Exploiting the Circulant Structure of Tracking-by-Detection with Kernels

  • João F. Henriques
  • Rui Caseiro
  • Pedro Martins
  • Jorge Batista
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7575)


Recent years have seen greater interest in the use of discriminative classifiers in tracking systems, owing to their success in object detection. They are trained online with samples collected during tracking. Unfortunately, the potentially large number of samples becomes a computational burden, which directly conflicts with real-time requirements. On the other hand, limiting the samples may sacrifice performance.

Interestingly, we observed that, as we add more and more samples, the problem acquires circulant structure. Using the well-established theory of Circulant matrices, we provide a link to Fourier analysis that opens up the possibility of extremely fast learning and detection with the Fast Fourier Transform. This can be done in the dual space of kernel machines as fast as with linear classifiers. We derive closed-form solutions for training and detection with several types of kernels, including the popular Gaussian and polynomial kernels. The resulting tracker achieves performance competitive with the state-of-the-art, can be implemented with only a few lines of code and runs at hundreds of frames-per-second. MATLAB code is provided in the paper (see Algorithm 1).


Support Vector Machine Fast Fourier Transform Kernel Matrix Polynomial Kernel Fourier Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • João F. Henriques
    • 1
  • Rui Caseiro
    • 1
  • Pedro Martins
    • 1
  • Jorge Batista
    • 1
  1. 1.Institute of Systems and RoboticsUniversity of CoimbraPortugal

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