TY - JOUR
T1 - Optical flow-based transport on image manifolds
AU - Nagaraj, S.
AU - Hegde, C.
AU - Sankaranarayanan, A. C.
AU - Baraniuk, R. G.
N1 - Funding Information:
This work was partially supported by the grants NSF CCF-0431150 , CCF-0728867 , CCF-0926127 , CCF-1117939 , ARO MURI W911NF-09-1-0383 , W911NF-07-1-0185 , DARPA N66001-11-1-4090 , N66001-11-C-4092 , N66001-08-1-2065 , AFOSR FA9550-09-1-0432 , ONR N00014-12-1-0579 , and LLNL B593154 .
PY - 2014/3
Y1 - 2014/3
N2 - An image articulation manifold (IAM) is the collection of images formed by imaging an object that is subject to continuously changing imaging parameters. IAMs arise in a variety of image processing and computer vision applications, where they support a natural low-dimensional embedding of the collection of high-dimensional images. To date IAMs have been studied as embedded submanifolds of Euclidean spaces. Unfortunately, their promise has not been realized in practice, because real world imagery typically contains sharp edges that render IAMs non-differentiable. Moreover, IAMs are also non-isometric to the low-dimensional parameter space under the Euclidean metric. As a result, the standard tools from differential geometry, in particular using linear tangent spaces to transport along the IAM, have limited utility. In this paper, we explore a nonlinear transport operator for IAMs based on the optical flow between images and develop new analytical tools reminiscent of those from differential geometry using the idea of optical flow manifolds (OFMs). We define a new metric for IAMs that satisfies certain local isometry conditions, and we show how to use this metric to develop new tools such as flow fields on IAMs, parallel flow fields, parallel transport, as well as an intuitive notion of curvature. The space of optical flow fields along a path of constant curvature has a natural multiscale structure via a monoid structure on the space of all flow fields along a path. We also develop lower bounds on approximation errors while approximating non-parallel flow fields by parallel flow fields.
AB - An image articulation manifold (IAM) is the collection of images formed by imaging an object that is subject to continuously changing imaging parameters. IAMs arise in a variety of image processing and computer vision applications, where they support a natural low-dimensional embedding of the collection of high-dimensional images. To date IAMs have been studied as embedded submanifolds of Euclidean spaces. Unfortunately, their promise has not been realized in practice, because real world imagery typically contains sharp edges that render IAMs non-differentiable. Moreover, IAMs are also non-isometric to the low-dimensional parameter space under the Euclidean metric. As a result, the standard tools from differential geometry, in particular using linear tangent spaces to transport along the IAM, have limited utility. In this paper, we explore a nonlinear transport operator for IAMs based on the optical flow between images and develop new analytical tools reminiscent of those from differential geometry using the idea of optical flow manifolds (OFMs). We define a new metric for IAMs that satisfies certain local isometry conditions, and we show how to use this metric to develop new tools such as flow fields on IAMs, parallel flow fields, parallel transport, as well as an intuitive notion of curvature. The space of optical flow fields along a path of constant curvature has a natural multiscale structure via a monoid structure on the space of all flow fields along a path. We also develop lower bounds on approximation errors while approximating non-parallel flow fields by parallel flow fields.
KW - Image articulation manifolds
KW - Optical flow
KW - Transport operators
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U2 - 10.1016/j.acha.2013.05.003
DO - 10.1016/j.acha.2013.05.003
M3 - Article
AN - SCOPUS:84892954990
VL - 36
SP - 280
EP - 301
JO - Applied and Computational Harmonic Analysis
JF - Applied and Computational Harmonic Analysis
SN - 1063-5203
IS - 2
ER -