Detecting branching structures using local Gaussian models

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Original citation:

Wang, L. and Bhalerao, Abhir (2002) Detecting branching structures using local

Gaussian models. University of Warwick. Department of Computer Science.

(Department of Computer Science Research Report). (Unpublished) CS-RR-385

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Loal Gaussian Models

Li Wang, Abhir Bhalerao

Department of Computer Siene

University of Warwik

Coventry CV4 7AL

November 26, 2001

Abstrat

This report presents a method of deteting branhing struture,

suh as blood vessels from retinal images, using a Gaussian

Inten-sity model. Features are modelled witha Gaussian funtion

parame-terisedbyposition,orientationand varianewithinsome spatial

win-dow. MultiplefeaturesaremodelledusingasuperpositionofGaussian

models. Anon-parametrilassier (k-means)isusedto luster

om-ponentsorrespondingtoeahfeature. Twodierentgroupsofimages

are usedto test the methodology: artiial images and images of the

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1 Introdution 1

2 Loal Linear Feature Estimation 2

2.1 A GaussianIntensity FeatureModel . . . 2

2.1.1 Feature Centroid estimation . . . 3

2.1.2 Orientation . . . 4

2.2 Multiple Linear FeatureEstimation . . . 5

3 Reonstrution and Hypothesis Testing 6 3.1 Feature Reonstrution . . . 6

3.2 Hypothesistesting . . . 7

4 Sale-Spae representation 8

5 Experimental Results and Disussion 9

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1 Sale spae representation and Juntion response at dierent

sale(The size of the irles reetthe detetion sales) . . . . 13

2 Examples of a GaussianIntensity Modelfor linearfeatures . . 14

3 Parameters of GaussianModel G(~x ). . . 14

4 Windowed Fouriertransform of aexample retinal image(a)

showing DFTMagnitude Spetraat dierentsales (b);();(d). 15

5 Clustering Approah using K-means (Dierent olour

repre-sents the omponents belong toeah feature infrequeny

do-main.) . . . 16

6 Hypothesistesting algorithm. . . 17

7 EstimationresultforeahhypothesisP 1

=0:97;P 2

=0:89;P 3

=

0:95 . . . 17

=0:22;P 2

=0:97;P 3

=

0:90 . . . 18

=0:70;P 2

=0:90;P 3

=

0:97 . . . 18

10 Hypothesistestinginareal retinalimageforN =64blok sizes 19

13 Labelthebranhpointindierentsaleofansynthetiimage

(the size of the irlesreet the detetion sales) . . . 22

14 Label the branh point in dierent sale of retina image (the

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Line, orner and branh detetion in digital images is widely used in many

omputer vision appliations suh as image registration, objet reognition

and motion analysis. We are interested in using suh methods for medial

image segmentation (e.g. blood vessel detetion inretinal images).

The detetion and measurement of blood vessels an be used as part of

the proess of automated diagnosisof disease [1℄. Thus a reliablemethod of

deteting bloodvessel struture in2D or3D tomographi images is needed.

The intersetions of the blood vessels reate juntions or orners whih are

important dominant points for the whole struture sine the information

about a shape isonentrated atthem.

Broadly speaking, orner detetion tehniques an be lassied into two

major ategories. The rst of these is boundary-based approahes that use

pre-segmentedontours(eg. [2℄),while theseondisbasedonthe analysisof

the rawgray-leveldata (eg. [3℄).

