This particular logo is easy, because:
- It's only three colors
- It's relatively simple vector art
- there's a simple outline around the shapes
- The goal is reconstruction, not preservation of a masterpiece
That means you don't need to light it very well and you don't need to worry too much about noise. Take a photograph straight down, and notice any sources of glare and shade them. Have someone hold a coat or a blanket. Don't worry about cranking up ISO to get the right exposure, and shoot wide open.
You could have someone trace this by hand (not really that hard), or you can do it semi-automatically, like this:
Take your image into Photoshop and Gimp. If the image isn't exactly straight-on, use the perspective correction tools to fix it. You can do this adjustment by eye — make sure the three arrows look balanced all around.
Then, push up the curves so you lose the unwanted detail (noise and dirt!) in the white, and maybe increase contrast overall. Like this:
Then, use the fill bucket tool with a wide threshold for similar colors to make the white area all perfectly white. You may need to use the eraser to get some leftover pixels, but overall should be pretty easy:
Same for the green. (This may work better if you select the green channel for matching in the bucket tool; I don't remember how that works in Photoshop offhand but it's easy in Gimp.) You could preserve the green color, but for this image I thought it easier to just go to black and white. (In fact, I converted to grayscale, but only after the cleanup, as the color channels make that work easier.)
You'd really want to clean that up better, but this will show you the approach. It would be much easier with a higher-resolution image, of course.
Now, for the next step, I switch to vector-editing program Inkscape (which is free and open source). Save your work as
JPG — it doesn't really matter), and
Import... the image into a new document. Then, from the
Path menu, choose
Trace Bitmap. You can play with the options there; here, I chose "Brightness Cutoff" of 0.8. That gave me this result:
Which is very rough, but that's because I'm starting with a "0.05 megapixel" image, and because I put very little effort into choosing the tracing parameters. Starting from the Canon 20D's 8 megapixel images will allow you to get a nice, smooth result mimicking the original very closely. You can add the green back, of course.
And, since the result is a vector image, not a photograph, it's a tiny file which can be reproduced at any size without blockiness.