Historically first training algorithms for cascades of classifiers were guided by constant per-stage requirements (constraints) imposed on false alarm and detection rates. Those constant values were calculated according to the geometric mean rule, implied by a pair of final requirements predefined for the whole cascade.

We provide and prove a theoretical result demonstrating that the presence of slack between the constant requirements and actual rates observed while learning, allows to introduce new relaxed requirements for each successive stage and still complete the training procedure successfully (with final requirements satisfied). The relaxed requirements can be met more easily, using fewer features. This creates a potential possibility to reduce the expected number of features used by an operating cascade — the crucial quantity we focus on in the paper. Taking advantage of the relaxation, we propose new stage-wise training algorithms that apply two approaches: uniform or greedy. They differ in the way the slack cumulated so far becomes “consumed” later on.

Reported experimental results pertain to cascades trained to work as face or letter detectors, with Haar-like features or Zernike moments being the input information, respectively. The results confirm shorter operating times of cascades obtained by the proposed technique, owing to the reduction in the number of extracted features.