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We started with labeling everything on the boundary and worked our way in. So he might have, in a typical scene, he might have tens or even hundreds of junctions to label and no easy way of dealing with it. And for each of those choices, whatever he's decided junction number 2 is has its own suite of possibilities. And so it becomes a simple depth-first search problem, right? So in actuality, as soon as Waltz-- he was my office mate at the time. As soon as he wrote this program, he kept looking over at the computer-- they were big in those days. So you looked over at the computer to see if the lights were still blinking.

Because he'd start this depth-first search program up and nothing would happen. Because the search base is exponential and much too big for an ordinary computer, or maybe even an ordinary universe. He has to come up with a new method for using all these labels that he's-- after about a year and a half's worth of hard work, with lots of paper on his desk in little blocks. After year and a half of hard work getting all these junction labels figured out, he then has to come up with a method for figuring out how to use them.

And so we don't know whether to think his biggest contribution was that label set or his method. But that will be convenient, since the line labels are here. Well, you can assume I'm looking at this through a window. So the edge of the window form boundary lines, and they exert no constraint whatsoever on what's behind them.

So this is a legitimate drawing to have to think about. Or do you get a firm sense that there's a unique interpretation of all those lines? I think there's a unique interpretation of all those lines. What I'm going to do is I'm actually going to solve this problem using Huffman's labels but Waltz's method. Because I can't simulate on the blackboard something with 50 line types and thousands of line junctions.

So I'm going to use Huffman's set to demonstrate Waltz's algorithm. So Waltz's algorithm involves starting out by plopping on some junction all of the possible labels that the answer has to be drawn from. So let me number these in the order that we're going to visit them.

And so far, I've just put down the three fork options that are resident on that first junction. And I have to take note of exactly what they do with the lines that come out of the junction. All I've done is copy the junction labelings from my library.

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And at this point, Waltz's algorithm says there's nothing else to do but go on to junction number two. And unfortunately, sadly, there are lots of labelings that have to be considered on junction number 2. But now, having copied those down, Waltz's algorithm looks around at the neighboring junctions and says, are any of the things that I just placed on junction two disallowed by what I've already placed on a neighboring junction? And it sees that these three arrows require the line that joins junctions 1 and 2 to be either minus or plus. So of the six possibilities, I can only keep the ones that are likewise content to put a plus on that line that joins the two.

So that means that the influence flowing from junction 1 eliminates that one, eliminates that one, eliminates that one, and eliminates that one. All the ones that try to put a boundary line on that line between 1 and 2 are disallowed. Now likewise, we could say, well, of the remaining ones, do they restrict what I can do at junction 1? So now, continuing on, we have to see what we can do at junction 3. So we have to copy exactly the same labels set as we had before.

And now we look down at junction 2 and say, well, what does that tell me about the three that I've just placed? If we look up from junction 2 to see what kind of constraints it puts on here, we have this one alive and this one alive. And they both but boundary lines-- I think I must have had this boundary line wrong, right? And that must be one that goes-- this minus goes up. So that means that something that tries to put a concave line there is gone. So the influence flowing up in this direction in the third step eliminates that guy and eliminates that guy, leaving only this guy.

But now, the thing I was worried about is you have to also at this point go down to 2 and see if there's any further constraint on what can survive down there, based on what has happened over here at junction 3. But this one goes down, which is not compatible with a survivor. So when I bring this down in step three, this guy is eliminated. So now I'm down to just one interpretation for what can be going on at vertex number 2. So now that I've made a change on vertex number 2, I have to also see if that causes a change at vertex number 1. And now I can see that the only possibility here is a minus, the label that's coming down from our survivor.

It's hard to do this by hand, but I've got three of the four things labeled. And even with just three of the four labeled, I'm down to a single interpretation for all of the junctions and the lines between them. We better deal with it, because for all we know, this is not a legal drawing in this world. I don't have to draw much here, because I know this is forced to be a plus now. And there's only one fork vertex with any pluses on it at all. So now I can come through and say, well, the only possible survivor is this one.

