Abstract. We demonstrate a technique allowing for constant-time calculation of low order Fourier moments, applicable in detection tasks. Real and imaginary parts of the moments can be used as features for machine learning and classification of image windows. The technique is based on a set of special integral images, prepared prior to the scanning procedure. The integral images are constructed as cumulative inner products between the input image and suitable trigonometric terms. Additional time invested in the preparation of such integral images is amortized later at the stage of scanning. Then, the extraction of each moment requires only 21 operations, regardless of the number of pixels in the detection window, and thereby is an O(1) calculation.As an application example, face detection experiments are carried out with detectors based on Haar-like features serving as opponents to the proposed Fourier-based detectors.
IntroductionConstant-time computational complexity is the most attractive complexity for a computer scientist. Unfortunately, favourable opportunities to apply algorithms of that complexity are rare -typically, they pertain to some selected data structures e.g. hash tables, Union-Find 1 [2] and constitute a narrow fragment of a larger software. Often, one deals in fact with a so-called amortized constant-time complexity. This means that in the company of essential operations, performed are also some auxiliary operations meant to guarantee the speed for the future.Not so long ago an algorithmic idea of that class has appeared in the field of computer vision and works remarkably well -namely, the idea of Haar-like features due to Jones (2001, 2004) [9,10]. Haar-like features are now commonly applied to detect objects (faces, people, vehicles, road signs, etc.)
in digitalThis work was financed by the National Science Centre, Poland. Research project no.: 2016/21/B/ST6/01495. 1 For strictness: the 'Find' operation in this data structure is of amortized complexity O(log * n) -iterated logarithm of n. Wherein log * 2 n is not greater than 5 for all quantities n observable in the universe; in particular, log * 2 2 65536 = 5.