I converted my comments to an actual answer:
If I understand your setup correctly, you're saying you have a line scan camera mounted to the top of the rotating head of a laser scanner, and all you're looking for is the one-time transform matrix between the scanner and camera coordinate frames. Is this correct? If so, why not just perform the test in a controlled environment where the features are easy to discern?
[The spectral camera doesn't really have] a 1d image if you have the encoder feedback from the Riegl. But yes, if you could generate a shape (easily detectable by the laser scanner) that also had some interesting spectral content (easily detectable by the hyperspectral camera), then you could compare encoder counts where the same feature were located in both systems.
As your spectral camera appears to cover the visible spectrum, you could use an exterior door (presumably the wall is one color, and outside is "noisy" colors), or a table, or an open window, or a or a monitor or any colored, physical object.
I don't know if you're trying to mass produce something, but probably the easiest way to do this is just to manually align the two outputs until you have good coherence. I don't know if this is the point of what you're trying to do or not, but again you have visible spectrum data from your line camera. You could segment out the RGB information from that data, use it to texture your point cloud, and then tweak the transform values until your texture overlay "fits" with the depth information you've got from the laser scanner.
Of course if you're mass producing the thing and/or you plan on constantly re-aligning the data, then you would probably want some way to automate this, but automation is not always the answer.
It would be really neat if it could align itself, but please please consider the time cost of just doing the alignment manually versus the time cost of writing a custom alignment algorithm for scratch. If you have a rigid coupling between the two sensors (your project doesn't seem to be hurting for money!!) then you shouldn't need to do the alignment more than once.