Adaptation of Multiscale Function Extension to Inexact Matching: Application to the Mapping of Individuals to a Learnt Manifold

This paper targets the specific issue of out-of-sample interpolation when mapping individuals to a learnt manifold. This process involves two successive interpolations, which we formulate by means of kernel functions: from the ambient space to the coordinates space parametrizing the manifold and reciprocally. We combine two existing interpolation schemes: (i) inexact matching, to take into account the data dispersion around the manifold, and (ii) a multiscale strategy, to overcome single kernel scale limitations. Experiments involve synthetic data, and real data from 108 subjects, representing myocardial motion patterns used for the comparison of individuals to both normality and to a given abnormal pattern, whose manifold representation has been learnt previously.

[1]  Hongyuan Zha,et al.  Adaptive Manifold Learning , 2004, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[2]  Hans Wackernagel,et al.  Multivariate Geostatistics: An Introduction with Applications , 1996 .

[3]  François-Xavier Vialard,et al.  Mixture of Kernels and Iterated Semidirect Product of Diffeomorphisms Groups , 2011, Multiscale Model. Simul..

[4]  Piet Claus,et al.  Toward understanding response to cardiac resynchronization therapy: left ventricular dyssynchrony is only one of multiple mechanisms. , 2009, European heart journal.

[5]  Ronald R. Coifman,et al.  Multiscale data sampling and function extension , 2013 .

[6]  J. Tenenbaum,et al.  A global geometric framework for nonlinear dimensionality reduction. , 2000, Science.

[7]  Alejandro F. Frangi,et al.  Temporal diffeomorphic free-form deformation: Application to motion and strain estimation from 3D echocardiography , 2012, Medical Image Anal..

[8]  Renaud Keriven,et al.  Projection onto a Shape Manifold for Image Segmentation with Prior , 2007, 2007 IEEE International Conference on Image Processing.

[9]  Ivor W. Tsang,et al.  The pre-image problem in kernel methods , 2003, IEEE Transactions on Neural Networks.

[10]  Pascal Vincent,et al.  Unsupervised Feature Learning and Deep Learning: A Review and New Perspectives , 2012, ArXiv.

[11]  Mikhail Belkin,et al.  Manifold Regularization: A Geometric Framework for Learning from Labeled and Unlabeled Examples , 2006, J. Mach. Learn. Res..

[12]  R. Coifman,et al.  Geometric harmonics: A novel tool for multiscale out-of-sample extension of empirical functions , 2006 .

[13]  Alejandro F. Frangi,et al.  Constrained manifold learning for the characterization of pathological deviations from normality , 2012, Medical Image Anal..

[14]  Alejandro F. Frangi,et al.  A spatiotemporal statistical atlas of motion for the quantification of abnormal myocardial tissue velocities , 2011, Medical Image Anal..

[15]  Pascal Vincent,et al.  Representation Learning: A Review and New Perspectives , 2012, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[16]  Ronald R. Coifman,et al.  Heterogeneous Datasets Representation and Learning using Diffusion Maps and Laplacian Pyramids , 2012, SDM.

[17]  Ross T. Whitaker,et al.  Manifold modeling for brain population analysis , 2010, Medical Image Anal..

[18]  Nicolas Le Roux,et al.  Out-of-Sample Extensions for LLE, Isomap, MDS, Eigenmaps, and Spectral Clustering , 2003, NIPS.