Abstract
Estimating smooth image warps from landmarks is an important problem in computer vision and medical image analysis. The standard paradigm is to find the model parameters by minimizing a compound energy including a data term and a smoother, balanced by a ‘smoothing parameter’ that is usually fixed by trial and error.
We point out that warp estimation is an instance of the general supervised machine learning problem of fitting a flexible model to data, and propose to learn the smoothing parameter while estimating the warp. The leading idea is to depart from the usual paradigm of minimizing the energy to the one of maximizing the predictivity of the warp, i.e. its ability to do well on the entire image, rather than only on the given landmarks. We use cross-validation to measure predictivity, and propose a complete framework to solve for the desired warp. We point out that the well-known non-iterative closed-form for the leave-one-out cross-validation score is actually a good approximation to the true score and show that it extends to the warp estimation problem by replacing the usual vector two-norm by the matrix Frobenius norm. Experimental results on real data show that the procedure selects sensible smoothing parameters, very close to user selected ones.
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References
Allen, D.M.: The relationship between variable selection and data augmentation and a method for prediction. Technometrics 16, 125–127 (1974)
Bishop, C.M.: Neural Networks for Pattern Recognition. Oxford University Press, Oxford (1995)
Bookstein, F.L.: Principal warps: thin-plate splines and the decomposition of deformations. IEEE Trans. Pattern Anal. Mach. Intell. 11(6), 567–585 (1989)
Bro-Nielsen, M., Gramkow, C.: Fast fluid registration of medical images. In: Visualization in Biomedical Imaging (1996)
Burrage, K., Williams, A., Erhel, J., Pohl, B.: The implementation of a generalized cross validation algorithm using deflation techniques for linear systems. Technical report, Seminar fur Angewandte Mathematik, July 1994
Christensen, G.E., He, J.: Consistent nonlinear elastic image registration. In: Workshop on Mathematical Methods in Biomedical Image Analysis (2001)
de Bruijne, M., Lund, M.T., Tankó, L.B., Pettersen, P.C., Nielsen, M.: Quantitative vertebral morphometry using neighbor-conditional shape models. Med. Image Anal. 11(5), 503–512 (2007)
Duchon, J.: Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces. RAIRO Anal. Numér. 10, 5–12 (1976)
Fischler, M.A., Bolles, R.C.: Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography. Comput. Vis. Graph. Image Process. 24(6), 381–395 (1981)
Fornefett, M., Rohr, K., Stiehl, H.S.: Radial basis functions with compact support for elastic registration of medical images. Image Vis. Comput. 19(1), 87–96 (2001)
Genant, H., Wu, C., van Kuijk, C., Nevitt, M.: Vertebral fracture assessment using a semiquantitative technique. J. Bone Miner Res. 8(9), 1137–1148 (1993)
Golub, G.H., von Matt, U.: Generalized cross-validation for large-scale problems. J. Comput. Graph. Stat. 6(1), 1–34 (1997)
Hawkins, D.M., Yin, X.: A faster algorithm for ridge regression of reduced rank data. Comput. Stat. Data Anal. 40(2), 253–262 (2002)
Kanatani, K.: Geometric information criterion for model selection. Int. J. Comput. Vis. 26(3), 171–189 (1998)
Maintz, J.B.A., Viergever, M.A.: A survey of medical image registration. Med. Image Anal. 2(1), 1–36 (1998)
Marsland, S., Twining, C.J., Taylor, C.J.: A minimum description length objective function for groupwise non-rigid image registration. In: Image and Vision Computing (2007)
Maybank, S., Sturm, P.: MDL, collineations and the fundamental matrix. In: British Machine Vision Conference (1999)
Modersitzki, J.: Numerical Methods for Image Registration. Oxford Science, Oxford (2004)
Nielsen, M., Johansen, P.: A PDE solution of Brownian warping. In: European Conference on Computer Vision (2004)
Nielsen, M., Johansen, P., Jackson, A.D., Lautrup, B.: Brownian warps: a least committed prior for non-rigid registration. In: Medical Image Computing and Computer-Assisted Intervention (2002)
Pilet, J., Lepetit, V., Fua, P.: Fast non-rigid surface detection, registration and realistic augmentation. Int. J. Comput. Vis. 76(2), 109–122 (2008)
Poggio, T., Rifkin, R., Mukherjee, S., Niyogi, P.: General conditions for predictivity in learning theory. Nature 458, 419–422 (2004)
Rifkin, R.M., Lippert, R.A.: Notes on regularized least squares. Technical Report MIT-CSAIL-TR-2007-025, MIT, May 2007
Rueckert, D., Sonoda, L.I., Hayes, C., Hill, D.L.G., Leach, M.O., Hawkes, D.J.: Nonrigid registration using free-form deformations: application to breast MR images. IEEE Trans. Med. Imaging 18(8), 712–721 (1999)
Szeliski, R., Coughlan, J.: Spline-based image registration. Int. J. Comput. Vis. 22(3), 199–218 (1997)
Torr, P.H.S.: Bayesian model estimation and selection for epipolar geometry and generic manifold fitting. Int. J. Comput. Vis. 50(1), 27–45 (2002)
Wahba, G.: Splines Models for Observational Data. SIAM, Philadelphia (1990)
Wahba, G., Wold, S.: A completely automatic French curve: fitting spline functions by cross-validation. Commun. Stat. 4, 1–17 (1975)
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Bartoli, A. Maximizing the Predictivity of Smooth Deformable Image Warps through Cross-Validation. J Math Imaging Vis 31, 133–145 (2008). https://doi.org/10.1007/s10851-007-0062-1
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DOI: https://doi.org/10.1007/s10851-007-0062-1