Almost anything is possible with hair but what you're asking to accomplish does have some definite challenges.  You point out that you want a "permanent" solution.

 

Unfortunately unless you were born with the appropriate natural hair texture to grow into a classic Afro like the photo you supplied (good photo by the way), no matter what you do to your hair (chemical perm or other treatments) your hair will eventually grow back to it's natural texture.

 

This is the mistake many people who get their hair chemically straightened make.  They assume because chemicals straighten their hair, it will be a permanent thing.  Well it will be permanent to the hair that has already extended from their roots but once the roots grow in, all bets are off.

 

I would suggest you take a copy of the photo to a hairdresser you trust and ask them what steps you would need to take to get your hair to look like the one in the image.

 

They can advise you of your various options and how long your hair would need to be to accomplish the look.

 

I can tell you that I knew a guy back in St. Louis who was able to achieve a similar look on his stick straight fine strands (with the help of a hairdresser) and he would maintain it with products.  He would get a perm every time his roots grew back in to a certain length.

 

The good news, with some work you can probably copy this look.  The bad news...you will have to continue to maintain it or else your hair will revert to your natural type and texture.

 

Hope that helps.