GLCM Correlation
The Correlation texture measures the linear dependency of grey levels on those of neighbouring pixels.GLCM Correlation can be calculated for successively larger window sizes. The window size at which the GLCM Correlation value declines suddenly may be taken as one definition of the size of definable objects within an image. People familiar with autocorrelation will see a resemblance to the information provided by them and by GLCM Correlation. Van der Sanden and Hoekman (2005) have demonstrated that GLCM Contrast is identical to semivariance, and GLCM Correlation provides almost identical information as provided by autocorrelation methods using Moran's I or Geary's C.
GLCM Correlation is quite a different calculation from the other texture measures described above. As a result, it is independent of them (gives different information) and can often be used profitably in combination with another texture measure. It also has a more intuitive meaning to the actual calculated values: 0 is uncorrelated, 1 is perfectly correlated.
Practical calculation note: Thanks to Sean Peters of Big Hammer Data for pointing this out.
When an image area is completely uniform, the GLCM variance is 0, just as the first-order image variance is 0. As a result, the denominator of the correlation equation becomes 0, and correlation becomes undefined. In my experience, this has occurred in near infrared images for lakes above a certain size. Provision for what to do in this case varies from software to software. If a software refuses to complete the correlation calculation, or returns error values in some pixels, it is possible that uniform window areas have not been provided for. In other cases, often the default undefined result value is 0. In the case of correlation, it would be more logical to have the undefined value set to 1, as the correlation among the original pixel values is perfect. Without access to software code, you can't do much about this problem: at least be aware if you have large uniform areas that the overall correlation results may be unexpected! If you are writing your own code or using open-source software, you might want to fix it.
Why GLCM variance is 0 when an area is uniform: This may appear obvious, but to be more rigorous here is the reason. In a normalized GLCM, a uniform image area has a single entry of 1 in the diagonal corresponding to the row and column headed by the original GL value in the pixels of the image (e.g. if all GL values are 2, "1" appears in position i = j = 2 in the GLCM). The mean then becomes μ = i = j (= 2 in the example given)When calculating the GLCM variance, Pij = 0 for every value except the single entry of 1. The formula collapses to = 1(i - μ)2, but since μ = i, this becomes 1 (i=i)2 = 0.
GLCM Correlation equation:
Exercise: Calculate the GLCM Correlation measure for the horizontal test image.