Gradient skewness tensors and local illumination detection for images

In this paper, we propose the definition of D$D$-eigenvalue for an arbitrary order tensor related with a second-order tensor D$D$, and introduce the gradient skewness tensor which involves a three-order tensor derived from the skewness statistic of gradient images. As we happen to find out that the skewness of oriented gradients can measure the directional characteristic of illumination in an image, the local illumination detection problem for an image can be abstracted as solving the largest D$D$-eigenvalue of gradient skewness tensors. We discuss the properties of D$D$-eigenvalues, and especially for gradient skewness tensors we provide the calculation method of its D$D$-eigenvalues and D$D$-characteristic polynomial. Some numerical experiments show its effective application in illumination detection. Our method also presents excellent results in a class of image authenticity verification problems, which is to distinguish artificial “flat” objects in a photograph.