In the ase of boundary-based orner detetion, the image is rst

pre-segmented. The Canny edge detetor and zero-rossing methods are

om-monly used to extrat the boundaries enountered [4℄. Some methods then

use alinkingstrategytoformhain odes. Ifa pointattheobjetboundary

makes disontinuous hanges in diretion or the urvature of the

bound-ary is above some ertain threshold, then that point is delared as a orner

point[5℄. Algorithmshavebeendevelopedtodetetornersalongthe

bound-ary by measuring the eigenvalues of ovariane metris to loate the orner

point [2℄. Other researhers have extended the Hough Transform to nd

pointsofhigh loalurvaturefromthe edge pixels. Thisis doneby

aumu-lating positions of `loalisation'points inthe Hough spae, ie. foreah edge

pixel,aloalisationpoint(similartotheentre of the radiusofurvature)is

omputed by movingaertaindistane away fromthe edgepixelorthogonal

to the edge diretion. Corners are then loated by using the intersetions

of the loalisation points [6℄. The main weakness of all these approahes is

that the performane of orner detetion relies on the suess or failure of

the pre-segmentation step.

Gray-levelornerdetetionmethodsanbedividedintotwogroups:

tem-plate based and geometry-based. A templatebased orner detetor uses the

similaritybetween agiventemplateofaspei angleandthe imagedata in

asub-windowtond theorners[7℄. Unfortunately,beauseofthe

omplex-ityofallpossibleornerstrutures,itisimpossibletodesigntemplateswhih

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ner positions. The produt of gradient magnitude and the rate of hange

of gradient diretion (urvature) with gradient magnitude are both used to

measure the 'ornerness'. A orner is delared if the `ornerness' is above

ertain threshold and the pixel is alsonominallyan edge point [8℄[3℄. Sine

these measures depend on seond order dierentialsof the image, the

algo-rithm is sensitive to noise. Furthermore, these approahes are only able to

detet step-edge orners and donot addressthe problem of linejuntions.

Someresearhershaveombinedsale-spaetheorywiththemeasuringof

loalurvaturetodetetjuntionpoints[1℄[9℄, wherethesignal issmoothed

by onvolution with Gaussian kernels of dierent width, then the loal

ur-vaturesare traked through dierent sales to loalise the orner point.

In this paper, weuse amultiple resolutionstrategy dierently exploiting

fromWilson'swork[10℄andisspeiallyaimedatndingbranhstruturesof

bloodvesselinmedialretinalimages. AGaussianintensitymodelisused to

representsimplelinearstrutures andtheMultiresolutionFourierTransform

(MFT) [10℄ [11℄is alsoused to estimateparameters of the model.

The report isorganised as follows: In setion 2,the Gaussian modeland

the algorithmof feature estimation is presented. Setion 3is devoted to the

feature reonstrution and hypothesis testing. Setion 4 reviews the main

steps in ageneral methodology for asale spae algorithmwhihis adapted

to the juntion detetion problem. For a more detailed desription about

sale-spae representation see [1℄[9℄ [12℄ and [13℄. Experimental results and

disussionarepresentedinSetion5. Conlusionsandideasonhowtoextend

the algorithm followin Setion6.

2 Loal Linear Feature Estimation

2.1 A Gaussian Intensity Feature Model

If an ideal linear feature is windowed by a smooth funtion w(), it an be

regarded as a 2-dimensional Gaussian funtion [14℄, examples of whih are

shown inFigure 1.

The 2-dimensionalGaussian funtionan bewritten inthe form:

G(~x)=(2) 1=2

jCj 1=2

exp( (~x ~) T

C 1

(7)

~x=(x;y) T

(2)

and ~ is the mean vetor and the ovariane matrix C = R T

VR , where V

is the diagonal matrix of varianes, V = 2 x1 0 0 2 y1

, R is the rotation

matrix, R =

os() sin()

sin() os ()

, is the angle to the x-axis. Figure 2

illustratesthe meaningof the parameters used in the model.

Beause of the omplexity of real images, the model learly an not be

used torepresent the whole image. However, itan be used ona small part

of the image suh as an image blok of size N N. In another words, we

an splitthe imageintoaset ofblokswithdierentsizes,thentrytotthe

modelin eah regionindividually.

