There are plenty of ways for recreating a logo, one is using vector magic as what I have written on my previews article about recreating raster into vector (but that wont work if the original copy really are in bad condition), second would be tracing it using vector base software such as illustrator and the third one would be recreating the logo using high resolution raster base software.
What we are going to do here is converting the logo into a vector after you produce the high resolution raster copy.
This tutorial is made for those who don't really know anything about illustrator but knowledgeable on Photoshop or should I say they are more convenient on photoshop.

I have here an example of a logo which in bad condition; I believe that this logo was hand drawn since it was created 1923, I'm about to recreate this logo digitally for the design of our high school reunion countdown.
Bring up the file on photoshop, press [Ctrl Alt I], then put the resolution up to 300 DPI

The image will go bigger and look heavier but don't panic coz we made that in purpose, create a set of palette by creating a new layer above, using rectangular marquee tool and eye dropper tool to pick our two main colors which is green and yellow.
(We do have our swatches panel but I prefer this style coz this will only be use for this project.)

Then we will be tracing the logo using pen tool, but before we do the tracing we should analyze the logo first and find some basic shapes, as much as possible avoid using pen tool for tracing basic shape.
The over all shape of this logo is much more like a five leaf clover, and you can create one by putting five circles together.

Using the Elliptical marquee tool hold down [Shift] while making an ellipse to create a perfect circle, then hold down [Space bar] to move them accordingly, and fill it with green.

Do the same but this time fill it with yellow.

And then another with green.

Now make a duplicate of that logo and put it on top temporarily.

We are now going to trace everything using the pen tool, consider tracing it 1 pixel contract coz they are blurred which mean there is an illusion of expansion which is caused by that blurred state.

After tracing the tower open up your path panel, [Ctrl click] the path to turn into a selection.

Create new layer and then fill it with yellow.

Create a new path and start tracing those waves, if you are not familiar with pen tool, you can start playing around those anchors, [Ctrl drag] the center anchor point to move the entire tangent line, and [Alt drag] those left and right anchors to extend them.

After you've done tracing, go to path panel, [Ctrl Click] the path then fill it with white, do the same procedure on the rest, as soon as you are done tracing everything turn off the visibility of our duplicate layer to preview our work.

The next step would be converting this logo into a vector.
You can convert this logo online on vectormagic.com, or you can purchase their desktop edition for more additional features.
Save the logo in PNG format, and then open the file on vector magic.

And since our new logo was done on a 300DPI I'm sure that vector magic can trace them perfectly so you can relax and select the Fully Automatic option.

Then save the file in .EPS and that's it you made a vector base logo from a high resolution raster logo.

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