Webkit uses 'css pixels', which is then recalculated to the resolution of the physical device - perhaps this is the problem here.
If you have a typical 4-inch mobile phone with a 480x800 screen, the css pixels can be 320x533 and the CSS aspect ratio is 1.5.
In this case, the 32 x 32px image should have at least 48 x 48 pixels in order to look good and sharp. Since some devices have a pixel ratio of 2 (iPhone 4/5), I usually use 2x large images and they look very good, almost as good as svg images (and there is no problem with the width of older browsers).
When doing this, you need to remember the image size
background:url('myimage64.png'); background-size:32px 32px;
or
<img src='myimage64.png' width='32' height='32'>
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