Notes on Expected Computational Cost of Classifiers Cascade: A Geometric View

A cascade of classifiers, working within a detection procedure, extracts and uses different number of features depending on the window under analysis. Windows with background regions can be typically recognized as negative with just a few features, whereas windows with target objects (or resembling them) might require thousands of features. The central point of attention for this paper is a quantity that describes the average computational cost of an operating cascade, namely — the expected value of the number of features the cascade uses. This quantity can be calculated explicitly knowing the probability distribution underlying the data and the properties of a particular cascade (detection and false alarm rates of its stages), or it can be accurately estimated knowing just the latter. We show three purely geometric examples that demonstrate how training a cascade with sensitivity / FAR constraints imposed per each stage can lead to non-optimality in terms of the computational cost. We do not propose a particular algorithm to overcome the pitfalls of stage-wise training, instead, we sketch an intuition showing that non-greedy approaches can improve the resulting cascades.