Since I spent a bigger amount of time reasarching on how to make the Validity fingerprint reader (VFS451) on my HP ProBook 4520s working together with Windows 10 and its "Hello" login function, I want to share my solution. Normally, I am only used to fiddle around with device drivers on Linux, but this time Microsoft left its customers without support.

I have read several forum and blog post about fingerprint reader issues on Windows 10, but nothing seemed to work. I tried to uninstall and reinstall the driver in the device manager, but without success. I also found different driver version on the internet and tried them. Some seemed to work for one scan, but failed on the second. Thus, I assume that with these drivers both scans didn't match, even the scan result was usable.
However, finally I found a driver that is working. The proper version is: 4.4.234.00.
Windows Hello could finally register my fingerprints and the login is working without problems.

You can find this drivers on the HP servers http://ftp.hp.com/pub/softpaq/sp58501-59000/sp58869.exe
After downloading and unpacking the driver, you can just manually install it in the device driver.
How you can do this, you can find here.

I hope this might help other people as well. If you have a question, you can write me.