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Image segmentation via $ L_1 $ Monge-Kantorovich problem

  • * Corresponding author: Wuchen Li

    * Corresponding author: Wuchen Li 
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  • This paper provides a fast approach to apply the Earth Mover's Distance (EMD) (a.k.a optimal transport, Wasserstein distance) for supervised and unsupervised image segmentation. The model globally incorporates the transportation costs (original Monge-Kantorovich type) among histograms of multiple dimensional features, e.g. gray intensity and texture in image's foreground and background. The computational complexity is often high for the EMD between two histograms on Euclidean spaces with dimensions larger than one. We overcome this computational difficulty by rewriting the model into a $ L_1 $ type minimization with the linear dimension of feature space. We then apply a fast algorithm based on the primal-dual method. Compare to several state-of-the-art EMD models, the experimental results based on image data sets demonstrate that the proposed method has superior performance in terms of the accuracy and the stability of the image segmentation.

    Mathematics Subject Classification: Primary: 68Uxx.


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  • Figure 1.  Overall framework of the proposed method

    Figure 2.  The corresponding features map obtained from the test image. (a) shows the intensity features map. (b) is the texture features map

    Figure 3.  Exemplar regions with bounding boxes of the test images. The blue bounding box is the foreground and the red bounding boxes are the background

    Figure 4.  Dynamical segmentation map. (a) shows the original image with bounding boxes. (b) is the corresponding $ 2D $ histogram contour of the blue bounding box. (c) is the evolution segmentation process. (d) illustrates the corresponding movement of $ 2D $ contours

    Figure 5.  Supervised model with comparison to LHBS [31] model. Row 1st shows the input images with boundary boxes. Row 2nd shows the LHBS model. Row 3rd shows the proposed method based on $ 2D $ feature histograms

    Figure 6.  Some segmentation examples of $ 3D $ histogram (RGB) features

    Figure 7.  The left figure shows that how we select the reference measure in each step. It is chosen as the red one, i.e. the delta measure supported at the median point of the blue histogram. The right figure demonstrates that the energy function decreases during these iterative steps

    Figure 8.  The first row shows the unsupervised model with $ 1D $ gray-intensity features, and the second row shows the unsupervised model with $ 1D $ texture features

    Figure 9.  Unsupervised model with $ 2D $ features (the gray-intensity and the texture)

    Figure 10.  Unsupervised model with comparison to the LHBS [31] model. The 1st row shows the original images. The 2nd row shows the segmentation results achieved by the LHBS model. The 3rd row shows the proposed method segmentation results.The 4th row shows the binary results of the LHBS model. The 5th row shows the binary results of proposed method. The 6th row shows the ground truth

    Figure 11.  The segmentation accuracy tested on the Microsoft GrabCut database. The magenta contour, and green contour are the segmentation accuracy of the LHBS[31] model, and our method, respectively

    Figure 12.  Some comparison examples on Berkeley segmentation data set. Top to bottom: test images, results of the NLAC [19] model, the LHBS [31] model, and our method, respectively

    Figure 13.  The Jaccard values of segmentation results on the 50 Berkeley data set images. The magenta contour, the green contour, and the blue contour are the Jaccard values of the proposed method, the LHBS [31] model, and the NLAC [19] model, respectively

    Figure 14.  The Hausdorff distances of segmentation results on the 50 Berkeley data set images. The magenta contour, the green contour, and the blue contour are the Hausdorff distances of the proposed method, the LHBS [31] model, and the NLAC [19] model, respectively

    Figure 15.  The average Jaccard index values of the LHBS model, the NLAC model, and our method, respectively

    Figure 16.  The average Hausdorff distances of the LHBS model, the NLAC model, and our method, respectively

    Table 1.  The JI values with different error rates of the cheetah image in Figure. 4

    Error rate $ 5\% $ $ 10\% $ $ 15\% $ $ 20\% $ $ 25\% $ $ 30\% $
    JI value 0.9273 0.8856 0.8365 0.7754 0.7146 0.6543
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