Reducing integrability error of color tensor gradients for image fusion

To overcome the difficulties in applying gradient-based operators to color images, Di Zenzo introduced the color tensor, an operator that provides a gradient field for multichannel images. An elegant application for this operator was developed in the domain of multichannel image visualization: Socolinsky and Wolff proposed to reintegrate Di Zenzo's gradient by solving a Poisson equation, yielding a greyscale representation of the multispectral contrast of the input image. Di Zenzo's gradients are, however, generally not integrable and some approximation must be introduced. Thus, the resulting image can suffer from artifacts such as the smearing of edges. In this paper, we focus on the integrability of Di Zenzo's gradients. We show that the integrability of the obtained field can be improved dramatically through a simple desaturation of the color image (as in the HSV color space). This result can be readily extended to multispectral images by defining an analogue to saturation. We present several results explaining what happens to color tensors as the saturation changes. Significantly we show that small changes of the saturation in the linear image space can result in large improvements in the integrability of tensor gradients calculated in logarithmic color space. This result is important for two reasons. 1) Log-differences are more perceptually meaningful. 2) In log-space we can operate with retinex algorithms, which are well known techniques for contrast enhancement. We propose that they can be used to “put back” any contrast that might be lost in the desaturation step and, more importantly, they can enhance contrast at the same time as reintegrating the gradient field because of their relation to partial differential equations. Finally, we evaluate our method psychophysically. Compared with other commonly used image fusion methods, experiments show that our data fusion using the Di Zenzo color tensor after desaturating the image and where a simple contrast boost is applied is strongly preferred.