Equivalence of Correlation Filter and Convolution Filter in Visual Tracking

TL;DR

This paper proves that correlation filters and convolution filters are mathematically equivalent in visual tracking under certain conditions, allowing flexible choice in tracker design and challenging the need for ideal response explanations.

Contribution

It establishes the first formal proof of equivalence between correlation and convolution filters in visual tracking, under specific assumptions.

Findings

Correlation and convolution filters have equal MMSE in tracking.

The equivalence holds when the ideal filter response is Gaussian and centrosymmetric.

This insight allows more flexible filter selection in tracker development.

Abstract

(Discriminative) Correlation Filter has been successfully applied to visual tracking and has advanced the field significantly in recent years. Correlation filter-based trackers consider visual tracking as a problem of matching the feature template of the object and candidate regions in the detection sample, in which correlation filter provides the means to calculate the similarities. In contrast, convolution filter is usually used for blurring, sharpening, embossing, edge detection, etc in image processing. On the surface, correlation filter and convolution filter are usually used for different purposes. In this paper, however, we proves, for the first time, that correlation filter and convolution filter are equivalent in the sense that their minimum mean-square errors (MMSEs) in visual tracking are equal, under the condition that the optimal solutions exist and the ideal filter response is Gaussian and centrosymmetric. This result gives researchers the freedom to choose correlation or convolution in formulating their trackers. It also suggests that the explanation of the ideal response in terms of similarities is not essential.