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Signal plus noise models in shape classification
t9
Pattern Recoonition, Vol. 27, No. 6, pp. 777 784, 1994 Elsevier Science Ltd Pattern Recognition Society Printed in Great Britain
Pergamon
0031-3203(94)E0001-2
SIGNAL PLUS NOISE MODELS IN SHAPE CLASSIFICATION R. H. GLENDINNING Defence Research Agency(DRA), Malvern, WR14 3PS, U.K. (Received 6 January 1993; in revised form 15 December 1993; received for publication 6 January 1994)
Abstract--In this study we show how a signalplus noiseformulationcan be used to improvethe performanceof auto-regressive shape classifiersin environments with time varying noise characteristics. Variations of this type may be due to changes in atmospheric conditions or the relative effect of discretization errors. This approach is compared to existing methodologies using a number of noisy templates in a cluttered environment. Typical applications are to aircraft identification and automatic signature validation. Shape
Classification Signalplus noise
Auto-regressiveprocess
1. INTRODUCTION Various time series models have been used to classify three-dimensional objects using the shape of their silhouette. There is ample evidence to demonstrate the practical value of this approach in classifying objects into a pre-determined number of classes. However, their performance can be significantly degraded by changes in the statistical characteristics of the imaging noise(i) in cluttered environments. Such variations may be due to changes in atmospheric conditions or the relative effect of discretization errors. In this study we show how the important class of auto-regressive shape classifiers can be modified to give improved performance in environments with time varying noise characteristics. Typical applications are to aircraft identification,(z) the classification of biological species, {3) automatic inspection of industrial processes ('*)and the automatic recognition of handwritten characters. (5~ We observe a sequence of points (Y~)sampled from the noise corrupted boundary of an object and decompose it into a signal component (Xi) and a noise component (qi) where Yi=Xi"['qi,
i = 1 ..... n.
(1)
Here (~/i)describes imaging noise which may be caused by discretization errors or changes in atmospheric conditions. In this paper we use the equal angle algorithm described by Dubois and Glanz{6) to collect this sequence of points, although there are several competing methods. (~) In the spirit of Grenander and K e e n a n F ) we express our knowledge about possible variations in the shape of (X~) by a circular auto-regressive process. Such variation may be due to changes in orientation or sampling effects. Our aim is to generate shape descriptors from the parameters of this stochastic process. The © Crown Copyright 1994.
Robustness
advantage of this approach is that noise characteristics can be estimated separately from the shape of the underlying object boundary. This "signal + noise" structure differs from earlier work where the process (Y/) is modelled directly. (4'6's 20) With one exception,(~6) low order univariate, bivariate or complex-valued auto-regressive processes are used. This means that the shape descriptors derived from the parameters of the model depend on the variance of the imaging noise which may vary over time. We have already seen that the latter may have a profound effect on the performance of such classifiers in environments with time varying noise characteristics. ") Noise corrupted auto-regressive models are well known and have been applied to a variety of problems.(21-24) Circular ARMA models which include noise corrupted auto-regressive processes as a special case have been used to describe noise corrupted object boundaries,(25) although the emphasis is on improved spectral estimation rather than classification. There appears to be no work on the use of such models in classifying objects in an environment with time varying noise characteristics, although the general approach is well known in other areas. A "signal + noise" structure also provides a natural means of generating nonGaussian models as we may use a Gaussian model for (Xi) and drawn (ql) from a long tailed distribution to describe the effects of atmospheric turbulence.(26) The classification of a three-dimensional object may be improved by an estimate of pose. One approach is to match a given silhouette to a library of admissible shapes which represent an object at varying orientations. This search typically uses Fourier descriptors, (27) although the shape descriptors used in this study can be used. 2. ESTIMATING THE SHAPE DESCRIPTORS
We observe a sequence (Yi) of points sampled from the noisy boundary of an object. We use a model of 777
778
R.H. GLENDINNING
the form Yi=Xi+tli,
i=1 ..... n
(2)
where 0/3 is a sequence of r a n d o m variables with zero mean and variance f2. Here (Xi) and (r/z) are mutually independent processes. In this study we assume that (X3 is drawn from the circular auto-regressive process P
Xi=~"~- ~
OjXti_sjl+Wi,
i=1 ..... n
(3)
j=t
2.2. Hi#her order Yule-Walker equations
where 1 < s~ < ... < sp < n is a sequence of integers, (w3 a sequence of independent normally distributed random variables with variance f 2 and I-x] is x interpreted periodically on the integers, 1,2,...,n. Put 0 = (01 . . . . . Op) and write X ~ CAR(O, f2,/~). A variety of shape descriptors t2°) can be defined using any rotation and scale invariant function of (0, f2,/z). These include (0), (0, f l ), f~ = #f~ ~and (0, f2 ), f2 = fix(1 - Y~_~~0j)# - ~, where a x2 is the variance of X~ and #(1 - 5:~= 10j) - ~ its mean. Note that a 2 can be written in terms of 0 and f2. Replacing (0, a 2, #) by appropriate estimates gives the corresponding "shape statistic".