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And now I have an interpretation for all of the junctions. You'd like to see this demonstrated to make sure I haven't made a mistake. So each of the places where a line is obscured has four possibilities, labeled E. The arrow junctions are labeled A. Or it could be something that we can think of as a step going up from left to right.

Yeah, but the stuff is creeping up from the lower left up to the upper right. Seems to be doing just fine until it hits the upper right-hand corner and discovers it can't label stuff. And what looked OK in the lower left is no good after all. So these results are kind of consistent with what we humans do when we look at these kinds of things.

So it's very likely that we, in our heads, do have some constraint propagation apparatus that we use in vision. But putting that aside, we can think about other kinds of intelligence different from human, that might use this kind of mechanism to solve problems that involve a lot of constraint in finding a solution.

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So here, we saw the constraint propagation activity at work on line drawing analysis. But next time, what we're going do is we're going to see at work in map coloring. People who do scheduling, because that turns out to be the same problem. Don't show me this again. This is one of over 2, courses on OCW. Find materials for this course in the pages linked along the left. No enrollment or registration.

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There's no signup, and no start or end dates. Knowledge is your reward. Use OCW to guide your own life-long learning, or to teach others. We don't offer credit or certification for using OCW. Made for sharing. Download files for later. Send to friends and colleagues. Modify, remix, and reuse just remember to cite OCW as the source. Lecture 7: Constraints: Interpreting Line Drawings. Flash and JavaScript are required for this feature. Instructor: Patrick H. Lecture 1: Introduction and Lecture 2: Reasoning: Goal Lecture 3: Reasoning: Goal Lecture 4: Search: Depth-Fi Lecture 5: Search: Optimal, Lecture 6: Search: Games, M Lecture 8: Constraints: Sea Lecture 9: Constraints: Vis Lecture Introduction to Lecture Learning: Ident Lecture 12A: Neural Nets.

Lecture 12B: Deep Neural Nets. Lecture Learning: Genet Lecture Learning: Spars Lecture Learning: Near Lecture Learning: Suppo Lecture Learning: Boosting. Lecture Representations Lecture Architectures Lecture Probabilistic I Lecture Model Merging, That was , I think. And we had to take four math courses, not just two. But most of us were men in those days. I think we only had 20 women in our class. Anyway, we had a quiz on Rayleigh scattering.

And I thought, well, this is pretty hard. And then I got my quiz back. I thought, well, I've been found out. I'm going to flunk out. My father will make me go to law school. I'll never attract anyone to marry. Horrible things will happen. Then the instructor announced the class average was I was two standard deviations above that. Some people say that the class average went down. Anyway, today we're going to study some stuff. Now we could, of course, do this in two ways. And anybody can do that. That's great. And you needed to learn some stuff that way.

So for example, there behind my outline is a line drawing. But most people would say there are two objects. And then his work was followed by Dave Huffman. And his work was followed up by Dave Waltz. So Guzman took a lot of pictures. So let me number these. Now I have an easier-to-deal-with picture. There are two links between 1 and 2 and 1 and 3.

One link between 2 and 3. One between 2 and 4. Two here. And one each from all of these. And you can see that that's a little bit too liberal. And the two-link theory says, oh, well, let's see. These three are pulled together into one object. So plainly, the two-link theory is too conservative.

So nothing new happens up here. So it pulls everything together like so. So that worked fine. Well, it didn't work fine. So we had two polar opposites of impressions there. He got an A, they say. So Huffman didn't like it. He thought it was a little bit too ad hoc. It was too heuristic. And so people began to say, well, why does it work?

And when does it not work? We'll reserve the word junction for something else. So this is in the drawing. That's in the world. That's abduction. And Huffman was a mathematician. So naturally, they approached the problem differently. That is to say, no screw cases.

So if you see a cube, it's going to look like this. And it's not going to look like this. So that's out. And this is in. So that's presumption number one. So you're going to have the intersection of three planes. And they don't have very much to do with the physical world. Which is concave and which is convex? You said something and you wrote the opposite. Thank you. So concave, convex, and boundary. So this down here, that's a concave line. And that would get a minus label. Oh, I don't know. Many times you see a boundary line.

