Mtg 18/26: Thu-13-Mar-2025

Outline for Today

Sampling and Assignments

Administration

Today

  • gaskets (and tetrahedra)
  • Assignments 2, 3 (and alternative project)
  • sampling

For Next Meeting

Wiki

Link to the UR Courses wiki page for this meeting

Media

Transcript

Audio Transcript

  • Okay, so since I didn't have the recording on right
  • away, so I want to talk about sampling and the rest of the
  • assignments, so we'll look at how to construct, sir pin ski
  • gaskets and tetrahedra. Look at those assignments, and then talk
  • about sound.
  • So I created an image here, and
  • anything you notice about the image?
  • The circles are pretty loaded, yeah,
  • I suppose
  • I are they bigger than they need to be? So if we think in two
  • dimensions and
  • so we have a circle, a unit circle at the origin, and then
  • an equilateral triangle and
  • is inscribed inside the circle, so When we translate to the Vert
  • towards the vertices and
  • I'm not doing a good job. Okay?
  • So they should be touching,
  • but not overlapping. You
  • Does that make sense? So Do
  • Yes, here they're overlapping. I
  • so what I did in this image was I, I made
  • an object instance that's floating in front and not to the
  • top, and then I transformed that three times to make the shape in
  • the background upon which the shadow is being cast and
  • it missed opportunity to call it that scare ginsky gasket. I
  • so you can have that.
  • So the way I've set up the pbrt file is I'm looking that I
  • position is two and y looking at
  • so we're just x is zero, y is two and Z is zero, and looking
  • at the origin, was the Z vector being the up direction.
  • So this is all COVID From the simple pbrt. And then I'm
  • including gasket dot PBR key, which defines the instance
  • called gasket. And then I'm scaling
  • I'm trying translating the object three times, and then I'm
  • just after that, I'm translating once more With the original,
  • with the original instance, not scale
  • So this is what the gasket object looks like.
  • So it's material, diffuse, material translate, and we're
  • scaling well. We say the sphere has a radius of point 03125,
  • so I believe There's 240 43 spheres. I
  • I have a little piece of code here. Let me show you where it
  • is on
  • the website and
  • so I didn't do very well with documentation. Sorry about that.
  • So we're doing, I'm defining a function gasket that will take a
  • level so it's, it's recursive function.
  • So if level is greater than zero, we're going to call the
  • function with the image
  • of the
  • of the point that we start with. And so in our case, the original
  • point is 000, because that's, that's this point here. And
  • and then I'm
  • outputting the details.
  • Nothing so
  • if levels gradients are up calling it, I'm recursively
  • calling it with each of the three transformations for the
  • vertices that correspond to the vertices of the triangle or the
  • gasket. And if I'm at the leaf node, then I,
  • then I print, then I Do the last transformation and print out The
  • scaling. So
  • Did That make sense? I
  • CS Third Way. The details of the transform of the program are in
  • this last little bit. So we defined three transformation
  • matrices so we scale from the coordinate of the vertex to the
  • origin, and then we scale by a half.
  • And then we scale, or translate back to the from the origin to
  • the sent fixed point.
  • Okay,
  • and then so we do object begin, and then SP is a starting point
  • which is 000,
  • and then we say starting point identity matrix, and then we're
  • giving it five levels. So I did this. There's room for
  • improvement in the code, but gives you an idea of How it can
  • proceed. I
  • so we start with zero, then We have the sphere, The unit
  • sphere, I
  • I'm curious why It's good. It's taking so long. I
  • maybe it's too close. The eye point is too Close. We'll see. C
  • I,
  • that's A bit too close. I
  • too far away. Let's Try That. I
  • click theory is going To be too Large. I
  • so it won't be very easy To see the Initial sphere, but do
  • It looks like they're really stuff in there, doesn't it? But
  • it seems to be correct. I
  • so the problem is with My diagram here, Isn't it? I
  • should we try? Let's try another intermediate size. Should have
  • added a command line argument for This. I
  • i was expecting the spirit to be still a little more visible on
  • the other side. Is there? Do I See it? This time? I
  • Okay, that's two. One
  • More, 1010.
  • So I think it's retracting from the sphere, because I just did
  • approximation, and I did four decimal places of the
  • transformation.
  • Okay, does that make sense? Is that helpful? I
  • so how do we make a tetrahedron out of this? I
  • Are we able to, like, instantiate points and stuff
  • like that? Because I know how to do it with just points, but with
