Exploiting the Circulant Structure of Tracking-by-Detection with Kernels

João F. HenriquesRui CaseiroPedro MartinsJorge Batista

article2012ECCV2,361 citations

Proposes a framework that exploits the circulant structure of densely sampled subwindows to train and evaluate non-linear kernel classifiers via the Fast Fourier Transform, enabling visual tracking at hundreds of frames per second.

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Visual object tracking is a critical technology in computer vision, underpinning applications in autonomous robotics, video surveillance, and human-machine interfaces. A dominant technique is tracking-by-detection, where an appearance model is continuously trained online to distinguish a target from its surroundings. However, conventional tracking-by-detection systems face a fundamental trade-off: processing thousands of candidate image patches around a target creates prohibitive computational bottlenecks, while randomly sampling only a small subset of patches degrades tracking accuracy.

The article evaluates whether the spatial overlap among all candidate image patches can be exploited mathematically to achieve both comprehensive sampling and ultra-fast execution. Specifically, it demonstrates how to train and evaluate complex non-linear classifiers across every candidate subwindow in an image without manually iterating through them.

The authors developed an analytical framework based on circulant matrices and Fourier analysis, proving that generating all cyclic shifts of an image creates structured data matrices. This mathematical property allows both model training and target detection to be computed as element-wise operations in the frequency domain using the Fast Fourier Transform. The researchers derived exact closed-form solutions for Kernel Regularized Least Squares (ridge regression) using non-linear kernels, such as Gaussian and polynomial kernels. They then validated the approach by benchmarking a prototype tracker against leading state-of-the-art trackers across 12 challenging video sequences featuring occlusions and scale variations.

The findings demonstrate substantial improvements in both computational efficiency and tracking precision. By moving from sparse sampling to dense sampling of all subwindows, the algorithm processes 64x64 pixel windows at 320 frames per second, compared to 10 to 25 frames per second for conventional online boosting and support vector machine approaches. In precision evaluations measuring target localization within a 20-pixel threshold, the proposed method achieved top-tier performance, matching or outperforming state-of-the-art trackers in 10 out of 12 benchmark video sequences. Furthermore, the complete tracker requires only a few lines of code and operates directly on raw pixel values without complex feature engineering or heavy numerical optimization.

These results show that tracking systems can break the traditional trade-off between algorithmic speed and localization accuracy. For engineering leaders and system designers, this approach significantly reduces hardware compute requirements, memory footprint, and power consumption while improving real-time reliability. It enables high-performance visual tracking on embedded, low-power edge devices where resource-intensive optimization algorithms are impractical.

Organizations developing computer vision systems should consider adopting circulant matrix and frequency-domain frameworks for real-time tracking pipelines. Practitioners can implement the core algorithm using standard Fast Fourier Transform libraries. When adapting the method, engineering teams should evaluate the main operational limitations highlighted in the article: the baseline formulation assumes a fixed target scale, lacks integrated failure detection or recovery mechanisms, and requires windowing techniques (such as cosine windowing) to suppress boundary artifacts caused by Fourier periodicity. Overall, confidence in the mathematical derivations and core tracking capabilities is high, though deploying the tracker in complex operational environments will require integrating scale-adaptation schemes and complementary motion models.

Institute of Systems and Robotics University of Coimbra fast fourier transforms fourier frames visual tracking tracking-by-detection circulant structure

Henriques et al (2012).pdf

Paper: Random Features for Large-Scale Kernel Machines, Ali Rahimi et al. (2007). Random Fourier features provide foundational theory for approximating kernel machines with linear methods, directly inspiring fast explicit mappings.

Paper: High-Speed Tracking with Kernelized Correlation Filters, João F. Henriques et al. (2014). This follow-up work extends the circulant tracking framework by incorporating multi-channel HOG features to significantly boost visual tracking accuracy.

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Abstract

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).

