Using Space Effectively: 3D

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Lecture on Nov 7, 2007



[edit] Readings

  • Chapter 2, Projection Systems. In Art and Representation. Willats (pdf)
  • Chapter 8: Marginal Distortions. In The Psychology of Perspective and Renaissance Art. Kubovy (pdf)
  • What object attributes determine canonical views? Blanz et al. (html) (pdf)

Optional Readings

  • Artistic Multiprojection Rendering. Agrawala et al. (html)
  • Automatic View Selection Using Viewpoint Entropy and its Application to Image-Based Modelling. Vázquez et al. (html)
  • Artistic Composition for Image Creation. Gooch et al. (pdf)
  • Map Projections in PDF

[edit] David Purdy - Nov 07, 2007 01:17:19 pm

Kubovy's book can be found online at

[edit] Robin Held - Nov 08, 2007 04:45:14 pm

In class we briefly discussed whether or not the visual system interprets real-life scenes as projected onto a planar or curved surface. I think the discussion is somewhat inapplicable to visual perception. The rays of light entering the eye are projected onto the retina, which, yes, is a curved surface. But this retinal projection is never perceptually flattened into a planar image, which I think was implied at some point. We just make judgments directly from rays of light that hit the retina. So, if we want to accurately reproduce a scene on a painting, we need to reproduce the rays of light that would have hit the viewer's eye, had s/he been at the original scene. Typically, we do so with a flat surface. The viewer's eye (note: SINGLE eye) must be located at the center of projection (COP) of the image in order for it to receive the correct bundle of rays. Since there is only ONE COP for an image, the bundle of rays will be incorrect for any other viewpoint, and certain objects will look distorted. This is why spheres, etc look distorted on some perspective images. If the eye is at the COP, those distorted shapes actually project to the image of a sphere on the retina. Also, it is crucial to use only one eye with a pinhole aperture to view the image correctly, because the visual system has processes that use the slant of a surface to interpret the imaged shapes. See this paper for an in-depth discussion:

[edit] Mark Howison - Nov 09, 2007 08:47:37 am

Willats observes on pg. 41 that for shape consistency, "we see the shape of a circle, set at an angle to the line of sight, as a compromise between its true shape as a circle and its projected shape." I think the effect is more subtle than this, and is informed by the visual context. For instance, in the image below, the free-standing ellipse on the left does not look circular to me. The ellipse on the right, however, appears circular because the horizontal lines add context that lead me to recognize the entire figure as a cylinder, and therefore the ellipse as a projected circle.

Image:Circle or ellipse.png

It should be possible to construct a figure in which two competing contexts create ambiguity between an ellipse and a projected circle. Here is an attempt:

Image:Cylinder or football.png

Is it an elliptical football or a cylinder with a circular top?

[edit] Ken-ichi - Nov 12, 2007 02:26:16 pm

Why is it necessary to break down projection systems beyond parallel and perspective? For all the variations of parallel projection systems Willats describes, all that seems to change is the relative positions of the object and the viewer. The only exception I can see in his examples is the isometric projection shown in Figure 2.16, in which the true object seems to have been oriented such that its projection makes it appear to be sitting flat on the ground, viewed from above, when in fact the projection rays are intercepting the picture plane at right angles.

[edit] N8agrin - Nov 13, 2007 11:00:38 pm

It's not clear to me based on Kubovy's discussion of perspective in Renaissance art whether those artists seemed to grasp the notion of true perspective and how it would distort a three dimensional space within a still life, or if they simply drew an image based on their mind's eye. Kubovy seems to imply that artists such as Uccello likely understood the power of perspective, but I wonder if they truly understood its implications or just developed their works based on 'what looked right'. Kubovy uses Raphel's The School of Athens as an example of an artist using two perspectives, where the background is drawn in perspective, and the people are draw in a somewhat orthogonal way (I believe, I'm having a hard time distinguishing between orthogonal and oblique. Regardless, Kubovy calls this a marginal distortion).

