Accounting for Aliasing in Correlation Filters : Zero-Aliasing and Partial-Aliasing Correlation Filters

Correlation filters (CFs) are well established and useful tools for a variety of tasks in signal processing and pattern recognition, including automatic target recognition and tracking, biometrics, landmark detection, and human action recognition. Traditionally, CFs have been designed and implemented efficiently in the frequency domain using the discrete Fourier transform (DFT). However, the element-wise multiplication of two DFTs in the frequency domain corresponds to a circular correlation, which results in aliasing (i.e., distortion) in the correlation output. Prior CF research has largely ignored these aliasing effects by making the assumption that linear correlation is approximated by circular correlation. In this work, we investigate in detail the topic of aliasing in CFs. First, we illustrate that the current formulation of CFs in the frequency domain is inherently flawed, as it unintentionally assumes circular correlation during the design phase. This means that existing CFs are not truly optimal. We introduce zero-aliasing correlation filters (ZACFs) which fix this formulation issue by ensuring that each CF formulation problem corresponds to a linear correlation rather than a circular correlation. By adopting the ZACF design modifications, we show that the recognition and localization performance of conventional CF designs can be significantly improved. We demonstrate these benefits using a variety of data sets and present solutions to the computational challenges associated with computing ZACFs. After a CF is designed, it is used for object recognition by correlating it with a test signal. We investigate the use of the well-known overlap-add (OLA) and overlap-save (OLS) algorithms to improve the computation and memory requirements of this correlation operation for high dimensional applications (e.g., video). Through this process, we highlight important tradeoffs between iii these two algorithms that have previously been undocumented. To improve the computation and memory requirements of OLA and OLS, we introduce a new block filtering scheme, denoted partial-aliasing OLA (PAOLA) that intentionally introduces aliasing into the output correlation. This aliasing causes conventional CFs to perform poorly. To remedy this, we introduce partialaliasing correlation filters (PACFs), which are specifically designed to minimize this aliasing. We demonstrate through numerical results that PACFs outperform conventional CFs in the presence of aliasing.