Thanks for doing this.
at least it's not just us?
“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).
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 1/60th of a second, given the parameters of 720p HD video. I will focus my attention on 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.
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.
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.
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.
Thanks for doing this.
The key point in this is the subtle move from keepinng the call on the field, unless unquestioned video evidence to now using video evidence alone(it seems) to make the final call. Wonder how the refs feel about this change?
Why did you only chose to use the camera that appears to be on or very near the one yard line. Shouldn't you also use the camera that was in the end zone to help prove your conclusion since it will give you a look that obviously appears to be a touchdown and may change your conclusion? This happened right in front of me and I was in row 8. Even the Ohio fans behind me thought it was a TD and were shocked when the call got overturned.
For one thing, I could not get any kind of accurate position information on the west sideline camera. I believe it was a mobile crane camera, but it does not appear clearly in the reverse angle, so I could only guess where it was filming from. Without more accurate placement data, the accuracy of the results would have been less. Also the zoom factor was lower and the ball is less visible, so it was a much less useful angle.
The end zone camera with Toussaint running toward it wouldn't help you determine the distance he had to the goal line. What would have helped validate his conclusion is if we knew the position of the camera on the other end of the goal line.
Oh. My. Word.
So who actually killed Kennedy?
A few things:
1. Why don't they just put a camera on the goal line? If that guy's standing a couple feet to his right, none of this is even a question because the camera angle is set up for these calls.
2. I'd love to see a real-time system built to do this kind of thing. Sounds like a lot of fun to put together.
3. Again. Oh. My. Good to get some of closure.
Ive always wondered myself why they couldn't set up a goal line gamera, it would just mean moving over the equipment by a foot. Another thing the networks don't do is sync up their videos and play multiple angles simultaneously. That way you could get a good view of when someone is down, and where the ball was at that time, assuming both were not clearly visible in the same shot.
Because it would just videotape the back of the ref's head. The ref stands there to make the call on the field.
Right, but you could put it above the ref, between the ref's legs, or heck, wireless cameras are cheap and small these days, just put it in the pylon!
The ref could even wear a camera on his hat.
how many angles you would need for a true 3D rendering? I've seen this stuff on CSI or something, and we know those shows don't lie.
You would need only 2 cameras assuming that you know the exact position of each camera. Each camera gives 2 dimensions, and as long as each camera isn't in the exact same position (practically impossible), then you can find where anything is in 3 dimensional space. Of course, if you want to get real exact, you would want to sync up the cameras so that the shutters open and close at the exact same time.
This is how they do the strike zone thing in baseball. Also, this is how humans have depth perception (two eyes, a couple inches apart), and 3-D movies are shot (the cameras have two lenses separated by a foot or less). If it seems like I know too much about this, my Ph.D. thesis involves two high frame-rate cameras used simultaneously to determine a particle's position in 3D space as a function of time.
Edit: Of course the accuracy of your method is limited by camera resolution.
Actually, you can infer a lot of 3D information from a single camera, since there are reference points in the frame whose exact 3d position is known (field markings). Two cameras give you a lot more information about the free moving objects and lets you construct 3d models of the scene, excluding only those areas that are hidden to both cameras. By incorporating additional information, like knowledge about human anatomy, you can make further inferences in constructing the models. This is similar to what the Xbox Kinect does.
Yes, you are correct. I felt like my post was getting too long to get into that, but since you brought it up...
It does, however, depend on your camera angle, as was mentioned and used in the post above. Also, imaging a football field and human beings on it differs greatly from imaging a sphere in relative darkness (as in my thesis).
However, my point stands that for an exact 3D rendering, you would need 2 cameras.
is a some-a-bitch. Lets get the guys in the engineering building to mount camera inside the pylons. Then we could mount one on the center-line of the goal post. This would give us a 3D view of the ball at any time. (Provided the camera can actually locate or "see" the ball).
These cameras would be powered by DC current from connectors located in the turf and supplied via cable under the turf. If a player hits the pylon, it will still break away and there will be zero risk of electrocution with only 12V of power.
This will, of course, make the pylons directional so someone will have to be responsible for ensuring the cameras are pointing in the correct direction. I will volunteer to be this person.
The reply ref will have to be versed in CAD to be able to manipule the angles around. It would take about 30 seconds to make a call.
Now, I am not the first person to have this idea. I just dont understand why they dont implement this. We have cameras as small as a pencil eraser. I just dont get it.
I don't know why but this is one of my favorite pieces of user-created content since Six-Zero unleashed "Wife Day." This really is just a brilliant analysis of what could have been a cirtical play for us.
Every season, it seems like there are 2-3 really close calls like this, some of which impact the outcome of a game and potentially a season. I hope that you will do something like this for any such plays that arise in the future.
Mods, during what will inevitably be a slow week, I suggest a bump to the front page on this.
he's in because, let's be real - no buckeye could ever understand the math that proves that he's short.
Awesome work. At the time I thought he was down, but even then I thought it was way too close to reverse.
