  Deep neural networks (DNN) abstract by demodulating the output of linear
filters. In this article, we refine this definition of abstraction to show that
the inputs of a DNN are abstracted with respect to the filters. Or, to restate,
the abstraction is qualified by the filters. This leads us to introduce the
notion of qualitative projection. We use qualitative projection to abstract
MNIST hand-written digits with respect to the various dogs, horses, planes and
cars of the CIFAR dataset. We then classify the MNIST digits according to the
magnitude of their dogness, horseness, planeness and carness qualities,
illustrating the generality of qualitative projection.
