Accurate Isosurface Interpolation with Hermite Data

Fuhrmann, Simon; Kazhdan, Michael; Goesele, Michael

doi:10.1109/3dv.2015.36

Cited by 17 publications

(8 citation statements)

References 23 publications

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“…Remarkably, (17) shows that we can incorporate the Eikonal equation ( 2) by a careful choice of covariance and a simple log-transformation, which justifies the name of Log-GPIS. Despite its simplicity, the log-transformation has a non-trivial benefit that the predicted distance will approach infinity as we query points further away from the surface unlike the standard GPIS, which will predict distance value of zero.…”

Section: B Log-gaussian Process Implicit Surfacesmentioning

confidence: 77%

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Faithful Euclidean Distance Field from Log-Gaussian Process Implicit Surfaces

Lee

Liu

et al. 2020

Preprint

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In this letter, we introduce the Log-Gaussian Process Implicit Surface (Log-GPIS), a novel continuous and probabilistic mapping representation suitable for surface reconstruction and local navigation. To derive the proposed representation, Varadhan's formula is exploited to approximate the non-linear Eikonal partial differential equation (PDE) of the Euclidean distance field (EDF) by the logarithm of the screen Poisson equation that is linear. We show that members of the Matérn covariance family directly satisfy this linear PDE. Thus, our key contribution is the realisation that the regularised Eikonal equation can be simply solved by applying the logarithmic function to a GPIS formulation to recover the accurate EDF and, at the same time, the implicit surface. The proposed approach does not require post-processing steps to recover the EDF. Moreover, unlike sampling-based methods, Log-GPIS does not use sample points inside and outside the surface as the derivative of the covariance allow direct estimation of the surface normals and distance gradients. We benchmarked the proposed method on simulated and real data against stateof-the-art mapping frameworks that also aim at recovering both the surface and a distance field. Our experiments show that Log-GPIS produces the most accurate results for the EDF and comparable results for surface reconstruction and its computation time still allows online operations.

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Section: B Log-gaussian Process Implicit Surfacesmentioning

confidence: 77%

“…Furthermore, we modified the original code to render the final mesh using the marching cubes method [28]. To increase the speed and accuracy of the extraction of marching cube vertices and faces, we generate the isosuface with Hermite data [17]. As discussed in Section III, we use Matérn 3/2 covariance.…”

Section: A Implementationmentioning

confidence: 99%

Faithful Euclidean Distance Field from Log-Gaussian Process Implicit Surfaces

Lee

Liu

et al. 2020

Preprint

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“…Furthermore, because our solver supports finite‐elements systems of any degree, and since the formulation of the Poisson system only requires computing first derivatives, we can implement SPR using degree‐one finite elements. (To extract a smooth isosurface, we use the gradients at the octree corners to define piecewise quadratic interpolants along the edges as in [FKG15]. Because degree‐one finite elements have derivative discontinuities at the corners, we define the gradients by averaging the left‐ and right‐sided derivatives.…”

Section: Resultsmentioning

confidence: 99%

An Adaptive Multi‐Grid Solver for Applications in Computer Graphics

Kazhdan

Hoppe

2018

Computer Graphics Forum

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A key processing step in numerous computer graphics applications is the solution of a linear system discretized over a spatial domain. Often, the linear system can be represented using an adaptive domain tessellation, either because the solution will only be sampled sparsely, or because the solution is known to be ‘interesting’ (e.g. high frequency) only in localized regions. In this work, we propose an adaptive, finite elements, multi‐grid solver capable of efficiently solving such linear systems. Our solver is designed to be general‐purpose, supporting finite elements of different degrees, across different dimensions and supporting both integrated and pointwise constraints. We demonstrate the efficacy of our solver in applications including surface reconstruction, image stitching and Euclidean Distance Transform calculation.

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“…To this purpose, several strategies can be considered depending on the information available on f . When both function values and derivatives are available, Hermite interpolation can be used, as advocated in [12]. It provides fast and accurate interpolated values as long as the local approximation is valid.…”

Section: Given the Unbounded Voronoi Tessellationmentioning

confidence: 99%

On Volumetric Shape Reconstruction from Implicit Forms

Wang

Hétroy-Wheeler

Boyer

2016

Computer Vision – ECCV 2016

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International audienceIn this paper we report on the evaluation of volumetric shape reconstruction methods that consider as input implicit forms in 3D. Many visual applications build implicit representations of shapes that are converted into explicit shape representations using geometric tools such as the Marching Cubes algorithm. This is the case with image based reconstructions that produce point clouds from which implicit functions are computed, with for instance a Poisson reconstruction approach. While the Marching Cubes method is a versatile solution with proven efficiency, alternative solutions exist with different and complementary properties that are of interest for shape modeling. In this paper, we propose a novel strategy that builds on Centroidal Voronoi Tessellations (CVTs). These tessellations provide volumetric and surface representations with strong regularities in addition to provably more accurate approximations of the implicit forms considered. In order to compare the existing strategies, we present an extensive evaluation that analyzes various properties of the main strategies for implicit to explicit volumetric conversions: Marching cubes, Delaunay refinement and CVTs, including accuracy and shape quality of the resulting shape mesh

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Accurate Isosurface Interpolation with Hermite Data

Cited by 17 publications

References 23 publications

Faithful Euclidean Distance Field from Log-Gaussian Process Implicit Surfaces

Faithful Euclidean Distance Field from Log-Gaussian Process Implicit Surfaces

An Adaptive Multi‐Grid Solver for Applications in Computer Graphics

On Volumetric Shape Reconstruction from Implicit Forms

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