The four robberies all happened across Burley and Headingley between 2.30pm and 6.30pm on February 9.
During one offence in Beechwood Crescent, a teenager was robbed while walking home at about 3.10pm.
The 19-year-old man was approached from behind by the offender, who grabbed his shoulder and chest before threatening him and pulling out a knife.
The victim managed to escape unharmed and with his property.
Police today (Friday) issued the E-Fit image of the offender and have appealed for any witnesses to get in touch.
Anyone with any information is asked to call police police via 101, quoting crime reference number 13180067324 or information can be given to Crimestoppers anonymously on 0800 555 111.