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We propose a new active contour model which integrates a local intensity fitting (LIF) energy with an auxiliary global intensity fitting (GIF) energy. The LIF energy is responsible for attracting the contour toward object boundaries and is dominant near object boundaries, while the GIF energy incorporates global image information to improve the robustness to initialization of the contours. The proposed model not only can provide desirable segmentation results in the presence of intensity inhomogeneity but also allows for more flexible initialization of the contour compared to the RSF and LIF models, and we give a theoretical proof to compute a unique steady state regardless of the initialization; that is, the convergence of the zero-level line is irrespective of the initial function. This means that we can obtain the same zero-level line in the steady state, if we choose the initial function as a bounded function. In particular, our proposed model has the capability of detecting multiple objects or objects with interior holes or blurred edges.

Implicit active contour models have been extensively studied and successfully used in image segmentation [

Unlike edge-based models that utilize typically an edge indicator depending on image gradient to perform contour extraction, region-based models usually use global and/or local statistics inside regions rather than gradient on edges to find a partition of image domain. They generally have better performances in the presence of weak or discontinuous boundaries than edge-based models. Early popular region-based models tend to rely on intensity homogeneous (roughly constant or smooth) statistics in each of the regions to be segmented. Therefore, they either lack the ability to deal with intensity inhomogeneity like the PC (piecewise constant) model [

To handle intensity inhomogeneity efficiently, some localized region-based models [

In this study, based on the PC model [

The remainder of this paper is organized as follows. Section

In [

Let

The solution of the CV model in fact leads to a piecewise constant segmentation of the original image

In order to improve the performance of the global CV model [

The solution of the RSF model leads to a piecewise smooth approximation of the original image

Given a positive constant

Keeping

Keeping

Like the CV and RSF models, our model is also implemented using an alternative procedure: for each iteration and the corresponding level set function

In the following, we first discuss the properties of

For the sake of simplicity, we state and prove the properties of

Let

(ii) The cases of

(iii) The significance of (

The proof of Theorem

Let

If

If

The proof of Corollary

We call the property in Theorem

Let

This theorem shows that the local force

The following result together with Corollary

Under the assumption of Theorem

If

If

The proofs of Theorem

In this section, we analyze the behavior of our model (

Due to the discrete nature of image, the continuous equation (

Let

If

If

We provide the proof of Corollary

Let

If

If

If

We provide the proof of Theorem

The significance of Theorem

Theorem

In applications, the initial function

If the curve

If the curve

We can also define a zero function as follows:

Next, we prove the fact that, with

If the initial function

We provide the proof of Theorem

By (

In tradition PDE-based methods, a certain diffusion term is usually included in the evolution equation to regularize the evolving function, but which increases the computational cost. Recently, [

The main procedures of the proposed algorithm can be summarized as follows.

Initialize the evolving function

Compute

Evolve the function

Regularize the function

Check whether the evolution is finished. If not, return to step 2.

In this section, we show experimentally that the proposed model not only can provide desirable segmentation results in the presence of intensity inhomogeneity but also allows for more flexible initialization of the contour compared to the RSF and LIF models.

In our numerical experiments, for our model, we choose the parameters as follows:

We will utilize two region overlap metrics to evaluate the performances of the three models quantitatively. The two metrics are the ratio of segmentation error (RSE) [

In the first example (Figures

Segmentations of our model for two real images with

Applications of our model to four images (slug, cell, ventriculus sinister MR, and real plane images). The curve evolution process from the initial contour (in the first column) to the final contour (in the fourth column) is shown in every row for the corresponding image.

Comparisons of three models. Rows 1–3: RSF model, LIF model, and our model with

In Figure

We choose the segmentation results of the RSF and LIF models as baseline foreground regions and then compute DSC values for the corresponding images. The RSE and DSC values for the four real images are given in Table

RSE and DSC for RSF and LIF models and our model.

RSF model | LIF model | |||
---|---|---|---|---|

RSE (%) | DSC (%) | RSE (%) | DSC (%) | |

Figure |
0.27 | 97.17 | 2.99 | 98.91 |

Figure |
2.71 | 96.37 | 0.12 | 99.13 |

Figure |
0.49 | 87.95 | 0.45 | 88.06 |

Figure |
0.16 | 97.66 | 0.76 | 98.91 |

The experimental results shown in Figure

Segmentation results of both models for a hand phantom. Upper row: LIF model. Lower row: our model with

In this study, we propose a new active contour model integrating a local intensity fitting (LIF) energy with an auxiliary global intensity fitting (GIF) energy. The LIF energy is responsible for attracting the contour toward object boundaries and is dominant near object boundaries, while the GIF energy incorporates global image information to improve the robustness to initialization of the contours. The proposed model can efficiently handle intensity inhomogeneity, while allowing for more flexible initialization and maintaining the subpixel accuracy. The utility model has the advantages of allowing for more flexible initialization of the contour and the capability of detecting multiple objects or objects with interior holes or blurred edges. But [

Clearly,

Now, we give the proof of Corollary

By (

Now, we give the proof of Corollary

(i) By Corollary

We prove (

From (

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve this paper. Besides, this work was supported by the Fundamental Research Funds for the Central University, Grant no. (CDJXS10100006).