Gaussian Mixture Model (GMM) clustering can also be used for quantizing the visual feature space. Unlike K-means, GMM use has not been as widespread in the computer vision community for partitioning visual feature spaces, although recent results in image classification are very promising [46], [37].
For GMM clustering, a set of L feature vectors X = {¯x1, ¯x2, ..., ¯xL}, where ¯xi ∈ RD, i = {1, 2, ..., L} derived from training videos is modeled by a mixture of m Gaus-sian distributions that are completely specified by the set of parameters Θ. Θ = {π1, ¯µ1, Σ1, ..., πm, ¯µm, Σm} comprises of prior probabilities πl∈ R+, mean values ¯µl∈ RD and covariance matrices Σl∈ RDxD, where l = {1, 2, ..., m}. A sample ¯xi, i = {1, 2, ..., L}, derived from the X set of feature vectors, is characterized by its density p(¯xi|Θ):
p(¯xi|πl, ¯µl, Σl) =
where pl, l = [1, 2, ..., m] is a probability density function (pdf) with parameters {π1, ¯µ1, Σ1, ..., πm, ¯µm,Σm}, prior probabilities πl ∈ R+, mean values ¯µl ∈ RD and covariance matrices Σl∈ RDxD for m Gaussian components. We make the assumption that the data is uncorrelated, leading to diagonal covariance matrixes, so the GMM is fully described by (2D + 1)m scalar parameters. A GMM is fit to the data X = {¯x1, ..., ¯xn}
Expectation Maximization (EM) is then initialized to learn these parameters. At the assignment step of the algorithm, soft data (i = {1, 2, ..., L}) to distribution (k = {1, 2, ..., m}) assignments are defined as follows:
qik = pk(¯xi) · πk Pm
l=1p(¯xi|¯µl, Σl) · πl
At the update step (Maximization step or M-step), the mixture parameter estimates are refined using the computed probabilities:
πk= 1 L
L
X
i=1
qik, µk= PL
i=1qikx¯i
PL i=1qik
Σk = PL
i=1qik(¯xi− ¯µk)(¯xi− ¯µk)T PL
i=1qik , k = {1, 2, ..., m}
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