By buying a better video camera, with a faster, better lens, and faster processor with more accurate/usable focus? A tripod might help, also. Plus shooting only on bright Sun-lit days of inanimate, high-contrast objects.
My personal opinion (so this is a good place to stop reading...
) is that video is designed to take advantage of the brains ability to 'fill in the blanks" and fool us into 'seeing' smooth movements, not take still image quality pictures. Since the shutter speed is pretty much fixed, the only variable is aperture. Unless there is lots of light, the lens may end up using the smallest f stop and have the largest depth of field which may make it harder to see the subject against the background (everything being similarly "in focus").
Even when "sharpening" an image with an image editing app, you are not actually adding information/detail. Usually all that happens in increasing some borders between contrasting colors. The higher the perceived contrast, the 'sharper' the image appears. I think you already know this by your use of "apparent resolution" although "resolution" is not always analogous to "sharpness." A 600dpi, out-of-focus image may be less 'sharp' than an in focus 180dpi image.
