Optimal approach classification

to image segmentation

DStewart, D.Blacknell, A.Blake, R.Cook and C.Oliver

Abstract: Many algorithms have becn proposed for the segmcntation and classification of SAR imagery Typically these arc heuristic in basis, and marc successful on somc types of imagery than others. Methods based on global optimisation techniques can achieve the optimal solution of the posed problem; the algorithm is then characterised by an objcctive function and thc chosen optimisation techniqnc. The anthors construct an appropriate objective function for SAR segmentation or classification, and show how to optimally quantify the relationship between the competing terms within it. This rcmoves the previous need for the introduction of hcuristics. Also described is an implementation bascd on simulated annealing, in which considerable attention has heen paid to details, and examples are given of the resulting algorithm’s performance upon real data.

1 Introduction

1. I Segmentation Segmentation dcpends on the introduction of a ‘cartoon’ model that asserts images arc made up of segments, separatcd by edges, in which some features, such as the radar cross-section (RCS) for cxample, are constant. A segmentation algorithm must determine the number and position of the segments, as well as their statistics, from the data and the adopted model.

This model oftcn includes both a statistical term, to assess the probability of the segmentation given the data, and an apriori term that expresses the particular propertics required in the final segmentation for the proposed appli- cation. For instance, it may be important to preserve resolution, e.g. fine structurcs for target detection [l] or to distinguish between regions with very small differences in RCS as in land usage monitoring [2].

Typically, a cost or objectivc function defines the optim- ality criterion; in fact i t has been claimed that all segmen- tation schemes, implicitly or othetwise, are minimising the same type of cost function (a Mumford-Shah energy) [3].

An algorithm based on the optimal minimisation of a simple objectivc function was described by Stewart et al. [4]. This algorithm was successful at producing good segmentations of SAR scenes. However, the rcgion edges produced were locally irregular, tending to follow thc speckle. This ‘crinkle’ was avoided via heuristics. In Section 2 we describe an extcnded objective function which overcomes this problem. This cost function consists

0 IBE, 2000

I,% Pmceeding,v online no. 20000400 Dol: 10.1049/ip-rsn:zoonn4nn Paper first receivcd 16th August 1999 and in rcviscd form 22nd February 2000 D. Stewart and R. Cook are with N.A Sofiwarc Ltd, Roscoe Mousc, 62 Roscoe Street, Liverpool, UK

D. Blacknell, A. Blake and C. Oliver 81% with DORA, Malvern, Worcs. WR14 3PS, UK

134

and

of two tcrms, a statistical one which expresscs the good- ness-of-fit of a scgmentation to the data, and another that depends upon thc length of the edgc or discontinuity sct. The relative strength of such terms has prcviously been set heuristically. Wc show here how to probabilistically quan- tify the relationship between thc tenns, and hence to remove the heuristic.

Segmentation is then the proccss of fitting o w model to the data with all possible positions of edge, number of regions and choicc of statistical parameters to find the one with the smallest cost. We achieve this using simulated anncaling, a global optimisation scheme which has proven to he very powerful in this and othcr applications. Geman and Geman [ 5 ] were among thc first to adapt (he technique to image processing, and there have heen many other applications since (e.g. Derin et al. 161, Kelly et al. [7], Manjunath et al. [XI and McConnell et al. [9]). The algorithm described in Stewart et al. [4] also incorporated simulated anucaling into the minimisation of the cost function; but again, a numbcr of heuristics were included. In Scction 3 we describc an enhanced implcmcntation in which the heuristic elements of Stcwart et al. [4] are largely removed, leading to an algorithm which has better performance and which requires substantially lcss tuning by the user.

7.2 Classification Classification involves putting each image pixel into one of a number of bins. This process differs from segmentation in that the pixels in a bin will not in gcneral form a single region with a closed boundary. They will, however, form a sct of such regions; thc cost functions we dcvclop can he uscd for either classification or segmentation,

2 Cost function

The cost function we use has two competing terms. The first (likelihood term) is derived from a single-point statis- tical model of SAR imagery whilst the sccond (shape term) relates to the behaviour of the boundaries between

IEE PmccRadnr: Sonor Nuvig., Yoi. 147, No. 3. l r n e 2000

segments or classes. Without this latter term, the segments in the optimal segmentation would simply follow the lines of speckle in the original image, whilst the optimal classification would just classify the speckle, i.e. bin thc data inlo the specificd number of classes. Minimising this cost function is then equivalent to maximising the like- lihood of the data fitting the segmenlation or classification with the appropriate degree of qualification upon region shapc. We write the cost function as

c = CA + (1)

where C,, denotes the likelihood term and C, thc shapc term. We define examples of these for thc instance of segmentation and classification of SAR data bclow. The scalar value s quantifies the relative weighting between the two terms, and has previously been set heuristically. We show how an appropriate value can be rigorously tlefincd, in general, and give the specifc formulation with respect to the likclihood and shape costs defined in the following Sections.

