First, here's the actual paper, since it clarifies what exactly he's suggesting and doesn't seem to be linked anywhere in the article.
It's not a suggestion that we start using non-square pixels for displays or cameras or scanners or what not, though he's certainly not being very clear about anything and the reporting on this is just making matters worse. What the paper proposes is a method where:
1) The image is split into 6x6 blocks
2) For each block, you go over the four rotations of the two following two-section masks:
The triangular mask:
The rectangular(ish) mask:
for a total of eight effective masks, and average the values under each section, resulting in two values, A and B.
3) For the mask and rotation that has the largest difference between A and B, you output the mask, the rotation, and the A and B values, resulting in 19 bits from a 6x6 (288 bits) block.
Though he talks of non-square pixels and whatnot, it's really just a compression algorithm. A really stupid one. Basically it's a bad variation of vector quantization, with lots of baffling details. Why 6x6 blocks? Why those specific masks? Why are you maximizing contrast instead of minimizing error like any sane person would do, WHY? There's no rationale given for any of these choices, not theoretical, not empirical, not even subjective.
The same sort of rigor extends to his comparison, where he compares his compression algorithm to, instead of, say, another compression algorithm, the image apparently simply downscaled and then scaled back up. And not even with a halfway decent resampling algorithm, but with nearest neighbour. Not to mention that the "non-square pixels" version has 2.375 times as many bits to work with. If he'd done a comparison to a reasonably modern compression algorithm like JPEG, the results would be much less favorable to him.
tl;dr Some old guy put together his My First Compression Algorithm kit and it's being treated like a revolution in graphics by ignorant reporters. Nothing to see here, move along.