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Using Symmetry in Robust Model Fitting

Using Symmetry in Robust Model Fitting
Using Symmetry in Robust Model Fitting

Using symmetry in robust model ?tting

Hanzi Wang *,David Suter

Department of Electrical and Computer Systems Engineering,Monash University,Wellington Road,Clayton,Vic.3800,Australia

Received 27August 2002;received in revised form 15April 2003

Abstract

The pattern recognition and computer vision communities often employ robust methods for model ?tting.In particular,high breakdown-point methods such as least median of squares (LMedS)and least trimmed squares (LTS)have often been used in situations where the data are contaminated with outliers.However,though the breakdown point of these methods can be as high as 50%(they can be robust to up to 50%contamination),they can break down at unexpectedly lower percentages when the outliers are clustered.In this paper,we demonstrate the fragility of LMedS and LTS and analyze the reasons that cause the fragility of these methods in the situation when a large percentage of clustered outliers exist in the data.We adapt the concept of ‘‘symmetry distance’’to formulate an improved regression method,called the least trimmed symmetry distance (LTSD).Experimental results are presented to show that the LTSD performs better than LMedS and LTS under a large percentage of clustered outliers and large standard variance of inliers.

ó2003Elsevier B.V.All rights reserved.

Keywords:Robust regression;Symmetry distance;Clustered outliers;Breakdown point

1.Introduction

One major task of computer vision,pattern recognition,machine learning,and related areas:is to ?t a model to noisy data (with outliers)(Fischler and Bolles,1981;Rousseeuw,1984;Rousseeuw and Leroy,1987;Zhang,1997;Danuser and Stricker,1998;Stewart,1999).It is common to employ ‘‘regression analysis’’to undertake such

tasks (Rousseeuw and Leroy,1987).The most common form of regression analysis is the least squares (LS)method,which can achieve optimum results under Gaussian distributed noise.But this method is extremely sensitive to outliers (gross errors or samples belonging to another structure and distribution).The breakdown point of an es-timator may be roughly de?ned as the smallest percentage of outlier contamination that can cause the estimator to produce arbitrarily large values.Mathematically,let Z be any sample of n data points ex 1;y 1T;...;ex n ;y n T,Z ?f z 1;...;z n g and z i ?f x i ;y i g .For m 6n ,the ?nite-sample break-down point of a regression estimator T can be written as (Rousseeuw and Leroy,1987,pp.10):

*

Corresponding author.Tel.:+61-395-159471;fax:+61-399-053454.

E-mail addresses:hanzi.wang@https://www.wendangku.net/doc/ba7196474.html,.au (H.Wang),d.suter@https://www.wendangku.net/doc/ba7196474.html,.au (D.Suter).

0167-8655/$-see front matter ó2003Elsevier B.V.All rights reserved.

doi:10.1016/S0167-8655(03)00156-9

Pattern Recognition Letters 24(2003)

2953–2966

e?n eT;ZT?min

m

;sup

Z?2Z m

k T neZ?Tk

?1

e1:1T

Because one single outlier is su?cient to force the LS estimator to produce arbitrarily large value,the LS estimator has a breakdown point of0%.

Since data contamination is usually unavoid-able(due to faulty feature extraction,sensor noise and failure,segmentation errors,etc.),there has recently been a general recognition that algorithms should be robust(Haralick,1986).Robust regres-sion methods are a class of techniques that can tolerate gross errors(outliers).Some robust methods also have a high breakdown point.

Several robust estimators with apparently high breakdown points have been developed during the past three decades.The least median of squares (LMedS)estimator and the least trimmed squares (LTS)estimator(Rousseeuw,1984;Rousseeuw and Leroy,1987)are the two most popular methods with claimed high breakdown points,and are currently used most widely in computer vision ?eld(Roth and Levine,1990;Bab-Hadiashar and Suter,1998;Ming and Haralick,2000,etc.).They are based on the idea that the correct?t will cor-respond to the one with the least median of re-siduals(for LMedS),or the least sum of trimmed squared residuals(for LTS).The essence of the argument claiming a high breakdown point for the LMedS is that if the uncontaminated data are in the majority,then the median of the residuals should be una?ected by the outliers,and thus the median residual should be a reliable measure of the quality of the?t.Likewise,since the LTS method relies only on(the sum of squares of)the h smallest residuals,for some choice of the para-meter h,it is thought that this should be robust to contamination so long as h data points,at least, belong to the true?t.Though the robustness of these methods,compared with patently non-robust methods such as Least Squares,is now widely demonstrated and acknowledged,it is per-haps less widely recognized that LMedS and LTS can breakdown at surprisingly low contamination if those outliers are clustered.Due to the a?ects of clustered outliers,the correct?t may not corre-spond to the?t with the least median of squared residual(for LMedS)or the least trimmed squared residuals(for LTS).It is worth mentioning that this phenomenon is not limited to the LMedS,and LTS.It also happens to most other robust esti-mators such as random sample consensus––RANSAC(Fischler and Bolles,1981),residual consensus estimator––RESC(Yu et al.,1994), adaptive least k th estimator––ALKS(Lee et al., 1998),etc.The mechanism of the breakdown in these robust estimators is similar to that of the LMedS and LTS(see Wang and Suter,2003).In this paper,we restrict ourselves to the better known and more widely applied LMedS and LTS. Making clear the reasons that cause the break-down of the LMedS and LTS will help understand the fragility of other robust estimators to clustered outliers.

