---
title: 'MKCFup: Dual Filter Updates in Tracking & Kalman'
url: https://www.emergentmind.com/topics/mkcfup
type: topic
---

# MKCFup: Dual Filter Updates in Tracking & Kalman

MKCFup denotes two distinct but high-impact methods within the correlation filter and robust Kalman filtering literature: (1) the Multi-Kernel Correlation Filter update for high-speed visual object tracking [1806.06418], and (2) the “update” step in the Maximum Correntropy Kalman Filter, a robust state-estimation methodology [1509.04580]. Both approaches exploit advanced kernel-based formulations and efficient alternating optimization. The following account details the MKCFup tracker for high-speed tracking and the Maximum Correntropy Kalman Filter update, including mathematical underpinnings, algorithmic schemes, evaluation, and performance insights.

## 1. Multi-Kernel Correlation Filter Frameworks

### KCF and MKCF Formulations

The Kernelized Correlation Filter (KCF) learns a discriminative function in a reproducing-kernel Hilbert space (RKHS) by minimizing a Tikhonov-regularized empirical risk:
\[
\min_{f\in\mathcal{H}_k}\;\frac{1}{2}\sum_{i=0}^{l-1}\left(f(\mathbf{x}_i)-y_i\right)^2 + \frac{\lambda_0}{2}\|f\|_k^2,
\]
where inputs are circulant shifts of a base patch, targets $y_i$ are Gaussian-shaped labels, and regularization parameter $\lambda_0>0$ penalizes RKHS norm $\|f\|_k$. By the Representer Theorem, $f(\cdot) =\sum_{i=0}^{l-1}\alpha_i\,k(\mathbf{x}_i,\cdot)$. Letting $\mathbf{K}$ be the kernel matrix, the dual problem is quadratic with closed-form solution; in the circulant setting, FFT diagonalization yields element-wise updates for efficient computation [1806.06418].

The Multi-Kernel Correlation Filter (MKCF) extends KCF to convex combinations of $M$ base kernels:
\[
k(\mathbf{x}_i,\mathbf{x}_j) = \sum_{m=1}^{M} d_m\,k_m(\mathbf{x}_i,\mathbf{x}_j),\qquad d_m \ge 0,~\sum_{m=1}^M d_m = 1.
\]
MKCF alternates between solving for the dual weights $\bm{\alpha}$ (fixed $\mathbf{d}$) and updating $\mathbf{d}$ (fixed $\bm{\alpha}$) within a convex quadratic program, but exhibits increased computational complexity and only moderate improvements over KCF [1806.06418].

## 2. Upper-Bound Reformulation and Decoupled Optimization

MKCFup introduces an upper-bound surrogate to the MKCF objective, fundamentally decoupling kernel contributions and suppressing negative interference:

\[
F(\bm{\alpha}, \mathbf{d}) \le U_F(\bm{\alpha}, \mathbf{d})=\frac{\mu}{2}\sum_{m=1}^M\left\|\mathbf{y}_c-d_m\,\mathbf{K}_m\bm{\alpha}\right\|^2 + \frac{\mu\lambda}{2}\sum_{m=1}^M d_m\,\bm{\alpha}^\top\mathbf{K}_m\bm{\alpha},
\]
with parameters $\mu=2M+1$, $\lambda=\lambda_0/\mu$, and $\mathbf{y}_c = \mathbf{y}/M$. This reformulation permits per-kernel loss evaluation, leading to more stable alternating block-coordinate optimization. Temporal adaptation and historical influence are controlled by per-kernel learning rates $\gamma_m$ and exponential weights $\beta_m^j$:

\[
F_p(\bm{\alpha}_p, \mathbf{d}_p) = \frac12\sum_{j=1}^{p}\sum_{m=1}^{M} \beta_m^j \left\| \mathbf{y}_c - d_{m,p}\mathbf{K}_m^j\bm{\alpha}_p \right\|^2 + \lambda d_{m,p} \bm{\alpha}_p^\top \mathbf{K}_m^j \bm{\alpha}_p,
\]
where $\beta_m^j$ reflects the influence of frames 1 to $p$ according to (per-kernel) exponential forgetting.

## 3. Alternating Block-Coordinate Algorithm

MKCFup employs a highly efficient alternating optimization per frame, summarized as:

1. **Initialize:** $d_{m,p}^{(0)}=1/M$ for $m=1\ldots M$.
2. **Alternate $T$ times (typically $T=3$):**
   - Fix $\mathbf{d}_p^{(t-1)}$, solve for $\bm{\alpha}_p^{(t)}$ via FFT-based elementwise updates in the Fourier domain, exploiting running accumulators for numerator and denominator terms per kernel.
   - Fix $\bm{\alpha}_p^{(t)}$, update $d_{m,p}^{(t)}$ in closed-form using current responses and running historical averages.

