Low Quality Raster Logo Reconstruction in Vector Format

Preparation

Place the the paper on the scanner and set the resolution to what you want. The color mode and scale of the image you want to be scanned.

Clean Up in Photoshop

For the clean up I adjusted the levels of the black and white of the image. Then I went and selected the black parts of the images and went over them with the brush tool. I did the same for the white parts and all the grays disappeared. 

Live Trace in Illustrator

For the live trace of the image I found there wasn't much of a difference in the normal ones. The image stayed the same for most of them. 

Logo Reconstruction Using the Pen Tool in Illustrator

For this I used adobe illustrator's pen tool to clean up the edges and make nice curves. I used the elliptical tool to make an oval and fill in for the background to make it yellow.

File Formats

I used the file format of the program I was working on it with to save it.

Advice

Don't skip out on the step and be careful about what you change.