Four mathematical modeling forms for correlation filter object tracking algorithms and the fast calculation for the filter

The correlation filter object tracking algorithm has gained extensive attention from scholars in the field of tracking because of its excellent tracking performance and efficiency. However, the mathematical modeling relationships of correlation filter tracking frameworks are unclear. Therefore, many forms of correlation filters are susceptible to confusion and misuse. To solve these problems, we attempted to review various forms of the correlation filter and discussed their intrinsic connections. First, we reviewed the basic definitions of the circulant matrix, convolution, and correlation operations. Then, the relationship among the three operations was discussed. Considering this, four mathematical modeling forms of correlation filter object tracking from the literature were listed, and the equivalence of the four modeling forms was theoretically proven. Then, the fast solution of the correlation filter was discussed from the perspective of the diagonalization property of the circulant matrix and the convolution theorem. In addition, we delved into the difference between the one-dimensional and two-dimensional correlation filter responses as well as the reasons for their generation. Numerical experiments were conducted to verify the proposed perspectives. The results showed that the filters calculated based on the diagonalization property and the convolution property of the cyclic matrix were completely equivalent. The experimental code of this paper is available at https://github.com/110500617/Correlation-filter/tree/main.