WEST ORANGE, N.J. — Nearly a year ago, a New York City police officer, Dalsh Veve, was flung from a fleeing car in Brooklyn and injured so grievously his family and colleagues did not know whether he would live or die.

But the officer hung on, despite a catastrophic brain injury and a grave prognosis. For his bravery, the police commissioner, James P. O’Neill, promoted him to detective in October.

Months passed. Detective Veve’s medical condition slowly improved as he was transferred from Kings County Hospital Center in Brooklyn to the Kessler Institute for Rehabilitation in West Orange, N.J. In time, he began to recognize visitors.

On Monday, his wife, Mathes Esther Veve, a nurse, escorted him out of the picturesque rehab facility and into a minivan for a trip home to continue his recovery at their house on Long Island, where the neighbors always hoped to see him again.