FACIAL BEAUTY AND FRACTAL GEOMETRY

Note IDSIA June

Jurgen Schmidhub er

IDSIA Corso Elvezia Lugano Switzerland

juergenidsiach httpwwwidsiachjuerg en

c

Copyright
J Schmidhub er All rights reserved

Abstract



What is it that makesaface b eautiful Average faces obtained by photographic or



 


digital blending are judged attractive but not optimally attractive digital exagger



ations of deviations from average face blends can lead to higher attractiveness ratings My

novel approach to face design however do es not involve blending at all Instead the image of

a female face with high ratings is comp osed from a fractal geometry based on rotated squares

and p owers of The corresp onding geometric rules are more sp ecic than those previously

used by artists such as Leonardo and Durer They yield a short algorithmic description of all

facial characteristics manyofwhich are compactly enco dable with the help of simple feature

detectors similar to those observed in mammalian brains This suggests that a faces b eauty

correlates with simplicity relative to the sub jective observers wayofencodingit

Each of test sub jects male age yr female age yr rated the articial

female face depicted in Figure as b eautiful and as more attractivethananyofthe six faces

in a previous study based on digital face blends The design principle is claried in Figure



First the sides of a square were partitioned into equal intervals then certain obvious interval

b oundaries were connected to obtain three rotated sup erimp osed grids thick lines in Figure

   

with rotation angles
ar csin and ar csin Higherresolution details of the fractal

grids were obtained by iteratively selecting two previously generated neighb ouring parallel lines

and inserting a new one equidistant to b oth for clarity some negrained detail is omitted in

Figure Shifted copies of circles also omitted for clarity inscribable in thicklined squares

of Figure scaled by powers of account for transitions between nonparallel face contours

such as facial sides and chin Toachieve a realistic impression some colorsgreyscales were rst

matched to those in the photograph of a real p erson then further edited digitally Figures



isolate imp ortant featuredening lines of the rst second and third selfsimilar grid resp ectively

Hundreds of alternative simple geometric designs I tried led to less satisfactory results

The face satises several ancient rules used by mathematically oriented artists such as Leonardo

da Vinci and Albrecht Durer eg it is symmetrical the distance between the eyes equals one

eyewidth or one nosewidth the tip of the nose is ab out half the way from chin to eyebrows



Certain facial measurements that correlate with attractiveness are automatic bypro ducts The

present featuredening rules however are much more sp ecic than those used in previous work

For instance they sp ecify compare Figures and The prolongations of the left eyebrows

lower and upp er edges meet the righteye lid and the righteyeshadows upp er edge resp ectively

The upp er edge of the left eyes nosenear interior and the lower edge of the righteyes lie on

the same line through the image center b etween the pupils Squares of equal size are formed

by lines dening a upp er edges of left eyebrow and left nostril left side of nasal ridge and left

forehead b upp er edge of left eyebrow lower edge of right nostril left part of nasal ridge and



In particular images herein may not b e used for any commercial purp oses eg as templates for computer

animation or plastic surgery without explicit written consentby the author

right facial b oundary c upp er edges of right upp er lip and the left eyes nosenear interior left

nasal ridge and right facial b oundary Rules such as ab ove symmetrically hold for opp osite

face parts Figure Figure shows how large diagonal squares with low fractal resolution

shap e additional imp ortant features such as left and right parts of lower lip and chin and certain

contours of eyes and eyebrows Figure also sp ecies many additional simple facial prop ortions

based on p owers of



The attractiveness of the face can b e explained in the spirit of a previous suggestion based on




the theory of minimum description length among several patterns classiedascomparable

by some subjective observer the subjectively most beautiful is the one with the simplest shortest

description given the observers particular method for encoding and memorizing it For instance

realistic representations of female faces must satisfy certain constraints concerning shap e and size

of dening features Among images sub jectively classied as b eing within the acceptable tolerances

the ones with compact co des will b e preferred

Tentatively accepting this hyp othesis since dierenthumans with facememorizing strategies

tuned by dierent sub jective exp eriences tend to have similar preferences we are led to ask what

is common among human faceenco ding algorithms Nob o dy really knows but at least it is well

known that early visual pro cessing stages of mammalian brains employ rotationsensitive edge and



bar detectors According to standard information theory such detectors could contribute to an

ecient co de of the particular face in Figure because they allow for compressed descriptions

of certain imp ortant features For example detectors that will be maximally resp onsiveto the

nearvertical eyebrow contours and parallel eye and mouth contours will resp ond minimally to

the orthogonal contours of nose and facial sides with high probability they will be either on

or o Lo osely sp eaking this corresp onds to one bit of information as opp osed to several bits

necessary for describing intermediate values and is reminiscent of results obtained by a recent

learning algorithm for articial neural nets that also favors co des based on features with sucient



but minimal information content

Similarly hyp othetical detectors measuring ratios of distances between parallel lines could

help to further compress the description of Figure b ecause the same distance ratios based

on p owers of recur over and over More generally dierent parts of the face are enco dable by

reusing descriptions of other parts thus allowing for compact algorithms describing the whole the


  

dening features of the face havelow algorithmic complexity given a description mo del

based on simple geometric feature detectors Hence the image maybeviewed as an example of



lowcomplexityart

Mathematicians nd b eauty in a simple pro of with a short description in the formal language

they are using Facial attractiveness may reect a similar correlation b etween b eauty and sub jec

tive simplicity Our results indicate that neither average faces obtained by blending nor certain

attractive digital caricatures thereof are as attractive as a particular face whose essential features

are compactly enco dable using a simple but novel geometric construction metho d Future analysis

of attractive face typ es other than the one in Figure may prot from examining the signicance

of other esp ecially dimensional fractal geometric patterns Generation of aesthetic faces by

artists mayalsoprovide clues as to howhuman face recognition works

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Figure Color image digital postscript version of articial female face designed according to

rules il lustrated in Figures

Figure Explanation of the short algorithm describing the essential features of Figure s low

complexity face based on superimposed grids The scheme enforces simple facial proportions



based on powers of The sides of the quadratic frame are partitioned into equal intervals

n

Large squares thick lines are responsible for basic contours smal ler ones by factors of

n for details Shifted copies of circles omitted for clarity inscribable in thicklined

squares of Figurescaledbypowers of account for transitions between nonparal lel facecontours

such as facial sides and chin



Figure Selectedfeaturedening lines of the grid rotatedbyar csin





Figure Selected lines of the grid rotatedbyar csin





Figure Selected lines of the grid rotatedby