Check the box on the ticket that indicates that you want a clerk's hearing. You will receive a notice in the mail from the court telling when date of the clerk's hearing. Prior to the hearing, take some pictures of the intersection. When you arrive in court that day, pay your $25 at the clerk's office and wait for your case to be called. The officer who wrote you the citation will not usually be at the clerk's hearing, and the police prosecutor will stand in the place of the citing officer. If you are found "responsible" by the clerk, you can appeal the matter to a judge for $50. The officer will have to appear for the judge hearing; if not, you will be found not responsible. Side note: clerk's sometimes are of the mindset that if the defendant shows up with a lawyer, the defendant is taking the matter seriously and/or has paid enough in legal fees, and may find the defendant "not responsible" for the traffic violation on that rationale alone.
The act of entering an intersection is either passing the limit-line or delineation of the perpendicular curbs if there is no limit-line. It sounds as if you "entered the intersection," on a green light or maybe yellow, but not on a red. Coming through an intersection last is not a violation, unless done on a red light. There are jurisdictions that have a separate violation for entering an intersection when the intersection was not clear. Clearly, use of the rear-view mirror to observe a violation is questionable at best. There are attorneys who specialized in traffic cases, and it would be prudent to consult one in your area. There are also several publications on how to fight a traffic ticket which are a wealth of knowledge. Many traffic cases are defeated on procedure over facts, however it sounds as if you have good facts to contest.