This is the image. I tried using
Polygonal Lasso Tool, but the edges don't appear to be OK.
Is there more precise way to remove the background and leave the letters and the logo icon on a transparent background?
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The issue you are faced here is a limitation of the GIF transparency which is limited to a single color from the palette. Semi-transparent colors are not possible.
"Transparency" of the original GIF file
To obtain an image with a transparent background but with keeping the shadows we can take the following steps (below done with GIMP but any other image processing application including Photoshop should be able to do this)
Convert the image from indexed to RGB
Remove the transparency alpha channel
Alpha channel removed and replaced by white
Replace white with an alpha channel
White replaced by transparency
Save in an image format supporting semi-transparent colors (here .png)
This will then lead to the desired transparency as seen below:
Transparent image on coloured background
Note that in the case here where the Image had grey scales not only in the shadows but also in the letters all these shades of grey will also have some transparency from the white alpha channel. This may in some cases not be wanted. We then will have to recreate the image.