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Lets say i have a picture. I send it to a server. While sending it to the server half way the upload gets canceled. Now the server has half or some of the picture. Now is there a way in which this can be processed and viewed. Are there any formats which support this? And are there any other factors which account as to why the picture can or cannot be processed?

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Yes, most formats work this way! For example JPEG can be decoded up to the point where the chunks are available:

enter image description here

Image 1 Scaled version of original JPEG picture at (1 048 198 bytes) and corrupted one (610 922 bytes) side by side. Original image by Maruf Mostafa image available here

This is easy for you to test take an image and just simply delete some bytes out of it (like with a text editor). Some files are easy just open them, like the JPEG in question that just opens up in many editors without a hitch.

PNG required some slight special handling to uncompress. It did't open in Photoshop or windows image preview. But was still relatively straightforward to open as the web browser happily opened even partial PNG files to the point it was defined to.

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