2.1 Likelihood term For thc likelihood term CA wc use the well known fact that thc gamma distribution providcs a reasonable approxima- tion to SAK data. Indeed, it provides an exact model for purely intensity segmentation (Oliver ef al. [2]), i.e. when the RCS is assumed to be constant. We suppose, therefore, that we havc an image containing r homogeneous regions or classes of Gamma distributed data. Let the data be represented by {x,li= 1,. . . . N } and let r j , pJ and v,, j = I , . . . , r, dcnote the indcxing scts, mcans and order paramctcrs of the appropriate regions or classes. Then the probability for the data fitting the distribution is given by

~

Let N . = ( I . l and let x/= I/NjCitr,xi and logxi= I/? ~j . l . ldg(xi) denote estimated mcan values i n the region. The log-likelihood is then calculated to he

A = -.V$log r(v,) - vj . log(vj) j= I + vj ' - (v, - I)% + vi , q g j ) (2)

Generally thc means and ordcr paraincters arc not known a priori so that we wish to obtain maximum likelihood estimates for them. These estiinatcs are

- - g. = x. .I J

'Y(5) - log(?) =e - log(%)

The maximum likelihood solutions can thus bc derived from the data and substituted into eqn. 2 to give the estimated solution for thc log likelihood. Since we want to maximise the likelihood of the data fitting our model, and our optimisation technique is designed to minimise a cost function, we multiply the log likelihood by - I to give the cost function

+ T . log(%) - (? - I ) G + l;) ( 3 )

If we are intercsted in intensity segmentation or classifica- tion alone, the statistics are determined purely by the speckle so that all regions correspond to a Gamma PDF of order L , where L is the number of looks of the SAR data, i.e. vj = L Vj. The likelihood cost term is then reduced to the forin

CA =a" +I,. EN, log(@ (4) j= I

where a. is a constant. Fig. 1 shows the minimisation of cost function 4, with a

fixed number of regions, for a typical SAR scene. It is cvident rrom thc result that the minimum is found by producing regions with long crinkly edges that follow trends within the spcckle. Indeed, without the restriction of a fixed number of regions the optimal result with respect to this cost function alone would be the original image, i.e. each pixel forming its own region. To counteract this effect it is neccssary to introduce another, competing, term into the cost function: thc shape tcrtn.

2.2 Shape term There have been many proposcd models, see for example Li [IO], to describc the typc of shape we expect scgments to have and, analogously, the connectivity we cxpect of classes. Most of these are expressions of the size of the boundary or cdge set. Here wc choose one of the most simple and definc the shape term to be given by

Fig. 1

166 P,uc.-Hndur. Sonar Noviz., WI, 147, NCI. 3, June ZUUO

vpical SAR image nnd the ve.ssuN ofminimisins co~ffirnctiorr 4 with nfrred number- ofmsiuns

135

where ri is the region or class label for a pixel i aud Nbhd(j) a defined neighbourhood, [n this paper we simply take the ncighbonrhood of a pixel to he the eight pixels nearcst to it. A larger neighbourhood may be dcsirable if the algorithm is to be designcd to segment more textural information. Alternatively, a differently shaped neighbour- hood may be more efficient at detecting linear features.

By minimising thc shape cost we arc thus minimising the quantily of cdge pixels i i i every ncighbourhood or the image. The straighter or smoother edgcs preferrcd by this shapc cost arc thus in direct competition with the more criukly ones, courtesy of the speckle, that the likelibood cost is liablc to pick. In fact, minimising an image into a specified numher of regions or classes with respcct to the shape cost function of eqn. 5 would simply produce a rcgnlar tesscllation. Without a rcstriction upou the number of regioiis, the solution would be given by an image with just a silrglc region or class. How we determine the rclative weighting of the two coinpcting terms is the subject of the next Section.

