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Computationally efficient eigenspace decomposition of correlated images characterized by three parameters

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Abstract

Eigendecomposition is a common technique that is performed on sets of correlated images in a number of pattern recognition applications including object detection and pose estimation. However, many fast eigendecomposition algorithms rely on correlated images that are, at least implicitly, characterized by only one parameter, frequently time, for their computational efficacy. In some applications, e.g., three-dimensional pose estimation, images are correlated along multiple parameters and no natural one-dimensional ordering exists. In this work, a fast eigendecomposition algorithm that exploits the “temporal” correlation within image data sets characterized by one parameter is extended to improve the computational efficiency of computing the eigendecomposition for image data sets characterized by three parameters. The algorithm is implemented and evaluated using three-dimensional pose estimation as an example application. Its accuracy and computational efficiency are compared to that of the original algorithm applied to one-dimensional pose estimation.

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Notes

  1. A colon (:) in an array argument is used here to specify that all entries in the corresponding dimension of that array are considered.

  2. Please note that we use indices such as l, m, and n (by convention), despite the fact that they are used elsewhere to denote other quantities, i.e., m and n are also used to denote the number of pixels and images, respectively. Context should prevent any confusion.

  3. The notation (1:i) here refers to the first i entries in the corresponding dimension of an array.

  4. Note that the image data set under consideration has LMN = 729 images in total. Hence, the corresponding right singular vectors will each have 729 elements. However, only nine of those 729 entries can be used to monitor the variation of images along any one parameter while keeping the other two parameters constant.

  5. A simple way of viewing this is that the images varying along γ n contain the “same” information in all of them, while the images varying along the other two parameters contain slightly different information in consecutive images.

  6. Note that because L = M = N = 9, there are five “real” frequencies (0 through 4) along all three parameters for this NER image data set.

  7. η,τ, and δ must be integers with no common factors.

  8. Due to the physical limitations of the robot, the range for the α l and β m parameters for the real objects was restricted to 60o.

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Acknowledgments

This work was supported in part by the National Imagery and Mapping Agency under contract no. NMA201-00-1-1003, through collaborative participation in the Robotics Consortium sponsored by the US Army Research Laboratory under the Collaborative Technology Alliance Program, Cooperative Agreement DAAD19-01-2-0012, and the Missile Defense Agency under the contract no. HQ0006-05-C-0035. Approved for Public Release 07-MDA-2783 (26 SEPT 07). The US Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation thereon. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the US Government. A preliminary version of portions of this work was presented at the IEEE Southwest Symposium on Image Analysis and Interpretation held at Denver, CO, USA, March 26–28, 2006.

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Authors and Affiliations

  1. Behavioral Recognition Systems, Inc., 2100 West Loop South, 9th Floor, Houston, TX, 77027, USA

    K. Saitwal

  2. Department of Electrical and Computer Engineering, Colorado State University, Fort Collins, CO, 80523-1373, USA

    A. A. Maciejewski

  3. Department of Electrical and Computer Engineering, Florida A & M—Florida State University, Tallahassee, FL, 32310-6046, USA

    R. G. Roberts

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  1. K. Saitwal
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  2. A. A. Maciejewski
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  3. R. G. Roberts
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Correspondence to A. A. Maciejewski.

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Saitwal, K., Maciejewski, A.A. & Roberts, R.G. Computationally efficient eigenspace decomposition of correlated images characterized by three parameters. Pattern Anal Applic 12, 391–406 (2009). https://doi.org/10.1007/s10044-008-0135-9

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