  Image segmentation is a fundamental and challenging task in image processing
and computer vision. The color image segmentation is attracting more attention
due to the color image provides more information than the gray image. In this
paper, we propose a variational model based on a convex K-means approach to
segment color images. The proposed variational method uses a combination of
$l_1$ and $l_2$ regularizers to maintain edge information of objects in images
while overcoming the staircase effect. Meanwhile, our one-stage strategy is an
improved version based on the smoothing and thresholding strategy, which
contributes to improving the accuracy of segmentation. The proposed method
performs the following steps. First, we specify the color set which can be
determined by human or the K-means method. Second, we use a variational model
to obtain the most appropriate color for each pixel from the color set via
convex relaxation and lifting. The Chambolle-Pock algorithm and simplex
projection are applied to solve the variational model effectively. Experimental
results and comparison analysis demonstrate the effectiveness and robustness of
our method.