Consider the following model which describes the boundary of our templates.
Xi=l.~-[-O1X[i_t]+O2Xti_2]+wi,
i= 1. . . . . n.
(9)
We estimate 0 using (6) and the following set of 12 over-determined H O Y W equations based on (9) x x x Ck--OlCk_l+O2Ck_ 2,
k - - 3 . . . . . 14.
(10)
Then we estimate p by
where 0j is an estimate of 0r Replacing c~ by the usual estimate gives
2.1. The Yule-Walker equations Typically ( 0 , f 2 , f 2 , # ) are u n k n o w n and must be estimated from (Y/). Here we use the Yule-Walker equations as maximum likelihood leads to a difficult optimization problem, t2a) First we introduce some notation. Let c~ = cov(Ytq, Yti+jl)J = 0 . . . . . n -- 1
(4)
and r~ = c~/c~. The corresponding quantities for (X 3 and (r//) are defined analogously. Then c~ = c~ + c], r~ = (r; + flrT)(1 + fl)- 1, fl = c[(c~)-i (5) where (1 + fl)- ~ is the signal accuracyJ 29) When (r/z) is a sequence of independent r a n d o m variables we have c ~ = c ~ ) + c ~ , c ~ = c ~ and r ~ = r ~ ( l + f l ) - l , j > l . (6) This simple relationship can be used to estimate (0, f~, f2,/~) through an appropriate set of estimating equations. These are derived from (3) by multiplying by Xtk ~, [k] < i and taking expectations. This gives P
C[x-k] = E OjctX-s~-M"4-EX[klWi,
this study we use higher order Yule-Walker equations (HOYW) which are a set of equations not containing c~. This approach has an extensive literature, t22'2s,30-36) Substantial improvements in performance have been reported using more than p H O Y W equations (the over-determined case), although the degree of improvement ~33'34~appears to be problem dependent. Further improvements can be expected in the circular case. t2 sj
Ik-I < i
(7)
j=l
where we assume that EXi = 0 without loss of generality. It is easy to see that the term containing wi in (7) tends to zero as n tends to infinity as the circular b o u n d a r y conditions have less effect as n increases. The first p equations are given by P
ct~k]= ~ OjCI~k_~A, k = l . . . . . p
(8)
j=l
and are known as the Yule-Walker equations (YW). A wide variety of techniques have been used to estimate (0, f~, f2, #) using equations of this form. In
f^2 x = (cY2_ 0tc~)02-1
(12)
2 a^2, = c~ - #x.
