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An active contour model for the segmentation of images with intensity inhomogeneities and bias field estimation.

Huang C, Zeng L - PLoS ONE (2015)

Bottom Line: The proposed model first appeared as a two-phase model and then extended to a multi-phase one.The experimental results demonstrate the advantages of our model in terms of accuracy and insensitivity to the location of the initial contours.In particular, our method has been applied to various synthetic and real images with desirable results.

View Article: PubMed Central - PubMed

Affiliation: Key Laboratory of Optoelectronic Technology and System of the Education Ministry of China, Chongqing University, Chongqing, 400044, China; Engineering Research Center of Industrial Computed Tomography Nondestructive Testing of the Education Ministry of China, Chongqing University, Chongqing, 400044, China.

ABSTRACT
Intensity inhomogeneity causes many difficulties in image segmentation and the understanding of magnetic resonance (MR) images. Bias correction is an important method for addressing the intensity inhomogeneity of MR images before quantitative analysis. In this paper, a modified model is developed for segmenting images with intensity inhomogeneity and estimating the bias field simultaneously. In the modified model, a clustering criterion energy function is defined by considering the difference between the measured image and estimated image in local region. By using this difference in local region, the modified method can obtain accurate segmentation results and an accurate estimation of the bias field. The energy function is incorporated into a level set formulation with a level set regularization term, and the energy minimization is conducted by a level set evolution process. The proposed model first appeared as a two-phase model and then extended to a multi-phase one. The experimental results demonstrate the advantages of our model in terms of accuracy and insensitivity to the location of the initial contours. In particular, our method has been applied to various synthetic and real images with desirable results.

No MeSH data available.


Segmentation of Li’s model.(a), (d) The original image with red initial contours. (b), (e) The final segmentation results of Li’s model. (c), (f) The bias field estimation of Li’s model.
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pone.0120399.g003: Segmentation of Li’s model.(a), (d) The original image with red initial contours. (b), (e) The final segmentation results of Li’s model. (c), (f) The bias field estimation of Li’s model.

Mentions: Let Ωx = {y:∣y−x∣ ≤ ρ} be the neighborhood of x in the image domain with a small radius ρ and Ωx∩Ωi represent the partition of Ωx produced by the i-th partition Ωi of the image. Based on assumption (A1), the value b(y) for all Ωx can be close to b(x); then, in the small region Ωx∩Ωi, the product of the bias field b(y) and image intensity J(y) can be an approximated by b(y)I(x) ≈ b(x)ci according to assumption (A2). By using the K-means clustering method, they considered all the N partitions of image, and defined a local energy function as follows:Ex=∑i=1N∫Ox∩ΩiKσ(x−y)/I(y)−b(x)ci/2dy(3)where Kσ(s) is a weighted function, which can be expressed as a Gaussian kernel function with standard deviation σ:Kσ(s)=12πσe−/s/2/2σ2,/s/<ρ0,otherwise(4)To find an optimal of the entire image domain Ω, an overall energy for all x is defined asE=∫∑i=1N∫ΩiKσ(x−y)/I(y)−b(x)ci/2dydxWhen considering the case in which N = 2 in Li’s model, after introducing the level set function ϕ(x), the overall energy can be written asE=∫∑i=12∫ΩKσ(x−y)/I(y)−b(x)ci/2dyui(ϕ(x))dxwhere u1(ϕ(x)) and u2(ϕ(x)) are the membership functions of each cluster, u1(s) = H(s), and H(s) is the heaviside function defined as (ε > 0), u2(s) = 1−H(s). For fixed ϕ, c1 and c2, the optimal bias field b can be computed by minimizing the local energy Ex in (3) as follows:b=Kσ∗∑i=12ciui(ϕ)Kσ∗∑i=12ci2ui(ϕ)(5)where ‘∗’ denotes the convolution operation. Similarly, the optimal c1 and c2 can be computed byci=∫(Kσ∗b)Iui(ϕ)dx∫(Kσ∗b2)ui(ϕ)dx,i=1,2(6)In this case, the image domain Ω is divided into two regions, Ω1 = {ϕ > 0} (objects) and Ω2 = {ϕ < 0} (background). Because the local image intensity information is embedded into the energy function, Li’s method can address some types of images with intensity inhomogeneity; however, it still has inherent drawbacks. From (5) and (6), the intensity means ci (i = 1, 2) and bias field b are related; thus, the estimation of c1 and c2 are critical for obtaining a better estimation of bias field b. However, when the object intensity is close to the background in the local region, the estimation of the bias field may be inaccurate, and thus, estimating the bias field using only the mean intensity in the local region is not sufficient. As shown in Fig. 2, Li’s method can obtain the correct segmentation (Fig. 2(b)) when the initial contour is located in the inner part of the object (Fig. 2(a)). However, when the initial contours contain both object and background (Fig. 2(d)), Li’s method fails to segment the object (Fig. 2(e)), and the false segmentation result leads to worse estimation of the bias field (Fig. 2(f)). Similar results can also be seen in Fig. 3. In other words, Li’s method may drop into local minimums [34] and is sensitive to the location of the initial contour; thus, the segmentation results and bias field estimation may be inaccurate in some cases.


An active contour model for the segmentation of images with intensity inhomogeneities and bias field estimation.