2.3 Relationship between the terms Traditiorlally the parameter s in the cost function I has been set as a constant or iteratively adjusted according to some fixed schedule as the algorithm proceeds (sce Li [ lo] for cxample). Fig. 2 gives minimisations for the cost function formed by adding togcther the likelihood cost and shape cost of eq. 4, and 5, respectively, with differcnt choices of constant weighting paranicter s. It is evidcnt, with an increase in s , that the effect of the shape term also increases, producing ‘blobbier’ looking segmenlations with smoother region boundaries.

We seek to avoid thc heuristic involved in choosing s. It is also likely that thc appropriate s will depend on local characteristics within the image. For example, whcu region sizes are larger our eslimates of thc regional statistics improve and we are more confident of the positioning of the regional boundary upon the basis of the likelihood cost alone. When regions are siiialler we are less confident of the positioning of an cdge on lhe basis of the statistical cost, for it may in this case be more prone to follow variations in the spcckle. To obtain an equivalent degree of edge smoothness therefore, we will require a lowcr value of s between large rcgions than between smaller ones.

2.3.1 Rigorous definition for the shape para- meter s: To optimally determine the shapc penally conslant s it is necessary to understand how the terms of the cost function interact. Simulatcd annealing is driven not by the absolute value of the cost function, but rather by the change in cost generated by a perturbation of the

segiiientation or classificalion. Thercfore wc are interested in the quantity

AC = AC, SAC,^

We procecd by detcrmining the distributions for the like- lihood differcnce and the shape cost differencc, for uniform background spcckle; wc can thcn quantify the extent to which the change in shape term doniinatcs the change in likelihood.

These distributions arc dependent both on the order parameters of the approximating Gamma distributions and thc number of regions or classes. It is also important to calculate thesc distributions for similar pcrtnrbations of the segincntation of classification to those that will be employcd in the intended algorithm. These are described in Section 3.

Given thc appropriatc distributions wc can calculate a probability of false alarm: the probability, given thcre are no rcal edges, that the edgc positions are wherc they arc on the strength of the change in likelihood rather than the change iii shape. This is dcnoted by p(s). Mathematically wc define

p(s) = Prob(AC < O~SAC,~ z 0) + Prob(AC > OlsAC, < 0)

( 6 ) + Prob(lAC1 z 0l.sACs = 0)

Thc first term in this sum qnautifies the likelihood that a perturbation of a segmentation or classification (of uniform background spcckle) will lead lo a rcduction in cosl evcn though the shape cosl increases. The second term gives a similar probability for when thc change in shapc cost is negative. Duc l o the discrctc nature of our choicc of shape tcrm, there arc instances when a scgmentation perturbation will givc iio changc in shapc cost, In that instance, the change in likelihood will always dominate and this is cxpressed in the final summand. In practice this third term may wcll be constant and we could then remove it. But if we wishcd to coinpare the erfccts of using a differcnt likelihood or shape cost function we would want to kcep thc same rclative weighting betwcen the two compcting terms, i.e. to have the same valuc ofp(s). In this instance it becomes necessary to keep all factors, constant or other- wise, in the formulation for p(s) .

2.3.2 Specific solution for the shape parameter s: Eqn. 6 connccts the probability p(s) with the actual shape parameter s. All other factors remaining constant, a change iu p(s) thus lcads to a chaiige in the relative wcighting, via s, belwccn the two competing terms in the cost function. We wish to iiivcrt this formula, to determinc s for particular values ofp(s), ordcr parameters and rcgioii size, To do lhis we need to calculatc the required distribu-

Fig. 2

136

tions, i.c. those for AG, and &. I n this Scction we assume CA to be thc intensity likelihood cost defined in eqn. 4. Thc method for the inorc gcncral cost likcliliood o f eqn. 3 is analogous and we comment npon it at the end of thc Section.

The distributions for both AC, and AC,s arc aiialytically intractable but they can be estimated via simulation. For AC, we fix an ordcr parameter, mean and two region sizes. Two rcgions, with these sizes, arc then gencratcd with values drawn from the appropriate Garnnia distribution and the resultant change in likelihood rccordcd for the pertur- bation method we intend to cmploy (which is dctailed in the next Section). Repeating this procedure inany timcs wc build up an enipirical approximation to thc PDF for AC,. The distribution for the changc in shape cost AC,y will be independent of the regional statistics, but it will depend on the number of regions or classes in the local nciglibonr- hood over which it is defined. We gcncrate random tcssela- tions with the rcqnircd region or class sizcs and then record the resultant change in shape costs when perturbing thc tcsscllation in thc fashion we intend to LISC. Typical distributions for AC, and ACs arc given in Fig. 3.

