Approximate thin plate spline mappings
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Approximate thin plate spline mappings. / Donato, Gianluca; Belongie, Serge.
In: Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2002, p. 21-31.Research output: Contribution to journal › Conference article › Research › peer-review
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TY - GEN
T1 - Approximate thin plate spline mappings
AU - Donato, Gianluca
AU - Belongie, Serge
N1 - Publisher Copyright: © Springer-Verlag Berlin Heidelberg 2002.
PY - 2002
Y1 - 2002
N2 - The thin plate spline (TPS) is an effective tool for modeling coordinate transformations that has been applied successfully in several computer vision applications. Unfortunately the solution requires the inversion of a p×p matrix, where p is the number of points in the data set, thus making it impractical for large scale applications. As it turns out, a surprisingly good approximate solution is often possible using only a small subset of corresponding points. We begin by discussing the obvious approach of using the subsampled set to estimate a transformation that is then applied to all the points, and we show the drawbacks of this method. We then proceed to borrow a technique from the machine learning community for function approximation using radial basis functions (RBFs) and adapt it to the task at hand. Using this method, we demonstrate a significant improvement over the naive method. One drawback of this method, however, is that is does not allow for principal warp analysis, a technique for studying shape deformations introduced by Bookstein based on the eigenvectors of the p × p bending energy matrix. To address this, we describe a third approximation method based on a classic matrix completion technique that allows for principal warp analysis as a by-product. By means of experiments on real and synthetic data, we demonstrate the pros and cons of these different approximations so as to allow the reader to make an informed decision suited to his or her application.
AB - The thin plate spline (TPS) is an effective tool for modeling coordinate transformations that has been applied successfully in several computer vision applications. Unfortunately the solution requires the inversion of a p×p matrix, where p is the number of points in the data set, thus making it impractical for large scale applications. As it turns out, a surprisingly good approximate solution is often possible using only a small subset of corresponding points. We begin by discussing the obvious approach of using the subsampled set to estimate a transformation that is then applied to all the points, and we show the drawbacks of this method. We then proceed to borrow a technique from the machine learning community for function approximation using radial basis functions (RBFs) and adapt it to the task at hand. Using this method, we demonstrate a significant improvement over the naive method. One drawback of this method, however, is that is does not allow for principal warp analysis, a technique for studying shape deformations introduced by Bookstein based on the eigenvectors of the p × p bending energy matrix. To address this, we describe a third approximation method based on a classic matrix completion technique that allows for principal warp analysis as a by-product. By means of experiments on real and synthetic data, we demonstrate the pros and cons of these different approximations so as to allow the reader to make an informed decision suited to his or her application.
UR - http://www.scopus.com/inward/record.url?scp=84949998361&partnerID=8YFLogxK
U2 - 10.1007/3-540-47977-5_2
DO - 10.1007/3-540-47977-5_2
M3 - Conference article
AN - SCOPUS:84949998361
SP - 21
EP - 31
JO - Lecture Notes in Computer Science
JF - Lecture Notes in Computer Science
SN - 0302-9743
T2 - 7th European Conference on Computer Vision, ECCV 2002
Y2 - 28 May 2002 through 31 May 2002
ER -
ID: 302056623