Huang C, Zeng L - PLoS ONE (2015)

Segmentation of Li’s model.(a), (d) The original image with red initial contours. (b), (e) The final segmentation results of Li’s model. (c), (f) The bias field estimation of Li’s model.
© Copyright Policy
Related In: Results  -  Collection

License
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getmorefigures.php?uid=PMC4383562&req=5

pone.0120399.g003: Segmentation of Li’s model.(a), (d) The original image with red initial contours. (b), (e) The final segmentation results of Li’s model. (c), (f) The bias field estimation of Li’s model.
Mentions: Let Ωx = {y:∣y−x∣ ≤ ρ} be the neighborhood of x in the image domain with a small radius ρ and Ωx∩Ωi represent the partition of Ωx produced by the i-th partition Ωi of the image. Based on assumption (A1), the value b(y) for all Ωx can be close to b(x); then, in the small region Ωx∩Ωi, the product of the bias field b(y) and image intensity J(y) can be an approximated by b(y)I(x) ≈ b(x)ci according to assumption (A2). By using the K-means clustering method, they considered all the N partitions of image, and defined a local energy function as follows:Ex=∑i=1N∫Ox∩ΩiKσ(x−y)/I(y)−b(x)ci/2dy(3)where Kσ(s) is a weighted function, which can be expressed as a Gaussian kernel function with standard deviation σ:Kσ(s)=12πσe−/s/2/2σ2,/s/<ρ0,otherwise(4)To find an optimal of the entire image domain Ω, an overall energy for all x is defined asE=∫∑i=1N∫ΩiKσ(x−y)/I(y)−b(x)ci/2dydxWhen considering the case in which N = 2 in Li’s model, after introducing the level set function ϕ(x), the overall energy can be written asE=∫∑i=12∫ΩKσ(x−y)/I(y)−b(x)ci/2dyui(ϕ(x))dxwhere u1(ϕ(x)) and u2(ϕ(x)) are the membership functions of each cluster, u1(s) = H(s), and H(s) is the heaviside function defined as (ε > 0), u2(s) = 1−H(s). For fixed ϕ, c1 and c2, the optimal bias field b can be computed by minimizing the local energy Ex in (3) as follows:b=Kσ∗∑i=12ciui(ϕ)Kσ∗∑i=12ci2ui(ϕ)(5)where ‘∗’ denotes the convolution operation. Similarly, the optimal c1 and c2 can be computed byci=∫(Kσ∗b)Iui(ϕ)dx∫(Kσ∗b2)ui(ϕ)dx,i=1,2(6)In this case, the image domain Ω is divided into two regions, Ω1 = {ϕ > 0} (objects) and Ω2 = {ϕ < 0} (background). Because the local image intensity information is embedded into the energy function, Li’s method can address some types of images with intensity inhomogeneity; however, it still has inherent drawbacks. From (5) and (6), the intensity means ci (i = 1, 2) and bias field b are related; thus, the estimation of c1 and c2 are critical for obtaining a better estimation of bias field b. However, when the object intensity is close to the background in the local region, the estimation of the bias field may be inaccurate, and thus, estimating the bias field using only the mean intensity in the local region is not sufficient. As shown in Fig. 2, Li’s method can obtain the correct segmentation (Fig. 2(b)) when the initial contour is located in the inner part of the object (Fig. 2(a)). However, when the initial contours contain both object and background (Fig. 2(d)), Li’s method fails to segment the object (Fig. 2(e)), and the false segmentation result leads to worse estimation of the bias field (Fig. 2(f)). Similar results can also be seen in Fig. 3. In other words, Li’s method may drop into local minimums [34] and is sensitive to the location of the initial contour; thus, the segmentation results and bias field estimation may be inaccurate in some cases.

Bottom Line: The proposed model first appeared as a two-phase model and then extended to a multi-phase one.The experimental results demonstrate the advantages of our model in terms of accuracy and insensitivity to the location of the initial contours.In particular, our method has been applied to various synthetic and real images with desirable results.

View Article: PubMed Central - PubMed

Affiliation: Key Laboratory of Optoelectronic Technology and System of the Education Ministry of China, Chongqing University, Chongqing, 400044, China; Engineering Research Center of Industrial Computed Tomography Nondestructive Testing of the Education Ministry of China, Chongqing University, Chongqing, 400044, China.

ABSTRACT
Intensity inhomogeneity causes many difficulties in image segmentation and the understanding of magnetic resonance (MR) images. Bias correction is an important method for addressing the intensity inhomogeneity of MR images before quantitative analysis. In this paper, a modified model is developed for segmenting images with intensity inhomogeneity and estimating the bias field simultaneously. In the modified model, a clustering criterion energy function is defined by considering the difference between the measured image and estimated image in local region. By using this difference in local region, the modified method can obtain accurate segmentation results and an accurate estimation of the bias field. The energy function is incorporated into a level set formulation with a level set regularization term, and the energy minimization is conducted by a level set evolution process. The proposed model first appeared as a two-phase model and then extended to a multi-phase one. The experimental results demonstrate the advantages of our model in terms of accuracy and insensitivity to the location of the initial contours. In particular, our method has been applied to various synthetic and real images with desirable results.

No MeSH data available.