[GRASS-user] g.transform: forward and reverse rmse

[Glynn Clements]

Maciej Sieczka wrote:

> > Note that, unless order=1, the two transformations will typically not
> > be exact inverses as the inverse of a polynomial containing quadratic
> > or higher terms is not itself a polynomial.
> 
> Even for order=1 I get rather huge differences between forward and 
> reverse. Normal? Order 1-3 examples for a single GCP set, g.transform -s 
> group=tmp format=idx,fd,rd:
> 
> # Order=1:
> 
> index forward reverse
>   0 140.159347 30.366192
>   1 75.549553 16.157024
>   2 20.384295 4.152418
>   3 88.853688 19.265900
>   4 88.257880 19.471545
>   5 28.997927 6.360764
>   6 96.951029 21.909275
>   7 98.251481 21.344657
>   8 75.633328 17.289931
>   9 55.503010 12.158829
>   10 52.191730 11.329603
>   11 121.973315 27.215642
>   12 69.859427 15.239316
>   13 137.981202 29.362434
> Number of active points: 14
> Forward:
> x[13] = 127.41
> y[11] = 119.24
> g[0] = 140.16
> RMS = 89.31
> Reverse:
> x[13] = 28.74
> y[11] = 25.14
> g[0] = 30.37
> RMS = 19.52

Note that the forward/reverse ratio only varies between 4.37 and 4.90,
so there's a high degree of correlation between the two. Each set of
numbers is measured in the corresponding coordinate system, so if you
have e.g. pixels for the source, metres for the destination, and a
resolution of 5 metres/pixel, you would expect the forward errors (in
metres) to be roughly 5 times the reverse errors (in pixels).

The above figures would make sense for a scale factor of around 4.5.

> # Order=2:

Ratio = 3.97 - 4.92

> # Order=3:

Ratio = 3.55 - 5.70

For higher-order transformations, I would expect more divergence if
the "exact" transformation is noticeably non-linear, as one of the
transformations will often be a better fit to a polynomial than the
other (e.g. for y = x^2 <=> x = sqrt(y), the former can be represented
exactly while the latter can only be approximated).

-- 
Glynn Clements <glynn at gclements.plus.com>