The key to salvaging the robustness of LMedS and LTS,even in the presence of clustered outliers, can be recognize that LMedS and LTS(and some other robust estimators such as M-estimators, ALKS,etc.)only depend upon a single statistical property of the data(the median of the squared residuals,or the sum of the smallest h residuals, respectively).If one expects some other property of the data,associated with the true?t,to hold, then one may seek to incorporate a measure of that property into the problem formulation.In this paper,we restrict ourselves to one such property––symmetry.

Symmetry is very common and important in our world.When we?t circles,ellipses,or any symmetric object,one of the most basic features in the model is symmetry.In our method,we intro-duce symmetry into the model?tting and thereby propose an improved method––the least trimmed symmetry distance(LTSD).The LTSD is in?u-enced not only by the sizes of the residuals of data points,but also by the symmetry of the data points and has applications where one is trying to?t a symmetric model(e.g.circle and ellipses).Experi-ments show that the LTSD method works better than LMedS and LTS under a large number of clustered outliers.

The main contributions of this paper are as follows:

1.We illustrate situations where LMedS and LTS

fail to correctly?t the data in the presence of

2954H.Wang,D.Suter/Pattern Recognition Letters24(2003)2953–2966

clustered outliers,and analyze the reasons that cause the breakdown of these two methods.

This provides an important cautionary note when employing these two robust estimators in situations where the outliers are clustered.

2.We introduce the concept of symmetry distance

(SD)into model?tting.The concept of SD in computer vision is not novel.However it is a novel concept in the?eld of model?tting.Based on Su et al.?s point SD(2001),we propose a novel symmetry distance and apply it to model ?tting.

3.We experimentally show that the proposed

method work better than LMedS and LTS under a large percentage of clustered outliers for both simulated and real data.

This paper is organized as follows:in Section2, several robust estimators are reviewed.The rea-sons why both LMedS and LTS fail to?t a model under a large percentage of clustered outliers are explored.In Section3a novel SD measure is given and our proposed method is developed in Section 4.Experiments demonstrating the utility of the approach(for circle?tting and ellipses?tting)are given in Section5.Finally,the contributions of this paper and future work are summarized in Section6.

2.Robust estimators

The classical linear model can be written in the form:

y i?x i1h1tááátx ip h pte iei?1;...;nTe2:1Twhere the error term,e i,is usually assumed to be normally distributed with mean zero and standard deviation.The aim of regression analysis is to es-timate h?eh1;...;h pTt from the dataex i1;...; x ip;y iT.

The ordinary LS method estimates^h by

min

^h X n

i?1

r2

i

e2:2T

where the residual of the i th datum:r i?y iàx i1^h1àáááàx ip^h p.

Although the LS method has low computational cost and high e?ciency when the data are Gaussian distributional,it is very sensitive to outliers.In order to reduce the in?uence of outliers,a number of robust estimators have been developed.Among these estimators,the maximum-likelihood-type estimators,i.e.the M-estimators,are best known (Huber,1973,1981).

2.1.M-estimators

The essence of the M-estimator is to replace the squared residuals r2

i

in(2.2)by another function of the residuals:

min

^h

X n

i?1

qer iTe2:3T

where qeáTis a symmetric,positive-de?nite func-tion with a unique minimum at zero.Di?erent choices of qer iTwill yield di?erent M estimators. Unfortunately,it has been proved that M-estima-tors have a breakdown point of at most1=ept1T, where p represents the dimension of parameter vector(Li,1985).This means that the breakdown point will diminish when the dimension of the parameter vector increases.

2.2.The repeated median method

Before the repeated median(RM)estimator,it was controversial whether it was possible to?nd a robust estimator with a high breakdown point.In 1982,Siegel proposed the RM estimator with a high breakdown point of50%(Siegel,1982).

