There is no rescaling required to send a 1920 x 1080 picture to a 1920 x 1080 display.
All current HD transmissions have 1920 x 1080 pixels.
Source and destination have a 1:1 pixel ratio. What you are doing is taking a 1920 x 1080 source, rescaling it to 1280 x 720 which discards video information. You then send it to the TV which has to invent the video data you have just thrown away. This makes no sense at all.
As I already said if you start off with a SD programme say a BBC one with 720 x 576 pixels, then you may well get a better picture by sending the original 576i content to the TV and letting the TV scaler produce the required 1920 x 1080 pixels. This will work if the TV has a better scaler than the box. The downside is that the epg and menu overlays will not be in HD and look decidedly fuzzy.
What you are doing though starting with a 576i source is getting the box to scale to 1280 x 720 and then the TV to 1920 x 1080. This will also slightly degrade the epg etc.