(13)
and
The residual variance a~ is estimated by t34) 2
dw-dx-
~
(14)
j=l
The extension of this approach to higher order models is immediate. Several other techniques have been proposed in the literature. These include variance deflation where the diagonal elements of the YW equations t22~are corrected by an estimate of the variance of ~/i (as c~ = c~ + c~). The limitations of the H O Y W equations and diagonal deflation are well known, t22'35'37~ An alternative approach describes a noise corrupted autoregressive process as an A R M A ( p , p) process. Then 0 is estimated using standard techniques, t38~ Such estimates may be statistically inefficient and improved estimates have been derived using non-linear regression techniques, t39~ Other techniques include matching the sample and theoretical auto-correlation function by least squares, t4°~ 3. THE CLASSIFICATION SCHEME
The "signal + noise" formulation is used to allocate noisy silhouettes to a n u m b e r of categories using the following scheme. (1) Model selection. The first step in our is to extract a n u m b e r of points from the of each template using an appropriate scheme.t t 3. t 4.41) Auto-regressive models of
algorithm boundary sampling increasing
Signal plus noise models in shape classification order are fitted to this sequence using the YW equations and the process repeated for a number of random rotations of each template. We choose an appropriate model order by examining the change in the average residual variance as parameters are added. (2) Training. We use a training set to estimate the mean and covariance matrix of the shape statistics for various a,2 covering a realistic range of values. The training set is made up of a number of realizations derived from each representative template. These may be real world (noisy) images, randomly rotated idealized templates corrupted by additive random noise or realizations from a CAR(O, tr2,1~) corrupted by additive random noise. The number of realizations used will depend on the variability of the shape statistics and is problem dependent. (3) Classification. In this study we classify noisy silhouettes using the shape statistic 0r. The expected value of this statistic is g which takes the unique value gk for the kth template. For cluttered environments we use a classification rule of the form: Allocate (Xi) to process k if
(~
-
gk)'Xk(~.~)
- 1(0~ -
gk)
< (0y -- gi)tZj(~r~)- 1(0y -- 9j), jv~k
(15)
(0r -
(16)
and Ok)'Zk(~)-
l(0k
-- Ok) <
Zp+l(q) 2
where the variance of r/i is a,z. Here the covariance matrix Ek(a,z) of 0y depends on a,.z To implement this rule we estimate ~ using the over-determined YW equations and find the nearest value in the training set. The corresponding "sample" means and covariance matrices are used to replace Okand Ek for all templates in the training set. The former is included as there are small variations in Ok as a~ varies. This algorithm can be modified~6) to account for large changes in apparent size. Each template being approximated to an appropriate degree of detail in each case, 4. T H E E X P E R I M E N T
In this section we compare the effectiveness of the "signal + noise" approach with the conventional one described in Dubois and Glanz. tr~ Our aim is to show that the "signal + noise" approach is less sensitive to variations in the variance of the imaging noise. We also study the robustness of the "signal + noise" approach to changes in the distributional shape of the imaging noise. Our comparisons are based on a cluttered environment classification rule as this gives a demanding trial for the relatively well separated templates used in this study. We use the shape descriptor (01, 02, f2) with the "signal + noise" formulation as it proved difficult to estimate f l for some templates using the overdetermined YW equations. This problem becomes progressively worse as the variance of the imaging noise increases and is due to negative estimates of the residual variance.
