How does Google Page Insights determine how much an image can be compressed?

Selected answers from the Dumb SEO Questions Facebook & G+ community.

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  Dragan Đurić: I do not know what measurement Google use, but when image is not larger than display size and properly compressed usually it does not appear on report. However, you can download zip with optimized images from link at the bottom of the Google PageSpeed Insights report page.
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  Gregor Grigorian: Whats the best way to compress images for website speed on blog posts and what not?
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  Neil Cheesman: I find http://optimizilla.com/ quite good... (FREE) and also shortPixel Plugin for Wordpress
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  Neil Cheesman: BUT - if anyone has a better solution I would be happy to hear about it
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  Gregor Grigorian: I usually do photoshop to be honest.
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  Gregor Grigorian: I tried testing it out, I don`t understand how to download the compressed image after >_> ugh. I will figure it out though, my head is fried.
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  Neil Cheesman: You either left click and download each image at a time.... OR download all and it downloads them zipped... you can also adjust the amount of compressing... right-hand side
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  Gregor Grigorian: Thats perfect Neil Cheesman thanks!
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  Sherbaz Khan: gtmetrix.com is a good alternative to Google Page Insights. try it as well :)
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  Neil Cheesman: I post quite a few images and Optimizilla seems to be as good as any paid option...
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  Alan Bleiweiss: DON`T FORGET: file size compression is often NOT enough, and GPSI ONLY evaluates file size compression. If you have a 4, 000 pixel wide image, stuffed into a 300 pixel wide block, all the file size compression in the world is not enough. If you have a 550 pixel wide image, at 300 DPI quality, stuffed into a 300 pixel wide block, file size compression often may work, yet it still may not be enough without first reducing the DPI and then resizing dimensionally.
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  Gregor Grigorian: Perfect, thanks Neil. This program is going to be a life saver in the long run with time !
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  Neil Cheesman: I have read elsewhere that Google have come out with a `tool` although not `official` yet...