The RM method can be summarized as fol-lowed:take any p observations(p is the dimension of the model),ex i1;y i1T;...;ex ip;y ipT.The parameter vector^h?e^h1;...;^h pTt can be calculated from this set of data points.The j th coordinate of this vector is denoted by h jei1;...;i pT.Then the RM estimator is de?ned as:

^h?med

i1

e...emed

i pà1

emed

i p

h jei1;...;i pTTT...Te2:4T

The RM estimator has succeeded in solving the problems with small p.But the time complexity of the RM estimator is Oen p log p nT,which is very high for multidimensional models and so the

H.Wang,D.Suter/Pattern Recognition Letters24(2003)2953–29662955

utility of the method is restricted to low dimen-sional models.Since this method is not widely used in the computer vision community,we do not consider it further in this paper.

2.3.The least median of squares method

Rousseeuw proposed the LMedS in1984.The LMedS method has excellent global robustness and claimed high breakdown point(also50%). Over the last two decades,LMedS has been growing in popularity.For example,Kumar and Hanson(1989)used the LMedS to solve the pose estimation problem;Roth and Levine(1990)em-ployed it for range image segmentation;Meer et al. (1990)applied it for image structure analysis in the piecewise polynomial?eld;Zhang(1997)used the least median squares in conic?tting;and Bab-Hadiashar and Suter(1998)employed it for optic ?ow calculation.

The LMedS method is based on the simple idea of replacing the sum in the least sum of squares formulation by a median.That is,the LMedS es-timate is given by

min

^h med

i

r2

i

e2:5T

A drawback of the LMedS method is that it esti-mates the parameters by solving this non-linear minimization problem.No explicit formula exists for the solution of this problem––the exact solu-tion can only be determined by a search in the space of all possible estimates.There are Oen pTp-tuples of data,and it takes Oen log nTtime to?nd the median of the residuals of the whole data for each p-tuple.Thus it costs Oen pt1log nTfor the LMedS method.In order to reduce the cost to a feasible value,a Monte Carlo type technique,as described next,is usually employed.

A p-tuple is‘‘clean’’if it consists of p good observations without contamination by outliers. One performs,m times,random selections of p-tuples,where one chooses m so that the probability (P)that at least one of the m p-tuples is‘‘clean’’is almost1.Let e be the fraction of outliers contained in the whole set of points.The probability P can be expressed as follows:

P?1àe1àe1àeTpTme2:6TThus one can determine m for given values of e,p and P by:

m?

loge1àPT

p

e2:7T

The relative e?ciency of the LMedS method is poor when Gaussian noise is present in the data. The LMedS has a low convergence rate of order nà1=3.Rousseeuw improved the LMedS method by further carrying out a weighted LS procedure after the initial LMedS.In this adaptation,a prelimi-nary scale estimate is found by:

S?1:4826e1t5=enàpTTM ie2:8Twhere M i is the median of the residuals returned by the LMedS procedure.

The weight function W i which will be assigned to the i th data point is usually given by:

W i?

1r2

i

6e2:5ST2

0r2

i

>e2:5ST2

e2:9T

The data points corresponding to W i?0are likely to be outliers.The data points having W i?1are inliers and will in?uence the weighted LS estimate:

min

^h

X n

i?1

W i r2

i

e2:10T

The LMedS method may be locally unstable when ?tting models to data.This means that an in?ni-tesimal change in the data can greatly alter the output(Thomas and Simon,1992).

2.4.The least trimmed squares method

The LTS method was introduced by Rousseeuw (1984),Rousseeuw and Leroy(1987)to improve the low e?ciency of LMedS.The LTS estimator can be expressed as

min

^h

X h

i?1

er2T

i:n

e2:11T

whereer2T

1:n

6ááá6er2T

n:n

are the ordered squared residuals,h is the trimming constant.

The method uses h data points(out of n)to estimate the parameters.The coverage value,h, may be set from n=2to n.The aim of LTS esti-mator is to?nd the h-subset with smallest LS

2956H.Wang,D.Suter/Pattern Recognition Letters24(2003)2953–2966

residuals and use the h-subset to estimate para-meters of models.The claimed LTS breakdown point isenàhT=n.When h is set to n=2,the LTS esti-mator has a claimed high breakdown value of50%.

The advantages of LTS over LMedS are:

?It is less sensitive to local e?ects than LMedS,

i.e.it has more local stability.

?LTS has better statistical e?ciency than LMedS (H€o ssjer,1994).It converges like nà1=2.

Due to these merits,LTS is preferred to LMedS (Rousseeuw and Aelst,1999;Ming and Haralick, 2000).

LTS is also implemented by using random sampling,because the number of all possible h-

subsetseC h

n Tgrows fast with n.There are two

commonly employed ways to generate a h-subset (Rousseeuw and Driessen,1999):

1.Directly generate a random h-subset from the

data points n.

2.Firstly generate a random p-subset.If the rank

of this p-subset is less than p,randomly add data points until the rank is equal to p.Next, use this subset to compute parameters^h jej?1;...;pTand residuals r iei?1;...;nT.Sort the residuals into j repe1Tj6ááá6j repehTj6ááá6 j repenTj,and h-subset is set to:H:?f pe1T;...;

pehTg.

