A Joint Learning Approach to Face Detection in Wavelet Compressed Domain

Algorithm 2

The proposed AdaBoost-based learning algorithm for face detection.

(i) Given example images (𝐱₁,𝑦₁),…,(𝐱_𝑛,𝑦_𝑛), where 𝑦_𝑖 takes the value 0 for negative examples or 1

   for positive examples, respectively.

(ii) Initialize weights 𝑤_1,𝑖=1/(2𝑚) for 𝑦_𝑖=0 or 𝑤_1,𝑖=1/(2𝑙) for 𝑦_𝑖=1, where 𝑚 and 𝑙 denote

   the total numbers of negative and positive images, respectively.

(iii) For 𝑡=1,…,𝑇

 (1) Estimate feature space warping function 𝑊_𝑗(𝑥_𝑖) via the sample distribution [𝑥_1,𝑗,𝑥_2,𝑗,…,𝑥_𝑛,𝑗]

   and weights 𝐰_𝐭

 (2) For each possible feature pair (𝑗,𝑘), map all training sample onto paired plane via

    warped feature value 𝑊_𝑗(𝑥_𝑖) and 𝑊_𝑘(𝑥_𝑖)

 (3) Apply ID3-like tree method to each axis of paired feature plane in rotation, and find

    the quantization function 𝑓_𝑗,𝑘(𝑥_𝑖) which will try to separate positive and negative

    samples into different bins.

 (4) Compute the conditional probability as the Bayesian classification result for each

    weak classifier 𝐶_𝑗,𝑘(𝑥_𝑖).

 (5) Estimate the error 𝜀_𝑗,𝑘 for each feature pair (𝑗,𝑘) as follows:

             𝜀_𝑗,𝑘=∑^𝑛_𝑖=1𝑤_𝑡,𝑖×𝑦_𝑖−𝐶_𝑗,𝑘𝑥_𝑖

 (6) Select the paired feature ℎ(𝑡) with minimum error 𝜀_𝑗,𝑘

              ℎ(𝑡)=argmin_𝑗,𝑘𝜀_𝑗,𝑘

 (7) Update the weights for all training samples as follows:

        𝑤_𝑡+1,𝑖=𝑤_𝑡,𝑖×𝛽^{1−|𝑦𝑖−𝑐ℎ(𝑡)(𝑥𝑖}_𝑡^)|,

    where 𝛽_𝑡=𝜀_ℎ(𝑡)/1−𝜀_ℎ(𝑡).

 (8) Normalize the weights by

            𝑤_𝑡+1,𝑖=𝑤_𝑡+1,𝑖∑^𝑛_𝑖=1𝑤_𝑡+1,𝑖

(iv) The final classifier is given by

         SC(𝑥)=facewhen^𝑇_𝑡=1𝛼_𝑡𝐶_ℎ(𝑡)1(𝑥)≥2^𝑇_𝑡=1𝛼_𝑡non-faceotherwise,

  where 𝛼_𝑡 = log1/𝛽_𝑡.