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confusing notation / incorrect definition
Nov. 30th, 2008 05:06 pmfrom Horn & Schunk (1981) "Determining Optical Flow", p.187:
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:
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).
<< 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 distancein 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).