To estimate the parameters of our model, namely [~;, x 1 2 , y 1 2 ℄, a

Windowed Fourier Transform is applied in eah blok before the feature

extration proess:

Y(~!)= X

w( ~ x 0

~x)y(~x)e ( j~x

0 ~ !)

(3)

where~! isthe frequeny o-ordinate,

~

!=(u;v) T (4) and w( ~ x 0

~x ) is a window funtion used to loalise the signal. In this

work, a osine square funtion isused for w().

Figure3showsthe magnitudespetraofthe windowed Fouriertransform

at dierent levels, ie. using a dierent sale window. For regions ontaining

a single feature, the orresponding spetral energy lies orthogonal to the

spatial orientation. For more ompliated regions like branh points, there

is a superposition of energies havinga less lear DFTstruture.

2.1.1 Feature Centroid estimation

If it is assumed that there isonly a single feature in one blok, the position

of the feature, i.e. the distane of its entre from the origin with respet to

the origin of the image plane, is linearly relatedto the phase spetrum [15℄.

(8)

where G(~!)is theFouriertransformof the spatialimage. Fora singlelinear

feature, the phase spetrum, (~!),an bemodelled as

(~!)= ~~! (6)

where ~ is the entroid vetor and an be alulated by taking the partial

derivatives of the phase in eah of the diretions. In the disrete ase, by

taking thediereneinphasebetween neighbouringoeÆients,theentroid

vetor of spatial position an be estimated as:

i = N 2 X ~ ! ^ (! i ) ^ (! i+1 ) (7) j = N 2 X ~ ! ^ (! j ) ^ (! j+1 ) (8) where N 2

isthe sampling interval.

2.1.2 Orientation

The MFT blok whih was modelled with Gaussian intensity proles may

be onsidered as having energy in an ellipse, entred on the origin. From

Borisenko and Tarapovs' work [16℄, a moment of inertia tensor T an be

alulated using the energy spetrumin plae of mass,

T = T 00 T 01 T 10 T 11 (9) T = X ~ !~! T ^ E(~!) jj~!jj (10)

where N is the size of the blok, the fator jj~!jj is used to redue the

greater emphasis to energy further away from the origin. ^

E(~!) is the

nor-malised energy ata given point (u;v)in the blok, ie.

^ E(~!)=

jE(u;v)j E sum (11) where E sum

(9)

tors of the matrix. Sine the orientation of maximum energy onentration,

,is dened asthe orientationof the majoraxis ofthe ellipse, itisindiated

by the diretion of the eigenvetor, ~e 1

, whih is assoiated with the largest

eigenvalue, 1

, i.e.

T~e 1 = 1 ~e 1 (12)

where~e 1

is dened as

~e 1 =(e 10 ;e 11 ) T (13)

The orientation an then be obtained from

^ =artan( e 11 e 10 ) (14)

2.2 Multiple Linear Feature Estimation

If more than one linear feature is presented in a blok, in order to perform

the estimation,itisneessary tosegmentthespetrumintoomponents

or-respondingtoeahfeature. Theomplete spetrumoftheregionismodelled

as the sum of the spetrumof eah luster:

G(~!)=jG(~!)jexp[ j(~!)℄= K X

l =1 jG

l

(~!)jexp[ j l

(~!)℄ (15)

The use of the multiple linearfeature modelallows regions ontaining

jun-tion pointsor orners.

Apartitioningmethod,K-means,isappliedtoseparatetheregionswhih

are ontributions from dierent features. K-means is anunsupervised,

non-hierarhial lustering method, whih is widely used in a number of image

proessing appliations [17℄ [18℄. It is an iterative sheme whih attempts

to both improve the estimation of the mean of eah luster, and re-lassify

eah sample tothe losest luster. Firstly, it piks randomly seleted initial

seeds whih are equal to the required number of lusters. Next, eah

om-ponent is examined and assigned to one of the lusters, depending on the

minimum distane. The entroid's positionof eah lusteris realulated at

eah iterationuntil nomore omponentsare hanginglass.

