3.3 Computational improvements
4.1.2 Two-dimensional spatial-frequency space
In 1978, Granlund presented a general picture processing operator which is a
two-dimensionalcounterpartoftheGEF[35],arguingthatasingleimageprocessingoperator
whichcoulddetectanddescribestructureatdierentlevelswasofinterest. However,the
eventthatincreasedthepopularityofGaboranalysisinthecomputervisioncommunity
wastheresultthatthetwo-dimensionalGEFresemblesthereceptiveeldsofsimplecells
in amammalianvisualsystem[80,24]. WhentheGEFsareusedaslters,theltercan
becentredat originandthe phaseand timeshift parameters, andt
0
, dropped. The
lteringwillbeconsideredincontinuousdomainforsimplicity.
The two-dimensionalGaborlter isa complexsinusoidalplane wavemodulatedby an
elliptical Gaussian probability density function (see Fig. 4.3). Several forms of Gabor
lters have been proposed for the two-dimensional case [35, 24]. Following Gabor's
formulationfortheone-dimensionalGEF,atwo-dimensionalGaborlter (x;y)canbe
dened as
isthefrequencyofthesinusoidalplanewave,istheanti-clockwiserotationof
theGaussianandtheplanewave,thesharpnessoftheGaussianalongtheaxisparallel
to thewave,and isthesharpnessalongtheaxisperpendiculartothewave. Itshould
–3 –2
Figure4.3: 2-DGaborlter. a)realpart;b)imaginarypart.
Gabor elementaryfunction buttheorientationoftheellipticalGaussian isthesameas
the orientationof theplanewave. Inaddition,theshifts inspace andphasehavebeen
droppedbecausethefunctionisusedasaconvolutionlter. Theresponsetoinputimage
(x;y)isthen
The lter in (4.6) can be normalised by xing the ratio of the frequency of the wave
and the sharpness values of the Gaussian, i.e., = f0
lter includes a constant number of waves. This formulation xes the behaviour of
the response regardless of the frequency and makes the DC-response identical for all
frequencies. It is desired that the DC-response is small, otherwise the average image
intensityaectstheresponse. Thiscanbecontrolledbysettingparameterlargeenough.
AnotherapproachistouseamodiedGaborlterwhereaGaussianwiththesame
DC-responseissubtractedfrom thelter[128]. TomaketheareaundertheGaussianunity,
anormalisationfactor
hastobeused. Thus,anormalisedltercanbepresentedas
(x;y)=
UsingFouriertransform,(4.8)canbepresentedin thefrequencydomainas
(u;v)=e
Thus,inthefrequencydomainthelterisarealGaussianwithcentroidatfrequencyf
0
_ 1 η
1 _ γ θ
Figure4.4: Gaborlterparametersinfrequencydomain.
SeveralGaborltersareusuallycombinedtoformalterbank. Thelterbankisusually
composedofltersinseveralorientationsandfrequencies,withequalorientationspacing
andoctavefrequencyspacing,aspresentedinFig.4.5,whiletherelativewidthsand
stayconstant. Thatis,
isthekthorientation,n
thenumberoforientations,f
l
thelthfrequency,f
max
themaximalfrequency,n
f
thenumberoffrequencies,ands(>1)theratiobetweentwo
consecutivefrequencies. Asshowninthegure,onlyahalf ofthefrequencyplaneneeds
tobecovered,becausetheinputtotheltersisassumedtoberealandthusitsfrequency
representationissymmetricandHermitian[13].
AsBoviketal.note,parameterselectionisanontrivialproblemhavingnosimplesolution
[12]. The most common criteria proposed in the literature include manual selection
(e.g., [102]),biologicalconsiderations(e.g., [73])and optimisation(e.g., [45,28]). If the
selection is based on the knowledge aboutprimate visual system, there is a drawback
that itis unclear,what has been thegoalof evolutionin thevisual system,that is, to
which purpose the visual systemhas evolved. On the other hand, optimisation based
selectionhasthedrawbackofbeingdependentonthetrainingset.
If the feature to be detected can be formulated analytically, the selection can also be
based on analysis ofthe lter response. Mehrotra et al.optimise thelter parameters
foredgedetectionbasedonthecriteriaproposedbyCanny[81]. Chenetal.selectsome
parametersbasedonexperimentalresultswhileothersareselectedbasedonmathematical
analysis [19]. Analytic parameterselection hasalso beenused in image representation
using GEFs [73]. InPublication IV, amethod for selecting Gaborlter parametersis
presentedfordirection sensitive edgedetectionin binary images. Themethod is based
ontheanalysisofthelterspatialsizeanditseect ontheangularaccuracy.
Despite theproblems of parameterselection,Gaborlters havea number ofattractive
qualities. First, theycanbe used to extractvarious kinds of visualfeatures, including
texture [12], edges[81], lines [19], and shapes[53]. Thisis illustratedin Fig.4.6where
the absolute responses ofa coinimage to ltersin several orientationsand frequencies
are shown. Also the similarity with the simple cells of the mammalian visual cortex
supports the representation power of the lters. Second, Gabor lter responses are
robust. It is known that the amplitudes of complex Gabor coecients are invariant
−1 −0.5 0 0.5 1
−1
−0.5 0 0.5 1
u
v
Figure 4.5: Abankof2-DGaborltersinfrequencydomain.
of the Gabor lters in spatial and frequency domains. While the shiftability requires
anonorthogonal,overcompleterepresentation[104], itprovidestherequiredrobustness.
In addition, shiftability makes it possible to interpolate responses both in spatial and
frequency coordinates. Furthermore, the Gaussian nature of the lters makes them
tolerant to noise [54]. However, the stability of the responses concerns primarily the
amplitudes of thecomplexlterresponses,while quiteoftenonlythereal orimaginary
partof thelter isused. Insomeapplications, such asedge detection,this isjustied,
but generallytherobustnessand smoothnessofthelterresponseislost.