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A novel image segmentation model with an edge weighting function

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Abstract

A variational model for image segmentation consists of a data term and a regularization term. Usually, the data term is chosen as squared \(\text{ L }_{2}\) norm, and the regularization term is determined by the prior assumption. In this paper, we present a novel model in the framework of MAP (maximum a posteriori). A new iteratively reweighted \(\text{ L }_{2}\) norm is used in the data term, which shares the advantages of \(\text{ L }_{2}\) and mixed \(\text{ L }_{21}\) norm. An edge weighting function is addressed in the regularization term, which enforces the ability to reduce the outlier effects and preserve edges. An improved region-based graph cuts algorithm is proposed to solve this model efficiently. Numerical experiments show our method can get better segmentation results, especially in terms of removing outliers and preserving edges.

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Notes

  1. http://www.biomecardio.com/matlab/medfilt3.html

Abbreviations

\(P\) :

Set of image pixels

\({{\varvec{I}}}(p)\) :

The feature vector of pixel \(p\)

\(I\) :

Feature matrix composed of the feature vectors of all pixels

\(l_p\) :

The label of pixel \(p\)

\(l\) :

Labeling function of the image

\({{\varvec{f}}}(i)\) :

The feature vector of the subregion in which all pixels are labeled with \(i\)

\(F=\{{{\varvec{f}}}(i),\quad i=1,2,\ldots ,n\}\) :

Union of the region features

\(\Pr \) :

Probability

\(\omega _p\) :

The weight of pixel \(p\) in the weighted \(\text{ L }_{2}\) norm used in the date term

\(W_{p,q}\) :

The penalty weight between pixel \(p\) and its neighboring pixel \(q\) in the regularization term

\(E_\mathrm{d}\) :

Energy functional corresponding to the data term and \(E_\mathrm{r}\) is that corresponding to the regularization term

\(\text{ num }_i\) :

Number of pixels within the subregion in which all pixels are labeled with \(i\)

\(I_L (p)\) :

Lightness of pixel \(p\)

\(I^\mathrm{CHF}\) :

Output of the first-order band-pass filter circular harmonic function (CHF) when \(I\) is applied as the input

\(\beta \) :

Threshold to distinguish whether a point is an edge point

\(\alpha \) :

Threshold to distinguish whether the neighborhood of a pixel contains edge points

\(\varphi _{pq}\) :

Direction of clique \(({p,q})\)

\(\arg (I^\mathrm{CHF})\) :

Argument of the complex value \(I^\mathrm{CHF}\)

\(v_m ({m=1,2,\ldots ,L})\) :

Small regions after the initial segmentation to the image, where \(L\) is the total number of the small regions

\(V=\{{v_{m,h} }|m\in \{{1,2,\ldots ,L}\}, h\in \{ {1,2,\ldots ,n-1}\}\}\cup \{ s\}\cup \{ t \}\) :

Set of nodes of the graph

\(n\) :

Number of subregions the image is to be partitioned into

\(s\) :

Source of the graph and \(t\) is the sink

\(\varvec{\varepsilon }\) :

Set of the edges of the graph

\(\varvec{\varepsilon }_D\) :

Set of the edges corresponding to the data term and \(\varvec{\varepsilon }_R \) corresponds to the regularization term

\(({a,b})\) :

Edge between the nodes \(a\) and \(b\)

\(c({a,b})\) :

Weight on the edge \(({a,b})\)

\(\text{ Num }_m\) :

Number of pixels in the small region \(v_m\)

\(M\cdot N\) :

Size of the original image

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Acknowledgments

This paper is supported by the National Natural Science Foundation of China (61271294), National Natural Science Foundation of China (60872138), National Youth Science Foundation of China (61105011), and National Natural Science Foundation of China (11101292).

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Authors and Affiliations

  1. School of Science, Xidian University, Xi’an, 710071, China

    Wen Juan Zhang, Xiang Chu Feng & Yu Han

  2. School of Science, Xi’an Technological University, Xi’an, 710012, China

    Wen Juan Zhang

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  1. Wen Juan Zhang
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  2. Xiang Chu Feng
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Correspondence to Wen Juan Zhang.

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Zhang, W.J., Feng, X.C. & Han, Y. A novel image segmentation model with an edge weighting function. SIViP 8, 121–132 (2014). https://doi.org/10.1007/s11760-013-0495-5

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