To define our measure, we introduce some notation. From Eq. (10), $G\u02dc(u,v)$ is the normalized DCT of $g(x,y)$ of size $N\xd7N$, and $\lambda j$, for $j=1,2,3$, is a vector along one of the three main orientations of the spectrum depicted in Fig. 3. We will restrict our study to angular partitions of the spectrum roughly equivalent to vertical, diagonal, and horizontal components of the image space. Our measure of anisotropy mainly consists in calculating a difference of weighted coefficients along these orientations. Let $G\u02dcj={G\u02dc(u,v):\theta =arctan(vu),\theta j\u2264\theta <\theta j+1,j=1,2,3}$ be the set of DCT coefficients located between $\theta j$ and $\theta j+1$ angles, for $\theta j\u2208{0\u2009\u2009deg,30\u2009\u2009deg,60\u2009\u2009deg,90\u2009\u2009\u2009deg}$. The function $\psi \lambda j(.)$ takes as input $Gj\u02dc\u2009$, performs orthogonal projection of all its elements along vector $\lambda j$, and averages the elements that after projection fall on the same discrete $(u,v)$ coordinates. With $\psi \lambda j(.)$ we seek to compact the information around the three main orientations in a one-dimensional vector of $N$ elements. To illustrate, let us compute $\psi \lambda 1(G1\u02dc)=[\psi \lambda 11,\psi \lambda 12,\u2026,\psi \lambda 1N]T$, where $G1\u02dc\u2009$ is the set of coefficients located between $\theta 1=0\u2009\u2009deg$ and $\theta 2=30\u2009\u2009deg$. In Fig. 3(b), we show the projection of the coefficient with coordinates (4,2) along $\lambda 1$. After projection, this coefficient has coordinates (4,1). Therefore, the element $\psi \lambda 14=mean[G\u02dc(4,1),G\u02dc(4,2)]$. Consequently, we can stack all $\psi \lambda j$ to form the following matrix, Display Formula
$\Psi =[\psi \lambda 11\psi \lambda 21\psi \lambda 31\psi \lambda 12\psi \lambda 22\psi \lambda 32\vdots \vdots \vdots \psi \lambda 1N\psi \lambda 2N\psi \lambda 3N].$
Note that the first element of each vector corresponds to the dc coefficient. This coefficient does not convey any directional information of the image; however, we decided to keep it in the matrix for the sake of completeness. To obtain a measure of anisotropy—the FM itself—from $\Psi $ we compute the variance of the weighted sum of the columns, computed as the matrix product $w\Psi $, Display Formula$Sa(g)=Var(w\Psi )=E[(w\Psi \u2212\mu )2],$(12)
where $w=[w1,w2,\u2026,wN]$, E is the expected value, and $\mu $ is the mean of the matrix product $w\Psi $. The vector $w$ can be regarded as a weighting procedure and with it we aim to achieve robustness to noise and illumination variation.