To define our measure, we introduce some notation. From Eq. (10), is the normalized DCT of of size , and , for , 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 be the set of DCT coefficients located between and angles, for . The function takes as input , performs orthogonal projection of all its elements along vector , and averages the elements that after projection fall on the same discrete coordinates. With we seek to compact the information around the three main orientations in a one-dimensional vector of elements. To illustrate, let us compute , where is the set of coefficients located between and . In Fig. 3(b), we show the projection of the coefficient with coordinates (4,2) along . After projection, this coefficient has coordinates (4,1). Therefore, the element . Consequently, we can stack all to form the following matrix, Display FormulaNote 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 we compute the variance of the weighted sum of the columns, computed as the matrix product , Display Formula
(12)where , E is the expected value, and is the mean of the matrix product . The vector can be regarded as a weighting procedure and with it we aim to achieve robustness to noise and illumination variation.