MGoCSI Report

“The Game” has come and gone, the good guys prevailed, and our collective mindset has mostly reoriented to the future, as visions of Sugar Bowls dance in our heads. So is there any reason to rehash the already much-debated issue of the Toussaint Touchdown Takeaway?

Of course there is, because such things live on in Michigan lore forever and sometimes, you have to beat a dead horse just for the sheer fun of it. Besides, I had an unsatisfied curiosity combined with some unaccustomed free time, so I set about to try to resolve the controversy once and for all.

This analysis may not appeal to you unless you are almost equal parts Michigan football fanatic and geometry geek, but in the end, I believe there is an important point to be made here. To discover my purpose, you will have to read on (or cheat and jump to the end).

The Evidence Speaks to Us

I start with Exhibit A below, a camera view that seems to suggest that Fitz planted his knee with the ball just short of the goal line. Almost certainly, it was this view that convinced the replay official to reverse the call on the field and overturn the touchdown. The shot appears to have been taken by a crane-mounted camera hovering about 10 feet in the air just beyond the goal line. It is not an ideal angle from which to make a definitive call. The knee may or may not be in contact with the ground and the relative position of the ball is distorted slightly by the angle. (Note: for formatting reasons, I am including scaled down versions of these screen shots; full resolution captures were used for the actual analysis. Click the photos for larger versions).

Exhibit A

I use Exhibit A not to attempt to resolve the issue at hand, but to call attention to the item highlighted in magenta. There is a cameraman clearly visible in the shot and it is his footage that will provide the basis for further analysis. We don’t know his name (Abe Zapruder?), but we have a very good idea of the physical position of his camera. The dashed boundary line he is standing very close to runs 12 feet outside the sideline. By analyzing statistical data on the average height of college cheerleaders, we can fairly accurately estimate the center of the lens to be 5’ 4” off the ground. I estimate his standing position to be 11 feet east of the side line and 2.5 feet south of the goal line. These estimates probably place the camera position reliably within an error sphere less than 1 foot in radius. This is important as we move forward with the analysis.

Let us move on to Exhibit B, which was definitively taken before Toussaint’s knee fell to earth, and Exhibit C, a shot in which he is definitely down. The time interval between these two shots is presumably 1/60^th of a second, given the parameters of 720p HD video. I will focus my attention on Exhibit C.

Exhibit B

Exhibit C

The time has come to let mathematics work its wonderful magic. Again, the viewing angle is not perfect, but because we were able to accurately determine the viewing position of the source camera, some surprisingly precise calculations are possible.

The dimensions and positions of the gridiron lines and hash marks are well known and presumably accurate. The only thing I am not quite sure of is the crown of the playing surface, which appears to be about 6-9 inches at midfield.

This allowed me to create a three-dimensional computer model of the playing surface and made it possible to determine the orientation of the camera (azimuth, elevation, zoom, and tilt) by matching the grid lines appearing within the frame with that of the rendered computer model.

Knowing this, we can now focus on the position of the ball within the frame. A more closely-cropped view is presented in Exhibit D. The projection of the ball in the frame spans about 40 pixels. Therefore we can determine its position within the 2 dimensional space of the video frame to an accuracy of about a quarter of an inch.

Exhibit D

The real world has the inconvenient habit of being three-dimensional, so there is one additional parameter required to ascertain the position of the ball relative to the plane of the goal line. This would be the distance from the camera to the ball, or alternatively, the perpendicular distance from the near (Zapruder) sideline to the ball. By examining other angles from the game video and observing grid lines, hash marks, and end zone lettering, this can be determined to be about 90 feet, plus or minus 2 feet.

The final calculation will be slightly sensitive to this distance, so I went ahead and determined the corresponding position of the ball over a range of two foot intervals between 88 and 92 foot distant from the sideline. The plot below (Exhibit E) shows a top down projection of the ball’s position relative to the goal line over the range of possible values. Due to the near perpendicular viewing angle from just off the goal line, the error contribution from this uncertainty is quite small (.3 inches per foot of error) and yet this is the largest source of potential error. Any imprecision in establishing camera position is largely cancelled by adjusting angles to precisely overlay grid line positions within the frame. I won’t bury you with an avalanche of error sensitivity equations; suffice it to say that I am confident that the final estimate of ball position relative to the goal line is accurate to within half an inch.

Exhibit E

Based on the best estimate of distance from the sideline (center ball), the results sadly report that Toussaint is holding the ball 2.5 inches short of the goal line with his knee clearly down. So, technically, the officials got the call right. Did the replay official have irrefutable evidence to overturn the call? Of course not! The ball was just inches from the goal line and he did not have the resources to make a definitive determination.

Conclusion

The play was so close that it was not humanly possible for an official on the field to make the call with complete certainty. The difference between touchdown and being down short of the goal line was a matter of inches and hundredths of a second. While the determination was ultimately correct, I think we can also safely claim that the replay official overstepped his authority by reversing the call on the field, based on the “irrefutable evidence” criterion and the limited technology available to him.

But my real point in all of this is to call attention to the fact that making an accurate determination is possible and current technology could accomplish this in real time, using techniques very similar to those employed to superimpose the first down line over the playing field, or track pitch trajectories in a baseball game. There is no need to put sensors in the ball or anything like that and accuracy within a fraction of an inch can be achieved. Higher frame rates and faster shutter speeds (super slow motion) improve the accuracy further.Multiple camera angles help as well. Cameras already have sensors to report their positions and orientations. I am calling on companies like SportVision that do enhanced sports graphics to develop the software to provide accurate ball positioning information to the fans and, dare I say, to the replay officials so that in the future, key plays like this can be accurately adjudicated.