The purpose of this work is to develop an image-based de-noising algorithm that exploits complementary information and noise statistics from multi-modal images, as they emerge in x-ray tomography techniques, for instance grating-based phase-contrast CT and spectral CT. Among the noise reduction methods, image-based de-noising is one popular approach and the so-called bilateral filter is a well known algorithm for edge-preserving filtering. We developed a generalization of the bilateral filter for the case where the imaging system provides two or more perfectly aligned images. The proposed generalization is statistically motivated and takes the full second order noise statistics of these images into account. In particular, it includes a noise correlation between the images and spatial noise correlation within the same image. The novel generalized three-dimensional bilateral filter is applied to the attenuation and phase images created with filtered backprojection reconstructions from grating-based phase-contrast tomography. In comparison to established bilateral filters, we obtain improved noise reduction and at the same time a better preservation of edges in the images on the examples of a simulated soft-tissue phantom, a human cerebellum and a human artery sample. The applied full noise covariance is determined via cross-correlation of the image noise. The filter results yield an improved feature recovery based on enhanced noise suppression and edge preservation as shown here on the example of attenuation and phase images captured with grating-based phase-contrast computed tomography. This is supported by quantitative image analysis. Without being bound to phase-contrast imaging, this generalized filter is applicable to any kind of noise-afflicted image data with or without noise correlation. Therefore, it can be utilized in various imaging applications and fields.