Although the?rst way is easier than the second, the h-subset yielded by the?rst method may con-tain a lot of outliers.Indeed,the chance of gen-erating a‘‘clean’’h-subset by method(1)tends to zero with increasing n.In contrast,it is easier to ?nd a‘‘clean’’p-subset without outliers by method (2).Therefore,method(2)can generate more (good)initial subsets with size h than method(1).

Like LMedS,the e?ciency of LTS can be im-proved by adopting a weighted LS re?nement as the last stage.

2.5.Factors a?ecting the achievements of LMedS and LTS

The LMedS method and the LTS method are based on the idea that the correct?t is determined by a simple situation:the least median of the squared residuals(for LMedS),or by the least sum of trimmed squared residuals(for LTS);and that such a statistic is not in?uenced by the outliers.

Consider the contaminated distributions as follows(Hampel et al.,1986;Haralick,1986):

F?e1àeTF0te He2:12Twhere F0is an inlier distribution,and H is an out-lier distribution.

The equation above is also called the gross error model.When the standard variance of F0is small ((1)and that of H is large or H is uniform dis-tributed,the assumptions leading to the robustness of LMedS or LTS,are true.However,when F0is ‘‘scattered’’,i.e.the standard variance of F0is big, and H is clustered distributed with high density, the assumption is not always true.

Let us investigate an example.In Fig.1,100 good data(inliers)with noise distribution F0(bi-variate normal with unit variance)were generated by adding the noise to samples of a circle with radius10.0and center ate0:0;0:0T.Then80clus-tered outliers were added,possessing a spherical bivariate normal distribution with one unit stan-dard variance and meane20:0;6:0T.As Fig.1 shows,both LMedS and LTS failed to?t the circle:LMS returned the result with a radius equal to7.4239and the center was located ate13:1294;

H.Wang,D.Suter/Pattern Recognition Letters24(2003)2953–29662957

8:6289T.The results obtained by LTS were:the radius equalled19.3069and the center was at e1:1445;10:1470T.

It is important to point out that the failure is inherent,and not simply an artefact of our im-plementation.Let us check the median of the re-siduals(for LMedS)and the sum of trimmed squared residuals(for LTS)and we will under-stand why LMedS and LTS failed to?t to the circle.The median of residuals of the perfect?t is 5.7928.However the median of residuals of?nal result by the LMedS method is5.1479.

In fact,during the searching procedure,the LMedS estimator consistently minimises the me-dian of the residuals,starting with initial?ts that have a larger median residual than the true?t, but successively?nding?ts with lower median resi-duals––proceeding to even lower median residuals than that possessed by the true?t(shown in Fig.

2).

The reason that LTS failed is similar.LTS?nds the?t with smallest trimmed squared residuals. The value of the least sum of trimmed squared residuals obtained is32.6378.However,the same statistic for the‘‘true?t’’is58.6688.Clearly,LTS has‘‘correctly’’,by its criterion,obtained a‘‘bet-ter’’?t(but in fact,the wrong one).The problem is not with the implementation but with the criterion.

Now,let us consider another example showing that the results of the LMedS and the LTS are a?ected by the standard variance of the inliers.We generated a circle with radius10.0and center at e0:0;0:0T.In addition,clustered outliers were added to the circle with meane20:0;6:0Tand unit standard variance.In total,100data points were generated.At?rst,we assigned100data to the circle without any outliers.Then we repeatedly moved two points from the circle to the clustered outliers until50data were left in the circle.Thus, the percentage of outliers changed from0%to 50%.In addition,for each percentage of clustered outliers,we varied the standard variance of the inliers from0.4to1.6with a step size of0.3.Fig. 3(a)illustrates one example of the distribution of the data,with38%clustered outliers and the standard variance of inliers1.3.

From Fig.3(b)and(c),we can see that when the standard variance of inliers is no more than 1.0,LMedS can give the right results under high percentage of outliers(more than44%).However, when the standard variance of inliers is more than 1.0,LMedS does not give the right result even when the percentage of outliers is less then40%. From Fig.3(b),we can see when the standard variance of inliers is0.4,the LTS estimator can correctly give the results even under50%clustered outliers;while when the standard variance of in-liers is1.6,LTS does not give the right results even when only30%of the data are outliers.

From the discussion above,we now see several conditions under which LMedS and LTS failed to be robust.A crucial point is:these methods mea-sure only one single statistic:the least median of residuals or the least sum of trimmed squared of residuals,omitting other characteristics of the data.If we look at the failures,we can see that the results lost the most basic and common feature of the inliers with respect to the?tted circle––sym-metry.