  • spheres, you'd have to, you'd either have to, again, use
  • points, or you have to start to collate the faces.
  • Can I meet? You're a little bit over complicating this. So how
  • do we get a tetrahedron instead of a triangle,
  • three extra, you'd have to do a
  • transform on the other like you do the one, and then you would
  • have to do a trend, basically a duplicate of it, but that
  • duplicate would be overall, transform to
  • be different. How to position different? Appear to be
  • different, position differently so it would come out. Basically
  • your object space
  • would be slightly different transformations.
  • So how about it? So what do we have now to make this triangle?
  • We got to replace all those spheres with the littles, except
  • we got to make it three dimensions, because it's still
  • only one dimension or two dimensions.
  • Now, yes, how many transformations are we using to
  • make this triangle? Two dimensions, three?
  • So how many transformations do we need to make a 3d object
  • form? Yes.
  • So we can think of the triangle being in the sphere at the
  • try not To confuse this. So this is just too deep. I
  • and then at this point we have
  • triangle. And then we want to put the other transformation
  • there,
  • but At the Top we're
  • I'm sorry I wrote that so small. So the fourth transformation is
  • at the center of the triangle at a different height.
  • So that means we can have, so this is one triangle here, and
  • then we have another triangle. And
  • so both triangles one and two and three and four.
  • So if we just take the triangle and copy it. You get the right
  • side looking properly bussing, but
  • it wouldn't have the interior structure. So that's why we're
  • iterating with four transformations instead of
  • three. Does that make sense? So what are some other ways you
  • could instead of using spheres, how could we describe the
  • tetrahedra shape the serpent ski tetrahedra He drawn
  • triangles.
  • Yeah, we Could do triangles, cubes, oh,
  • here so we can define
  • the shape and then transform it instead of A sphere
  • How About cylinders? I
  • or you could do boxes as well, I guess so. Originally I did
  • spheres.
  • One is 35 years old, and
  • if you do cubes, you might have the opposite problem, where
  • there's gaps and they're not actually touching, just because
  • where the points are around the cube. I Right? Because, maybe
  • not because the triangle, but if we actually use the triangle,
  • there would be the height would be less than the width, so you
  • end up getting like, so you have to, like, sort of account for
  • that little gap and you face together, so maybe you wouldn't
  • have that same cubic gap. It's like the soft mother triangle is
  • smaller than the width of the side. All
  • the lengths are the same. All the sides are the same
  • length, but the interior height is different than that side
  • length. You have, I think, root three for a length of two, and
  • then the height of the pyramid is root two. I've done all these
  • calculations already because I made it out of points. So
  • points. So then it's, it's easy to do points instead of spheres
  • instead of points, because we Just placing this under the
  • sphere at the point. Okay?
  • So is 10 levels of recursion. Much
  • it gets a little worried. I
  • so what do we see? Let me zoom in here. I
  • blurring surface key. That's good. I
  • What was your answer?
  • How did You describe them first? Oh, okay. Do
  • so we have jagged edges from aliasing. I
  • So where we have,
  • where we have blurring. When we look at the image at Greg,
  • intended resolution. We don't see this. I
  • It looks fine there for that resolution, but once
  • you hit so many recursions, you can't really see the next ones
  • anyway, regardless of what
  • resolution. So 10 is probably too many, because you get to the
  • point where you just can't see the recursion anymore.
  • Yeah. So there are a couple ideas here
  • the larger screen. Do
  • so,
  • so we see a staircase effect, but also the different colors of
  • the pixels indicate
  • that we're
  • applying some anti aliasing here to compete. The blur is it's
  • compensating for the AVIC And
  • so question is, what size is sphere? I
  • so that question makes sense, so what? How big a sphere do we
  • need, so that we're making the most of our display, but not
  • adding
  • adding extra detail that, what kind of can we think of an
  • expression to or attest to
  • do that I use,
  • like a radius of half the
  • height short, yeah, Okay, so Let's Say you
  • so what are the world coordinates that we have In this
  • image? And
  • so We can figure out what
  • how much world how much spaces and world coordinates in the
  • image, and We divide that so we figure out how much we
  • how Much of that space is covered by A Pixel i
