In some ways, the shooting death of Mr. Hill was defined by contrasting images of two men in uniform: Mr. Hill, a 27-year-old Air Force veteran who had bipolar disorder and who had struggled with trauma sustained while serving in conflict; and Mr. Olsen, who had been with the DeKalb County Police Department for seven years and whose lawyers described him as “a good cop who had to make a tough decision in those few seconds.”

On March 9, 2015, Mr. Olsen was responding to 911 calls from Mr. Hill’s neighbors, who had reported him walking around undressed and acting bizarrely in public areas of his suburban apartment complex east of Atlanta. Lawyers in court said that Mr. Hill ran toward Mr. Olsen, who is white, and did not stop after Mr. Olsen shouted for him to do so.

As the verdict was read on Monday, Mr. Olsen, who resigned from the police force after he was indicted in 2016, stared ahead and appeared visibly relieved when he heard that he had been found not guilty of murder and the associated counts. But his face stiffened and his eyes twitched when the rest of the verdict was read. His wife shook and sobbed, and had to be led out of the courtroom.

Mr. Hill’s relatives, including his mother, Carolyn Giummo, also wiped away tears.Later, his family declined to talk to reporters, but earlier in the day, after prosecutors objected to Mr. Olsen’s release on bond, Ms. Giummo testified to her family’s pain.

“It’s been four years that we’ve been waiting for this,” she said, describing her eagerness for Mr. Olsen to be behind bars. “I just feel like it’s time now.”