Topics

fast fourier transforms fourier frames gaussians matlab object detection tracking system closed-form solution tracking algorithm tracking method

Keywords

visual tracking tracking-by-detection circulant structure adaptive correlation filters circulant matrices kernel trick Fast Fourier Transform Gaussian kernel

1 Introduction
1.1 Previous Work
1.2 Contributions
2 Learning with Dense Sampling
2.1 Regularized Risk Minimization
2.2 Circulant Matrices
2.3 The Kernel Matrix with Dense Sampling
2.4 Efficient Kernel Regularized Least Squares solution
2.5 Fast Detection
3 Fast Computation of Non-linear Kernels
3.1 Dot-Product Kernels
3.2 Radial Basis Function Kernels
3.3 The Linear Case
4 Experiments
4.1 Pre-processing
4.2 Training Outputs
4.3 Overview
4.4 Evaluation
5 Conclusion
References
Appendix A.1: Dense Sampling KRLS Derivation
Appendix A.2: Derivation of Fast Detection Formula
Appendix A.3: Generalization of Circulant Forms

Knowls

Knowl 1 — Circulant and Block-Circulant Structure of Kernel Matrices Under Dense Sampling
theoretical result
Let $x \in \mathbb{R}^n$ be a base signal or one-dimensional image patch, and let $P$ denote the cyclic permutation matrix that shifts a vector by one element. Dense sampling constructs the dataset of all $n$ cyclic shifts $x_i = P^i x$ for $i = 0, \dots, n-1$ .

Theorem: If a kernel function $\kappa(x, x')$ is unitarily invariant—that is, $\kappa(x, x') = \kappa(U x, U x')$ for any unitary matrix $U$ —then the $n \times n$ kernel matrix $K$ with elements $K_{ij} = \kappa(P^i x, P^j x)$ is a circulant matrix $C(k)$ . The elements satisfy $K_{ij} = \kappa(x, P^{j-i} x)$ , depending only on $(j - i) \bmod n$ , and $K$ is fully characterized by the generating vector $k \in \mathbb{R}^n$ where $k_i = \kappa(x, P^i x)$ .

Unitarily invariant kernel families that yield circulant kernel matrices include:
- Dot-product kernels: $\kappa(x, x') = g(\langle x, x' \rangle)$ for any function $g$ (e.g., linear and polynomial kernels).
- Radial Basis Function (RBF) kernels: $\kappa(x, x') = h(\|x - x'\|^2)$ for any function $h$ (e.g., Gaussian kernels).
Generalization to 2D Images: For a 2D image $X \in \mathbb{R}^{n \times n}$ cyclically shifted along rows by $i$ and columns by $i'$ ( $P^i X P^{i'}$ ), the resulting $n^2 \times n^2$ kernel matrix is a Block-Circulant with Circulant Blocks (BCCM) matrix, entirely generated by an $n \times n$ matrix $K'$ with entries $K'_{i i'} = \kappa(X, P^i X P^{i'})$ .
Knowl 2 — Closed-Form Frequency-Domain Solution for Dense Kernel Regularized Least Squares
equation
In Kernel Regularized Least Squares (KRLS / Ridge Regression), given training samples $x_i = P^i x$ ( $i = 0, \dots, n-1$ ) obtained from all cyclic shifts of a base sample $x \in \mathbb{R}^n$ , and a continuous target regression vector $y \in \mathbb{R}^n$ , the dual parameter vector $\alpha \in \mathbb{R}^n$ minimizes the regularized quadratic loss: $\min_{\alpha} \|K \alpha - y\|^2 + \lambda \alpha^T K \alpha$ where $\lambda > 0$ is the regularization parameter.

Because the kernel matrix $K = C(k)$ and the identity matrix $I = C(\delta)$ (where $\delta = [1, 0, \dots, 0]^T$ ) are circulant, the matrix inversion $(K + \lambda I)^{-1} y$ is diagonalized by the Discrete Fourier Transform (DFT), yielding the exact closed-form solution: $\alpha = \mathcal{F}^{-1}\left( \frac{\mathcal{F}(y)}{\mathcal{F}(k) + \lambda} \right)$ where $\mathcal{F}$ denotes the Discrete Fourier Transform, $\mathcal{F}^{-1}$ denotes the Inverse Discrete Fourier Transform, $k \in \mathbb{R}^n$ is the kernel autocorrelation vector with entries $k_i = \kappa(x, P^i x)$ , and the division is performed element-wise.

