Wikipedia:Reference desk/Archives/Mathematics/2020 September 26

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September 26

Derivation of belief state transition function

The formula about belief state transition function in Partially observable Markov decision process § Belief MDP is

$\tau (b,a,b')=\sum _{o\in \Omega }\Pr(b'|b,a,o)\Pr(o|a,b)$

(F1)

So ... is F1 derived from

$P(A\mid C)=\sum _{n}P(A\mid C\cap B_{n})P(B_{n}\mid C)$

(F2)

which is one of statements in Law of total probability?

If I understand correctly, F2 can be rewritten as

$\Pr(A|C)=\sum _{U\in \Upsilon }\Pr(A|C,U)\Pr(U|C)$

(F3)

where $\Upsilon$ is a partition of the sample space, $U$ is just an element in set $\Upsilon$ ($U$ is NOT the universal set. I was running out of symbols.). Now, replace $A$ by $b'$, replace $C$ by pair "$b,a$", replace $U$ by $o$ and replace $\Upsilon$ by $\Omega$ in F3 so we get

$\Pr(b'|b,a)=\sum _{o\in \Omega }\Pr(b'|b,a,o)\Pr(o|b,a)$

(F4)

Furthermore,

$\tau (b,a,b')=\Pr(b'|b,a)$

(F5)

so we get F1 by combining F4 and F5. Is the derivation correct? - Justin545 (talk) 18:45, 26 September 2020 (UTC)

Ellipse in perspective

I was wondering: a circle in a perspective is always an ellipse (circle ⊂ ellipse). Is an ellipse in perspective also always an ellipse? I'm thinking that, since any ellipse can be gotten by viewing at least some circle in perspective (shouldn't be hard to prove from original statement), viewing an ellipse in perspective is also viewing a circle shown in perspective, in perspective. But is a perspective of an object in perspective still a perspective of the original object? 93.142.121.167 (talk) 22:34, 26 September 2020 (UTC)

It is almost always a conic section – possibly degenerate – but not necessarily an ellipse. One plane that is relevant is the plane containing the ellipse, which I'll call the "ellipse plane". Another one is the plane through the oculus of the projection that is parallel to the picture plane. If this parallel plane intersects the ellipse in two distinct points, you get a hyperbola, and if it merely touches it, you get a parabola. If the ellipse coincides with the parallel plane, you get nothing at all, and otherwise, if the oculus lies in the ellipse plane, you get a straight line or line segment.) This is also so if the ellipse was a circle, so the introductory sentence is not correct. If the parallel plane and the ellipse are disjoint, and the oculus does not lie in the ellipse plane, you do indeed get an ellipse.  --Lambiam 23:30, 26 September 2020 (UTC)

Ah, I think I get it now. (Sorry about the conic part, I should've added "if it's a closed curve"). I had to draw a diagram of projections into a parabola/hyperbola - those (and cases where the ellipse is pretty close to the plane thru the viewpoint) were actually the cases where I figured it wouldn't be an ellipse. Parabolas and hyperbolas reach outside the FOV and that's where rectilinear perspective stops being a good model for what the eye really sees, so it defeated my intuition.

What do you think about the second question? If you create (e.g. paint) a perspective projection of an object, and then consider a perspective projection of that picture, is there always some angle from which the original object would look like this projection² ? 93.142.121.167 (talk) 03:41, 27 September 2020 (UTC)

An additional aspect in which the mathematical perspective projection goes beyond the concept of perspective in painting and photography is that points "behind" the oculus are also projected. For example, a photo of a pair of rails will have them meet at the horizon and then vanish; they are two line segments forming the legs of a wedge. The perspective projection of two (infinite) parallel lines is (except for some anomalous cases) a pair of equally infinite but intersecting lines, minus the point of intersection, as if, on a photo, you see the rails behind the camera too, but above the horizon.

For the second question, anomalies apart, the answer is yes (modulo a scale factor, a proviso that is not needed if the thing being painted may be in front of the canvas). It is irrelevant for the problem whether the second painting is of an existing painting, or of anything representational for that matter. All that matters is that it is an image of a "flob" (= flat object); it could be a drawing of a sign bearing the legend "TRESPASSERS WILL", or a photograph of some pressed and dried leaves. The flob, while being painted, was situated in a plane in 3-space, which I'll call the flob plane. The canvas was in the picture plane. The painter's oculus was somewhere else. After the flob has been painted, we leave the freshly created painting where it is but remove the flob. After the paint has dried, we reverse the roles of the planes: the flob plane becomes the new image plane, and the image plane becomes the new flob plane. So the painting, which is a second flob, remains where it is, but a blank canvas is now mounted in the plane where before the original flob resided. When painted on the new canvas, the image of the second flob will be as the original one. (Some linear scaling with the oculus as the point of origin of either the second flob or the blank canvas may be needed to respect real-world constraints that are mathematically irrelevant.)  --Lambiam 13:04, 27 September 2020 (UTC)

Thanks, that helped. Sorry, read it earlier but forgot to post. 93.142.115.132 (talk) 19:35, 29 September 2020 (UTC)