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Active contour model (ACM) is a powerful segmentation method based on differential equation. This paper proposes a novel adaptive ACM to segment those intensity inhomogeneity images. Firstly, a novel signed pressure force function is presented with Legendre polynomials to control curve contraction. Legendre polynomials can approximate regional intensities corresponding to evolving curve. Secondly, global term of our model characterizes difference of Legendre coefficients, and local energy term characterizes fitting evolution curve of interested region. Final contour evolution will minimize the energy function. Thirdly, a correction term is employed to improve the performance of curve evolution according to the initial contour position, so wherever the initial contour being in the image, the object boundaries can be detected. Fourthly, our model combines the advantages of two classical models such as good topological changes and computational simplicity. The new model can classify regions with similar intensity values. Compared with traditional models, experimental results show effectiveness and efficiently of the new model.

Image segmentation [

The edge ACM utilizes image gradient to guide the contours toward the boundaries of desired objects, and segmentation result relies on the location of the initial contour. Region model applies intensity and texture information to guide curve evolution. Chan-Vase (CV) [

Intensity inhomogeneity image segmentation is still a challenging problem. Li et al. propose a local binary fitting (LBF) [

This paper proposes a novel Legendre polynomial approximation with adaptive global energy based on our previous model [

This rest of this paper is organized as follows. In Section

Zhang et al. propose the SBGFRLS method [

The model utilizes region information to stop the curve evolution, which is more efficient and less sensitive to noise. SBGFRLS model and CV model cannot deal with images with intensity inhomogeneity because they all consider global image intensities merely.

Mukherjee and Acton [

The model approximates foreground and background by computing

Motivated by the SBGFRLS model, we propose a novel ACM based on Legendre polynomial. The energy function of our model includes two parts, the local term and the global term:

The local energy term

The balloon force

We can infer from the above equation that the highest degree of the 1D basis is

For each category of images, a correction term will decide the curve to evolve from inside to outside or from outside to inside, so that the initial contour being anywhere in the image can detect the object boundaries. At the same time, a new SPF can effectively drive curve to stop contours at weak edge, even images in the presence of inhomogeneity intensity.

The main role of global energy affects the speed and accuracy of the evolution curve. Therefore, it is necessary to choose an appropriate global term. Most of parameters are selected manually, but we design an adaptive parameter strategy based on the difference between the foreground and background. The foreground and the background can be modeled by a set of Legendre basis functions in our model and can be represented in a lower dimensional subspace. When the gray value of background is less than the gray value of foreground, a novel adaptive global term can be written by utilizing Legendre basis functions as follows:

Then, the final proposed model is as follows:

Initialization:

Initialize the level set function

Compute

Evolve the level set function according to Eq. (

Let

Regularize the level set function

Check whether the evolution of the level set has converged. If not, return to stage d

The step (e) serves as an optional segmentation procedure.

In this section, the experimental results of our proposed model will be presented on a series of synthetic and real images. All experimental are implemented in Matlab R2014a on a 3.30-GHz PC. The initial contour can be chosen as rectangle, ellipse, and multiball manually according to image. The correction term

Figure

Segmental results for images with noise: (a) CV model segmentation results with noise image variance of 0.1, 0.2, and 0.3; (b) SBGFRLS model segmentation results with noise image variance of 0.1, 0.2, and 0.3; (c) Our model segmentation results with noise image variance of 0.1, 0.2, and 0.3.

The corresponding Dice values of the segmental results in Figure

Figure

The comparisons of LBF model, LIC model, L2S model, LSACM model, and our proposed model on segmenting images with intensity inhomogeneity: (a) results of LBF model; (b) results of LIC model; (c) results of L2S model; (d) result of LSACM model; (e) result of our proposed model.

The corresponding DICE values of the segmental results in Figure

LBF | LIC | L2S | LSACM | Ours |
---|---|---|---|---|

0.6198 | 0.9061 | 0.8532 | 0.7893 | 0.9348 |

0.9654 | 0.8994 | 0.8823 | 0.9033 | 0.9311 |

0.9759 | 0.8669 | 0.8238 | 0.9035 | 0.9366 |

Figure

Detected contour of regions of interest by LCV model, LSACM model, and our model: (a) results of LCV model; (b) results of LSACM model; (c) results of our model.

Running time and the DICE value for images shown in Figure

Image | CPU running time | ||
---|---|---|---|

LCV | LSACM | Ours | |

First | 1.9375 | 37.2031 | 2.4375 |

Second | 7.1719 | 35.8125 | 7.7031 |

Image | The DICE value | ||

First | 0.4708 | 0.9745 | 0.9757 |

Second | 0.9607 | 0.9966 | 0.9968 |

Figure

The comparisons of LSACM model, LBF model, LIC model, and our model on segmenting images with the intensity inhomogeneity: (a) results of LSACM model; (b) results of LBF model; (c) results of LIC model; (d) results of our model.

Figure

The process of segmentation using our model: (a) initial contour; (b) final contour, 20 iterations; (c) final contour, 40 iterations; (d) final contour, 100 iterations; (e) initial contour; (f) final contour, 60 iterations; (g) final contour, 120 iterations; (h) final contour, 240 iterations.

Figure

Comparison results of our model with CV, LBF, and L2S models: (a) initial contours; (b) final segmentation results using the CV model; (c) final segmentation results using the LBF model; (d) final segmentation results using the L2S model; (e) final segmentation results using our model.

A novel adaptive segmentation model for images in the presence of low contrast, noise, weak edge, and intensity inhomogeneity is proposed in this paper. Regions are represented by a set of Legendre basis function, so Legendre polynomials are introduced to deal with intensity inhomogeneous image segmentation problem. The local and global information are all considered, and GAC model and SBGFRLS model are combined in our model. Our model has good topological changes and computational simplicity. The evolution direction can be chosen adaptively according to the parameter

In this appendix, we deduce the corresponding coefficients

Let

Then, we perform

Let

However, computing the coefficient vectors needs a matrix inversion step. Here, we concretely show that the coefficient vectors

All experimental images come from reference literatures, and we also point out the source one by one in the manuscript.

The authors declare there is no conflict of interest.

All authors typed, read, and approved the final manuscript.

This paper is partially supported by the Natural Science Foundation of Shandong Province of China (ZR2017MA012), the Natural Science Foundation of Guangdong Province (2018A030313364), the Special Innovation Projects of Universities in Guangdong Province (2018KTSCX197), the Science and Technology Planning Project of Shenzhen City (JCYJ20180305125609379), the Natural Science Foundation of Shenzhen (JCYJ20170818091621856), and the China Scholarship Council Project (201508440370).

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