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I ran an image I grabbed off Google through TinyPng's website. The image was compressed, something like 21%. Out of curiosity I put it in again and it was compressed again, by a much lesser value. I kept going, in total 19 times, before the files was no longer getting smaller. Leaving this broken looking version of a cat. enter image description here

I then grabbed another picture of a cat off Google and ran it through TinyPng, the very first time showed there was 0% compression. Yet there was no artifacts or discolorations as shown in the first picture.

Why was the second image not able to be compressed? Why was the first image stopped from compressing more to the point of being a bunch of garbled pixels?

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By using the same compression - JPG, I presume, you have met the following limit:

Every new compression-opening turn introduces new errors (=fake detail = JPG artifacts) as much as the compression reduces the need of the storage.

JPG compression shaves off small details making the image smoother and sharp limits rounder. In addition here and there, mostly near sharp edges, there grows fake details, when image is opened in image viewer.

Image viewers routinely try to find the artifacts, reduce them and sharpen the image to compensate the introduced unsharpness. This is again more error, altough it visually may look accaptable and to be an enhancement..

In yor open your final image after several open, save as jpg, open, save as jpg, open... turns, the image is quite crappy. In every turn the more real details are lost and new fake detail is inserted instead.

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