Seattle, November 20, 2022
Hello again, morphmetters,
Jim's comment about the "traffic policeman" approach
to mosquito wing vein landmarking echoes many
older discussions in foundations of measurement. The best place
for that "policeman" is the center of the largest circle that
can be inscribed inside the boldface Helvetica Y-shape where the
veins are seen to branch. (This would be the triple-point
of the medial axis of that outline, see my 1991 textbook,
Section 3.5.) Such an operational definition is fine as long
as the data are imagined as curving lines of infinitesimal
thickness -- the original article on medial axes (Blum 1963)
argues that the human visual system actually carries out this
sort of optimization all by itself. But the approach fails
completely under higher magnification, when these "curves" turn out
instead to be curving cliffs of gray-scale, for which the
location of "the" line is a matter of image gradients
(second derivatives). As the
information in grayscale images is isomorphic to
curving surfaces in 3D space, the same critique applies to
actual surface data as well. This is an example of a point the
Dutch physicist Jan Koenderink made in his great treatise
"Solid Shape" (MIT Press, 1990), where he says that inspection
of EVERY observation that uses a machine requires
attention to two crucial parameters, the outer scale of
that machine's spectrum and its inner scale. Our visual systems
do this automatically, or, to use Bessel's own term,
"involuntarily," more's the pity -- in the Mach band process,
we see edges where there aren't any, only gradients.
Which brings up another fundamental point, the difference
between observations based on first-order derivatives with respect
to time or space and observations based on derivatives
of second order or higher.
(There's a nice comment on this distinction in Wigner's great
essay "The Unreasonable Effectiveness of Mathematics in the
Natural Sciences," 1960.) The Hoffmann article I cited in my
previous email notes that Bessel measured the time of transit
of a star across the zenith by setting a grid of evenly spaced
lines in his telescope, starting a metronome nearby, and
visualizing the location of the star with respect to the
line at the two consecutive click times of which one just preceded
the crossing, the other coming just after. The corresponding
time measurement might be precise to about a tenth of a second,
Hoffmann claims, given what we know about the human perceptome.
But this is a _linear_ phenomenon, the crossing of a line --the
first derivative is nonzero. Compare it to any of the usual
definitions of landmarks in the textbooks or the journals, which
refer to second derivatives (extremes of position of
a curve with respect to some coordinate axis, or tangents to a curve)
or even third derivatives (extremes of curvature, as at Glabella).
Orbitale, the "bottom" of the outline of the optic socket, is
a particularly notorious example of this ambiguity; another is
the once-standard "Frankfurt Horizontal" for registration of
skulls in comparative studies, as it is defined
by a line tangent to two separate "curves" that are actually
oblique cliffs in space, one contact "point" at Orbitale and the other
at the auditory meatus. (If you don't see the problem, just
pick up an actual skull with one hand while you hold a ruler
with the other.) I got my start in craniofacial biology
over forty years ago in a paper with Robert Moyers on the meaninglessness
of the Frankfurt Horizontal ("The Inappropriateness of
Conventional Cephalometrics," 1979), but the problem had been
acknowledged for decades before _that._
This apercu, in turn, echoes an old claim of the psychologist
Edwin Boring (1886-1968), quoted in the Hoffmann article I cited
yesterday, who claimed (back in the era when biometrics dealt
with continuous measures rather than discrete genomes) that
that field reduced to psychometrics, not vice versa. His
argument (a polemic, really), which you can find in Harry Woolf's
proceeding "Quantification" of 1961, makes fascinating reading today.
Particularly relevant are two of his "generalities":
(.) "A new line of investigation is established and
everyone joins in the new pursuit." And,
(.) "Quantification is favored by the desire of investigators
to claim the prestige of science for their research."
And he concludes,
"So opinion -- sometimes within scientific awareness, sometimes
in spite of scientific unawareness, operates to aid or hinder
progress, and you may not know which it is, any more than you know
at the moment whether progress has got itself into a blind alley or
is going straight with a clear course ahead. And that is the
way that quantification in science comes about, is pushed ahead or
held back ..."
I recommend Woolf's book warmly to any of you who might
have some interest in history of science.
For instance, it includes the first appearance in print
of Thomas Kuhn's argument about "paradigms" that turned into
the all-time best seller in philosophy of science, his
"Structure of Scientific Revolutions." And please forgive these
occasional lectures of mine on history of science, especially
their prevailing theme of the irony that "everything
important was already said before, but nobody
listened" -- they are so much fun to write!
Fred Bookstein
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