Higher Order Local Autocorrelation Features

It is well known that the autocorrelation function is shift-invariant. Its extension to higher orders has been proposed in [12]. The Nth-order autocorrelation functions with N displacements ${\bf a}_1, \ldots,{\bf a}_N$ are defined by

$$x^N(m) = \int P^m_{<{\bf r}>} P^m_{<{\bf r}+{\bf a}_1>} \cdots P^m_{<{\bf r}+{\bf a}_N>} d{\bf r},$$ (5)

where functions $P^m_{<{\bf r}>}$ denotes the mth order PARCOR coefficient of pixel $<{\bf r}>=<x,y>$.

Since the number of these autocorrelation functions obtained by the combination of the displacements over the PARCOR images P^m are enormous, we must reduce them for practical application. First, we restrict the order N up to the second (N=0,1,2). We also restrict the range of displacements within a local $3 \times 3$ window, the center of which is the reference point. By eliminating the displacements which are equivalent by the shift, the number of patterns of the displacements are reduced to 25.

  

Figure 2: Local mask patterns for computing HLAC features.

Fig.2 shows the patterns, where the symbol ``*'' represents ``don't care''. The number of features is then 35 including all combinations of up to second order autocorrelations. We call these features Higher Order Local Autocorrelation (HLAC) Features features [6,7]. Since these features are obviously invariant to the shift in PARCOR image, the gesture recognition system becomes robust to changes of the position of the person within a image frame.