I am an artist who paints from photos. I would like to use clips from video of films and TV shows. I wouldn't copy the exact image, but if you were to put the two images together, you would see a strong similarity. Do I need permission?
Copyright law gives creators certain exclusive rights. Among these is the right to create derivative works. Derivative works include, for example, a sequel to a movie that you made, or a second book based upon the leading character in your first book.
By your description, your paintings could be considered "derivative," that is, based on or derived from other original works. The copyright owner (in this case, the videographer or movie studio) has a right to control the making of derivative works.
This can be a fairly common issue for artists. Several years ago, artist Jeff Koons based one of his sculptures on a photograph he had purchased in a gift shop. The photographer later sued Koons and was awarded several hundred thousand dollars. Painter Robert Rauschenberg also had to pay damages when a photographer sued over his use of a photograph in one of Rauschenberg's collages. So, if your use is discovered and a court determines it is an infringement of the original video, you could face liability.
Importantly, a mere similarity is not infringement. It's impossible to say how much copying or similarity is permitted, since these determinations are done on a case-by-case basis. But if, as you say, there is a strong similarity between your work and the original photograph, liability is more likely.
However, a legal principle known as "fair use" may provide a basis for your defense. Under fair use, you are permitted to copy an image if you do so for commentary, criticism, or parody. As an artist, you could certainly argue that your painting was a commentary on the original film.
Of course, as a practical matter, it might be unlikely that your painting will be discovered by the copyright owner who created the original video. The examples mentioned above were lawsuits against big-name artists, and that's probably no coincidence. If your work slips under the copyright owner's radar screen, then you're unlikely to have a problem. Unless you are advertising the painting, or otherwise attracting attention, it is relatively improbable that a videographer would discover a local artist's single use of a video frame.