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In the old days of analogue electronics it wasn't feasible to store or process video signals on the fly. At best you could shove them into ever narrower pipes and mangle the quality of the signal as it goes through your equipment. This is essentially what happens to broadcast and composite video - the colour channel is violently combined with the luminosity channel for transmission, then haphazardly separated at the other end. Since the vertical resolution of a TV was a constant factor, the broadcaster was limited to mangling things in a horizontal fashion - the chroma channel was badly beaten to be sure, but only in the horizontal dimension. Colours were smeared and stretched, detail was lost, and cheap electronics finished off whatever clarity the signal had by the time it arrived at your television.
Enter the digital age, a whole new era of signal mangling and video abuse. Efficient video compression, required for applications like DVD, demands no small amount of computer processing power. Segments of the video signal must be held in memory long enough to be processed and encoded in a frighteningly compact manner, and this is where you learn about a technique called "Chroma Subsampling". Sampling is the term used to describe periodic checks of a signal to learn its value at a certain point. Imagine your video stream as a conveyor belt whipping past you at high speed. Sampling means you take a reading of that signal at regular intervals, for example every 15,000th of a second. Sub-sampling is taking samples less often, resulting in a reduction in space required to store the data. Obviously if you take fewer measurements less space is required to store them.
Have a look at the following image:
The image on the left is a JPG made with standard 2x2 chroma subsampling, which is what most programs use, meaning that the colour data was sampled at half the resolution of the center image. The JPG in the center has both the luma + chroma sampled at 1x1 pixels, but actually uses a higher level of compression than the left image (25% vs 15%), resulting in a similar filesize. The one on the right is the original, uncompressed .bmp image. As you can see the center JPG is much more faithful to the original than the left JPG is - a result solely attributable to the difference in subsampling. Compression isn't nearly as important as the subsampling, except at the extremes.
This technique is applied not only to DVD recordings (And, incidently, the American HDTV standard) but to things like jpg images on your computer. Instead of compressing each line horizontally, resulting only in a drop in horizontal resolution, only every second line is sampled, resulting in a 50% reduction in vertical resolution as well. There's an incredibly detailed article on the way this can deleteriously affect your DVD playback on this page.