Symmetry is considered a pre-attentive feature that enhances recognition and reconstruction of

2958H.Wang,D.Suter/Pattern Recognition Letters24(2003)2953–2966

shapes and objects(Attneave,1995).Symmetry exists in many man-made and natural objects.In the next section,we will introduce the concept of SD into robust regression methods and propose an improved method,called the LTSD,by which the better performance is acquired even when data include clustered outliers.

3.The symmetry distance

Symmetry exists almost at all places of the world.A square,a cube,a sphere,and a lot of geometric patterns show symmetry.Architecture usually displays symmetry.Symmetry is also an important parameter in physical and chemical processes and is an important criterion in medical diagnosis.Even we human beings show symmetry, (for instance,our faces and bodies are roughly symmetrical between right and left).One of the most basic features in the shapes of models we often?t/impose on our data, e.g.circles and ellipsis,is the symmetry of the model.Symmetric data should suggest symmetric models and data that is symmetrically distributed should be pre-ferred as the inlier data(as opposed to the out-liers).For decades,symmetry has widely been studied in computer vision community(Bigun, 1988;Marola,1989;Nalwa,1989;Kirby and Sirovich,1990;Reisfeld et al.,1992;Zabrodsky

H.Wang,D.Suter/Pattern Recognition Letters24(2003)2953–29662959

et al.,1995;Su and Chou,2001).More detailed de?nitions of symmetry can be found in(Zab-rodsky,1993).We demonstrate here that symme-try can also be used as a feature to enhance the performance of robust estimators when?tting models with symmetric structure.

3.1.The symmetry distance

The exact mathematical de?nition of symmetry (Miller,1972;Weyl,1952)is insu?cient to describe and quantify symmetry found both in the natural world and in the visual world.

Su and Chou(2001)proposed a SD measure based on the concept of‘‘point symmetry’’.Given n points x i;i?1;...;N and a reference vector C (e.g.the centroid of the data),the point SD be-tween a point x j and C is de?ned as follows:

d sex j;CT?min

i?1;...;N

and i?j kex jàCTtex iàCTk

j i

e3:1T

From(3.1)we can see that the point SD is non-negative de?nition.In essence,the measure tries to‘‘balance’’data points with others symmetric about the centroid––for example,x i?e2Càx jTexists in the data,d sex j;CT?0.

However,according to(3.1),one point could be used repeatedly as the‘‘balancing point’’with re-spect to the center.This does not seem to properly capture the notion of symmetry.

In order to avoid one point being used as a ‘‘symmetric point’’more than one time by other points,we re?ne the point SD between a point x j and C as follows:

D sex j;CT?min

i?1;...;N

and i?j

and i2R kex jàCTtex iàCTk

k x jàC ktk x iàC k

e3:2T

where R is a set of points that have been used as ‘‘symmetric point’’.

Based on the concept of‘‘point SD’’,we pro-pose a non-metric SD.Given a pattern x consisted of n points x1;...;x n and a reference vector C,the symmetry distance of the pattern x with respect to the reference vector C is:SD nex;CT?

1

n

X n

i?1

D sex i;CTe3:3T

When the SD of a pattern is equal to0.0,the pattern is perfect symmetric;when the SD of a pattern is very big,the pattern has little sym-metry.

4.The proposed method

We proposed a new method,which couples the LTS method with the symmetry distance measure. Besides residuals,we also choose SD as a criterion in the model?tting.For simplicity,we call the proposed method the LTSD(least trimmed SD). Mathematically,the LTSD estimate can be written as:

^h?arg min

h;C

SD hex;CTe4:1T

Only h data points with the smallest sorted resi-duals are used to calculate the SD.The estimated parameters correspond to the least SD.

The speci?c procedures of the proposed method is given as follows:

Step1.Set repeat times(RT)according to equa-tion(2.7).Initialise h with?entpt1T=

2 6h6n.If we want LTSD to have a

high breakdown point,say50%,we can

set h?entpt1T=2.

Step2.Randomly choose p-subsets,and extend to

a h-subset H1by the method(2)in Section

2.4.

https://www.wendangku.net/doc/ba7196474.html,pute^h1by LS method based on H1.

Compute symmetry distance SD1based

on^h1and H1using(3.3)in Section3and

using the centre of the?t(circle or ellipse)

as the reference vector C.Decrement RT

and if RT is smaller than0,go to step4,

otherwise,go to step2.We calculate the

parameters^h based on h-subset instead

of p-subset in order to improve the statis-

tical e?ciency.

Step4.Finally,output^h with the lowest SD.