For an $n \times n$ 2D image, replacing 1D transforms with 2D Fourier transforms solves KRLS for all $n^2$ dense subwindows in $\mathcal{O}(n^2 \log n)$ time, compared to $\mathcal{O}(n^4)$ or $\mathcal{O}(n^6)$ for naive solvers.
Knowl 3 — Fast Non-Linear Kernel Computation at All Subwindows via Fourier Transforms
equation
Evaluating a non-linear kernel between a base signal $x \in \mathbb{R}^n$ and all cyclic shifts of another signal $x' \in \mathbb{R}^n$ , defined as the vector $k$ with elements $k_i = \kappa(x, P^i x')$ for $i = 0, \dots, n-1$ , is computed simultaneously via the Discrete Fourier Transform without looping over subwindows:
1. Dot-Product Kernels: For $\kappa(x, x') = g(x^T x')$ : $k^{\text{dp}} = g\left( \mathcal{F}^{-1}\left( \mathcal{F}(x) \odot \mathcal{F}^*(x') \right) \right)$ For a polynomial kernel $\kappa(x, x') = (x^T x' + c)^d$ : $k^{\text{poly}} = \left( \mathcal{F}^{-1}\left( \mathcal{F}(x) \odot \mathcal{F}^*(x') \right) + c \right)^d$
2. Radial Basis Function (RBF) Kernels: For $\kappa(x, x') = h(\|x - x'\|^2)$ : $k^{\text{rbf}} = h\left( \|x\|^2 + \|x'\|^2 - 2 \mathcal{F}^{-1}\left( \mathcal{F}(x) \odot \mathcal{F}^*(x') \right) \right)$ For a Gaussian RBF kernel $\kappa(x, x') = \exp\left( -\frac{1}{\sigma^2} \|x - x'\|^2 \right)$ with bandwidth $\sigma$ : $k^{\text{gauss}} = \exp\left( -\frac{1}{\sigma^2} \left( \|x\|^2 + \|x'\|^2 - 2 \mathcal{F}^{-1}\left( \mathcal{F}(x) \odot \mathcal{F}^*(x') \right) \right) \right)$
where $\odot$ denotes element-wise product, $\mathcal{F}^*$ is the complex conjugate of the DFT, and functions $g$ , $h$ , $\exp$ , and $(\cdot)^d$ operate element-wise. For $n \times n$ 2D images, using 2D FFTs computes the kernel at all $n^2$ locations in $\mathcal{O}(n^2 \log n)$ time.
Knowl 4 — Fast Classifier Detection at All Subwindows via Frequency-Domain Correlation
equation
Given learned dual coefficients $\alpha \in \mathbb{R}^n$ and training base patch $x$ , the regression responses for all cyclic displacements $z_i = P^i z$ ( $i = 0, \dots, n-1$ ) of a test image patch $z \in \mathbb{R}^n$ are evaluated as: $\hat{y}_i = \sum_{j=0}^{n-1} \alpha_j \kappa(P^i z, P^j x)$ In matrix form, $\hat{y} = C(\bar{k}) \alpha$ , where $\bar{k} \in \mathbb{R}^n$ is the kernel vector with elements $\bar{k}_i = \kappa(z, P^i x)$ . Exploiting the convolution property of circulant matrices, all responses across the entire displacement grid are evaluated simultaneously in the Fourier domain: $\hat{y} = \mathcal{F}^{-1}\left( \mathcal{F}(\bar{k}) \odot \mathcal{F}(\alpha) \right)$ where $\mathcal{F}$ and $\mathcal{F}^{-1}$ denote the Discrete Fourier Transform and its inverse, and $\odot$ represents element-wise multiplication. For an $n \times n$ patch, the 2D FFT evaluates responses over all $n^2$ candidate locations in $\mathcal{O}(n^2 \log n)$ time.
Knowl 5 — Circulant Kernel Tracker Algorithm
algorithm
The visual tracker optimizes a Kernel Regularized Least Squares classifier online using closed-form FFT operations on dense cyclic image patches.
```
% MATLAB code for visual tracking using a Gaussian kernel
% x (current frame template) and z (next frame search patch) are M-by-N matrices,
% pre-processed with a cosine window. y is a Gaussian regression target.

function alphaf = training(x, y, sigma, lambda)
