X-ray-scattering tensor tomography requires dark-field measurements in the projection domain that are both unbiased and rotation equivariant. We show that, in the context of circular-grating-based tensor tomography, the commonly used cell-based chord retrieval (CCR) cannot satisfy either requirement. From the two-dimensional convolution model of circular gratings, we derive a closed-form expression demonstrating that CCR introduces an illumination-coupling term that directly contaminates the measured dark-field amplitude. This illumination-dependent contribution is fixed in the grating frame and therefore does not co-rotate with the sample, producing orientation-dependent errors and breaking rotation equivariance. In contrast, global two-dimensional Fourier transform retrieval cancels illumination through the convolution theorem and is inherently rotation equivariant. Experiments with a laboratory circular-grating-based tensor tomography system confirm both effects. Illumination coupling is thus an intrinsic limitation of CCR and prevents it from achieving rotation-faithful quantitative anisotropic dark-field imaging.

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• Imaging & optical processing
• Tomography
• Amorphous materials
• Tissues
• Medical X-ray imaging
• Small-angle x-ray scattering
• X-ray imaging
• X-ray tomography