Dual Correlation Filter (DCF) Overview

• Dual Correlation Filter (DCF) is a discriminative visual tracking method that leverages circulant matrices to model cyclic shifts, enabling efficient FFT-based solutions.
• It utilizes a dual formulation of kernelized ridge regression to capture multi-channel and nonlinear features at computational costs comparable to linear methods.
• The DCF approach underpins many modern trackers, balancing high speed and accuracy while influencing subsequent methods like KCF and deep-feature tracking.

A Dual Correlation Filter (DCF) is a class of discriminative visual tracking models that exploits the mathematical properties of circulant matrices to train a multi-channel (or kernelized) correlation filter in the frequency domain, providing highly efficient and accurate object tracking. The central innovation lies in the dual (kernel) formulation of ridge-regression under circulant sampling, combined with fast closed-form solvers via the FFT, enabling the use of rich multi-channel or nonlinear features at a computational cost comparable to the linear case. The DCF framework has profoundly influenced the evolution of high-speed, high-accuracy correlation-filter trackers, and continues to serve as the backbone for modern kernelized and deep-feature correlation-filter methods.

1. Mathematical Foundations and Circulant Structure

The DCF framework emerges from recognizing that dense cyclic translations of an image patch can be jointly modeled as a circulant matrix. For a base sample $x \in \mathbb{R}^n$, all its cyclic shifts generate a matrix $C(x)$:

$X = C(x), \qquad X_{ij} = x_{((j - i) \mod n) + 1}$

Such matrices are diagonalized by the discrete Fourier transform (DFT) matrix $F$:

$C(x) = F\, \text{diag}(F x)\, F^H$

This property enables reformulating linear ridge regression or kernel ridge regression on all shifted samples with closed-form solutions in the frequency domain, reducing complexity from $O(n^2)$ to $O(n \log n)$ per frame (Henriques et al., 2014).

2. From Linear to Kernelized Ridge Regression: The Dual Formulation

In DCF tracking, the target is to regress a filter $w$ such that for cyclic shifts $x_i$ of a training patch $x$, the response $f(x_i)$ approximates a Gaussian-shaped target $y_i$. The standard primal regularized least squares solution is:

$w = (X^T X + \lambda I)^{-1} X^T y$

The dual (kernel) form, leveraging the kernel trick, is:

$\alpha = (K + \lambda I)^{-1} y$

where $K_{ij} = \kappa(x_i, x_j)$ for a positive-definite kernel $\kappa$. For circulation-invariant kernels (e.g., linear, polynomial, Gaussian), $K$ is circulant and diagonalizable under the DFT (Henriques et al., 2014).

The solution becomes fully element-wise:

$\hat{\alpha} = \frac{\hat{y}}{\hat{k} + \lambda}$

where hats denote DFTs of the respective vectors, and all divisions are element-wise.

3. Multi-Channel Extension and DCF Specifics

The DCF, as introduced by Henriques et al., denotes the multi-channel (e.g., HOG-channel) linear kernel extension, where $x$ and $z$ have $C$ channels. The kernel correlation becomes:

$$k^{xz} = \text{IFFT}\left( \sum_{c=1}^C \hat{x}_c^* \odot \hat{z}_c \right)$$

with the dual solution:

$$\hat{\alpha} = \frac{\hat{y}}{\sum_c \hat{x}_c^* \odot \hat{x}_c + \lambda}$$

This enables efficient filter learning and detection with multi-channel descriptors, with runtime dominated by FFTs and elemental operations, supporting several hundred frames per second with rich features. The dual-form DCF, therefore, allows for the practical realization of robust, high-speed learning and detection in visual tracking (Henriques et al., 2014).

4. Relationship to Kernelized Correlation Filter (KCF) and Modern Trackers

The DCF serves as the direct precursor to the Kernelized Correlation Filter (KCF), which introduced the explicit application of nonlinear kernels (typically RBF/Gaussian) in the same circulant/DFT analytical framework. Both DCF and KCF achieve $O(n \log n)$ complexity for per-frame training and detection, with the difference that KCF generalizes to any shift-invariant kernel, while DCF denotes the linear (multi-channel) case (Chen et al., 2015, Henriques et al., 2014).

