Based on a good deal of general viewing, and of course, walking up close to the screen to look at some of the finer points of image noise, and running a couple of tests, let's just say that the BenQ W1500 Projector passes easily.
Like most DLP projectors, including the W1070 and W1080ST we also reviewed, I found mosquito noise - that general background noise, to be more visible than on other technology projectors. Of course, the only other technology in the price range is 3LCD, as LCoS start a lot higher price wise.
Panning was very smooth on 24fps movies without any fancy noise reduction.
But most importantly, there was no heavy "noise" that can show up in close ups of faces, and some types of gradients. That type of noise tends to make the image look like there isn't enough color palette, and/or too much contrast. This is an area where the BenQ's do great, and by comparison, the Optoma DLP's suffer significant issues. Acer falls somewhere between the two, but is definitely more acceptable than those Optomas (like the HD25-LV).
In this image gallery, the same closeup of Jennifer Lawrence from The Hunger Games is shown first from the W1500, then the Optoma HD25-LV which definitely shows some noise problems when you look at her cheeks, third is a nice expensive $3500 LCoS projector (the JVC X35) which does really well. I didn't take this image with the Pro Cinema 4030, but here's a similar image from Epson's $999 HC2030. It too also has less of a problem than the DLP projectors.
It should be noted that both DLP projectors, the BenQ and Optoma had Brilliant Color engaged, which can add to the noise, but even without Brilliant Color both showed more such noise than any of the non-DLP competition as well.
Before you panic, while this type of noise is something worth noting, it is going to be less noticeable when you are viewing a movie, rather than a paused image. Even so, the Optoma of this group, is still visibly inferior to the others, while the BenQ's issue is only slightly noticeable when viewing, and probably not an issue unless it specifically bugs you.