779
Our comments are based on the ability of each scheme to classify 1000 noisy versions of the hand sketched templates presented in Fig. 1. We generate test and training data by extracting a number of points from the boundary of each randomly rotated template using the equal angles unwrapping algorithm and adding random noise to the radial components. We call this sequence a "centroid profile". This approach was used for comparability with earlier work/6) In our experiments we implement the "signal + noise" approach using 160 angles and the conventional approach with 64. This gives an even handed comparison as the robustness of the conventional approach appears to deteriorate as the number of angles increases where as model fit improves which favours the "signal + noise" approach. All random numbers used in this study were generated by the algorithm described in Wichmann and Hill~42~ using the repeatable seed (40273, 579, 3271). Normal deviates were generated using the Box-Muller method. 4.1. Training Each template was randomly rotated about the centre of gravity of its vertices and its boundary sampled using the equal angles unwrapping algorithm. (6) Autoregressive models of increasing order were fitted to the centroid profile of 1000 randomly rotated versions of each template using the YW equations. By examining the decrease in the residual sum of squares as more complex models are fitted we decide that a model with p = 2 provides a reasonable fit to most templates. The "signal + noise" classifier and the conventional approach were trained on 1000 randomly rotated versions of each template. The resulting centroid profiles were corrupted by random normally distributed noise with zero mean and variance a,.2 This process was repeated for tr2 taking the values 1, 2. . . . . 15. The mean and variance of the shape statistics are presented in Table 1 for or,-2 _ 1. The "signal + noise" estimates are derived from a set of 12 over-determined YW equations. The corresponding means and co-variance matrices were collected and used in our classification scheme. On the basis of the templates used in this study we saw that there can be substantial differences between the HOYW and YW estimates of 01 and 02 even if they are calculated from the same centroid profile. This is a consequence of the approximate nature of our model. However the use of a large number of estimating equations generally leads to a substantial reduction in the variance of the parameter estimates. 4.2. A comparison with varying conditions In this section we examine the robustness of the "signal + noise" approach and the corresponding autoregressive scheme to changes in the distributional properties of the noise (~/i).Such changes occur in imaging systems whose noise characteristics vary over time. This may be due to changes in atmospheric conditions
780
R.H. GLENDINNING
1
3
2
3
aircraft shapes
4
machine parts Fig. 1. The templates.
Table 1. The mean and variance of the shape statistics using the "signal + noise" formulation (HOYW) with 160 angles and the Dubois and Glanz approach (YW) using 64. Each template is corrupted by normally distributed noise with variance tr~ = 1.0. The sample variance x 106 is given in parentheses HOYW
YW
Template
01
02
1 Shuttle
1.7998 (17) 1.8925 (8) 1.6782 (3469) 1.7912 (16)
-0.8200 (15) --0.9158 (6) --0.7116 (2762) -0.8195 (15)
1.7625 (198) 1.8635 (344) 1.8341 (59) 1.8850 (4)
-0.7723 (187) -0.8762 (303) -0.8521 (53) -0.9465 (5)
2 F 16 3 B1 4
DC 10
1 part 2 part 3 part 4 part
f2 Aircraft 0.4522 (3) 0.3217 (2) 0.4477 (98) 0.4802 (4) Machine parts 0.6426 (5) 0.7501 (7) 0.4656 (7) 0.3714 (6)
01
02
fl
1.6391 (2180) 1.3867 (2221) 1.1591 (4006) 1.4904 (1246)
-0.7370 (2142) --0.5684 (2377) -0.3312 (3322) -0.6433 (1232)
0.9774 (7313) 1.4848 (11,059) 0.8309 (13,931) 0.9919 (2579)
1.5305 (22,085) 1.5486 (4677) 1.3798 (16,901) 0.9936 (976)
-0.5774 (20,526) -0.6186 (3998) -0.4910 (13,850) -0.5166 (520)
0.3826 (2894) 0.4089 (1208) 0.7359 (4612) 2.1961 (3828)
Signal plus noise models in shape classification
781
