37-Issue 5

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https://diglib.eg.org/handle/10.2312/2632598

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Browsing 37-Issue 5 by Subject "Computer graphics"

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    Interactive Curve Constrained Functional Maps
    (The Eurographics Association and John Wiley & Sons Ltd., 2018) Gehre, Anne; Bronstein, Michael M.; Kobbelt, Leif; Solomon, Justin; Ju, Tao and Vaxman, Amir
    Functional maps have gained popularity as a versatile framework for representing intrinsic correspondence between 3D shapes using algebraic machinery. A key ingredient for this framework is the ability to find pairs of corresponding functions (typically, feature descriptors) across the shapes. This is a challenging problem on its own, and when the shapes are strongly non-isometric, nearly impossible to solve automatically. In this paper, we use feature curve correspondences to provide flexible abstractions of semantically similar parts of non-isometric shapes. We design a user interface implementing an interactive process for constructing shape correspondence, allowing the user to update the functional map at interactive rates by introducing feature curve correspondences. We add feature curve preservation constraints to the functional map framework and propose an efficient numerical method to optimize the map with immediate feedback. Experimental results show that our approach establishes correspondences between geometrically diverse shapes with just a few clicks.
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    Kernel Functional Maps
    (The Eurographics Association and John Wiley & Sons Ltd., 2018) Wang, Larry; Gehre, Anne; Bronstein, Michael M.; Solomon, Justin; Ju, Tao and Vaxman, Amir
    Functional maps provide a means of extracting correspondences between surfaces using linear-algebraic machinery. While the functional framework suggests efficient algorithms for map computation, the basic technique does not incorporate the intuition that pointwise modifications of a descriptor function (e.g. composition of a descriptor and a nonlinearity) should be preserved under the mapping; the end result is that the basic functional maps problem can be underdetermined without regularization or additional assumptions on the map. In this paper, we show how this problem can be addressed through kernelization, in which descriptors are lifted to higher-dimensional vectors or even infinite-length sequences of values. The key observation is that optimization problems for functional maps only depend on inner products between descriptors rather than descriptor values themselves. These inner products can be evaluated efficiently through use of kernel functions. In addition to deriving a kernelized version of functional maps including a recent extension in terms of pointwise multiplication operators, we provide an efficient conjugate gradient algorithm for optimizing our generalized problem as well as a strategy for low-rank estimation of kernel matrices through the Nyström approximation.
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    Sensor-aware Normal Estimation for Point Clouds from 3D Range Scans
    (The Eurographics Association and John Wiley & Sons Ltd., 2018) Comino Trinidad, Marc; Andujar, Carlos; Chica, Antonio; Brunet, Pere; Ju, Tao and Vaxman, Amir
    Normal vectors are essential for many point cloud operations, including segmentation, reconstruction and rendering. The robust estimation of normal vectors from 3D range scans is a challenging task due to undersampling and noise, specially when combining points sampled from multiple sensor locations. Our error model assumes a Gaussian distribution of the range error with spatially-varying variances that depend on sensor distance and reflected intensity, mimicking the features of Lidar equipment. In this paper we study the impact of measurement errors on the covariance matrices of point neighborhoods. We show that covariance matrices of the true surface points can be estimated from those of the acquired points plus sensordependent directional terms. We derive a lower bound on the neighbourhood size to guarantee that estimated matrix coefficients will be within a predefined error with a prescribed probability. This bound is key for achieving an optimal trade-off between smoothness and fine detail preservation. We also propose and compare different strategies for handling neighborhoods with samples coming from multiple materials and sensors. We show analytically that our method provides better normal estimates than competing approaches in noise conditions similar to those found in Lidar equipment.