confusing notation / incorrect definition

[gusl]

from Horn & Schunk (1981) "Determining Optical Flow", p.187:

<< We will derive an equation that relates the change in image brightness at a point to the motion of the brightness pattern. Let the image brightness at the point (x,y) in the image plane at time t be denoted by E(x,y,t). Now consider what happens when the pattern moves. The brightness of a particular point in the pattern is constant, so that
dE/dt = 0. >>

What does this mean?? The natural interpretation is that E is a function of image coordinates.

First of all, since E is a function of (x,y,t), I interpret dE/dt as a partial derivative, i.e. how much E(x,y,t) changes as we marginally increase t:




I've just spent >1 hour frying my brain on this silly thing.

What does "The brightness of a particular point in the pattern is constant" mean? Is this an existential or a universal statement?

My classmate's interpretation, which I think is correct, is that the above is a universal statement. To quote Appendix A:

Consider a patch of the brightness pattern that is displaced a distance in the x-direction and in the y-direction in time . The brightness of the patch is assumed to remain constant so that

I believe this expresses the constancy statement from the top: "The brightness of a particular point in the pattern is constant". The points in the pattern are moving through points in coordinate space, but if you keep track of same pattern point as it moves, its brightness will not change.

So in dE/dt = 0, E must refer to a function of *pattern* coordinates, rather than image coordinates. This contradicts the way E is defined on the first paragraph quoted.

Obviously, if E referred to image coordinates, then dE/dt = 0 would mean that we are in a boring situation where the image never changes (and thus there would be no paper for us to read).

Comments

[bhudson.livejournal.com]

My interpretation: There's an image plane, upon which the pattern is projected. The pattern is constant; its projection changes with time.

[bhudson.livejournal.com]

Oh wait. Now I see how that doesn't make any sense at all. Nevermind.

[darius.livejournal.com]

The only interpretation that came to mind was bhudson's, but that doesn't make sense of dE/dt = 0 to me.

Feynman would try and derive equations for optical flow himself and then use his answer to check/make sense of this. For what it's worth -- me not being Feynman.

[bhudson.livejournal.com]

Another interpretation is that I know from personal experience that academics sometimes are in the office late on a Sunday working desperately on a paper, and some things slip through that don't make any sense.

[puellavulnerata.livejournal.com]

My intepretation is that E is not a partial derivative, but includes x and y as functions of time defining the motion of the pattern over the image plane.

Hence the notation




rather than

[gustavolacerda.livejournal.com]

So for each point p, you would define its path in time in image coordinates x_p(t), y_p(t)? That sounds reasonable. But I believe this paper is referring to whole patterns (e.g. geometric figures), rather than single points.

[puellavulnerata.livejournal.com]

Yeah, and each point of the pattern moves along a path like that, with x and y coordinates in the image plane varying as functions of time, so E(x,y,t) defines a projected image varying over time, and d/dt(E(x(t),y(t),t)), where x(t) and y(t) define the path of a particular point of the pattern, represents the same point of the pattern as it moves over the image plane, and thus dE/dt must be zero (but the partials are all non-zero).

[spoonless.livejournal.com]

Right, that would be my interpretation too. They say "the brightness of a particular point in the pattern is constant." They should have emphasized it more, but the key point is that it's a point in the pattern they're talking about, not a point on the screen. The brightness of points on the screen change with time, while the brightness of any point in the pattern is constant, since it's a fixed pattern (moving over the screen).

[jcreed.livejournal.com]

Yeah multivariate calculus is a tarpit of terrible notation and ambiguity.

I would describe the situation like this.

An image is a function i : R² → R.

Suppose the image "moves around" in space according to a function m : R → R². That is, at time t, the image is displaced by the vector m(t).

Suppose we have a sensor sitting at p : R². At time t, it detects brightness i(p-m(t)).

So the time-derivative of the brightness our sensor gives is computed by the chain rule.
(d/dt)(i(p1-m1(t), p2-m2(t))) = -m1'(t) i₁(p-m(t)) - m2'(t) i₂(p-m(t))
where a subscript n indicates partial derivatives with respect to the nth argument. That expression on the right is exactly negative the dot product of the gradient of i with m'(t).