Convexity Shape Prior for Level Set based Image Segmentation Method

Yan, Shi; Tai, Xue–Cheng; Li, Jun; Huang, Haiyang

doi:10.48550/arxiv.1805.08676

Cited by 4 publications

(10 citation statements)

References 34 publications

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“…Let φ be a signed distance function of an object. Then in the level set method, a simple linear constraint φ 0 for the signed distance function φ can force the segmented region to be convex [13]. This is because the curvature κ = div( ∇φ |∇φ| ) in the implicit representation of the curves would be reduced to κ = φ when |∇φ| = 1 in terms of the proposition of signed distance function.…”

Section: Introductionmentioning

confidence: 99%

Convex Shape Prior for Deep Neural Convolution Network based Eye Fundus Images Segmentation

Liu,

Tai,

Luo

2020

Preprint

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Convex Shapes (CS) are common priors for optic disc and cup segmentation in eye fundus images. It is important to design proper techniques to represent convex shapes. So far, it is still a problem to guarantee that the output objects from a Deep Neural Convolution Networks (DCNN) are convex shapes. In this work, wepropose a technique which can be easily integrated into the commonly used DCNNs for image segmentation and guarantee that outputs are convex shapes. This method is flexible and it can handle multiple objects and allow some of the objects to be convex. Our method is based on the dual representation of the sigmoid activation function in DCNNs. In the dual space, the convex shape prior can be guaranteed by a simple quadratic constraint on a binary representation of the shapes. Moreover, our method can also integrate spatial regularization and some other shape prior using a soft thresholding dynamics (STD) method. The regularization can make the boundary curves of the segmentation objects to be simultaneously smooth and convex. We design a very stable active set projection algorithm to numerically solve our model. This algorithm can form a new plug-and-play DCNN layer called CS-STD whose outputs must be a nearly binary segmentation of convex objects. In the CS-STD block, the convexity information can be propagated to guide the DCNN in both forward and backward propagation during training and prediction process. As an application example, we apply the convexity prior layer to the retinal fundus images segmentation by taking the popular DeepLabV3+ as a backbone network. Experimental results on several public datasets show that our method is efficient and outperforms the classical DCNN segmentation methods.

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Section: Introductionmentioning

confidence: 99%

Convex Shape Prior for Deep Neural Convolution Network based Eye Fundus Images Segmentation

Liu,

Tai,

Luo

2020

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“…To impose the convexity constraint, we require the Laplacian of the SDF is non-negative at the given set. The equivalence of these two conditions was proved in [32] and [25]. The second model we introduced is for the cases with outliers or noise, where the convex hulls do not have to enclose all the given points.…”

Section: Introductionmentioning

confidence: 99%

Convex hull algorithms based on some variational models

Li,

Luo,

Tai

et al. 2019

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Seeking the convex hull of an object is a very fundamental problem arising from various tasks. In this work, we propose two variational convex hull models using level set representation for 2-dimensional data. The first one is an exact model, which can get the convex hull of one or multiple objects. In this model, the convex hull is characterized by the zero sublevel-set of a convex level set function, which is non-positive at every given point. By minimizing the area of the zero sublevel-set, we can find the desired convex hull. The second one is intended to get convex hull of objects with outliers. Instead of requiring all the given points are included, this model penalizes the distance from each given point to the zero sublevel-set. Literature methods are not able to handle outliers. For the solution of these models, we develop efficient numerical schemes using alternating direction method of multipliers. Numerical examples are given to demonstrate the advantages of the proposed methods.

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“…In [2], an Euler's elastica energy-based model was studied for convex contours by penalizing the integral of absolute curvatures. Recently, the continuous methods or curvature-based methods were developed further in [24,37]. For a given object in 2-dimensional space, it was proved in [37] that the non-negative Laplacian of the associated signed distance function [32] (SDF) is sufficient to guarantee the convexity of shapes.In [24], the authors also proved that this condition is also necessary.…”

mentioning

confidence: 99%

“…Recently, the continuous methods or curvature-based methods were developed further in [24,37]. For a given object in 2-dimensional space, it was proved in [37] that the non-negative Laplacian of the associated signed distance function [32] (SDF) is sufficient to guarantee the convexity of shapes.In [24], the authors also proved that this condition is also necessary. In addition, instead of solving negative curvature penalizing problems like [2,33,38], these two papers incorporated non-negative Laplacian condition as a constraint into the involved optimization problem.…”

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A level set representation method for N-dimensional convex shape and applications

li¹,

Luo²,

Tai³

et al. 2020

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In this work, we present a new efficient method for convex shape representation, which is regardless of the dimension of the concerned objects, using level-set approaches. Convexity prior is very useful for object completion in computer vision. It is a very challenging task to design an efficient method for high dimensional convex objects representation. In this paper, we prove that the convexity of the considered object is equivalent to the convexity of the associated signed distance function. Then, the second order condition of convex functions is used to characterize the shape convexity equivalently. We apply this new method to two applications: object segmentation with convexity prior and convex hull problem (especially with outliers). For both applications, the involved problems can be written as a general optimization problem with three constraints. Efficient algorithm based on alternating direction method of multipliers is presented for the optimization problem. Numerical experiments are conducted to verify the effectiveness and efficiency of the proposed representation method and algorithm.

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Convexity Shape Prior for Level Set based Image Segmentation Method

Cited by 4 publications

References 34 publications

Convex Shape Prior for Deep Neural Convolution Network based Eye Fundus Images Segmentation

Convex Shape Prior for Deep Neural Convolution Network based Eye Fundus Images Segmentation

Convex hull algorithms based on some variational models

A level set representation method for N-dimensional convex shape and applications

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