an explicit link between surface shape and projection behavior by suggesting use of projection surfaces parameterized by arc-length, binding rendering behavior to surface modeling. This allows us to create new panorama variations beyond the conventional for creating panoramas of CG environments as well as for resampling panoramas created from cameras. Further we describe an interface for composing these panoramas and show how this technique lends itself to controlling distortion and composition of panoramic projections. Additionally we provide details on rendering these projections