This blog amazes me every day. While its a shame he was not in, I agree it was impossible to have indisputable evidence. Thanks for taking time away from your family to do this.
Thanks for putting the time in on these calculations. I always thought the replay ref got it right about the ball being short of the GL, but I agree with you that, at the time, the evidence was not irrefutable - hence, ANGAR! Then there was the "place the ball on the 1 foot line" - hence, MOAR ANGAR!!!!
This post is just one more example of my MGoBlog is THE GREATEST COLLEGE BLOG ON THE PLANET (TGCBOTP?).
There, fixed it for you.
* The greatest Blog in the universe.
...By analyzing statistical data on the average height of college cheerleaders...
You will henceforth be known as "that creepy guy with the tape measure", won't you?
How do you explain your google search history to wife/girlfriend after that sort of research?
Not only did the replay official overturn the call, they placed the ball on the one foot line. Bad enough they overturned the call, but the ball was obviously a lot closer to the goal than a foot when the knee touches.
line for mistake number 2 (not enough to over turn the call being mistake #1). The spot is the real mystery to me.
That said, we won so the hell with it. Kinda nice winning the last gameof the season isn't it?
That's why this is our greatest rivalry, you can feel good the rest of the year.
It is being played January 3, in New Orleans.
Regardless, a bowl game loss does not get to me the same way as an Ohio loss.
But if they spotted the ball accurately at the 2.5 inch line, that would be evidence on its face that they did not have enough evidence to overturn the call. It may be a policy that any overturned touchdown will be placed at least 12 inches out. Eye-spotting is probably +/- 12 inches most of the time anyway, so how silly when they slide a credit card between the ball and the marker. Could you count bricks to cross-check the cheerleader height? Not as fun, i'm sure.
Thank you for doing this. I always thought he looked short from both camera angles, but so many people disagreed that it was not worth arguing about. It's good to know that I'm not completely crazy.
Such cool work.
Even though your conclusion is contrary to the hopes of the Michigan fanbase, the fact that you put in the work and effort to get there, and still posted your findings speaks volumes of your analytic abilities and your character, and is a testament to the Michigan fanbase as a whole. Well done.
This is the most complete block of evidence since this Seinfeld episode:
That is one magic loogie.
I agree that the evidence for Fitz’s TD was not irrefutable, given the information available to the replay official at the time. On the other hand, the replay official seems to have considered your Exhibit A sufficient proof. Fair-minded fans should want the calls to be correct, and in the end this call was exactly that: correct.
Hemingway's catch wasn't a matter of geometry. It was a matter of deciding if he had possession when he went out of bounds.
What? Are you talking about the call on the field or what the replay officials ruled? You know he didn't use the camera that was in the end zone, in this analysis, that would would have shown a TD. The call on the field was correct. The replay official botched the call and that comes from Mike Pereira former head of NFL officials.
He also thinks that the Junior Hemingway non-TD call vs. Iowa was correct. (To be precise: he thought the call on the field could have gone either way, and once it was ruled incomplete, the booth official did not have the evidence to overturn him.) I think Pereira is a pretty good guy, but people seem to forget that after the Iowa game he thought the officials got it right.
Regarding the Fitz TD, we now know (thanks to the above diary) that he, in fact, didn’t score. And I think fair-minded fans should not want a TD that didn’t, in fact, happen.
The question whether evidence is irrefutable is itself a judgment call upon which reasonable minds can differ. I am not going to get too upset at an official making a call that turns out, upon detailed analysis, to have been factually correct, even if there is no way he could actually have had that analysis available to him at the time.
I'm not sure I understand your comment. The point of this diary entry is proving that the call on the field was incorrect, and that Fitz was in fact down short of the goal line. The person above is simply saying that it is better that the final result (no TD) matched the facts (no TD), even though the replay official was wrong to overturn the initial incorrect call on such inconclusive evidence. Yes, the replay official botched the call, but reached the correct outcome in doing so.
This is mgoblog fergodsakes!
...ready for the, "Pylon Cam?" There are helmet cams, suspended cams, goalpost cams, cameras in the dirt (baseball), etc. I mean, I know it is really only a matter of time until there are tiny microchips and sensors all over the football and alarms will sound once the ball crosses the plane, but until then, I vote for Pylon Cam.
And great work, WolverineBlue. That was quite impressive and fun to read.
I think the pylon mounted cam would have shown a great view of Huyge's knee/thigh and been even less useful in determining the ball's position. The football's position, I mean, not Huyge's.
the ball crossed the goal line there would be a lot of meaningless buzzers
You combined two of my favorite things, math and Michigan football.
Reminds me of the extra credit problem we were given in calculus to find the height of a dodecahedron in terms of one of it's sides...great fun.
Excellent analysis, excellent post. Thanks!
I referred this blog post to my son ( recently accepted by the M engineering program) as an example of the value an M degree would confer.
This is great work. I was actually considering doing a (much simpler) analysis of this myself, but never got around to it, so I'm incredibly grateful that you did.
I vote for an invisible fence. Shock collar on the ref and a chip in both ends of the football. I bet they wouldn't miss an TD calls.