Image segmentation using graph cuts have become very popular in the last years. These methods are computationally expensive, even with hard constraints (seed pixels). We present a solution that runs in time proportional to the number of pixels. Our method computes an ordered region growing from a set of seeds inside the object, where the propagation order of each pixel is proportional to the cost of an optimum path in the image graph from the seed set to that pixel. Each pixel defines a region which includes it and all pixels with lower propagation order. The boundary of each region is a possible cut boundary, whose cut measure is also computed and assigned to the corresponding pixel on-the-fly. The object is obtained by selecting the pixel with minimum-cut measure and all pixels within its respective cut boundary. Approaches for graph-cut segmentation usually assume that the desired cut is a global minimum. We show that this can be only verified within a reduced search space under certain hard constraints. We present and evaluate our method with three cut measures: normalized cut, mean cut and an energy function. © Springer-Verlag Berlin Heidelberg 2006.4179 LNCS 138149Wu, Z., Leahy, R., An optimal graph theoretic approach to data clustering: Theory and its applications to image segmentation (1993) IEEE Trans. on Pattern Analysis and Machine Intelligence, 15, pp. 1101-1113Cox, I.J., Rao, S.B., Zhong, Y., Ratio regions: A technique for image segmentation (1996) Intl. Conf. on Computer Vision and Pattern Recognition (CVPR), pp. 557-564Wang, S., Siskind, J.M., Image segmentation with minimum mean cut (2001) Intl. Conf. on Computer Vision (ICCV), 1, pp. 517-525Sarkar, S., Boyer, K.L., Quantitative measures of change based on feature organization: Eigenvalues and eigenvectors (1996) Intl. Conf. on Computer Vision and Pattern Recognition (CVPR), pp. 478-483Shi, J., Malik, J., Normalized cuts and image segmentation (2000) IEEE Trans. on Pattern Analysis and Machine Intelligence, 22, pp. 888-905Wang, S., Sinkind, J.M., Image segmentation with ratio cut (2003) IEEE Trans. on Pattern Analysis and Machine Intelligence, 25, pp. 675-690Yuri, Y., Boykov, M.P.J., Interactive graph cuts for optimal boundary & region segmentation of objects in N-D images (2001) Intl. Conf. on Computer Vision (ICCV), 1, pp. 105-112Boykov, Y., Kolmogorov, V., An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision (2004) IEEE Trans. on Pattern Analysis and Machine Intelligence, 26, pp. 1124-1137Kolmogorov, V., Zabih, R., What energy functions can be minimized via graph cuts (2004) IEEE Trans. on Pattern Analysis and Machine Intelligence, 26, pp. 147-159Fowlkes, C., Belongie, S., Malik, J., Efficient spatiotemporal grouping using the nyström method (2001) Intl. Conf. on Computer Vision and Pattern Recognition (CVPR), pp. 231-238Carballido-Gamio, J., Belongie, S.J., Majumdar, S., Normalized cuts in 3D for spinal MRI segmentation (2004) IEEE Trans. on Medical Imaging, 23, pp. 36-44Ford, L., Fulkerson, D., (1962) Flows in Networks, , Princeton University PressGreig, D., Porteous, B., Seheult, A., Exact maximum a posteriori estimation for binary images (1989) J. Royal Statistical Society, Series B, 51, pp. 271-279Kohli, P., Torr, P.H., Efficiently solving dynamic markov random fields using graph cuts (2005) Intl. Conf. on Computer Vision (ICCV), 2, pp. 922-929Falcão, A., Stolfi, J., Lotufo, R., The image foresting transform: Theory, algorithms, and applications (2004) IEEE Trans. on Pattern Analysis and Machine Intelligence, 26, pp. 19-29Falcão, A., Udupa, J., Miyazawa, F., An ultra-fast user-steered image segmentation paradigm: Live-wire-on-the-fly (2000) IEEE Trans. on Medical Imaging, 19, pp. 55-62Leitao, H., Stolfi, J., A multiscale method for the reassembly of two-dimensional fragmented objects (2002) IEEE Trans. on Pattern Analysis and Machine Intelligence, 24, pp. 1239-1251Falcão, A., Bergo, F., Interactive volume segmentation with differential image foresting transforms (2004) IEEE Trans. on Medical Imaging, 23, pp. 1100-1108