Abstract-The core component of most modern trackers is a discriminative classifier, tasked with distinguishing between the target and the surrounding environment. To cope with natural image changes, this classifier is typically trained with translated and scaled sample patches. Such sets of samples are riddled with redundancies -any overlapping pixels are constrained to be the same. Based on this simple observation, we propose an analytic model for datasets of thousands of translated patches. By showing that the resulting data matrix is circulant, we can diagonalize it with the Discrete Fourier Transform, reducing both storage and computation by several orders of magnitude. Interestingly, for linear regression our formulation is equivalent to a correlation filter, used by some of the fastest competitive trackers. For kernel regression, however, we derive a new Kernelized Correlation Filter (KCF), that unlike other kernel algorithms has the exact same complexity as its linear counterpart. Building on it, we also propose a fast multi-channel extension of linear correlation filters, via a linear kernel, which we call Dual Correlation Filter (DCF). Both KCF and DCF outperform top-ranking trackers such as Struck or TLD on a 50 videos benchmark, despite running at hundreds of frames-per-second, and being implemented in a few lines of code (Algorithm 1). To encourage further developments, our tracking framework was made open-source.

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).

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