Actually, visually the stairstep method makes sense to me. I have spent a lot of time testing different methods of resizing and I am convinced that stair step interpolating works very well, especially with images that contain lots of small details like scenics and wildlife. I think the key to it is to first prepare the photo by making it as clean as possible by removing as much noise as you can and even applying a very small amount of sharpening. Basically you are making the original sized photo look as good as possible before you upsize it. It depends on the amount that you upsize in each step. I find that either 5 or 10% increases in size work the best and even that can depend on the particular image. 10% is very safe and still looks better than upsizing in one step. It's not a huge night and day difference but the larger you go the more significant it is. One time, for the fun of it, I tried upsizing in 1% increments. Not only did it take forever it did not look very good. So there is a sweet spot for each image. Genuine Fractals has its place for sure but, again, it depends on the image. It has a nasty habit of producing razor sharp edges where they need not be and for smoothing away fine details where they should remain. Its great for graphic design oriented and mechanical type images. I stick to the stairstep interpolation for the majority of my scenic and wildlife photos because I am not interested in sharpening the edges anymore than they are but rather maintaining a consistent and realistic quality throughout the image as it is enlarged. Genuine Fractals, on the other hand, can tend to give the image a plastic and artificial quality on close inspection. This is just my opinion, of course, and you should use the method that you think produces the best results.
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