Table Of ContentLecture Notes in Statistics 206
Editedby
P.Bickel,P.Diggle,S.Fienberg,U.Gather,I.Olkin,S.Zeger
Forfurthervolumes:
http://www.springer.com/series/694
Marlos A.G. Viana • Vasudevan Lakshminarayanan
Dihedral Fourier Analysis
Data-Analytic Aspects and Applications
123
MarlosA.G.Viana VasudevanLakshminarayanan
UniversityofIllinoisatChicago UniversityofWaterloo
EyeCenter SchoolofOptometry
ChicagoIllinois UniversityOntario
USA Canada
ISBN978-1-4614-5561-5 ISBN978-1-4614-5562-2(eBook)
DOI10.1007/978-1-4614-5562-2
SpringerNewYorkHeidelbergDordrechtLondon
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To IngramOlkin
Preface
ThetheoryandapplicationsofdihedralFourieranalysispresentedinthesenotesare
aimedatthestudyofexperimentaldataindexedby,orassociatedwith,thepointsin
adihedralsymmetryorbitsuchastheimagesofaretina’svisualfield.Theimages
◦
are iterated 90 rotationsand/orhorizontalreflections of each other, and together,
the resulting set of images gives a symmetry orbit generated by any one of the
images.Theorbithasthesymmetriesoftherotations{1,r,r2,r3}=C andreversals
4
{h,rh,r2h,r3h} = C h that together define the dihedral group D = C ∪C h.
4 4 4 4
Similarly, the vector field illustrated in the diagram below gives a D orbit when
4
the rotations in C and reversals in C h are represented with the (homomorphic)
4 4
substitutions (cid:2) (cid:3) (cid:2) (cid:3)
cosφ−sinφ 1 0
r(cid:4)→R= , h(cid:4)→H= ,
sinφ cosφ 0 −1
where φ=2π/4, and applied to the position (q) and the direction (p) of a given
fieldvector(q,p),thusdefiningthedihedralorbitshownbelow
vii
viii Preface
2
1
0
1
2
2 1 0 1 2
O ={(τq,τp); τ∈D }.
4
Here a reversal was defined as a counterclockwiserotation precededby an x-axis
reflection.Notethatanytwodistinct-colorvectorsareline-reflectionimageofeach
other,whereasanytwovectorsofthesamecoloraredouble-reflection(orrotation)
imageofeachother.Inthatorbit,forexample,wemaywanttoevaluatethegradients
xτ=[∇f(τq)]·τp, τ∈D4,
ofaresponsefunction f evaluatedatτqinthedirectionofthevectorτpalongeach
fieldvectorintheorbitO,orthebehaviorofthecenterofmass
Xσ= ∑τ∑mττmστFτ, σ∈D4,
inthehomogeneousfield
Fτ=(τq,p); τ∈D4,
withfixeddirection pandallocatedmassesmτ.
The dihedral orbits generated by D , as planar transformations, are simply
n
the 2n symmetries of an n-sided regular polygon for n > 2, whereas the planar
transformations in D = {1,r,h,rh} describe the symmetries of a 180◦ rotation
2
Preface ix
(r) and of two line reflections (h, rh=v) along the vertical and horizontal axes,
respectively.Weoftenwrite,then,D ={1,r,h,v}.
2
Theorbitsillustratedabovearegeneratedbyapplyingthesymmetriesτ∈D to
4
any one of the eight components in the orbit, so that they are independent of the
orbit’sgeneratororinitialcondition.
It is precisely this arbitrariness that is characteristic of the resulting invariants
in any dihedral analysis. They are always invariant (in a way to be made precise
throughout the text) to the different choices in initial conditions. Obviously, the
image
resultingasthesuperposition1+r+h+voftheelementsintheD orbit
2
remainsthesameforallD transformations,whereasthesuperposition1−r−h+v
2
x Preface
eitherstaysthesame,whentransformedby1orv,orbecomescolor-reversedwhen
transformedbyrorh.
Every orbit O is to be understood, in the present data-analytic context, as a
framework, a set of labels for experimental observations, so that the analysis of
any data indexed by a dihedral orbit consists of systematically determining the
summaries of the data allowed by the orbit invariants, suggesting their statistical
analysis,andproposingeventualbroad-rangeinterpretations.
Thepointsinthedataspaceforanydihedralanalysiswilloftenbeexpressedas
thesymboliclinearcombinations
x=∑xττ,τ∈Dn
τ
in which the 2n coefficients xτ are the data of interest along the orbit. The xτ
are naturally real (R) scalar measurements, eventually extracted from complex
coefficients. In a few cases, the xτ will be points in the ring R of n×n real data
matrices. We write, respectively, CD or RD to indicate the dihedral (group)
n n
algebra and the dihedral (group) rings. They are complex vector spaces endowed
with a multiplication induced by the dihedral multiplication. Details will be
presentedinChap.2.
As it will turn out, every dihedral algebra shall correspond to a direct sum of
irreducible(matrix)linearsubspaces
M (C)⊕...⊕M (C)(cid:8)CD
n1 nm n
ofdimensionsn2,...,n2 thatisisomorphictotheoriginaldatavectorspaceCD ,in
1 m n
dimensionequaltotheorder(2n)ofD .Inthedihedralringcase,thepointsinthe
n
linearsubspacesarematricesovertheringR.
