Sale-Spae Theory in
Computer Vision
1
Sale-Spae Theory in
Computer Vision
Tony Lindeb erg
Royal Institute of Tehnology
Stokholm, Sweden
KLUWER ACADEMIC PUBLISHERS
Boston/London/Dordreht
Foreword
The problem of
sale
p ervades b oth the natural sienes and the vi-
sual arts. The earliest sienti disussions onentrate on visual p er-
eption (muh like to day!) and o ur in Eulid's (
. 300 B.C.)
Optis
and Luretius' (
. 100{55 B.C.)
On the Nature of the Universe
. A very
lear aount in the spirit of modern \sale-spae theory" is presented
by Bosovitz (in 1758), with wide ranging appliations to mathemat-
is, physis and geography. Early appliations o ur in the artographi
problem of \generalization", the entral idea b eing that a
map
in order
to b e useful has to be a \generalized" (oarse grained) representation of
the atual terrain (Miller and Voskuil 1964). Broadening the sop e asks
for progressive summarizing. Very muh the same problem ours in the
(realisti) artisti rendering of senes. Artisti generalization has been
analyzed in surprising detail by John Ruskin (in his
Mo dern Painters
),
who even desrib es some of the more intriate generi \sale-spae sin-
gularities" in detail: Where the anients onsidered only the merging of
blobs under blurring, Ruskin disusses the ase where a blob splits o
another one when the resolution is dereased, a ase that has given rise
to onfusion even in the modern literature.
It is indeed lear that
any
physial observation of some extended quan-
tity suh as mass density or surfae irradiane presupp oses a sale-spae
setting due to the inherent graininess of nature on the small sale and its
apriious artiulation on the large sale. What is the \right sale" does
indeed depend on the problem, i.e., whether one needs to see the forest,
the trees or the leaves. (Of ourse this list ould be extended indenitely
towards the mirosopi as well as the the mesosopi domains, as has
b een done in the popular lm
Powers of Ten
(Morrison and Morrison,
1984)). The physiist almost invariably manages to pik the right sale
for the problem at hand
intuitively
. However, in many mo dern applia-
tions the \right sale" need not b e obvious at all, and one really needs a
prinipled mathematial analysis of the sale problem.
In appliations suh as
vision
the front end system has to pro ess
the radiane funtion blindly (sine no meaning resides in the photons
as suh) and the problem of nding the right sale b eomes espeially
aute. This is true for biologial and artiial vision systems alike. Here
a prinipled theory is mandatory and an
a priori
b e exp eted to yield
imp ortant insights and lead to mehanisti mo dels. The mo dern sale-
spae theory has indeed led to an inreased understanding of the low level
i