2960H.Wang,D.Suter/Pattern Recognition Letters24(2003)2953–2966

5.Experimental results

In this section,we will show several examples using the proposed method to?t a model with symmetrical structures.Circle?tting and ellipse ?tting have been very popular topics in the com-puter vision?eld.One of the obvious characteris-tics of circles and ellipses is that they are symmetric. We?rst present an example of circle?tting,to provide insights into the proposed method.We then present a relatively more complicated exam-ple of ellipse?tting.The results are compared with those of the LMedS method and the LTS method.

5.1.Circle?tting

In Fig.4,about45%clustered outliers were

added to the original circle data.Since LMedS and LTS only rely on the residuals of the data points, their results were a?ected by the standard variance of the inliers and percentages of the clustered out-liers.Therefore,they failed to?t the circle under high percentage of clustered outliers(see Fig.1). However,because the LTSD method considers the symmetry of the object,this enables LTSD?nd the right model(see Fig.4):the true centre and radius of the circle are respectivelye0:0;0:0Tand 10.0;by the LTSD method,we obtained centre eà0:23;0:01Tand radius10.06.

Another example showing the advantages of the proposed method is given in Fig.5(a)(corre-sponding to Fig.3(a)).From Fig.5(a),we can see that when the outliers are clustered,the LMedS and LTS broke down under very low percentages of outliers,in this case,they both broke down under38%outliers!In comparison to the LMedS and LTS methods,the proposed method gives the

H.Wang,D.Suter/Pattern Recognition Letters24(2003)2953–29662961

most accurate results.The proposed method is a?ected less by the standard variance of the inliers and the percentages of the clustered outliers.Fig. 5(b)shows that the radius found by the LTSD method in circle?tting(true radius is10.0)chan-ged less under di?erent standard variance of the inliers and percentages of clustered outliers.In comparison to Fig.3(b)and(c),the?uctuation of the radius found by the LTSD method is smaller. Even when50%clustered outliers exist in the data and the standard variance of inliers is1.6,the re-sults did not(yet)break down.However,both the LMedS and the LTS broke down.

5.2.Ellipse?tting

Ellipses are one of most common and important primitive models in computer vision and pattern

recognition,and often occur in geometric shapes, man-made and natural scenes.Ellipse?tting is a very important task for many industrial applica-tions because it can reduce the data and bene?t the higher level processing(Fitzgibbon et al.,1999). Circles may be projected into ellipses under per-spective projection.Thus ellipses are frequently used in computer vision for model matching (Sampson,1982;Fitzgibbon et al.,1999;Robin, 1999,etc.).In this subsection,we apply the pro-posed robust method to ellipse?tting.

A general conic equation can be written as follows:

ax2tbxytcy2tdxteytf?0e5:1Tea;b;c;d;e;fTare the parameters needed to?nd from the given data.When b2<4ac,the equation above corresponds to ellipses.

The ellipse can also be represented by its more intuitive geometric parameters:ex cos hty sin hàx c cos hày c sin hT2

A2

t

eàx sin hty cos htx c sin hày c cos hT2

B2

?1

e5:2Twhereex c;y cTis the center of the ellipse,f A;B g are the major and minor axes,and h is the orientation of the ellipse.

The relation betweenea;b;c;d;e;fTandex c;y c; A;B;hTis(Robin,1999):

x c?

beà2cd

4acàb2

y c?

bdà2ae

4acàb2

f A;B g?2

???????????????????????????????????????????????????????

à2f

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Table1

Comparison of the estimated parameters by the LTSD,LTS,and LMedS methods in ellipse?tting under40%clustered outliers

x c y c Major axis Minor axis h(°)

True value0.00.010.08.00.0

The LTSD method)0.125)0.1459.7607.810 6.355

The LTS method19.786 5.162 3.328 3.03534.129

The LMedS method9.560 5.20811.757 3.679)3.307

2962H.Wang,D.Suter/Pattern Recognition Letters24(2003)2953–2966

It is convenient to ?nd ea ;b ;c ;d ;e ;f T?rst by the given data and then convert to ex c ;y c ;A ;B ;h T.As illustrated in Fig.6and Table 1,200data were generated with 40%clustered outliers.The outliers were compacted within a region of radius 5and center at e20:0;5:0T.The ellipse had a stan-dard variance 0.8,major axis 10.0,minor axis

8.0,

Fig.7.Fitting a mouse pad:(a)a mouse pad with some ?owers;(b)the edge image by using Canny operator and (c)the results obtained by the LTSD,LTS and LMedS

methods.

Fig.8.Fitting the ellipse in a cup:(a)a real cup image;(b)the edge of the cup by applying Prewitt operator and (c)com-parative results obtained by the LTSD,LTS and LMedS methods.

H.Wang,D.Suter /Pattern Recognition Letters 24(2003)2953–29662963

centere0:0;0:0T,and orientation to horizon direc-tion is0.0°.The results of LTS and LMedS were seriously a?ected by the clustered outliers.How-ever,the LTSD method worked well.