1. Initialise k = 2 or k = 3 lasses, hoosing k pixels' oordinates as

initial entroidsatrandomfromtheimage. Makesurethatthepairwise

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2. Using the phase gradient i;j

, onvert eah phase spetrum oeÆient

into a spatial vetor ~ P i;j

. The sampling interval is 2

N

where N

repre-sents the size of thewindow. Thespatial positionis alulated by

~ P i;j = N 2 ~ i;j (16)

Then, ompare the distane between eah pixel and eah lass entre

and assign oeÆient to the lass to whih it islosest.

3. Realulate the entroid for eah lass.

4. Repeat from step 2 until the movement of lass entre is lower than a

ertain threshold t m

(we use t m

=2 for 128128 image).

Dierent olours are used in Figure 4 to show the lustering approah

of the K-means algorithm in given window whih ontain 2 and 3 features.

After lassifying the omponents belonging to eah feature, the parameters

of eah feature an be individually estimated using equations(5){(14).

After the parameters of Gaussian model orresponding to eah feature

have been estimated, the orner points, q(~x) an be loalised as the

inter-setion of eah feature, denoted as A l ;m

, i.e. 8~x2A l

\A m

where i6=j and

A l ;m 2[A 1 ;A 2 ;A 3 ℄.

3 Reonstrution and Hypothesis Testing

3.1 Feature Reonstrution

If it ispossibleto synthesis the loalspetrum using the estimated

parame-ters, thefeature modelan bereonstruted. Someresearhers [19℄[20℄ have

usedthemagnitudespetrumderivedfromthedataandtheestimatedphase

spetrum to generate the synthesised spetrum. In this paper, we use both

theestimatedphaseandmagnitudespetrumtoreonstrutthemodel. Sine

the eigenvalues,denotedL 1

;L 2

alulatedpreviously,areinversely relatedto

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the synthesised spetrum, G(~!), an begenerated using the model

parame-ters,

~

G(~!)=jG 0

(~!)jexp[ j( 0

(~!)℄ (19)

where the estimated phase spetrum, 0

(~!),is given by

0 (~!)= 0 (!~ i )+ 0 (!~ j ) (20) and 0 (! i )=[ X ~ ! ^ (! i ) ^ (! i+1 ) ℄! i (21) 0 (! j )=[ X ~ ! ^ (! j ) ^ (! j+1 ) ℄! j (22)

By taking an inverse DFT of ~

G(~!) $ Y 0

(~x), the model reonstrution

an bediretly ompared with the data,Y(~x) totest the goodness of t.

3.2 Hypothesis testing

One the parameters havebeen estimated,the auray of the hypothesisis

hekedandthemosttmodelshouldbeusedtorepresenttheorresponding

data. In this work, we apply a probabilisti approah to test the model t.

Theprobabilitythatasynthesised data ~ Y 0

tsthe originaldata ~

Y isdenoted

as P(G K

j ~

Y),where G K

;K =1;2;3 represents the hypothesis model, ie.

G k =G K ( k ;C k

;k) (23)

As noted in [21℄, there are several kinds of algorithms whih ould be

usedforthefeaturemathing. Themostommonly-usedistheinnerprodut

method. Given the model, a likelihoodof the data an beapproximated by

P( ~ YjG

K )=

Y Y 0

jjYjjjjY 0

jj

(24)

whih is simply a normalised inner produt of the data with the estimated

model. It islearthat whenthe synthesised spetrumisexatlythe sameas

real spetrum, the value of P will be maximum and equal to 1. The more

aurate thereonstrutionP !1measuringhowwellthefeaturemodelts

the atualdata is used in agiven region.

The above method an be applied for K = 1;2 and 3 features

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k

ertain threshold, denoted as t r

, the blok is onsidered as not ontain any

likely model, G k

. Otherwise, the hypothesis with the maximum orrelation

results, P max

, hosen from P( ~ YjG

K

), gives G max

as the best feature model

for the region. Figure5 is anoverview of the algorithm.