    k = dgk(x, x, sigma);
    alphaf = fft2(y) ./ (fft2(k) + lambda);
end

function responses = detection(alphaf, x, z, sigma)
    k = dgk(z, x, sigma);
    responses = real(ifft2(alphaf .* fft2(k)));
end

function k = dgk(x1, x2, sigma)
    c = fftshift(ifft2(fft2(x1) .* conj(fft2(x2))));
    d = x1(:)' * x1(:) + x2(:)' * x2(:) - 2 * c;
    k = exp(-1 / (sigma^2) * abs(d) / numel(x1));
end
```
Implementation and Hyperparameters:
- Search window: Cropped at twice the target dimensions centered at the previous position.
- Pre-processing: Modulated by a 2D cosine window to remove boundary discontinuities.
- Label target: 2D Gaussian $y$ with spatial bandwidth $s = \frac{\sqrt{mn}}{16}$ for an $m \times n$ target.
- Hyperparameters: Gaussian kernel bandwidth $\sigma = 0.2$ , regularization parameter $\lambda = 10^{-2}$ .
- Model adaptation: Appearance template $x$ and Fourier coefficients $\alpha_f$ are updated via linear interpolation with learning rate $\eta = 0.075$ : $x \leftarrow (1 - \eta)x + \eta x_{\text{new}}, \quad \alpha_f \leftarrow (1 - \eta)\alpha_f + \eta \alpha_{f, \text{new}}$
- Operating Speed: Operates at 320 FPS for a $64 \times 64$ window.
Knowl 6 — Cosine Windowing and Continuous Gaussian Regression Targets for Boundary and Ringing Suppression
model/method
To train and evaluate frequency-domain circulant classifiers on bounded image patches without artifacts, two pre-processing formulations are employed:
1. Cosine Windowing: Because discrete Fourier operations assume periodic boundary conditions, discontinuities across opposite edges introduce high-frequency Fourier noise. Raw pixel values $x^{\text{raw}}_{ij}$ for an $n \times n$ patch are windowed by: $x_{ij} = \left(x^{\text{raw}}_{ij} - 0.5\right) \sin\left(\frac{\pi i}{n}\right) \sin\left(\frac{\pi j}{n}\right), \quad \forall i, j \in \{0, \dots, n-1\}$ which smoothly forces pixel intensities to zero at the patch boundaries.
2. Continuous Gaussian Labeling: Instead of discrete binary labels ( $\pm 1$ ), the training regression target $y \in \mathbb{R}^{n \times n}$ is defined as a continuous Gaussian peak centered at the target coordinates $(i', j')$ : $y_{ij} = \exp\left( -\frac{(i - i')^2 + (j - j')^2}{s^2} \right), \quad \forall i, j \in \{0, \dots, n-1\}$ where $s = \frac{\sqrt{mn}}{16}$ is the spatial bandwidth for an $m \times n$ object. This continuous label distribution minimizes ringing artifacts in the Fourier domain and produces smooth displacement response maps.
Knowl 7 — Equivalence of Linear Dense KRLS to MOSSE Correlation Filters
theoretical result
For a linear kernel $\kappa(x, x') = \langle x, x' \rangle = x^T x'$ , the classifier weight vector $w \in \mathbb{R}^n$ can be expressed explicitly in the primal space rather than dual coefficients $\alpha$ . Under dense sampling using all cyclic shifts of an image $x \in \mathbb{R}^n$ with continuous targets $y \in \mathbb{R}^n$ , the regularized least squares solution for $w$ in the Fourier domain is: $w = \mathcal{F}^{-1}\left( \frac{\mathcal{F}(x) \odot \mathcal{F}^*(y)}{\mathcal{F}(x) \odot \mathcal{F}^*(x) + \lambda} \right)$ where $\mathcal{F}^*$ denotes the complex conjugate of the Fourier transform, $\odot$ is element-wise multiplication, and division is element-wise with regularization $\lambda > 0$ .