Next,we will apply the LTSD method to real images.

5.3.Experiments with real images

The?rst example is to?t an ellipse in an image of a mouse pad,shown in Fig.7.The edge image was obtained by using Canny operator with threshold 0.07.In total,310data points were in the edge image (Fig.7(b)).The clustered outliers,due to the?ower, occupy50%of the data.Three methods(the LTSD, LTS and LMedS)were applied to detect the mouse pad edge.As shown in Fig.7(c),both LTSD and

LTS correctly found the edge of the mouse pad. However,LMedS fails to detect the edge of the mouse pad.This is because under the condition that the standard variance of inliers is small,the statis-tical e?ciency of LTS is better than LMedS.

Fig.8shows the use of the LTSD method to?t an ellipse to the rim of a cup.Fig.8(a)gives a real cup image.After applying the Prewitt operator, the edge of the cup is detected is shown in Fig. 8(b).We can see there is a high percentage(about 45%)of clustered outliers existing in the edge im-age,external to the rim of the cup(the ellipse we shall try to?t),mainly due to the?gure on the cup. However,the rim of the cup has a symmetric elliptical structure.Fig.8(c)shows that the LTSD method correctly?nds the ellipse in the opening of the cup,while both the LTS and the LMedS fail to correctly?t the ellipse.

5.4.Experiments for the data with uniform outliers

Finally,we investigated the characteristics of the LTSD under uniform outliers.We generated 200data points with40%uniform outliers(see Fig.

9).The ellipse had a standard variance0.5,major axis10.0,minor axis8.0,centere0:0;0:0T,and orientation to horizon direction h is0.0°.The uniform outliers were randomly distributed in a rectangle with left upper cornereà20:0;20:0Tand right lower cornere20:0;à20:0T.We repeated the performance100times and the averaged results were shown in Table2.We can see the LTSD method can also work well in uniform outliers. 6.Conclusion

The fragility of traditionally employed robust estimators:LMedS and LTS,in the presence of clustered outliers has been demonstrated in this paper(a similar story applies more widely––see Wang and Suter,2003).These robust estimators can break down at surprisingly lower percentage of outliers when the outliers are clustered.Thus this paper provides an important cautionary note

Table2

Comparison of the estimated parameters by the LTSD,LTS,and LMedS methods in ellipse?tting with40%randomly distributed outliers

x c y c Major axis Minor axis h(°)

True value0.00.010.08.00.0

The LTSD method)0.1070.12110.0058.024)1.119

The LTS method)0.0090.1209.8777.959)2.117

The LMedS method0.0070.0029.9878.062)0.981

2964H.Wang,D.Suter/Pattern Recognition Letters24(2003)2953–2966

to the computer vision community to carefully employ robust estimators when outliers are clus-tered.We also proposed a new method that in-corporates SD into model?tting.The comparative study shows that this method can achieve better performance than the LMedS method and the LTS method especially when large percentages of clus-tered outliers exist in the data and the standard variance of inliers is large.The price paid for the improvement in?tting models is an increase of the computational complexity due to the complicated de?nition of symmetry distance.It takes about Oen2Ttime to compute SD for each p-subset.The time complexity of the proposed method is also related the times that how many p-subsets are needed(according to equation(2.7)).The pro-posed method can be applied to other symmetric shapes?tting and other?elds.

Unfortunately,our method was especially de-signed for spatially symmetric data distributions. For inlier distributions that are not spatially symmetric(including structures that,though they may be symmetric,have large amounts of missing or occluded data so that the visible inliers are not symmetric),the LTSD is not a good choice. However,the LTSD does provide a feasible way to greatly improve achievements of conventional estimators––the LMedS and the LTS,especially, when the data contain inliers(with symmetry)with large variance and are contaminated by large percentage of clustered outliers.

Further work includes?nding a more e?cient and simple de?nition of SD by which the compu-tational complexity can be reduced moderately. We will also extend our horizon to other features of the model studied so that the new method can be used more widely(data with di?erent types of symmetry and reducing the limits required in that symmetry).

Acknowledgements

This work is supported by the Australia Re-search Council(ARC),under the grant A10017082. The authors would like to thank reviewers for their valuable comments.The authors also thank Alireza Bab-Hadiashar for his helpful suggestions.References