4 Sale-Spae representation

The basi idea behindsale-spae representation isto separate out

informa-tion atdierentsales [12℄. Any imagean beembeddedinaone-parameter

family representation whih derived by onvolving the original image F(~x)

with Gaussian kernels of inreasing varianet.

S(~x;t)=F(~x)G(~x ;t) (25)

where G(~x ;t)denotes the Gaussian kernel whih an be writtenas

G(~x;t)= 1 2t e x 2 1 +x 2 2 2t (26)

Underthis representation, fora 2Dimage, the multi-salespatial

deriva-tivesan bedened as

S ~ x

n(~

x;t)=F(~x)G ~ x

n(~

x;t) (27)

where G ~x

n denotes aderivative of some order n.

After the whole stak of imagesis obtained, we an then extrat orners

atdierentsales. AsstatedinKithen'swork,theorneranbedetetedby

measuring the urvatureof level urves, i.e. the hange ofgradientdiretion

along anedge ontour. Oneof thespeialhoieis tomultiplytheurvature

by the gradient magnitude raisedto the powerof three [9℄, whih is:

k =S 2 x 2 S x 2 1 2S x 1 S x 2 S x 1 x 2 +S 2 x 1 S x 2 2 (28)

One implementation result of this algorithm is shown in Figure 6.

Fig-ure 6(a) shows an original retina image as well as the images whih have

been smoothed by onvolution with Gaussian kernels of dierent widths.

The result of 50strongest orner response k 2

afterapplying equation (28)is

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The reonstrution results and deision algorithmwere tested using several

syntheti and real images. Figure 7, 8, 9 show the reonstrution results

and the orrelation values P k

of artiial images for eah hypothesis. In

gure 7(a), there is only one feature in the blok. We an see that the

maximum orrelationvalueP 1

isderived fromone feature hypothesis, whih

is the best tted model. Similarly,on two and three features respetively in

gure 8(a), 9(a), it an be seen that maximum orrelation values are both

from the best tted hypothesis.

Results for eah hypothesis ona real retinalimage are illustrated at

dif-ferent sales in gure 10, 11, 12. In gure 10(b), 11(b), 12(b), one feature

hypothesisisusedatdierentsales,(eg. 6464,3232and1616).

Sim-ilarly,resultsfromtwo featuresand threefeatureshypothesises are shown in

gure 10()(d); 11()(d); 12()(d). The last image of gure 10, 11, 12show

the resultsfromhoosing the best tmodelbasedon the orrelationtesting

in eahblok.

A syntheti image of the basi omponents of a blood vessel, shown in

gure 13, was used to test the algorithm at dierent sales: (6464 and

3232). The regions whih ontain a orner are emphasised based on the

deision model, i.eif two orthree features hypothesis wasused in the blok,

the geometri interset of these features ould then be found. The size of

the irles in Figure 13 reets the detetion sale. It an be seen that the

regions inludingjuntion pointsare labelled aurately.

In gure 14, the same test algorithm as used in gure 13, is applied

to \pik up" the regions whih may ontain a orner or juntion point in

a real retinal image. Comparing the result whih was given in gure 6,

we an onlude that using the Gaussian model a greater number of the

juntion points or orners are deteted than by the method of urvature

measuring. The urvature method fails to nd many of the branhes at

smallsales, althoughthis ould perhaps be improved by parameter tuning.

Our estimator, however, is still aeted by the noise and the omplexity in

the real image so some failures or false-positives our in the retinal image

as it does not attempt to ombine information aross sales. Also, it does

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This work uses and extendsthe ideas previously presentsby Davies, Wilson

and Calway [19℄ [22℄. Itsmain ontributionisthat weapply a superposition

ofGaussianmodelsanduseasynthesisedmagnitudetoreonstrutthedata.