I started thinking about this because it is so difficult for the mind to disassociate an object, no matter its perspective, with its absolute shape. What I mean by this is that regardless of what angle we look at a regular rectangular painting on a wall, we can perceive it as a rectangle even though it may actually be a trapezoid from our current perspective. Given this, it might make sense for an artist, such as Raphel, to paint a larger all encompassing scene in a perspective fashion, such as the background in School of Athens, while each individual in the painting as a more orthogonal projection, given that we tend to look at individual's faces carefully and focus our attention on their details as opposed to the space they occupy. This became more obvious to me when I read Willat's piece and came upon figure 2.4, the telephoto image of an airplane and a truck. Willat's point is to demonstrate a real-life orthogonal perspective, but I had a hard time thinking about the truck in the photo as anything other than a truck existing in three dimensions, even though the telephoto lens has already flattened the image, distorting the otherwise perception of perspective that might have otherwise been captured.

[edit] James Andrews - Nov 13, 2007 11:26:07 pm

Mark -- I think the shape of the circle comment was more just to answer, in the case that you do interpret it as a 'angled' view, what shape do you see? So it's not trying to say that every ellipse we see is interpreted as a tilted circle (that would be a pretty absurd claim, I think ...), but rather that if we do see an ellipse as a tilted circular shape, the shape we perceive will be a compromise between the true 3D circle and the actually-visible 2D ellipse.

[edit] David Jacobs - Nov 14, 2007 01:02:08 am

N8agrin - The situation when looking at pictures isn't that simple. True, in normal circumstances it's very easy for a human to determine that a picture on a wall is supposed to be transformed in some manner to get the (sometimes) intended frontoparallel view (looking straight at it from the center). We can construct the transformation because we've got a lot of prior assumptions that simplify the manner. Because we can look with both eyes, it's easy to see that the picture frame is slanted away from us, rather than being of an odd shape. Even with a single eye and pinhole aperture it's reasonably easy to do the same if you assume the picture frame to be rectangular. In the context of viewing graphics on a computer screen, however, it's much more difficult because there are conflicting cues, you may be looking at a frontoparallel surface (the monitor), but be viewing a picture in which the intended CoP (if there is one) is somewhere off the normal axis of the monitor. Robin or Amanda could explain this a bit better, but I know the question of picture perception is an open research topic.

Also, since we're talking about perspective, I have to bring up [The Ambassadors], because it's just that cool.

[edit] Willettw - Nov 14, 2007 01:50:00 pm

In addition to some of the more practical cartographic projections we saw today that balance shape preservation and area preservation to varying degrees, there are an awful lot of projections out there that are less useful but still awfully fun to look at. One nice example I've always been drawn to is Waterman's Butterfly Projection which projects the map onto the sides of a semiregular polyhedron whose vertices are drawn from the intersections in a sphere packing layout. Although it doesn't preserve either area or angles perfectly, this projection is actually quite legible and I've always found it very aesthetically appealing.


Original image copyright © 1996, Steve Waterman

Image:800px-Waterman Butterfly Map 1996.jpg

wikipedia link

A gallery of other polyhedral projections here is also pretty fascinating.

[edit] Amanda Alvarez - Nov 17, 2007 05:47:34 pm

I think the use of two projections in the School of Athens stems from an understanding (probably not explicit at the time of painting) that the viewer perceives frontoparallel (parallel to picture plane, facing viewer) and receding, non-frontoparallel elements differently. Now we understand that people compensate differentially for different viewing angles and pictorial content. (This is articulated more in the paper cited by Robin above: Vishwanath et al. 2005). Portions of the scene that vary in depth (floor, walls, ie. non-frontal elements perpendicular to the picture surface) are drawn in perspective, while frontal or near-frontal elements (people) are drawn orthogonally. Frontal vs. non-frontal elements are compensated for in different ways by the viewer; this is manifest in the pointing phenomenon, where an element pointing out of a picture (eg. a finger) seems to follow you as you move in front of the picture, whereas frontal elements in the picture retain the direction they are facing. This ties in with shape constancy; however I think the invariance or constancy of perception has less to do with familiarity than with the fact that geometry, optics, and evolution have constrained the visual system to (try to) compensate when disambiguating information is present (as in near-frontal or 'rectangular' cases). When the estimates that we base the compensation process on become too uncertain (ie. the COP is too far out as in Holbein's 'The Ambassadors'), we simply cannot compensate. Julian Beever's pavement drawings are another example of anamorphic art, where the COP is way off from a normal projection.

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