Once reasonable approximating distributions have been found, it remains to determine bow thcir paranictcr values change with respect to lhe variation in rcgion sizes and order parameters of the simulated data. This change can bc quite marked, as the plot in Fig. 4 indicates for variation in rcgion size.

0.18-

0.16-

0.14-

0.12

0.5 t ' - 1 . 5 - -1.0 -0.5 0 0,5 1.0 1.5

x a

1 0 b

0.8r

0.6

0.5

0 0 L 50 1bO 150- 200' 250 300

Note that, in the classification casc, a class is niodelled as a union of disjoint rcgions. Thc class statistics are thus calculated ovcr thc whole class, yet the changc in cost, i.e. the change in likelihood and shape costs, is defined locally. Therefore, wlicn evaluating the change i n cost for a classitication perturbation, wc takc the region size to hc the thc size of the appropriatc region, rather than that of the

- 0 0.1 0.2 0.3 0.4 0.5 0.6

shape penalty a

2.0

1.6

L 4 6 8

region size I1

137

whole class, whilst slill estimating the regional statistics ovcr the wholc class.

In what follows we make thc simplifying assumption that both regions involved in the appropriate segmentation or classification perturbation arc of the same s i x R. This cnahles us to approximate the change in likelihood dislri- hution with one that is symmetric around zero. Io this case we find that a mirrored exponential distribution well approximates the distribution for AC,

Note that the distribution is independent of the order parameter of the simulated data. The effect of this para- meter is fully incorporated within the likelihood cost tcrm itself although in general this will not bc the case. Wc can

also find the discrele probabilities for ACx by reference to thc appropriatc points of the approximating Gaussian

wilh (i = 3 . Although the goodncss of this approximation also depends on R the approxiination is sufficient for our pnrposcs except for when we desire R to be either very small or vety large. This then gives the approximating distribution for to be

Substituting eyn. I and 9 into eqn. 6 we can solve f o r s to find

- log@’(.?))

f i s =

where p ’ ( ,~ ) is monotonically related lo the originally dcfincdp(s). Fig. 5 show the shape penalty plotted against

Fig. 7 Top X-band SAR and intcnsily mncaling sogmcntalion Hottom: Edge n i q a n d ratio iniage

138

the actual p(s) values, for cerlain I<, and against region size for fixed values for p(s), respectively.

The formulation for the cost function with likelihood term given by eqn. 3 is more complicated. Thc distribution for the likelihood difference is then dependent on the order parameter of thc simulated data as well as thc region or class size R . Since the explicit formulation is both lengthy and of little intrinsic interest we omit it here.

2.3.3 Results: Ilaving a univcrsal valiie fur p(s ) will give the same degree of shape effect for whatever SAR imagery fits our models and for whatever the values of order parameter and region or cliiss size.

First we consider the results for thc intensity scginenta- tion algorithm with the shape parameter given by eq. IO. In this instancc we can also determinc what constitutes the best choice for p ( s ) by examining the residual image rormcd from dividing tlic original data by the scgmenta- tion. This image should ideally look likc pure speckle with a gamma PUP with order parameter equal to the effcctive number of looks (ENL) of the SAN sensor. In the example

of Pig. 6, the original image had 4.8 effective looks. So, using the formulation

3,: = J i j E for the ratio statistic r (Oliver et al. [2]) the expected standard deviation for the intensity ratio image would be 0.456. The value for the residual imagc in Fig. 6 is 0.45 for

Keeping p(s) = 0.025 should therefore give segmenta- tions of other imagery with equally good residuals. The expectcd standard deviation for tlic intensity ratio images of Fig. 7 would be 0.65 and the actual value is 0.64. Once again the lack of structure within the ratio image also implies the segmentation has been effective, but the hints of some linear features in the ratio image suggest that there remains room for improvement in the choice of objective function (particularly within the shape term), for scenes in which such features are important.

Wc use the same value forp(s) for an intensity annealing classification algorithm. 'Thc only difference between this and the segmentation routine is the removal of the require-

p(s)=0.025.

