I will agree with @Antone88 that it is a complex one.
In terms of web design, if you are given images at their sizes and for example, you want to make them bigger, it will be impossible to do so without losing image quality. That would still be the same case even if you used graphic software like Photoshop too.
So, the options to solve this is either:
1. If they are your images
You retake the pictures at its largest size. Once you do, duplicate all the images but resize them for different screen devices. Also, do image optimisation on each image.
2. If the images are not your (e.g. they are images stocks)
Well, it also goes down to a few options. You either will need a new picture if you do not want to change the layout or if you want to keep the images, you may have to change the layout.
I do have one idea, but I am not sure if this will work. If the image is a background image that takes the whole screen device, you can add a transparent colour overlay on top of the image. However, if that will cover the pixelated areas or not is questionable, but you have nothing to lose.