Abstract
Manifold learning has been successfully applied to a variety of medical imaging problems. Its use in real-time applications requires fast projection onto the low-dimensional space. To this end, out-of-sample extensions are applied by constructing an interpolation function that maps from the input space to the low-dimensional manifold. Commonly used approaches such as the Nyström extension and kernel ridge regression require using all training points. We propose an interpolation function that only depends on a small subset of the input training data. Consequently, in the testing phase each new point only needs to be compared against a small number of input training data in order to project the point onto the low-dimensional space. We interpret our method as an out-of-sample extension that approximates kernel ridge regression. Our method involves solving a simple convex optimization problem and has the attractive property of guaranteeing an upper bound on the approximation error, which is crucial for medical applications. Tuning this error bound controls the sparsity of the resulting interpolation function. We illustrate our method in two clinical applications that require fast mapping of input images onto a low-dimensional space.
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References
Aronszajn, N.: Theory of reproducing kernels. Trans. AMS (1950)
Bach, F.R., Jenatton, R., Mairal, J., Obozinski, G.: Optimization with sparsity-inducing penalties. Foundations and Trends in Machine Learning (2012)
Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM Journal on Imaging Sciences (2009)
Belkin, M., Niyogi, P.: Laplacian eigenmaps and spectral techniques for embedding and clustering. In: NIPS (2002)
Bengio, Y., Paiement, J.F., Vincent, P., Delalleau, O., Roux, N.L., Ouimet, M.: Out-of-sample extensions for LLE, Isomap, MDS, eigenmaps, and spectral clustering. In: NIPS (2004)
Bhatia, K.K., Rao, A., Price, A.N., Wolz, R., Hajnal, J., Rueckert, D.: Hierarchical manifold learning. In: Ayache, N., Delingette, H., Golland, P., Mori, K. (eds.) MICCAI 2012, Part I. LNCS, vol. 7510, pp. 512–519. Springer, Heidelberg (2012)
Cawley, G.C., Talbot, N.L.C.: Reduced rank kernel ridge regression. Neural Processing Letters (2002)
Donoho, D.L., Grimes, C.: Hessian eigenmaps: New locally linear embedding techniques for high-dimensional data. PNAS (2003)
Drucker, H., Burges, C.J.C., Kaufman, L., Smola, A.J., Vapnik, V.: Support vector regression machines. In: NIPS (1997)
Georg, M., Souvenir, R., Hope, A., Pless, R.: Manifold learning for 4d ct reconstruction of the lung. In: CVPR Workshops (2008)
Gerber, S., Tasdizen, T., Joshi, S., Whitaker, R.: On the manifold structure of the space of brain images. In: Yang, G.-Z., Hawkes, D., Rueckert, D., Noble, A., Taylor, C. (eds.) MICCAI 2009, Part I. LNCS, vol. 5761, pp. 305–312. Springer, Heidelberg (2009)
Hamm, J., Davatzikos, C., Verma, R.: Efficient large deformation registration via geodesics on a learned manifold of images. In: Yang, G.-Z., Hawkes, D., Rueckert, D., Noble, A., Taylor, C. (eds.) MICCAI 2009, Part I. LNCS, vol. 5761, pp. 680–687. Springer, Heidelberg (2009)
van der Maaten, L.J.P., Postma, E.O., van den Herik, H.J.: Dimensionality reduction: A comparative review. Tilburg University Technical Report (2008)
Natarajan, B.K.: Sparse Approximate Solutions to Linear Systems. SIAM J. Comput. (1995)
Rohde, G.K., Wang, W., Peng, T., Murphy, R.F.: Deformation-based nonlinear dimension reduction: Applications to nuclear morphometry. In: ISBI (2008)
Rohlfing, T., Maurer Jr., C.R., O’Dell, W.G., Zhong, J.: Modeling liver motion and deformation during the respiratory cycle using intensity-based free-form registration of gated MR images. In: Medical Imaging: Visualization, Display, and Image-Guided Procedures (2001)
Roweis, S.T., Saul, L.K.: Nonlinear dimensionality reduction by locally linear embedding. Science (2000)
Saunders, C., Gammerman, A., Vovk, V.: Ridge regression learning algorithm in dual variables. In: ICML (1998)
Schölkopf, B., Smola, A.J.: Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. MIT Press (2001)
Suzuki, K., Zhang, J., Xu, J.: Massive-training artificial neural network coupled with laplacian-eigenfunction-based dimensionality reduction for computer-aided detection of polyps in ct colonography. IEEE TMI (2010)
Tenenbaum, J.B., de Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science (2000)
Wachinger, C., Mateus, D., Keil, A., Navab, N.: Manifold learning for patient position detection in MRI. In: ISBI (2010)
Wachinger, C., Yigitsoy, M., Navab, N.: Manifold learning for image-based breathing gating with application to 4D ultrasound. In: Jiang, T., Navab, N., Pluim, J.P.W., Viergever, M.A. (eds.) MICCAI 2010, Part II. LNCS, vol. 6362, pp. 26–33. Springer, Heidelberg (2010)
Zhang, Q., Souvenir, R., Pless, R.: On Manifold Structure of Cardiac MRI Data: Application to Segmentation. In: CVPR (2006)
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Chen, G.H., Wachinger, C., Golland, P. (2013). Sparse Projections of Medical Images onto Manifolds. In: Gee, J.C., Joshi, S., Pohl, K.M., Wells, W.M., Zöllei, L. (eds) Information Processing in Medical Imaging. IPMI 2013. Lecture Notes in Computer Science, vol 7917. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38868-2_25
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DOI: https://doi.org/10.1007/978-3-642-38868-2_25
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