Tracy Smart was found guilty in May of threatening Alistair Doull, of Pilton Drive North, by sending the image of the house ablaze along with a fire engine, which suggested his home would be set alight.
Smart was caught after a DNA swab taken from her by police was matched with a sample lifted from the envelope in which the picture arrived, which had been sent in May 2010.
Edinburgh Sheriff Court heard that the relationship between Smart and her neighbours was a “bad one” prior to the letter being sent.
Smart maintained to the police that she was innocent, and her lawyers submitted an appeal following her conviction.
They argued that prosecutors had failed to establish when or how the DNA had been deposited on the envelope. But following an appeal court hearing, judge Lady Paton refused the appeal and said the DNA “provided a particularly accurate means of identification”.