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Re-initialization Free Level Set Evolution via Reaction Diffusion
TL;DR: By successfully applying diffusion to LSE, the RD-LSE model is stable by means of the simple finite difference method, which is very easy to implement.
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Abstract: This paper presents a novel reaction-diffusion (RD) method for implicit active contours, which is completely free of the costly re-initialization procedure in level set evolution (LSE). A diffusion term is introduced into LSE, resulting in a RD-LSE equation, to which a piecewise constant solution can be derived. In order to have a stable numerical solution of the RD based LSE, we propose a two-step splitting method (TSSM) to iteratively solve the RD-LSE equation: first iterating the LSE equation, and then solving the diffusion equation. The second step regularizes the level set function obtained in the first step to ensure stability, and thus the complex and costly re-initialization procedure is completely eliminated from LSE. By successfully applying diffusion to LSE, the RD-LSE model is stable by means of the simple finite difference method, which is very easy to implement. The proposed RD method can be generalized to solve the LSE for both variational level set method and PDE-based level set method. The RD-LSE method shows very good performance on boundary anti-leakage, and it can be readily extended to high dimensional level set method. The extensive and promising experimental results on synthetic and real images validate the effectiveness of the proposed RD-LSE approach.
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TABLE II: Iterations (Iter) and CPU time (in seconds) by RD and GDRLSE methods. The values in bold represent the best results. 
Fig. 4: The GAC model implemented by the proposed RD method on an image with interior boundary. (a) Initial level set function. (b) Final level set function. (c) Testing image. Blue circle represents the initial contour. There are three regions: A, B and C. (d) Middle slices of level set function during LSE. The red solid line represents the middle slice of the final level set function, which is piecewise constant in each region (A, B or C). We set Δt1=0.1 and Δt2=0.001. ![Fig. 9: Segmentation results on a synthetic image (downloaded from [29]). From left to right: results by re-initialization method [8], RD, GDRLSE1, GDRLSE2 and GDRLSE3. The red curves represent the initial contours, and the blue solid curves represent the final contours. We set parameters Δt1=0.1, Δt2=0.001, ν=0.5, α=0.2.](/figures/figure9-1-1ygomp3cqee0.png)
Fig. 9: Segmentation results on a synthetic image (downloaded from [29]). From left to right: results by re-initialization method [8], RD, GDRLSE1, GDRLSE2 and GDRLSE3. The red curves represent the initial contours, and the blue solid curves represent the final contours. We set parameters Δt1=0.1, Δt2=0.001, ν=0.5, α=0.2. 
Fig. 1: Left: different diffusion rates; Right: profiles of the two commonly used Dirac functional δ1,ρand δ2,ρ. ![Fig. 10: Segmentation results on a real image with noisy background (downloaded from [46]). From left to right: results by re-initialization method [8], RD, GDRLSE1, GDRLSE2 and GDRLSE3. Top row: initial contours (red curves) and final contours (blue curves). Bottom row: final LSFs. We set the same parameters Δt1 =0.1, μ=0.5×2552, ν=0, λ1=λ2=1 for all the methods, and set parameter Δt2=0.1 for our RD method. For GDRLSE methods, we set parameter α=0.2.](/figures/figure10-1-1g3d9dzxxmsb.png)
Fig. 10: Segmentation results on a real image with noisy background (downloaded from [46]). From left to right: results by re-initialization method [8], RD, GDRLSE1, GDRLSE2 and GDRLSE3. Top row: initial contours (red curves) and final contours (blue curves). Bottom row: final LSFs. We set the same parameters Δt1 =0.1, μ=0.5×2552, ν=0, λ1=λ2=1 for all the methods, and set parameter Δt2=0.1 for our RD method. For GDRLSE methods, we set parameter α=0.2. 
Fig. 3: LSE force analysis for RD and GDRLSE methods. (a) The possible forces at different positions for GDLRSE methods. The red arrows represent the LSE force F, while the blue arrows represent the regularization force FR. (b) In the RD method, only the LSE force F (denoted by red arrows) drives the zero level set evolve because we set the time step for the diffusion term small enough to prevent the zero level set moving.
![Fig. 9: Segmentation results on a synthetic image (downloaded from [29]). From left to right: results by re-initialization method [8], RD, GDRLSE1, GDRLSE2 and GDRLSE3. The red curves represent the initial contours, and the blue solid curves represent the final contours. We set parameters Δt1=0.1, Δt2=0.001, ν=0.5, α=0.2.](/figures/figure9-1-1ygomp3cqee0.webp)
![Fig. 10: Segmentation results on a real image with noisy background (downloaded from [46]). From left to right: results by re-initialization method [8], RD, GDRLSE1, GDRLSE2 and GDRLSE3. Top row: initial contours (red curves) and final contours (blue curves). Bottom row: final LSFs. We set the same parameters Δt1 =0.1, μ=0.5×2552, ν=0, λ1=λ2=1 for all the methods, and set parameter Δt2=0.1 for our RD method. For GDRLSE methods, we set parameter α=0.2.](/figures/figure10-1-1g3d9dzxxmsb.webp)
Citations
A Level Set Approach to Image Segmentation With Intensity Inhomogeneity
TL;DR: The proposed level set method can be directly applied to simultaneous segmentation and bias correction for 3 and 7T magnetic resonance images and demonstrates the superiority of the proposed method over other representative algorithms.
427
Weighted Level Set Evolution Based on Local Edge Features for Medical Image Segmentation
TL;DR: Evaluation results show that the proposed method leads to more accurate boundary detection results than the state-of-the-art edge-based level set segmentation methods, particularly around weak edges.
An improved edge-based level set method combining local regional fitting information for noisy image segmentation
Cheng Liu,Weibin Liu,Weiwei Xing +2 more
TL;DR: This paper proposes an improved edge-based level set method combining local regional fitting information by applying the proposed variable regional coefficient and the improved ESF to the energy function of level set function and shows that the method is efficient and robust.
118
A Variational Approach to Simultaneous Image Segmentation and Bias Correction
TL;DR: An efficient iterative algorithm is proposed for energy minimization, via which the image segmentation and bias field correction are simultaneously achieved and the smoothness of the obtained optimal bias field is ensured by the normalized convolutions without extra cost.
A convex variational level set model for image segmentation
Yongfei Wu,Chuanjiang He +1 more
TL;DR: It is proved that the value of the unique global minimizer for the energy functional is within the interval -1, 1 for any image, and equals to 1 in the object and -1 in the background for an ideal binary image.
73
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