So that's a boundary line. So three kinds of lines, four kinds of labels. And there's some things left out. What's left out? Cracks are left out. Shadows are left out. So let's have a little vocabulary before I go on. And I'll try to stick to it. And I'll try to stick to that vocabulary. All right?

## The Interpretation of Line Drawings with Contrast Failure and Shadows - Semantic Scholar

Yes, Christopher? It depends on which side of the object the stuff is on. So one, two, three assumptions. A little bit of vocabulary. So we have the possibility of making a complete catalog. This is so simple. But maybe it won't take a couple of years. Everything else is excluded. So we don't consider that case. If we don't fill any the octants, there's no junction. There's no vertex. So here's a fork-style junction. Now that's not the only way you can see that.

And here's another way. There's an L-style junction, right? And both of those are boundary lines. Are there any more? There is one more that's a little different, though. I can hold this guy up like so. Two boundaries, and the barb is convex. So in this particular case, I've got that. I've got that. And I've got a plus there.

### Line Drawing Interpretation

I happen to be able to reverse this, though. And here's the seven octants filled case. So another fork-style junction looks like so. Now we have a couple of other possibilities here. We might have five octants filled with stuff. So that means there are three octants that we can look from. Did I get that right? So I'm looking at it from this perspective.

It's an arrow. So it's an L. And this one is a boundary. And that's concave. Well, we're off and running. But we still have an awful lot to go. Think about the junctions that it can produce. I think I'll do that for you. So let me think about how that's going to work out. I might as well not hide that from you. And this one's going to be minus. And now we've got two more that are just like that. Look like so.

## Line Drawing Interpretation

And you say, oops. And the answer is sure. So that takes care of that. And then there's one more of these fork-style junctions. And that's plus, plus, minus that derives from this case. And they look like, let's see, plus, then plus. I'm having to think this through as I go. And then-- and that's it. Well, what about the T's? Now let's see. What about two, four, and six?

But they will have more than three faces. They'll have six faces. Like that. So we're going to ignore those. So we went to a lot of work there. But what have we discovered? There's nothing else in this world. So that's a very powerful constraint. So now let's see what we can do with it. Yet we'll use it any way. And I'll ask you the question. Can you build one of those? I don't know. Let's give it a shot. You look confused. That's commonly the case. So I know instantly that there must be a plus on the shaft. So we can come back here and take all these arrows here. And label them with plus lines on their shafts.

Now a line can't change its nature along its length. So what else can we do? Here deep inside is a fork-style junction. It's got convex markers on both of those two lines. And now we're done. We've labeled everything. Except-- look at this. What about that guy? Is there one of those in my catalog? You can't make it. You can't construct it. If it passes the test, does that mean it's possible? It's a necessary but not sufficient condition. For example, we could put a line like so. You feel better about it now? Let's see.

## Line Drawing Interpretation in a Multi-View Context

Is there such a junction label? We lose. It doesn't help. You think you can make it, but you can't. Let me show you the next example, Christopher. Consider this example. Can you make that? Your intuition is yes. So let's label it. Oh, I've already lost. But you feel like you can make it. So what's wrong? What's wrong is-- what, Elliott?

So you can make it. But not with three faces.

So some of these look like you could make it. And we can carry that idea back here. You can make this OK. But this junction, you've got two in the back and two here. So it has four faces. Same idea. So that's Guzman's contribution. Email Address. Sign In. Access provided by: anon Sign Out. Line-drawing interpretation: a mathematical framework Abstract: The author reports on the progress toward a mathematical theory of line-drawing interpretation. A working framework is developed, and a variety of tools and techniques are demonstrated within it.

He reviews related work and discusses his assumptions. He presents a projective mapping which he expects will find use in the extension of results derived for orthographic projection to perspective projection. The results render unnecessary the assumption of stability under perturbation of the viewed surface, which was required in previous work based on Whitney's results.