Patrick Lam, 58, and his wife, Timmy, 48, were attacked after three men wearing balaclavas and brandishing knives forced their way into their home in Barnton Avenue West.
Officers said the couple had “done everything in their power” to identify possible suspects, but drew a blank.
Detective Inspector Gordon McCreadie, who is leading the inquiry, believes the victims may have “come in at the wrong time” and been picked out by the gang who made off with £15,000 of jewellery, designer handbags and other items, as well as £5000 in cash.
The inquiry team has identified two vehicles caught on CCTV leaving the cul-de-sac just a minute apart at the time the robbery ended on March 9.
DI McCreadie said investigators believe the suspects were in the vehicles, and appealed for anyone who has seen the cars in the Capital to come forward.
He said: “It may have been that the suspects were looking at different areas and the victims just came in at the wrong time. It is unclear why they were chosen.
“If anyone else saw these vehicles in their area we’d ask them to contact police. The suspects might have been trawling elsewhere for potential victims before targeting this couple.
“The victims have been really strong and cooperative, and have done everything in their power to think about why this happened to them and they can’t identify anything.”
The Lams, who are of Chinese descent and run a successful retail business, were left “terrified” by the raid.
One of the vehicles caught on camera is described as a dark-coloured saloon while the other is a silver-coloured people carrier with blacked-out windows.
The suspects were all wearing dark clothing. Two of them are described as being about 6ft tall and spoke with foreign accents, the other with a Scottish accent.
DI McCreadie added: “The CCTV inquiries have been very extensive, with officers viewing footage from pubs, shops, private homes as well as city council cameras. We’ve also been checking cameras on the front of Lothian Buses.
“All together we’ve been looking at around 150 hours of footage from about 30 cameras.
“From the cameras on the road itself, we have images of two cars leaving Barnton Avenue West around the time that the robbery concluded. The vehicles were seen travelling south from Barnton junction.”
Officers are also appealing to traders across the country to be aware of anyone trying to sell on the stolen items, which included Louis Vuitton and Hermes handbags.
DI McCreadie said: “Some of these handbags are of high value, and such items being offered for sale would be quite rare. If anyone is offered these items we would ask them to contact police.”