Articulated Human Motion Tracking Using Sequential Immune Genetic Algorithm

Algorithm 1

ISOMAP-based manifold reconstruction.

Input: The training data set {𝑥_𝑖∣𝑥_𝑖∈𝑋,𝑖=1,…,𝑙}.

Output: The mappings 𝑔∶𝑋→𝑌,  𝑦=𝑔(𝑥) and 𝑓∶𝑌→𝑋,  𝑥=𝑓(𝑦).

Step 1: (preparing)

(1) Using the ISOMAP algorithm to compute the low-dim vector {𝑦_𝑖∣𝑦_𝑖∈𝑌,𝑖=1,…,𝑙} for the

original input vector {𝑥_𝑖∣𝑥_𝑖∈𝑋,𝑖=1,…,𝑙};

(2) Construct the following matrixes:

𝑋_𝑖=(𝑥_𝑖1−𝑥_𝑖,…,𝑥_𝑖𝑙𝑖−𝑥_𝑖)∈𝑅^{𝑛×𝑙𝑖}, 𝑌_𝑖=(𝑦_𝑖1−𝑦_𝑖,…,𝑦_𝑖𝑙𝑖−𝑦_𝑖)∈𝑅^{𝑑×𝑙𝑖}, where {𝑥_𝑖𝑗∣𝑗=1,…,𝑙_𝑖} is the

𝜀-neighbor of 𝑥_𝑖.

(3) Compute 𝑄_𝑖=𝑋_𝑖𝑌^𝑇_𝑖(𝑌_𝑖𝑌^𝑇_𝑖)^𝐺∈𝑅^𝑛×𝑑, where (𝑌_𝑖𝑌^𝑇_𝑖)^𝐺 is the generalized inverse matrix of 𝑌_𝑖𝑌^𝑇_𝑖.

Step 2: (manifold reconstruction)

(1) Mapping from original space to latent space: 𝑔∶𝑋→𝑌,  𝑦=𝑔(𝑥).

Given a high-dim pose vector 𝑥₀, the corresponding low-dim vector 𝑦 can be computed as:

(1.1) Find the nearest neighbor of 𝑥₀ in {𝑥_𝑖∣𝑖=1,…,𝑙}, set it to be 𝑥_𝑠;

(1.2) The low-dim vector correspondence to 𝑥₀ can be formulated as:

𝑦=𝑔(𝑥₀)=𝑦_𝑠+𝑄^𝑇_𝑠(𝑥₀−𝑥_𝑠);

(2) Mapping from latent space to original space: 𝑓∶𝑌→𝑋,  𝑥=𝑓(𝑦).

Given a low-dim pose vector 𝑦₀, the corresponding high-dim vector 𝑥 can be computed as:

(2.1) Find the nearest neighbor of 𝑦₀ in {𝑦_𝑖∣𝑖=1,…,𝑙}, represented as 𝑦_𝑠;

(2.2) The high-dim vector correspondence to 𝑦₀ can be formulated as:

𝑥=𝑓(𝑦₀)=𝑥_𝑠+𝑄_𝑠(𝑦₀−𝑦_𝑠).