This allows ustoderive a likelihood,P( ~ YjG

K

), toselet amodelG k

, whih

models a juntion with K =1;2;3 branhes. By using anexpliit Gaussian

intensity model to represent linear features with some width, it gives us a

simple representation oflinear andbranhingstrutures likebloodvessels in

medial images. The model and estimation readily extends to3D [23℄.

Thealgorithmhas been tested onboth syntheti (lean)imagesand real

(noisy) images. Weomparedourresultsagainstasalespaesheme[1℄[9℄.

Theresults,showninFigure13and14,demonstratethatthe juntionpoints

anbeorretlydetetedinanartiialimage. However, duetotheinuene

of the noise, loalisationerrors stillexistfor real data.

This approah is still in its initial stages. The next step is to onsider

some ways of simultaneous tting super-posed models to redue the eets

of noiseto getbetterauray ofthe loalisation[14℄. Anotherdevelopment

would be generalising the model inluding a lassier in order to expliitly

label the juntion. Furthermore, a neighbourhood linking strategy to trak

vesselsbetweenthebranhpointouldbeemployedtoextrattheentiretree

struture. Wemodelthedataoverarangeofwindowsizessoasale-seletion

strategy ould be usefullyapplied toonrm/selet ahypothesis [17℄.

Aknowledgements

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[1℄ M. E.Martinez-Perez, A.D. Hughes,A.V. Stanton, S.A. Thom,A.A.

Bharath,andK.H.Parka, \Retinalbloodvesselsegmentationbymeans

of sale-spaeanalysis and regiongrowing," inProeedingsof the

Inter-national Conferene on Image Proessing, 1999,vol. 2,pp. 173{176.

[2℄ D. M.Tsai,H.T. Hou,andH.J.Su, \Boundary-basedornerdetetion

using eigenvalues of ovariane matries," Pattern Reognition Letters,

vol. 20,pp. 31{40, 1999.

[3℄ Z. Zheng, H. Wang, and E. K. Teoh, \Analysis of gray level orner

detetion," Pattern Reognition Letters, vol. 20,pp. 149{162,1999.

[4℄ F. Mokhtarian and R. Sulomela, \Robust image orner detetion

through urvature sale spae," IEEE Trans. on PAMI, vol. 20, no.

12, pp. 1376{1381, 1998.

[5℄ H. C. Liu and M. D. Srinath, \Corner detetion from hain-ode,"

Pattern Reognition, vol.23,pp. 51{68,1990.

[6℄ E.R.Davies, \Appliationofthegeneralisedhoughtransformtoorner

detetion," IEE Proeedings, vol.135, no. 1,pp. 49{54, 1988.

[7℄ R.Mehrotr, S.Nihani,and N.Ranganathan, \Cornerdetetion,"

Pat-tern Reognition, vol.23(11), pp. 1223{1233, 1990.

[8℄ J.A.Noble, \Findingorners," ImageandVisionComputing, vol.6(2),

pp. 121{128,1988.

[9℄ T. Lindeberg, \Juntiondetetionwith automatiseletionofdetetion

sales andloalizationsales," inPro. 1st International Confereneon

Image Proessing, Nov. 1994, vol.1, pp. 924{928.

[10℄ R. Wilson, A. D. Calway, E.R.S. Pearson, and A. Davies, \An

intro-dution to the multiresolution fourier transform and its appliations,"

Teh.Rep. RR170, University of Warwik, UK, January1992.

[11℄ A. H. Bhalerao, Multiresolution Image Segmentation, Ph.D. thesis,

University of Warwik, U.K., 1991.

[12℄ T. Lindeberg, \Sale-spaetheory: Abasitoolforanalysingstrutures

at dierent sales," Journal of Applied Statistis, vol. 21, no. 2, pp.

(16)

of Computer Vision,vol. 30,no. 2, 1998.