This establishes that the Minimum Output Sum of Squared Error (MOSSE) correlation filter derived from classical signal processing corresponds directly to Kernel Regularized Least Squares (Ridge Regression) trained on all dense cyclic translations of a single image using a linear kernel.

Knowl 8 — Precision Evaluation on Benchmark Tracking Sequences

data/table

Tracking performance measured as distance precision at a threshold of 20 pixels (percentage of video frames where the predicted center location is within 20 pixels of the ground truth) across 12 benchmark video sequences. The dense circulant kernel tracker is compared against MILTrack, Struck, original MOSSE, and an enhanced linear baseline (MOSSE2):

Sequence	MILTrack	Struck	MOSSE	MOSSE2	Proposed method
coke11	0.61	0.97	0.71	0.71	1.00
faceocc	0.46	0.96	0.21	1.00	1.00
faceocc2	0.69	0.95	0.53	0.93	1.00
surfer	0.98	0.97	0.37	0.99	0.99
sylvester	0.90	0.95	0.78	0.90	1.00
tiger1	0.83	0.94	0.26	0.30	0.61
tiger2	0.93	0.91	0.25	0.22	0.63
dollar	0.82	0.96	0.39	1.00	1.00
girl	0.31	0.95	0.83	0.99	0.59
david	0.56	0.92	0.77	0.34	0.49
cliffbar	0.89	0.44	0.37	0.56	0.97
twinings	0.98	1.00	0.20	1.00	0.93

The proposed method matches or outperforms the improved linear filter (MOSSE2) in 10 of the 12 sequences and achieves perfect precision (1.00) on 5 sequences (coke11, faceocc, faceocc2, sylvester, and dollar). Dense cyclic sampling provides robust negative sample coverage (both distant background distractors and slight object shifts), yielding highly localized responses while running at over 300 FPS.

Coverage note — None was omitted; all core theoretical theorems, closed-form equations, non-linear kernel computations, the complete tracking algorithm, pre-processing techniques, and experimental benchmark results have been captured.

References

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Citation

MLA

Henriques, J. F., et al. “Exploiting the Circulant Structure of Tracking-by-Detection with Kernels”. Lecture Notes in Computer Science, Springer Berlin Heidelberg, 2012, pp. 702–15, https://doi.org/10.1007/978-3-642-33765-9_50.

APA

Henriques, J. F., Caseiro, R., Martins, P., & Batista, J. (2012). Exploiting the Circulant Structure of Tracking-by-Detection with Kernels. In Lecture Notes in Computer Science (pp. 702–715). Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-33765-9_50

Chicago

Henriques, J. F., R. Caseiro, P. Martins, and J. Batista. 2012. “Exploiting the Circulant Structure of Tracking-by-Detection with Kernels”. In Lecture Notes in Computer Science. Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-33765-9_50.

Harvard

Henriques, J.F. et al. (2012) “Exploiting the Circulant Structure of Tracking-by-Detection with Kernels”, Lecture Notes in Computer Science. Springer Berlin Heidelberg, pp. 702–715. Available at: https://doi.org/10.1007/978-3-642-33765-9_50.

Vancouver

1. Henriques JF, Caseiro R, Martins P, Batista J (2012) Exploiting the Circulant Structure of Tracking-by-Detection with Kernels. In: Lecture Notes in Computer Science. Springer Berlin Heidelberg, pp 702–715

BibTeX

@inbook{Henriques_2012, title={Exploiting the Circulant Structure of Tracking-by-Detection with Kernels}, ISBN={9783642337659}, ISSN={1611-3349}, url={http://dx.doi.org/10.1007/978-3-642-33765-9_50}, DOI={10.1007/978-3-642-33765-9_50}, booktitle={Computer Vision – ECCV 2012}, publisher={Springer Berlin Heidelberg}, author={Henriques, João F. and Caseiro, Rui and Martins, Pedro and Batista, Jorge}, year={2012}, pages={702–715} }

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Exploiting the Circulant Structure of Tracking-by-Detection with Kernels

Abstract

Topics

Keywords

Table of Contents

Knowls

Knowl 1 — Circulant and Block-Circulant Structure of Kernel Matrices Under Dense Sampling

Knowl 2 — Closed-Form Frequency-Domain Solution for Dense Kernel Regularized Least Squares

Knowl 3 — Fast Non-Linear Kernel Computation at All Subwindows via Fourier Transforms

Knowl 4 — Fast Classifier Detection at All Subwindows via Frequency-Domain Correlation

Knowl 5 — Circulant Kernel Tracker Algorithm

Knowl 6 — Cosine Windowing and Continuous Gaussian Regression Targets for Boundary and Ringing Suppression

Knowl 7 — Equivalence of Linear Dense KRLS to MOSSE Correlation Filters

Knowl 8 — Precision Evaluation on Benchmark Tracking Sequences

References

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