You guys are not quite answering Carl's question...
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The behind-the-lines message is "we don't need PS so probably you won't either" but that's not telling Carl how to resolve the problems he's used to solve in a particular way with another software.
Carl, there's something you should keep in mind. Sometimes a particular workflow in PS doesn't have a parallel process in PI. What I mean is, yes you can probably solve that problem in PI, but the tools and the workflow will likely be different. And sometimes, trying to replicate the steps will lead you to frustration because there are things you can do with PS that you cannot do with PI, and viceversa, if what you try is to replicate step-by-step the workflow. That however is different from saying that the problem cannot be solved using either software. Often times PI offers an alternative way to do it, and often times it gives much better results.
For example, I could try to describe how to do that "selecting the area of interest", creating a new image, blending it with the "darken mode" (which is basically a min(img1,img2) operation in PixelMath), etc. but... would that be the best way to do it in PI? Probably not.
And people have pointed you to the right tools: Deconvolution and the MorphologicalTransformation (MT) tool.
Now the "trick" is learning how to use those tools to achieve the goals of fixing the problems these tools can solve, and again, usually delivering superior results. For example you mentioned the Minimum filter... Once you get to know how the MT tool works, you'll realize it's like a maximum/minimum filter on steroids, and LOTS of them! In fact, saying that probably doesn't do justice to the MT tool.
But new tools take time to learn, practice, etc. It's not difficult, but I see many people getting into PI coming from PS and because after a few hours they cannot do in PI what they already know well how to do in PS, they give up. That's unfortunate but it's their loss. Anyway I don't want to turn this into a lecture (I hate to lecture almost as much as being lectured
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), since you just want a straight answer, so I'd suggest you to experiment with these two tools, always use a star mask with them for this purpose like Carlos pointed out, protecting everything but the stars (easy to do with the StarMask tool), and if you get lost with all the parameters, since there's still no official page with the PI documentation, in the meantime you can use this "Unofficial PI Reference Guide"
http://blog.deepskycolors.com/PixInsight/ and of course, Harry's videos for several good introductory (and not so introductory) video tutorials:
http://www.harrysastroshed.com/pixinsighthome.html