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细菌性病害是植株由于受到细菌侵染而引起的一种病害,一般表现出坏死、腐烂、萎蔫、畸形的特点。 软腐病 您瞧这颗大白菜,叶球直接露出来了,叶柄基部和根茎处的心髓部组织已经完全腐烂,充满了灰黄色的粘稠物,还散发出很大的臭味,这就是软腐病的典型症状。 软腐病又叫做烂疙瘩、烂葫芦、腐烂病、水烂病等,发生极为普遍。它的主要特点是轻轻一掰,植株就倒了,病部呈黏滑软腐状.并伴有恶臭味。小白菜、菜心等白菜类蔬菜发生软腐病时,症状与大白菜基本相似。 拿大白菜来说,大白菜定植后直到形成心叶的这个过程是长外叶的过程,这个过程中软腐病不会发生。而当植株外叶即将罩严地面的时候,大白菜渐渐进入壮心期,这时软腐病开始发生,从壮心开始至收获的整个过程中都有发病的可能,如果在这个时期,一开始植株外围的叶片在烈日下表现出萎蔫,但早晚尚能恢复,慢慢儿地外叶不能恢复的话,那您就要注意了,这有可能是得软腐病的早期症状。 黑腐病 您瞧这棵大白菜,从叶片的边缘往两侧和里边扩展,形成“V”字形黄褐色枯斑,病斑的周边呈淡黄色,这就是黑腐病的症状。以后,病原菌还会沿着叶脉向里扩展,形成大块黄褐色病斑或网状黑脉,并感染叶柄。大白菜一般在莲座期以后容易得这种病,它也是由细菌引起的。

蔬菜病虫害防治

蔬菜——不用农药怎么防止病虫害? ?A+ ?A- 2017-04-05 10:01:47农产信息网关注 说实话,作物病虫害防治不用农药很不现实,比较麻烦且费人力,但要有想学习如何不用农药来防治病虫害的朋友可以来看一下。 蔬菜虫害是蔬菜种植户们非常头疼的问题,若是用传统的农药喷洒方式解决虫害,会因为农药残留影响蔬菜的品质。这里和大家分享一下蔬菜虫害的科学防治方法。

一、伴生植物法: 1.青椒和大蒜间作。由于大蒜有一种特殊气味,能使为害青椒的害虫闻之即逃,避免青椒受害。 2.番茄和甘蓝套种。番茄的叶片会散发一种特殊的气味,可驱赶走为害甘蓝的菜青虫和蚜虫。除此之外,这两种蔬菜吸收的营养有很强的互补性,能充分发挥地力。 3.葱头与胡萝卜间作。它们各自散发的气味能驱走相互间的害虫。若单一种植胡萝卜,为防止虫害,可在地内或四周种上几棵葱头,这也能起到驱虫的作用。 并非所有的蔬菜都可以间作,如甘蓝和芹菜、黄瓜和番茄等不宜间作在一起,因为它们各自的分泌物能抑制对方的生长。 这种方法对适用于种植户和家庭小面积种植者。 二、自然材料治虫

1.草木灰液治虫。草木灰10千克对水50千克浸泡24小时,取滤液喷洒可有效地防治蚜虫、黄守虫。若葱、蒜、韭菜受种蝇、葱蝇的蛆虫危害,每亩沟施或撒施草木灰20~30千克,既治蛆又增产。 2.红糖液防治病。害红糖300克溶于500毫升清水中,加入10克白衣酵母,置于温室或大棚内,每天搅拌1次,发酵15~20天,待其表面出现白膜层为止。然后将此发酵液再加入米醋、烧酒各100克,对入100千克水。每隔10天1次,连喷4~5次,防治黄瓜细菌性斑点病和灰霉病有良好效果。 3.兔粪治地老虎每10千克水加兔粪1千克,装入瓦缸内密封沤15~20天,用时搅拌均匀,浇于瓜菜根部,可防治地老虎。 4.尿洗合剂治菜蚜用洗衣粉、尿素、水按1∶4∶400的比例制成混合液,可防治菜蚜,杀虫率达90%以上。 5.猪胆液治病虫10%浓度的猪胆液加适量小苏打、洗衣粉,能防治茄子立枯病、辣椒炭疽病,能驱赶长豆角、四季豆、瓜类等蔬菜上的蚜虫、菜青虫、蜗牛等多种害虫。稀释液可保持10天有效。 6.大蒜、番茄叶巧杀红蜘蛛用大蒜(捣烂成泥状)2份,水1份混拌均匀,取其滤液喷治。或用新鲜的番茄叶(捣烂成浆)加清水2倍并浸泡5小时然后取滤液喷洒果树、花木或蔬菜,都可有效将红蜘蛛杀死。 7.糖醋、烂果诱捕金龟子选用熟烂酸臭的无花果、烂西瓜等,与糖醋液(红糖、醋、水比为1:3:16),一起放入陶钵,支撑分布在果园或菜园中,每2—3天收集钵中的金龟子即可。 8.三合板涂漆聚捕微型害虫在较大的三合板两面涂上橙黄色油漆,干后再涂一层机油、黄油混合油,分布挂在果园或菜园中,蚜虫、白粉虱、美洲斑潜蝇等害虫就会自投罗网。1周后更换涂刷油漆、混合效果更好。

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