  • So conceptually, as we find out how much world space is being
  • covered by a pixel, and then,
  • and then, we find the relationship between the size of
  • the sphere and the size of the pixel and
  • so should the sphere be the same size as The pixel, or should it
  • be half as Big or
  • so I went to lead This into a discussion about sampling if
  • I so we think about it in terms of a signal. How
  • many samples do we need to take to reconstruct that signal?
  • Does that question make sense?
  • Let's think about CD audio, compact disc audio. You still
  • familiar with compact discs?
  • Okay, yeah,
  • is that what C stands for? Yes. Oh, yeah, well, oh, good God,
  • the answer
  • might be all
  • over. I mean, I knew the Fauci so just didn't know that what CS
  • stand
  • for, yeah, I once asked somebody what an LP was. This is a few
  • years ago, and someone and they said, LP, it's a lumbar
  • puncture. They said they were watching house at the time, if
  • you remember that show. Anyway, LP stands for long play, 33 and
  • a half third revolutions per minute on the turntable. Anyway,
  • I digress a little bit. See
  • answering machine all over again. I
  • so this CD audio standard is the sample at 44.1 kilohertz.
  • Can people hear up to 44.1 kilohertz.
  • They tried really hard.
  • Most People can hear around 22 i
  • So have you ever heard of Nyquist?
  • Sometimes it's also, have you heard of Shannon Claude Shannon?
  • Anyway? Sometimes it's referred to as Nyquist Shannon, but in
  • the text, it's called Nyquist. So Nyquist, the Nyquist sampling
  • rate is the rate we need to reconstruct the signal. So if we
  • sample it at twice the highest frequency, then we can
  • reconstruct the sound, or you can reconstruct a signal. If
  • it. So there are a couple quiz questions here that we can bring
  • In
  • to Review And
  • I so there is a question, Is it useful to think of pixels as
  • squares, true or false? Anyone think it is useful? So I said,
  • covered by a pixel. So I'm just thinking about the area that
  • we're sound from which we're drawing samples. So there's no
  • geometry with the pixel. The value we compute for the pixel
  • is a sample, and if it's seen with the ray tracer, we might
  • have more than one sample per pixel that we average together
  • to come up with the value that we assign to that the image
  • pixel. So
  • So Elvie Ray Smith, who was at Pixar, I believe, made an
  • emphatic argument for pixels not being squares. That was back in
  • 1995 I think so that reference is only 30 years old. So
  • so why don't we just increase the number of samples per pixel
  • when we're ray tracing?
  • At some point, you're going to be just taking way too long, but
  • you're also not going to be seeing any gains from additional
  • sampling.
  • So yeah, so as we take more samples, the time goes up and we
  • get better images, but there's still noise in them.
  • So is there if we didn't have to worry about the time? Let's say
  • we had a super duper computer that made that first point not
  • so important. I
  • Is there a number of samples we could take that would say, Okay,
  • Well, this is enough to reconstruct the image.
  • Yeah, I
  • try that one again. I'm
  • so the Nyquist works in the audio case because people can't
  • hear, generally people can't hear above 22,000 hertz. So that
  • gives us a limit on the frequency that we need to
  • reconstruct get the signal of the audio. But in the case, when
  • we have things in computer graphics, if it's we may not be
  • band limited, So that means we can never take enough samples. I
  • CS, does That make sense? I
  • I said, Yes, where do you get the signal?
  • Or reconstruct is
  • not band limited. So if it's not, if there isn't a maximum
  • frequency that we can double the i
  • Okay, and So about the assignments
  • I've linked up things, so there's a place to get a
  • description, and I'll work on bit more detail for the
  • descriptions,
  • but they're there in your courses, and they link Back to
  • my website,
  • so I'll film the details, but the idea is linked to A small
  • version of the Web of The picture. I
  • it Okay, so keep on filling them in. Well, it won't be a
  • continual process, but I'll get more detail. So I fixed the
  • links and by Tuesday, well, we can discuss final versions of
  • the assignment with the rubrics and so forth. Does that sound
  • okay? So they're both. The two assignments are due on April 8.
  • Cut off date will be April 15.
  • Okay, so we're out of time. Thank you for today. You
  • does the did that? Exploration of gaskets and pbrt help clarify
  • some things, very polite. And let me go on about that. I think
  • you're all very polite anyway. Anyway, thanks for today. Have a
  • good weekend. See you on Tuesday. Bye.

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