Table 2. The percentage of correctly classified aircraft using a cluttered environment classifier with q = 0.01. Each template is corrupted by normally distributed noise with variance a.. 2 The conventional scheme is trained on templates corrupted by normally distributed noise with variance = 1 Template
Gn2
Model
1.0
Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz
2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
1 97.5 98.5 96.3 86.7 94.5 63.9 92.8 39.4 89.9 23.8 88.5 11.5 88.7 6.2 88.1 2.6 86.8 1.1 87.3 1.1
2
3
4
Average
98.3 99.7 93.0 96.6 86.5 90.4 75.1 84.3 67.6 75.2 61.3 66.9 59.8 62.3 62.7 49.1 67.7 45.0 71.2 37.5
99.7 96.8 100.0 96.6 99.9 94.8 99.7 95.4 99.7 94.0 99.8 92.3 99.7 89.8 99.6 89.7 99.3 87.8 99.2 83.1
97.6 99.0 96.0 96.0 91.3 92.5 90.2 84.8 91.3 77.0 91.3 67.7 91.7 59.9 92.1 52.2 91.8 43.5 92.7 36.7
98.3 98.5 96.3 94.0 93.1 85.4 89.4 76.0 87.1 67.5 85.2 59.6 85.0 54.6 85.6 48.4 86.4 44.3 87.6 39.6
Table 3. The percentage of correctly classified machine parts using a cluttered environment classifier with q = 0.01. Each template is corrupted by normally distributed noise with variance a 2. The conventional scheme is trained on templates corrupted by normally distributed noise with variance = 1 Template 2
Model
1
2
3
4
Average
1.0
Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz
89.1 97.5 94.7 94.8 97.1 91.5 98.5 85.7 98.4 80.3 97.3 69.8 98.0 62.0 98.4 53.7 97.9 45.9 98.4 37.1
100.0 99.7 99.7 97.5 99.2 94.9 98.4 90.7 99.7 85.3 99.7 77.7 99.3 75.5 99.5 66.7 99.4 60.6 99.7 58.8
98.9 96.8 97.2 98.6 94.2 94.4 94.2 90.1 94.4 86.4 95.6 76.2 94.8 71.2 97.5 64.6 95.3 56.0 95.7 53.2
99.2 99.3 94.1 96.7 89.9 91.9 80.5 84.0 76.0 75.8 77.8 67.2 74.6 56.9 73.8 56.4 76.1 52.0 73.5 43.0
96.8 98.3 96.4 96.9 95.1 93.2 92.9 87.6 92.1 81.9 92.6 72.7 91.7 66.4 92.3 60.3 92.2 53.6 91.8 48.0
2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
or the relative effect ofdiscretization errors. In Tables 2 and 3 we examine the effect of incorrectly specifying the noise variance a~2 on the performance of each scheme. Imaging noise is represented by a sequence of indep e n d e n t normally distributed r a n d o m variables with zero mean and variance tr.. 2 The conventional autoregressive classifier is trained on 1000 noisy templates
with tr.2 1.0. N o t e that the variance of the imaging noise is different in the test and training sets when a 2 > 1.0. The results presented in this section are based on a cluttered e n v i r o n m e n t classification rule with q = 0.01. We see from Tables 2 and 3 that the use of a "signal + noise" formulation substantially improves the performance of an auto-regressive classifier when there are =
782
R.H. GLENDINNING
large changes in the variance of the imaging noise a 2. The conventional approach may give better results when the true value of tr 2 is known. This is due to the fact that tr2 is estimated in our approach and is therefore subject to random errors and bias. The results for small tr2 are generally untypical of the "signal + noise" approach as they are influenced by our treatment of estimates of tr 2 lying near zero which are allocated to %2 = 1.0. Significant improvements were obtained using the synthetic templates used in Glendinning. ") In the later experiment we used 128 angles for the "signal + noise" approach and 32 for the conventional classifier. 4.3. Robustness Next we examine the robustness of the "signal + noise" approach to changes in the distributional form of the imaging noise. In Tables 4 and 5 we consider the case where the "signal + noise" classifier is trained on templates corrupted by independent normally distributed noise with variance in the range 1-15. These are used to classify templates corrupted by
Table 4. The percentage of correctly classified aircraft using a cluttered environment classifier with q =0.01 and the "signal + noise" formulation. Imaging noise is drawn from the density f(x)= (1 -p)~b(0, 0.2)+ p~b(0,60.2), with p = 0.05 and variance 0.2 Template 0"72
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
