visualizing a tensor field

I am working with Soledad Villar (JHU) and others on making generalizations of convolutional operators (and image-based non-linear functions based on those convolutions) that can deal (correctly) with input data that contain vectors and tensors. That is, tensor convolutions of tensor images. Anyway, one of the problems is: How do you visualize a tensor field or an image of tensors? I implemented a possible solution, pictured below: You make a figure that has no symmetry, and you take that figure through the tensor! That only works for 2-tensors of course. 3- and 4-tensors? I'm at a loss.