Weighted averages on surfaces

Panozzo, Daniele; Baran, Ilya; Diamanti, Olga; Sorkine‐Hornung, Olga

doi:10.1145/2461912.2461935

Cited by 54 publications

(50 citation statements)

References 36 publications

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“…We compared our method to the Weighted Averages on Surfaces framework of [Panozzo et al 2013]. We ran their code on the pair of SCAPE models shown in Figure 7 (bottom row) and Figure 8, and provided their method with the same input landmarks we used as input (total of 20 pairs of points).…”

Section: Comparisons With Other Methodsmentioning

confidence: 99%

“…Lifted Bijections for Low Distortion Surface Mappings • 69:9 Figure 12: Comparison of our algorithm to Weighted Averages (WA) on Surfaces [Panozzo et al 2013] using the same input landmarks (right, in black). Left column: textured source model.…”

Section: Comparisons With Other Methodsmentioning

confidence: 99%

“…Although able to produce good correspondences between pairs of surfaces, all of these methods do not produce a continuous bijective map between the surfaces. Recently, Panozzo et al, [2013] used generalized weighted averages on surfaces to define mappings between surfaces. However, this is done without guaranteeing bijectivity.…”

Section: Previous Workmentioning

confidence: 99%

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Lifted bijections for low distortion surface mappings

2014

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Figure 1: The algorithm presented in this paper generates low-distortion bijective mappings between surfaces from a sparse set of landmarks (visualized as colored spheres here). The maps are visualized by transferring the texture of the visible part in the left mesh of each pair to the right mesh, using the computed mappings. For example, the right pair shows a mapping of a horse to a giraffe; note how the map stretches gracefully at the neck area. AbstractThis paper introduces an algorithm for computing low-distortion, bijective mappings between surface meshes. The algorithm recieves as input a coarse set of corresponding pairs of points on the two surfaces, and follows three steps: (i) cutting the two meshes to disks in a consistent manner; (ii) jointly flattening the two disks via a novel formulation for minimizing isometric distortion while guaranteeing local injectivity (the flattenings can overlap, however); and (iii) computing a unique continuous bijection that is consistent with the flattenings. The construction of the algorithm stems from two novel observations: first, bijections between disk-type surfaces can be uniquely and efficiently represented via consistent locally injective flattenings that are allowed to be globally overlapping. This observation reduces the problem of computing bijective surface mappings to the task of computing locally injective flattenings, which is shown to be easier. Second, locally injective flattenings that minimize isometric distortion can be efficiently characterized and optimized in a convex framework. Experiments that map a wide baseline of pairs of surface meshes using the algorithm are provided. They demonstrate the ability of the algorithm to produce high-quality continuous bijective mappings between pairs of surfaces of varying isometric distortion levels.

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Section: Comparisons With Other Methodsmentioning

confidence: 99%

Section: Comparisons With Other Methodsmentioning

confidence: 99%

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Lifted bijections for low distortion surface mappings

2014

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“…While robust and globally shape-aware, these are not guaranteed to be true distance metrics. Another hybrid distance was proposed in [Panozzo et al 2013], where geodesics between sampled vertices are embedded in Euclidean space using multi-dimensional scaling (MDS). The embedding is interpolated to the entire mesh by solving a biharmonic equation, and Euclidean distances in the embedding space provide a distance measure on the entire mesh.…”

Section: Related Workmentioning

confidence: 99%

“…In this case, however, the optimal optimization objective is exactly the sum of squared geodesic distances from the barycenter to each of the input points. In this way, this strategy reveals an alternative to [Panozzo et al 2013] for averaging sets of points on a surface.…”

Section: Distance To Featuresmentioning

confidence: 99%

Earth mover's distances on discrete surfaces

et al. 2014

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We introduce a novel method for computing the earth mover's distance (EMD) between probability distributions on a discrete surface. Rather than using a large linear program with a quadratic number of variables, we apply the theory of optimal transportation and pass to a dual differential formulation with linear scaling. After discretization using finite elements (FEM) and development of an accompanying optimization method, we apply our new EMD to problems in graphics and geometry processing. In particular, we uncover a class of smooth distances on a surface transitioning from a purely spectral distance to the geodesic distance between points; these distances also can be extended to the volume inside and outside the surface. A number of additional applications of our machinery to geometry problems in graphics are presented.

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