139

ment for conncctivity. Recall that thc shape parameter s is in fact calculated using the region sizes of the connected segments of a class whilc the statistics are calculated ovcr the whole class. Thc result, on the same SAR imagc as in Fig. 6 is giveu in Fig. 8. Since there are only eight classes, we would expect there to be niorc evidence of structure within the ratio image and this is obviously thc case. Also, the standard deviation of the ratio image is 0.72 which is further away from the expectcd value of 0.65 than in the segmentation case. Unfortunately there is no way to numerically compare the result with that of the segmenta- tion, but a visual inspection implies thc edges arc of an equivalent smoothness and the number of regions are similar (1289 in the segmentation and 1100 in the classi- fication).

Finally, we use p(s) = 0.025 for an annealing segmenta- tion (using the likelihood cost in eqn, 3) to produce the result in Fig. 9. Once again there is no accepted mcthod for dctermining the effectiveness of the result although the edges appear to have a similar degrce of smoothness.

We have now defincd an optimal cost function for the segmentation or classification of an image given that the data fits both our statistical and shape models. It remains to provide an optimisation technique which is capable of finding the minimum of this cost function and an imple- mentation that allows it to do so.

3 Implementation

3.1 Simulated annealing Simulated annealing is a broad term applicd to a class of techniques for finding the global minimum of a given function, called the objective or cost function, over a configuration space S. In our context, S represents the space of all possiblc segmentations or all possible classi- fications of a particular image.

Simulated annealing algorithms proceed by randomly changing from one state in the configuration space, say si, to another, si. By evaluating the change in cost, ACii = C(sj) - C(si), the algorithm decides, via a probabil- istic acceptance criterion, whether to accept the new Configuration or kcep the current onc. The acccptance criterion is dependent on the valuc of' a parameter T, known as the temperature, which itself is updatcd aRer a fixed number of configuration changes have bcen consid- cred. Thc criterion is such that a configuration change entailing an increase in cost is more likely to be accepted a1 a higher temperature than at a lower one. By slowly decrcasing the value of we avoid being trapped in local minima and will rcach the optimal global solution.

The parameters within the simulated annealing algo- rithm need careful tuning for the particular application, in particular the behavionr of the temperature Z Howcver, there exist methodologies for setting lhe parametcrs which work reasonably well in practice and in the following outline of the algorithm we give lhose as espoused by Aarts et al. [ I l l .

3.2 Outline of the algorithm The implementation dcscribed here kecps two main considerations in mind: (i) To ensure the succcss ofthe annealing it is necessary to providc the capability to visit, in thcory, all points of thc configuration space. (ii) To avoid the use of heuristics as far as possible.

I40

There are many possible ways of perturbing segmenta- tions and classifications: see for example Oliver et al. [2] and Morel et al. [3]. For the results that follow, we employ a free topological model and alter the segmentation on a pixel by pixel basis. By making the smallest admissible changes, wc expect to achieve results of the highest possible resolution. However, the implementation is still able (potentially) to visit all points of the configuration space within a reasonable exccution time, and we believe that the algorithm is robust enough not only to provide the solution for a particular image, but of doing so with all SAR imagery. Its stability is demonstrated by the unim- portance of the method of initialisation, i.e. whatever segmcntation or classification we initialise with, the algo- rithm will produce a comparable answer.

1. Initialise the program. In the casc of segmentation wc start with the scgmentation of every pixel belonging to its own rcgion. For classification, we randomly assign pixels to classes. Fig. 10 demonstrate how the number of regions decreases after using this particular method of initialisa- tion. 2. For a number of outer iterations: (a) Set the temperature T The initial temperature, To, is set by monitoring the evolution of the system during a number of perturbations before the aclual optimisation starts. Inilially To = 0 and this cstimate is updated via the formula

Thc stages involved are:

where m1 and m2 denotc the number of pcrmutations o M n e d so far with AC, 5 0 and AC, > 0, rcspectively; A d + ) is the averagc value of those AC, values for which AC, > 0. The constant x is the ratio between the number of accepted perturbations and the total number generated. Initially we wish to accept most new configurations to allow the algorithm access to all the configuration space, and so 1: is set to bc close to one.

