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(Click
here for the overview page on the
complexity-based theory of beauty and on
Low-Complexity Art.)

**Note IDSIA-28-98, June 7, 1998 **

**Jürgen Schmidhuber
IDSIA, Corso Elvezia 36, 6900 Lugano, Switzerland
**

Copyright © by J. Schmidhuber. All rights reserved.

**Abstract.**
What is it that makes a face beautiful? Average faces obtained
by photographic^{1-4} or digital^{5} blending are judged
attractive^{1-5} but not optimally attractive^{6,7} - digital
exaggerations of certain deviations from average face blends can lead
to higher attractiveness ratings^{7}. My novel approach to
face design does not involve blending at all. Instead, the
image of a female face with high ratings is composed
from a fractal geometry based on rotated squares and powers of 2.
The corresponding geometric rules are much more specific than those
previously used by
artists such as Leonardo
and Dürer. They yield
a short algorithmic description of all facial characteristics, many of
which are compactly encodable with the help of simple feature detectors
similar to those observed in mammalian brains. This suggests
that a face's beauty correlates with simplicity relative to the
subjective observer's way of encoding it.

Each of 14 test subjects (10 male, age 26-37 yr; 4 female, age 16-46 yr)
rated the artificial female face depicted in Figure 1 (below)
as ``beautiful'' and as more attractive than any
of the six faces in a previous study based on
digital face blends^{7}.
The design principle is clarified in Figure 2.
First the sides of a square were partitioned
into 2^{4} equal intervals; then
certain obvious interval boundaries
were connected to obtain three
rotated, superimposed
grids (thick lines in Figure 2)
with rotation angles
+- arcsin 2^{-3}
and arcsin 2^{0} =
45^{o}.
Higher-resolution details of the fractal^{8} grids were obtained
by iteratively selecting two previously generated, neighbouring,
parallel lines and inserting a new one
equidistant to both (for clarity, some
fine-grained detail is omitted in Figure 2).
Shifted copies of circles
(also omitted for clarity)
inscribable in thick-lined squares of Figure 2,
scaled by powers of 2,
account for transitions between non-parallel face contours
such as facial sides and chin.
To achieve a realistic impression,
some colors were first matched to those in
the photograph of a real person, then further edited digitally.
Figures 3-5 isolate important feature-defining
lines of the first, second, and third self-similar^{8}
grid, respectively.
Hundreds of alternative simple geometric designs I tried led
to less satisfactory results.

The face satisfies several ancient
rules used by mathematically oriented artists
such as Leonardo da Vinci and Albrecht Dürer, e.g.: it is symmetrical;
the distance between the eyes equals one eye-width or one nose-width;
the tip of the nose is about half the way from chin to eyebrows.
Certain facial measurements
that correlate with attractiveness^{9,10}
are automatic by-products.
The present feature-defining rules, however, are
much more specific than those used in previous work.
For instance, they specify (compare Figures 2 and 3):
(1) The prolongations of the left eyebrow's lower
and upper edges meet the right eye lid
and the right eyeshadow's upper edge, respectively.
(2) The upper edge of the left eye's nose-near interior
and the lower edge of the right eye's lie on the same line through the
image center between the pupils.
(3) Squares of equal size are formed by
lines defining (3a) upper edges of left eyebrow and left nostril,
left side of nasal ridge, and left forehead;
(3b) upper edge of left eyebrow, lower edge of right nostril,
left part of nasal ridge, and right facial boundary;
(3c) upper edges of right upper lip and the left eye's
nose-near interior, left nasal ridge, and right facial boundary.
Rules such as (1-3) above symmetrically hold for opposite face
parts (Figure 4). Figure 5 shows how large
diagonal squares (with low fractal resolution)
shape additional important features, such as left and right parts
of lower lip and chin, and certain contours of eyes and eyebrows.
Figure 2 also specifies many additional simple facial proportions
based on powers of 2.

The attractiveness of the face could be explained
in the spirit of a previous suggestion^{11}
based on the theory of minimum description length^{12-16}:
**among several patterns classified as ``comparable''
by some subjective observer, the subjectively most beautiful
is the one with the simplest (shortest) description,
given the observer's particular method for encoding and
memorizing it.**
For instance, realistic representations of female
faces must satisfy certain constraints concerning
shape and size of defining features. Among images
subjectively classified as being within the acceptable
tolerances the ones with compact codes will be preferred.

Tentatively accepting this hypothesis,
since different humans with face-memorizing strategies
tuned by different subjective experiences tend to have similar
preferences,
we are led to ask: what is common among human face-encoding
algorithms? Nobody really knows,
but at least it is well-known that early visual processing stages of
mammalian brains employ rotation-sensitive edge and bar detectors.
According to standard information theory^{17} such detectors
could contribute to
an efficient code of the particular face in Figure 1, because they
allow for compressed descriptions of certain important features.
For example, detectors that will be maximally responsive
to the near-vertical eyebrow contours and parallel eye and mouth contours will
respond minimally to the orthogonal contours of nose and
facial sides -- with high probability
they will be either on or off.
Loosely speaking, this corresponds to one bit of information
as opposed to several bits necessary for describing intermediate
values, and is reminiscent of results obtained by a recent learning
algorithm for artificial neural nets that also favors codes based on
features with sufficient but minimal information content^{18,19}.

Similarly, hypothetical detectors measuring
ratios of distances between parallel lines could help
to further compress the description of Figure 1, because
the same distance ratios (based on powers of 2)
recur over and over.
More generally, different parts of the face
are encodable by reusing descriptions of other parts,
thus allowing for compact algorithms describing the whole:
the defining features of the face have low
algorithmic complexity^{12-14},
given a description model^{15,16}
based on simple geometric feature detectors.
Hence the image may be viewed as an example of
low-complexity art^{11}.

Mathematicians find beauty in a simple proof
with a short description in the formal language they are using.
Facial attractiveness may reflect a similar correlation
between beauty and subjective simplicity.
Our results indicate that neither average faces
obtained by blending nor certain attractive, digital
caricatures thereof^{7} are as attractive as
a particular face whose essential features
are compactly encodable using a simple but novel
geometric construction method. Future analysis
of attractive low-complexity face types other than the one in Figure 1 may
profit from examining the significance of other,
especially 3-dimensional,
fractal geometric patterns.
Generation of aesthetic faces by artists may also provide
clues as to how human face recognition works.

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