On building fast kd-Trees for Ray Tracing, and on doing that in O(N log N)

Wald, Ingo; Havran, Vlastimil

doi:10.1109/rt.2006.280216

Cited by 263 publications

(207 citation statements)

References 10 publications

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“…A third option is to choose the splitting location such that the two subnodes have equal mean density. This method has shown to give good results in computer graphics ray-shooting simulations (Wald & Havran 2006).…”

Section: Implementation In Skirtmentioning

confidence: 99%

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Hierarchical octree andk-d tree grids for 3D radiative transfer simulations

Saftly

Baes

Camps

2014

A&A

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Context. A crucial ingredient for numerically solving the three-dimensional radiative transfer problem is the choice of the grid that discretizes the transfer medium. Many modern radiative transfer codes, whether using Monte Carlo or ray tracing techniques, are equipped with hierarchical octree-based grids to accommodate a wide dynamic range in densities. Aims. We critically investigate two different aspects of octree grids in the framework of Monte Carlo dust radiative transfer. Inspired by their common use in computer graphics applications, we test hierarchical k-d tree grids as an alternative for octree grids. On the other hand, we investigate which node subdivision-stopping criteria are optimal for constructing of hierarchical grids. Methods. We implemented a k-d tree grid in the 3D radiative transfer code SKIRT and compared it with the previously implemented octree grid. We also considered three different node subdivision-stopping criteria (based on mass, optical depth, and density gradient thresholds). Based on a small suite of test models, we compared the efficiency and accuracy of the different grids, according to various quality metrics. Results. For a given set of requirements, the k-d tree grids only require half the number of cells of the corresponding octree. Moreover, for the same number of grid cells, the k-d tree is characterized by higher discretization accuracy. Concerning the subdivision stopping criteria, we find that an optical depth criterion is not a useful alternative to the more standard mass threshold, since the resulting grids show a poor accuracy. Both criteria can be combined; however, in the optimal combination, for which we provide a simple approximate recipe, this can lead to a 20% reduction in the number of cells needed to reach a certain grid quality. An additional density gradient threshold criterion can be added that solves the problem of poorly resolving sharp edges and strong density gradients. Conclusions. We advocate the use of k-d trees and the proposed combination of criteria to set up hierarchical grids for 3D radiative transfer. These recipes are straightforward for implementing and should help to develop faster and more accurate 3D radiative transfer codes.

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Section: Implementation In Skirtmentioning

confidence: 99%

“…Nowadays they are the most popular space partitioning structure in computer graphics applications (e.g. MacDonald & Booth 1990;Havran 2000;Reshetov et al 2005;Wald & Havran 2006;Shevtsov et al 2007;Zhou et al 2008). To the best of our knowledge, k-d trees have not been used in the context of astrophysical radiative transfer simulations.…”

Section: Introductionmentioning

confidence: 99%

Hierarchical octree andk-d tree grids for 3D radiative transfer simulations

Saftly

Baes

Camps

2014

A&A

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“…A Kd tree which is a space-partitioning data structure for organizing triangles in a k-dimensional space is designed in order to fast the intersection between ray and 3D model to a reasonable time. Constructing a efficient Kd tree for triangular meshes is also time consuming, so a O(N log 2 N ) approach is adopted and more details can be found in (Wald and Havran, 2006). However, there are several reasons that the reconstructed shadows may not exactly match with shadows in the image.…”

Section: Shadow Reconstructionmentioning

confidence: 99%

Shadowdetection From VHR Aerial Images in Urban Area by Using 3d City Models and a Decision Fusion Approach

Zhou

Gorte

2017

Int. Arch. Photogramm. Remote Sens. Spatial Inf. Sci.

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ABSTRACT:In VHR(very high resolution) aerial images, shadows indicating height information are valuable for validating or detecting changes on an existing 3D city model. In the paper, we propose a novel and full automatic approach for shadow detection from VHR images. Instead of automatic thresholding, the supervised machine learning approach is expected with better performance on shadow detection, but it requires to obtain training samples manually. The shadow image reconstructed from an existing 3D city model can provide free training samples with large variety. However, as the 3D model is often not accuracy, incomplete and outdated, a small portion of training samples are mislabeled. The erosion morphology is provided to remove boundary pixels which have high mislabeling possibility from the reconstructed image. Moreover, the quadratic discriminant analysis (QDA) which is resistant to the mislabeling is chosen. Further, two feature domains, RGB and ratio of the hue over the intensity, are analyzed to have complementary effects on better detecting different objects. Finally, a decision fusion approach is proposed to combine the results wisely from preliminary classifications from two feature domains. The fuzzy membership is a confidence measurement and determines the way of making decision, in the meanwhile the memberships are weighted by an entropy measurements to indicate their certainties. The experimental results on two cities in the Netherlands demonstrate that the proposed approach outperforms the two separate classifiers and two stacked-vector fusion approaches.

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“…For a detailed description on building kd-trees see [4] by Wald and Havran. Although they propose an O(nlogn) algorithm -the theoretical lower bound for comparison-based sorting algorithms -they did not apply low-level optimization techniques or other approaches to speed-up the method.…”

Section: Algorithm 1: Recursive Kd-tree Constructionmentioning

confidence: 99%

“…Such works have been done by Wald & Havran [9] in the field of acceleration constructions and new approaches rendering in kd-tree. Shevtson [10] has introduced an algorithm called min-max binning.…”

Section: Previous Workmentioning

confidence: 99%