| Summary: | Image articulation manifolds (IAMs) arise in a wide variety of contexts in image processing and computer vision applications. IAMs are a natural nonlinear model for image ensembles generated by the variation of imaging parameters (scale, pose, lighting etc.). In the past, IAMs have been studied as being embedded submanifolds of higher dimensional Euclidean spaces. However, this view suffers from two major defects: lack of a meaningful metric and reliance on linear transport operators via tangent vectors. Recent work in the area indicates the existence of better nonlinear transport operators for IAMs, with optical flow based transport being a prime candidate. In this thesis, we provide a detailed theoretical analysis of optical flow based transport on IAMs. In particular, we develop new analytical tools reminiscent of differential geometry to handle the apriori data driven nature of IAMs using the notion of optical flow manifolds (OFMs). We define an appropriate metric on the IAM via a metric on the corresponding OFMs that satisfy certain local isometry conditions and we show how to use this new metric to develop a host of mathematical tools such as optical flow fields on the IAM, parallel fields and parallel transport as well as an intuitive notion of "optical curvature". We show that the space of optical flow fields along a path of constant optical curvature has a natural multiscale structure. We also consider the question of approximating non-parallel flow fields by parallel flow fields.
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