Later trackers extend or build upon the DCF/KCF duality in various ways:

• MKCF (Multi-Kernel Correlation Filter): Combines multiple kernels as a convex sum, but suffers mutual interference and high computational cost ($\sim 30$ FPS) (Tang et al., 2018).
• MKCFup: Introduces upper-bound decoupling for convex kernel mixture optimization, preserving the FFT-based $O(n \log n)$ regime and achieving $\sim 150$ FPS and AUC competitive with much slower state-of-the-art CF trackers (Tang et al., 2018).
• DCFNet: Embeds a DCF (dual-form correlation filter layer) within a Siamese deep network, with all learning and backpropagation steps carried out in the frequency domain, enabling end-to-end feature learning while retaining $O(D MN \log (MN))$ per-frame complexity (Wang et al., 2017).

5. Algorithmic Implementation and Computational Properties

A typical DCF tracker processes each new frame as follows:

1. Detection: Compute the response map (in Fourier domain) for the candidate patch $z$:

$$\text{response}(z) = \mathcal{F}^{-1}\left( \hat{k}^{xz} \odot \hat{\alpha} \right)$$

where $\hat{k}^{xz}$ is the DFT of kernel correlations between $x$ and cyclic shifts of $z$, and $\mathcal{F}^{-1}$ is the inverse FFT.

2. Training/Update: Crop a new positive patch at the updated position, recompute or exponentially update the model parameters using the same FFT-based closed-form as above.

The per-frame cost is $O(n \log n)$ for $n$-pixel patches and all channels, supporting high real-time rates even for multi-channel features (Henriques et al., 2014, Chen et al., 2015).

6. Practical Impact and Evolution

DCF and its dual formulation underpin a large body of visual tracking literature:

• Empirical results demonstrate that DCF trackers can achieve $\sim 280$ FPS with raw pixels and $\sim 292$ FPS with HOG features, outperforming or matching more complex trackers at much lower cost (Henriques et al., 2014).
• Successive methods such as MKCFup further close the accuracy gap with sophisticated CF trackers (e.g., ECO, SRDCF), but at a fraction of their computational cost (Tang et al., 2018).
• Deep-learning variants (e.g., DCFNet) inherit and exploit the analytic DCF solution, supporting over 60 FPS with end-to-end optimized features, and outpacing conventional HOG-based trackers in both accuracy and speed (Wang et al., 2017).

A summary table illustrates the empirical trade-offs (as reported in (Tang et al., 2018)):

Tracker	Precision@20px	AUC	FPS (CPU)
KCF	76.7%	56.4%	297
MKCF	77.0%	57.2%	30
MKCFup	83.5%	64.1%	150
ECO_HC	84%	64%	39
SRDCF	80%	60%	6

This suggests that DCF-based formulations currently provide one of the best efficiency-accuracy trade-offs in real-time visual tracking.

7. Limitations and Extensions

Despite their success, DCFs have several limitations:

• Boundary effects: Standard DCF and KCF formulations assume periodic (circular) boundary conditions, causing artifacts when the target's search area is not significantly larger than the object.
• Scale and Rotation Handling: Extensions like DSST introduce a separate scale filter, RKCF learns a companion filter on a cyclic HOG vector for rotation compensation with negligible overhead (Hamdi et al., 2017).
• Multi-Kernel Discriminability: While MKCF and its successors ameliorate the one-kernel limitation, interference and computational overhead must be mitigated through careful optimization design (e.g., convex upper-bounds).

A plausible implication is that future DCF research will focus on further mitigating boundary effects, fusing spatial regularization or non-periodic basis sets, and integrating learned features and kernel mixtures, all while retaining the analytic, FFT-based efficiency that distinguishes the DCF paradigm.