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Saliency detection has been an intuitive way to provide useful cues for object detection and segmentation, as desired for many vision and graphics applications. In this paper, we provided a robust method for salient object detection and segmentation. Other than using various pixel-level contrast definitions, we exploited global image structures and proposed a new geodesic method dedicated for salient object detection. In the proposed approach, a new geodesic scheme, namely geodesic tunneling is proposed to tackle with textures and local chaotic structures. With our new geodesic approach, a geodesic saliency map is estimated in correspondence to spatial structures in an image. Experimental evaluation on a salient object benchmark dataset validated that our algorithm consistently outperformed a number of the state-of-art saliency methods, yielding higher precision and better recall rates. With the robust saliency estimation, we also present an unsupervised hierarchical salient object cut scheme simply using adaptive saliency thresholding, which attained the highest score in our F-measure test. We also applied our geodesic cut scheme to a number of image editing tasks as demonstrated in additional experiments.