[14℄ A. Bhalerao and R. Wilson, \Estimating loaland global image

stru-ture using a gaussian intensity model," Medial Image Understanding

and Analysis,2001.

[15℄ A. Papoulis, Signal Analysis, MGraw-Hill,New York, 1977.

[16℄ A. I. Borisenko and I. E. Tarapov, Vetor and Tensor Analysis with

Appliations, DoverPubliations, New York, 1979.

[17℄ A. Davies and R. Wilson, \Curve and orner extrationusing the

mul-tiresolution fourier transform," Teh.Rep. RR 202, University of

War-wik, UK, November 1991.

[18℄ B.Kovesi, J.M.Bouher,andS.Saoudi, \Stohastik-meansalgorithm

for vetor quantization," Pattern Reognition Letters, vol. 22, pp. 603{

610, 2001.

[19℄ A. Davies, Image Feature Analysis using the Multiresolution Fourier

Transform, Ph.D. thesis, University of Warwik, UK, August 1993.

[20℄ C. T. Li, Unsupervised Texture Segmentation Using Multiresolution

Markov Random Fields,Ph.D.thesis,UniversityofWarwik,U.K,1998.

[21℄ P.Smith,D.Sinlair,R.Cipolla,andK.Wood,\Eetiveorner

math-ing," in BritishMahine Vision Conferene,U.K., 1998.

[22℄ R. Wilson, A.D. Calway,and E.R. S.Pearson, \Ageneralized wavelet

transformforfourieranalysis: themultiresolutionfouriertransformand

its appliation to image and audio signal analysis," IEEE Tran. IT,

Speial Issue on Wavelet Representations, 1992.

[23℄ A. Bhalerao and R. Wilson, \A fourier approah to 3d loal feature

estimation from volume data," in British Mahine Vision

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t =1

t =4

(a)Sale-spae Representation (b)50 strongest juntion response k 2

Figure 1: Sale spae representation and Juntion response at dierent

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Image Model Image Model

Figure 2: Examples of aGaussian Intensity Model for linearfeatures

G(~x)

~

y

1

x

1

0 x

y

(19)

forN =64bloksizes

()WindowedFouriertransform forN =32bloksizes

(d)WindowedFouriertransform forN =16bloksizes

Figure 4: Windowed Fouriertransform of aexample retinal image (a)

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(2ndIteration) (4th Iteration)

(d)OriginalImage (e) Classied Region (2ndIteration)

(f) Classied Region (4th Iteration)

Figure 5: Clustering Approah using K-means (Dierent olour represents

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DFT

Spectrum

IDFT

K-means Synthesised

Model Estimation

Y

3 1

2 P

k

^ Y 1

^ Y 2

Y 0

^ Y 3

Figure 6: Hypothesistesting algorithm

(a) Original Image

(b) 1 feature hypothesis

() 2 feature hypothesis

(d) 3 feature hypothesis

Figure 7: Estimation result for eah hypothesis P 1

= 0:97;P 2

= 0:89;P 3

=

(22)

Image hypothesis hypothesis hypothesis

= 0:22;P 2

= 0:97;P 3

=

0:90

(a) Original Image

(b) 1 feature hypothesis

() 2 feature hypothesis

(d) 3 feature hypothesis

= 0:70;P 2

= 0:90;P 3

=

(23)

() 2featurehypothesis (d)3featurehypothesis

(e)bestttedhypothesis

(24)

()2featureshypothesis (d)3featureshypothesis

(25)

()2featureshypothesis (d)3featureshypothesis

(26)

point for N = 64 blok sizes

()Labelthebranhpoint forN=32bloksizes

(d) Branh point detet throughthedierentsale

Figure 13: Label the branh point in dierent sale of an syntheti image

(27)

bloksizes

()LabelthebranhpointforN =32 bloksizes

(d) Branh point detet through the dierentsale

Figure14: Labelthe branh pointindierentsale of retinaimage (the size