The officer, Eder Loor, 28, walked onto a stage at a hospital auditorium and spoke slowly but clearly as he thanked the doctors who saved his life. A long scar that stretched across the left side of his head bore testimony to his injury. “I don’t know what to say, but I’m really happy for the medical team here,” he said.

The officer’s recovery from an attack that could have ended his life has been dramatic. The hospital said he showed “excellent balance and endurance” and went jogging Tuesday in Central Park.

On April 17, Officer Loor was escorting an emotionally disturbed man to a hospital when the man, Terrence Hale, drew a knife and plunged it into the officer’s temple and down through his brain to the base of his skull, missing inflicting fatal injury by a fraction of a millimeter, the police and doctors have said.

Officer Loor, who is also an emergency medical technician, pulled the blade out of his head and started to apply pressure before paramedics arrived. Mr. Hale has been charged with attempted murder.

The officer said he did not remember most of what had happened, but Dr. Joshua B. Bederson, the surgeon who treated him, said Officer Loor’s quick thinking and the fast decisions by the hospital’s staff helped save his life.

Despite Officer Loor’s progress, doctors said, he will have a long road to a full recovery. Dr. Kristjan T. Ragnarsson, the chairman of the hospital’s rehabilitation department, said that Officer Loor had some numbness on the left side of his face and low energy, but that those problems should be temporary.

The officer will start his rehabilitation next week, consisting of physical therapy, endurance training and neurological training, Dr. Ragnarsson said.

Officer Loor, a six-year veteran of the police force, said he did not know if he would return to patrolling the streets.

His pregnant wife, Dina Loor, said his release was well timed: Wednesday was their daughter’s fifth birthday.

“That is all she wanted,” Ms. Loor said.