f (z) = w T z

m i n w Σ i (f (x i) - y i) 2 + λ | | w | | 2

w = (X T X + λ I) - 1 X T y

w * = (X H X + λ I) - 1 X H y

High-Speed Tracking with Kernelized Correlation Filters

X = F H d i a g (x^) F

x^= F (x)

xi in linear regression corresponds to a vector x=[x1,...,xn]
So, we can use circulant matrice X to represent the vector x.

X H X = F H d i a g (x^*) F F H d i a g (x^) F

X H X = F H d i a g (x^* \cdot x^) F

However, this formula isn’t used in the code. The following one is used.

w = Σ i α i φ (x i)

φ T (x) φ (x') = k (x, x')

k is the kernel function (Gaussian or Polynomial)

f (z) = w T ϕ (z)

w = Σ i α i ϕ (x i)

f (z) = [Σ i α i ϕ (x i)] ϕ (z) = Σ i [α i ϕ (x i) ϕ (z)]

f (z) = Σ i α i k (z, x i)

then we need to compute α

α = (K + λ I) - 1

z represent the current frame

K z = C (k x z)

use circulant matrice K to represent the kernel correlation kxz
High-Speed Tracking with Kernelized Correlation Filters

summary

image patch => hog feature
hog feature => fft
kf = gaussian_correlation(xf, xf, kernel.sigma); (compute the kernel correlation of the first frame)
alphaf = yf ./ (kf + lambda); (compute α
kzf = gaussian_correlation(zf, model_xf, kernel.sigma); (second frame)
response = real(ifft2(model_alphaf .* kzf));
as illustrated in the formula

High-Speed Tracking with Kernelized Correlation Filters