After that, the temperature updating formula is

where T, is the temperature at the nth iteration, and o(T,,-,) is tlle (cstimated) standard deviation of the values of the cost function at temperature T,-,. The coostanl 6 specifies the cooling rate and is a small positive number. Fig. I I shows how the cost function decreases over time for a typical SAR image for various values of 6. There is obvious competition between the goodness of the final result and the time taken to achieve it. Since these considerations may be application dependent, it is impor- tant to allow thc cooling parametcr to be set by the user of the routines. (b) For a number of inncr iterations: (i) Perturb the currcnt segmentation or classification. In the segmcntation case this involves moving a pixel between neighboilring segments, while in the classification case it is simply a random change of the pixel's class label. Regions are allowcd to split into 1wo or more parts and even to be merged into other regions. (ii) Calculate s from p(s) by inverting eqn. 6. This will depend on the region sizes of the relevant segments and possibly also upon the looks of the data. (iii) Use s to evaluate the change in cost, AC, In the instance where the cost hnclion requires second moment estimates, it will he neccssaiy to make a simplifying

IllB Pmc.-Rarlad Sonar Nnvig. E/, 147, No. 3, June 2000

Top: AirSAR lmsgc which i s also the illgoiitliiin's iiiiliill scgmcntution Middle: Pait of the way towards coiivcrgciicc Boltom: Final solution

-50000

-130000' ' 100 200 300 400 500 600 700

Value of ihe cosiJi,nclion over itemlion.? J ~ r s e v o n l valum q f i k iterations

Fig. 11 coolingpnrameler 6 (cqn. 12)

~ . h=0.25

. . . . . 6=0.75

.. ..... h= I

1EE Pm.-Rado,: Sonar Novig., Ynl. 147, No 3, s i n ? 2001)

6=0.5

assumption for when regions or classes contain only one pixel. (iv) Refer to the acceptancc criteria. The acceptancc prob- ability most widely used equals one iT AC< 0 and CX~(-AC/T,~) if AC> 0. (v) If the new configuration is rejected, reinstate the old nnc. (c) Stop the algorithm when the quantity

falls below some threshold. IIcrc C(7;,) is the smoothed average cos1 at temperatore T,, and the above formula is based 011 an extrapolation (towards a temperature of zero) of these values over iterations. Hence the stopping critcrioo will stop the algorithm when the rate of improvement of the minimisation falls below an appropriate threshold. Pig. 12 demonstrates how thc stopping crilerion is applied for the examples of Fig. I I ,

One of the problems with the stopping critcrion of eqn. 13, is its oscillating bchaviour with respect to the number of iterations; a result of this was that the algorithm of Stewart ef al. [4] was rim for a fixed iiumber of itcrations,

141

0.014r 4 Conclusions

Effective segmentation and classification algorithms are characterised by objectivc fuiictions with two or move compcting terms. The prohlcm is then defined to bc the selection of thcsc tcrms, a deterniination of the relative weightings between them and an implementation of an algorithm that successfiilly minimises the fmiction.

Typically a failure to optimally define thc objective function leads to difficulties when implementing an algo- rithm to minimise it. In this paper we have rigorously defined tlic relative weighting For an objective function with two competing t e r m and in a way that naturally extends to functioiis with more terms. We havc also given an algorithm, based upon simulatcd annealing, which has thc required freedom in which to produce a global solution at the highest rcsolution. The methodology presented here is thus optinial once the constituent terms of the cost function have hecn chosen. With this work we havc hecn interested in SAR segmentation and classification. The results demon- strate that our choices for the objective function constituent terins were suitablc fur RCS reconstruction. With particular user requirements or specific featurc extraction, these terms will change, but tlic relative weighting formulation and algorithmic implementation will remain optimal.

5 References

I , 0.012 k t .

50 100 150 200 250 300 iterations

a -soooor

YI

0

I

8-90000 - - - - _ _ _ -100000

-110000

-120000 ---____- -1300001 ' L - A - - I d

100 200 300 400 500 600 700 iterations

b Fig. 12 a Difference between calculating stopping paranreter diroctly 60m sinootlictl cost V ~ I ~ U D S and ~ ~ s i a g n best-fit line - hest fit ~~~ rlilrct

Helruviuw of the stoppina witerion (eyn, !.?)

,s = 0.5 . . . . . , s=0.75 . ... .. , 6 = 1

rathcr than stopping automatically. By calculating a best fit line to determine the local gradient, a much smoothcr value o f t is obtaincd, as evidenced in Fig. 12, and the current implementation will run to convergence. 3. Post-processing. The segmentation may contain more than one indcpeudent quantity, for cxample mean and order. It is necessary to providc appropriate display options for the quantity within each scgment. Also, depending on the application, it may be desirable to apply other post- processing, for example to employ a final merging routine to merge adjacent segments which, though their differences are statistically significant, need not bc diffcrentiated in terms of a particular application.

5

6

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8

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