Probably not as your mug shot is a public document not owned by you. Very creative thinking though.
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No, you were not the photographer.
J Charles Ferrari Eng & Nishimura 213.622.2255 The statement above is general in nature and does not constitute legal advice, as not all the facts are known. You should retain an attorney to review all the facts specific to your case in order to receive advise specific to your case. The statement above does not create an attorney/client relationship. Answers on Avvo can only be general ones, as specific answers would require knowledge of all the facts. As such, they may or may not apply to the question.
In Utah, there's a new law (House Bill 408), which prohibits county sheriffs from giving mugshots to people who will post it online and demand a fee for removing it. To get around that law, it seems that many websites offer to remove your picture "for free" if you ask them, (they just charge a ridiculous amount to do it quickly, calling it "expedited service.") You might want to look at the websites fine-print, to see if there's a free take-down option.
Also, many of these sites will take down your photo for free if the charges have been dismissed or expunged, so that's a good reason to talk to a local defense attorney about your case if you haven't already. Good luck!