Research reports

Years: 2025 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

Optimal image alignment with random projections of manifolds: algorithm and geometric analysis

by E. Kokiopoulou and D. Kressner and P. Frossard

(Report number 2009-41)

Abstract
This paper addresses image alignment based on random measurements. Image alignment consists in estimating the relative transformation between a query image and a reference image. We consider the specific problem where the query image is not given exactly, but rather provided in a compressed form with linear measurements captured by a vision sensor. According to the theory behind compressed sensing, image alignment can still be performed effectively in this case, provided that the number of measurements is sufficiently large. We cast the alignment problem as a manifold distance minimization problem in the linear subspace defined by the measurements. We then show that, when the reference image is sparsely represented over parametric dictionaries, the corresponding objective function can be decomposed as the difference of two convex functions (DC). Thus the optimization problem becomes a DC program, which in turn can be solved globally optimally by, e.g., a cutting plane method. The quality of the solution is typically affected by the number of random measurements and the condition number of the manifold that describes the transformations of the reference image. We show that the manifold condition number remains bounded in our image alignment problem, which means that the relative transformation between two images can be determined optimally in a reduced subspace.

Keywords:

@Techreport{KKF09_422,
  author = {E. Kokiopoulou and D. Kressner and P. Frossard},
  title = {Optimal image alignment with random projections of manifolds: algorithm and geometric analysis},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2009-41},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2009/2009-41.pdf },
  year = {2009}
}

Disclaimer
© Copyright for documents on this server remains with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use. The administrators respectfully request that authors inform them when any paper is published to avoid copyright infringement. Note that unauthorised copying of copyright material is illegal and may lead to prosecution. Neither the administrators nor the Seminar for Applied Mathematics (SAM) accept any liability in this respect. The most recent version of a SAM report may differ in formatting and style from published journal version. Do reference the published version if possible (see SAM Publications).