There is no set amount of manipulation or alteration of a copyrighted piece of work that will get you to a point where you can consider the work to be your own or 'safe' from infringing the original copyright.
Many people will tell you that there are rules regarding a certain number of 'points of difference' (I've heard 7 and 5 quoted for this number) which need to exist between the original and the copy in order for copyright to not apply. This is categorically not the case.
There are also theories regarding the percentage of your design that is made up of elements borrowed from copyrighted material. The idea being that if only a small percentage of your design is identifiable as being part of the source material then that's acceptable. This is also not true.
The copyright rules for this kind of work are quite simple: If there is anything left in your creation that would allow anyone to identify the source material, then you have infringed the copyright of the person who created the original image. End of story. Regardless of whether you have distorted, blurred, recoloured, flipped, inverted or completely redrawn the original; if it still looks like the same thing (or part thereof), then it's the same thing.
That said, there are images that are in the public domain to which copyright does not apply so using these as your start point would be a safe route. Also, if you have manipulated and worked on your design to the point that nothing of the original remains, then you should be fine.