Detection in the next frame utilizes the estimated weights and filter response:
\[
y(\mathbf{z}) = \sum_{m=1}^M d_{m,p}^* \cdot \mathrm{IFFT}\left[\mathrm{FFT}(k_m^p)^* \odot \mathrm{FFT}(\alpha_p^*)\right].
\]
All operations remain in the Fourier domain, maintaining low computational overhead.

## 4. Feature Extraction, Kernels, and Historical Adaptation

MKCFup adopts two base kernel types:
- **Color Name (CN) features:** 13-D, projected to 4-D by PCA.
- **HOG features:** 9 orientation bins over $4\times4$ cells, PCA-reduced to 4-D.

Both use Gaussian base kernels
\[
k_m(\mathbf{x},\mathbf{x}') = \exp\left(-\frac{\|\mathbf{x}-\mathbf{x}'\|^2}{\sigma_m^2}\right),
\]
with cross-validated kernel widths for color ($\sigma_\mathrm{color}=0.515$, $\sigma_\mathrm{HOG}=0.6$ for color images; reduced for grayscale). Per-kernel historical learning rates are $\gamma_\mathrm{color}\approx0.0174$, $\gamma_\mathrm{HOG}\approx0.0173$, and initial weights $d_{m,1}=1/2$ adapt automatically thereafter [1806.06418].

## 5. Empirical Performance and Implementation Considerations

On OTB2013, MKCFup achieves 83.5% precision@20px and 64.1% AUC at 150 fps, outperforming both KCF (70.9% precision/50.7% AUC/297 fps) and classic MKCF (76.7%/57.0%/30 fps). The table below summarizes empirical metrics:

| Tracker  | Precision@20px |   AUC   |   fps   |
|----------|----------------|---------|---------|
| KCF      | 70.9%          | 50.7%   | 297     |
| MKCF     | 76.7%          | 57.0%   | 30      |
| fMKCF    | 78.6%          | 58.0%   | 50      |
| MKCFup   | 83.5%          | 64.1%   | 150     |

Key practical considerations:
- **Search region:** 2.5× object bounding-box (same as KCF)
- **Hann window** applied to suppress high-frequency FFT artifacts
- **Gaussian response label** matching feature patch shape
- **Regularization:** $\lambda_0 = \mu\cdot\lambda$, $\lambda\approx1.0$
- **Scale estimation:** fDSST applied post-tracking

## 6. Theoretical and Practical Advantages

MKCFup’s upper-bound reformulation suppresses kernel mutual interference, enabling more discriminative per-feature optimization. Temporal weighting via $\beta_m^j$ and per-kernel $\gamma_m$ enables adaptive memory length per feature. All updates rely on FFT-based operations, facilitating real-time throughput ($\sim$150 fps).

The search region size is optimized for small inter-frame translations (offset ratio $\tau<0.6$), avoiding unnecessary background and increasing robustness in “small-move/high-speed” scenarios. Empirically, MKCFup consistently exhibits superior accuracy and computational efficiency over both linear kernel methods (KCF) and standard MKL-based correlation filters (MKCF) [1806.06418].

## 7. MKCFup in the Maximum Correntropy Kalman Filter

In robust state estimation, “MKCFup” also refers to the update step in the Maximum Correntropy Kalman Filter (MCKF) [1509.04580]. Here, the MMSE criterion of classic Kalman filtering is replaced with the Maximum Correntropy Criterion (MCC), leading to a fixed-point update of the posterior mean:
\[
\hat{x}_{k|k} = \hat{x}_{k|k-1} + K(k)[y(k) - H(k)\hat{x}_{k|k-1}]
\]
with modified covariances and gain reflecting data-driven residual weights:
\[
K(k) = P_{k|k-1} H^T(k)[H(k)P_{k|k-1}H^T(k) + R(k)]^{-1},
\]
where $P_{k|k-1}$ and $R(k)$ depend on the diagonal correntropy matrices $C_x$, $C_y$ induced by
\[
G_\sigma(e) = \exp\left(-\frac{e^2}{2\sigma^2}\right).
\]
The fixed-point equations down-weight outliers via $G_\sigma(e_i)$, yielding robustness to heavy-tailed or impulsive noise. Convergence is guaranteed for sufficiently large $\sigma$. For Gaussian noise, MCKF reduces to the classical Kalman update. This robustification incurs a marginal increase in computational complexity (a few fixed-point iterations per timestep), yielding superior empirical error distributions in non-Gaussian settings [1509.04580].

Source: https://www.emergentmind.com/topics/mkcfup