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Quarter Pixel Motion Estimation
Quarter pixel (often also called as q-pel or qpel) refers exactly to what it states, to a quarter of a standard pixel.
But what makes it so often-used term in video technology is that several advanced video encoding methods, such as certain MPEG-4's flavors use so-called motion estimation to improve the video quality and when the technical specs refer to how precise each compression mechanism's motion estimation routines are, they typically are far more accurate than simply one pixel, but rather come down to sizes of a quarter pixel.
Quarter-Pixel Motion Compensation allows the calculations used to detect motion (for prediction purposes) to be more accurate than MPEG-2. Although obviously you can't store less than a whole pixel, that doesn't mean movement in your video will be in 1 pixel increments. By increasing the precision to 1/4 of a pixel (from MPEG-2's Half Pixel precision) prediction is improved, sometimes meaning better quality at a comparable bitrate. Older standalone DVD players with MPEG-4 ASP support won't playback files encoded with Qpel, but it shouldn't be a problem for newer models.
Quarter Pixel Precision (Qpel) for Motion Estimation is used to more accurately track movement across the video frame. Although entire pixels must be stored, when calculating locations for a moving object the motion from one frame to the next may be less than a single pixel. MPEG-2 uses Half Pixel Precision.