Core 11 min 2 of 3

Scale, shift, and the alignment protocol

A two-parameter fit stands between the model output and every number in the results table.

What this repairs

Comparing depth metrics across papers without knowing whether the alignment was scale-only, affine, in depth space, or in disparity space.

Assumes

What the network actually predicts

If a model's output is only defined up to an unknown transformation, the transformation has to be estimated before any error can be computed. For an affine-invariant model the estimation is a two-parameter least-squares fit:

$\min_{a,\,b} \sum_i \left(a\,\hat{d}_i + b - d_i^{\text{gt}}\right)^2,$

which is an ordinary straight-line fit, usually performed in disparity space because that is where the ambiguity was introduced. That is all "scale and shift alignment" is. It sounds mystical in a paper and it is a line fit.

Median scaling, and why it is not least squares

$s^{\star} = \operatorname{median}_i\!\left(\frac{Z_i^{\text{gt}}}{\hat{Z}_i}\right)$

The common "median scaling" protocol: one parameter, no shift, chosen as a median rather than a mean.
$s_{\text{LS}} = \frac{\sum_i \hat{Z}_i Z_i^{\text{gt}}}{\sum_i \hat{Z}_i^{2}}$

The least-squares scale, for comparison. It is dominated by the largest depths, which are exactly the least reliable ones.
$s^{\star} \ne s_{\text{LS}} \;\text{ whenever the ratio distribution is skewed}$

And it always is. The median is robust to the far-field tail, which is why the protocol chose it — a robustness decision hiding inside what looks like a normalisation step.

Check yourself

Why does adding a shift parameter make a model look better even when its predictions are unchanged?

Show answer

Because you have given the fit an extra degree of freedom to absorb error. Affine alignment can never do worse than scale-only alignment on the same predictions — it optimises over a strictly larger family. The improvement is real as a measure of relative structure and it is not evidence that the model improved. Comparing a scale-aligned number to an affine-aligned one is comparing two different questions.