A new variational approach based on level-set function for convex hull problem with outliers

Lingfeng Li; Shousheng Luo; Xue-Cheng Tai; Jiang Yang

doi:10.3934/ipi.2020070

Article Contents

2021, Volume 15, Issue 2: 315-338. Doi: 10.3934/ipi.2020070

This issue Previous Article Imaging junctions of waveguides Next Article Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal

A new variational approach based on level-set function for convex hull problem with outliers

1.
Department of Mathematics, Hong Kong Baptist University, Hong Kong, China
2.
Department of Mathematics, Southern University of Science and Technology, Shenzhen, China
3.
School of Mathematics and Statistics, Data Analysis Technology Lab, Henan University, Kaifeng, China
4.
Henan Engineering Research Center for Artificial Intelligence Theory and Algorithms, kaifeng, China

^* Corresponding author: sluo@henu.edu.cn
^* Corresponding author: sluo@henu.edu.cn

Received: November 30, 2019

Revised: August 31, 2020

Early access: November 2020

Published: April 2021

Shousheng Luo: supported by the Programs for Science and Technology Development of He'nan Province (1921 02310181).Xue-Cheng Tai: supported by RG(R)-RC/17-18/02-MATH, HKBU 12300819, and NSF/RGC grant N-HKBU214-19 and RC-FNRA-IG/19-20/SCI/01

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Abstract

Seeking the convex hull of an object (or point set) is a very fundamental problem arising from various tasks. In this work, we propose a variational approach based on the level-set representation for convex hulls of 2-dimensional objects. This method can adapt to exact and inexact convex hull problems. In addition, this method can compute multiple convex hulls simultaneously. In this model, the convex hull is characterized by the zero sublevel-set of a level-set function. For the exact case, we require the zero sublevel-set to be convex and contain the whole given object, where the convexity is characterized by the non-negativity of Laplacian of the level-set function. Then, the convex hull can be obtained by minimizing the area of the zero sublevel-set. For the inexact case, instead of requiring all the given points are included, we penalize the distance from all given points to the zero sublevel-set. Especially, the inexact model can handle the convex hull problem of the given set with outliers very well, while most of the existing methods fail. An efficient numerical scheme using the alternating direction method of multipliers is developed. Numerical examples are given to demonstrate the advantages of the proposed methods.

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Mathematics Subject Classification: Primary:68U05, 68U10, 35A15.

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Figure 1. Convex hulls of an object with outliers. (A) is the real convex hull. (B) and (C) are the results by the quickhull algorithm and the proposed algorithm, respectively. Here the boundaries of the results are colored in red

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Figure 2. (A) shows different level-sets of the SDF of a leaf with periodic the boundary condition and (B) shows the surface plot of the SDF

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Figure 3. Original images taken from the dataset

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Figure 4. Convex hulls corresponding to the images in Figure 3 found by Algorithm 1

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Figure 5. (A), (B) and (C) plot the profiles $ \phi $ at the 0th, 200th and 400th iterations. (D), (E) and (F) plot the corresponding level-set curves of $ \phi $ at the 0th, 200th and 400th iterations

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Figure 6. (A) shows the convex hulls result when $ c = 10 $. (B), (C) and (D) show the corresponding function values of $ \phi $, level-set curves of $ \phi $ and value of $ \Delta\phi $. (E) shows the convex hulls result when $ c = 30 $. (F), (G) and (H) show the corresponding function values of $ \phi $, level-set curves of $ \phi $ and value of $ \Delta\phi $

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Figure 7. Convex hulls of multi-objects by choosing proper values of $ c $

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Figure 8. Convex hulls in object detecting tasks. (A) and (C) are images of bus and cars with segmented masks. (B) and (D) are the associated convex hulls computed by our proposed methods

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Figure 9. Images with outliers and their convex hulls corresponding to the images in Figure 3

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Figure 10. (A), (B), (C) and (D) plot the function value of $ \phi $ at the 0th, 200th, 800th and 3200th iterations when computing the convex hull of the owl image. (E), (F), (G) and (H) show the corresponding level-set curves of $ \phi $ at the 0th, 200th, 800th and 3200th iterations

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Figure 11. Experiment results of the convex layers method. Each row displays three layers of one object

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Figure 12. Convex hulls of occluded objects with a large amount of outliers

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Figure 13. Convex hulls of camera with different $ \lambda $s

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Figure 14. Convex hulls of the aircraft with different $ \lambda $s and landmarks. Figure (D) is obtained by adding two landmarks on the end of blades

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Figure 15. Convex hulls of a set of isolated points. (A) is the exact solution and (B) is the approximation given by our method when the data contains noise

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Table 1. The relative errors of Algorithm 1

name	eggs	frog	aircraft	moth	tendril
error	1.28%	1.51%	1.13%	0.92%	0.73%
name	owl	boat	castle	cart
error	0.61%	1.08%	0.63%	0.79%

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Table 2. The Relative Errors of Algorithm 2.

name	eggs	frog	aircraft	moth	tendril
error	1.28%	4.79%	9.63%	3.33%	4.39%
name	owl	boat	castle	cart
error	2.73%	6.85%	3.79%	3.93%

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Table 3. Experiment results of using different objective functions

objective functional	(15)	(16)
average iteration number	576	679
average accuracy	1.45%	1.44%
average time	6.01s	7.30s

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