Implementation details

It is common for those wishing to emulate biological vision systems to implement the local energy model using Gabor filters. The (odd-symmetric) Gabor function is a sine wave modulated by a Gaussian, as shown in figure 10. Its quadrature partner, in the spatial domain, is not simply a cosine modulated by a Gaussian, since that fails to satisfy the condition of identical sums-of-squares norm. However, it is simple enough to design a quadrature pair in the Fourier domain by simply shifting the signal by 90^o, and then back-transforming into the spatial domain.

 

Figure 10: The odd symmetric Gabor filter.

The odd and even symmetric filters are extended to 2D by a Gaussian spreading function. Local energy is implemented by computing an energy function in each of a number of orientations, usually between six and ten. Thus, for each orientation i, we compute

$$E_i(x) = \sqrt{{M_{e,i} * f(x)}^2 + {M_{o,i} * f(x)}^2},$$

and then

$$E(x) = \sum_{i} E_i(x).$$

A wavelet implementation of local energy has been given by Xie [17] and a wavelet implementation of phase congruency has been given by Kovesi [4]. Venkatesh gave the first 2D implementation of local energy in two dimensions in her Ph.D. thesis [16].