[From Bill Powers (921221.0630)]
Morning thoughts on perceptual invariants.
The following is a diagram of the relationships in my post of
921220.1745. At the bottom is the v1=v2 space with a circle of
radius R plotted in it, and two sets of "lines of indifference"
due to two perceptual functions, p1 = v1 + v2 and p2 = v1 - v2.
Some added thoughts about this were in my head when I woke up.
P3 = R^2
···
>
=====================
> (p1-a)^2 + (P1-b)^2 |
====================
/ \
/ \
/ \
/ \
* * *
* * * *
* *
* *
* * * *
sine * * cosine
* *
* * * *
*< *
P1 | P2 |
> >
--------- --------
> v1 + v2 | | v1 - v2 |
--------- --------
\ /
\ /
> \ \ \
> \ / \ / /
> \ / / \ / \ /
> \ / \ / / / \ /
>\ /\ * \* * \ / /
> \ */ / */ \ / \ /
>/ \*/ \ / \ / * / / \
V1 | / \ /\ .____R___* \ / \
> / * / \/ \ * / \
>/ */ \ / \ / *\ / \
> / * / \ /\ * / \ \
> / /* *\/* \/ \ \
------------------------------------
V2
If v1 increases while v2 decreases by the same amount (lines
slanting down and to the right), P1 remains constant. If v1 increases while v2
increases by the same amount (lines slanting
up and to the right), P2 remains constant.
A point moving around the circle plotted with *** will generate a
phase-shifted sine wave in P1 and a phase-shifted cosine wave in
P2. As long as a point remains somewhere on the circumference
of the circle, whether moving or not, it will generate values of
P1 and P2 with amplitudes in a quadrature relationship (sine and
cosine of the same angular variable).
If now a second level of perceptual function is added which
computes the sum of the squares of P1 and P2 suitably offset by
subtraction of constants, the resulting perceptual signal P3 will
have a magnitude proportional to the square of the radius of the
circle in v1-v2 space. As the point in v1-v2 space moves around
the circumference of the circle, P3 will remain constant. So all
points on the circumference of the circle will produce the same
magnitude of perception at level 2.
If, on the other hand, a point lies off the plotted circle, which
is to say on the circumference of a larger or smaller circle, the
perceptual signal P3 will become larger or smaller accordingly.
We can therefore say that the perception P3 is a perception of
the radial distance of a point from a center a,b without regard
for the direction from that point. It can also be called the
perception of size of a circle, as all points on a circle of one
size will give rise to the same magnitude of P3, while points on
different-sized circles will give rise to correspondingly
different mangnitudes of P3.
Note that the location of the point from which all radial
distances are computed is determined by the constants a and b in
the perceptual function, not by the location of the plotted
circle in v1-v2 space. To perceive an invariant with respect to
direction from a center, the constants a and b must be adjusted
to fit the location of the circle, or the circle must be shifted
to the location specified by a and b. A shift in a and b might
correspond to a control process by which attention (or gaze) is
shifted to the centroid of a figure, with the effect of exposing
any invariance with respect to rotation that might thus be
created.
The lines of indifference are suggestive of the line-detectors
found by Hubel and Wiesel. These line-detectors would represent
the outputs of perceptual functions of the form P = k1*v1 +
k2*v2. Implied is an underlying orthogonal v1-v2 coordinate
system in visual perception (if vision is what we are talking
about here) at the lowest level, where v1 and v2 are perceived.
These relationships must be richly suggestive to a mathematician.
Unfortunately I am not a mathematician. Note that the space
involved does not have to be visual space, nor do the coordinates
have to be orthogonal in any geometric sense. All that is needed
is that v1 be capable of varying independently of v2. And of
course this treatment could be extended to spaces of any
dimensionality.
All of this gives me a faint sense of encouragement about the
idea that perceptions may have something fundamental to do with
the universe for which they stand. On the other hand, for all I
know they prove once and for all that the universe is totally
defined by the way our peceptual functions are organized and we
will never know its basic nature, if it has a basic nature. The
mathematical reasoning required here simply exceeds my abilities.
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Best to all,
Bill P.