A homeless man "cold-bloodedly" chased and shot two policemen to death Friday before he was killed by other officers in a noontime gun battle recorded by a student television crew.

The man, a transient, grabbed the gun of an officer questioning him outside a doughnut shop, then chased the officer and opened fire, wounding him and then fatally shooting him in the head, police and witnesses said.The homeless man then mortally wounded a second officer before other police officers arrived and killed him a long series of shots from pistols and shotguns, witnesses and police officials reported.

The second officer died about six hours after the shooting, after undergoing three hours of surgery at the San Jose Medical Center, doctors said.

Coroners officials withheld the officers' names until their families could be notified.