Average
1
2
3
4
p = 0.05
p = 0.0
97.7 96.9 95.1 92.7 90.6 90.2 87.3 88.6 85.1 85.3
98.3 93.1 85.1 74.1 68.0 62.2 61.7 62.7 67.2 69.1
99.8 99.9 99.9 99.5 99.8 99.6 99.7 99.5 99.3 99.4
97.4 96.3 91.8 90.5 90.3 91.2 91.3 91.7 92.3 92.1
98.3 96.6 93.0 89.2 87.2 85.8 85.0 85.6 86.0 86.5
98.3 96.3 93.1 89.4 87.1 85.2 85.0 85.6 86.4 87.6
Table 5. The percentage of correctly classified machine parts using a cluttered environment classifier with q = 0.01 and the "signal + noise" formulation. Imaging noise is drawn from the density with variance 0.~2 and p = 0.05 given by f(x) = (1 -- p)t~ (0, 0"2) + pq~ (0, 60. 2 )
Template 0.2 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
1
2
3
88.1 100.0 98.8 92.9 9 9 . 8 97.6 96.8 9 9 . 3 94.7 98.1 98.9 94.3 98.6 9 9 . 3 94.2 97.1 99.7 95.3 97.9 9 9 . 3 94.9 98.6 9 9 . 5 97.2 98.5 9 9 . 5 95.4 98.0 99.6 95.9
Average 4
p = 0.05
p = 0.0
99.4 94.6 89.7 80.8 76.8 76.8 74.4 73.3 76.5 74.7
96.6 96.2 95.1 93.0 92.2 92.2 91.6 92.1 92.5 92.2
96.8 96.4 95.1 92.9 92.1 92.6 91.7 92.3 92.2 91.8
non-normal noise from the following distribution: "}
f(x) = (1 - p) ~b(0, tr 2) + p~b(0, ka 2), k = 6, p = 0.05 (17) where ~b(/~,~r2) is the normal density with mean # and variance a 2. This density is symmetric about zero and has "fatter" tails than a normal distribution with matched mean and variance. In our experiments we use values of a~ in the range 1-10. In each case a2 =
~ ( 1 + (k - 1)p)- 1. F r o m Tables 4 and 5 we see that the "signal + noise" approach is relatively insensitive to the distributional shape of the imaging noise. This is consistent with the behaviour of the conventional approach described in Glendinning.I ~) 5. S U M M A R Y
In this study we compared the performance of a "signal + noise" formulation with an auto-regressive shape classifier. ~6) This work was motivated by the study presented in Glendinning ") which describes the limitations of the conventional approach in environments with time varying noise characteristics. We have shown that a "signal + noise" formulation gives a substantial improvement in performance when there are large variations in the variance of the imaging noise tr2. The conventional approach may be preferable when the true value of tr 2 is known. Both approaches are equally insensitive to changes in the distributional form of the noise. F r o m Table 1 we see that the use of an over-determined set of H O Y W equations leads to a significant reduction in the variance of the parameter estimates, although the "signal + noise" formulation is generally more sensitive to model inadequacies which may result in biased estimates. F o r example, the approximate bias in our estimates of tr~2 are about (0.9,2.3, - 0 . 9 , - 5 . 9 ) for the machine parts and (1.2, - 6 . 5 , 1.5, - 2 . 4 ) for the aircraft. This clearly limits the applicability of this approach. N o difficulties were encountered in the numerical solution of the over-determined Y W equations, although their use generally resulted in an additional computational burden. For the aircraft templates we estimated an increase of 37% over the conventional approach when tr~2 = 1.0. We have seen that this approach is relatively insensitive to variations in the distributional shape of the imaging noise. When this is not the case, robust estimators "a'15) may be used, although they require extensive optimization in this case. Computationally less demanding estimates may be obtained by replacing the covariances in the H O Y W equations by an appropriate robust estimator, t43) The problem of object classification in time varying environments is often encountered in the study of radar signals. In this case we estimate the structure of the noise separately using background clutter344) Other approaches to this problem include suppressing
Signal plus noise models in shape classification
noise at the pre-processing stage by spatial s m o o t h i n g or direct filtering of the centroid profile using a n app r o p r i a t e filter. Similar ideas are used to e n h a n c e noisy speech in n a t u r a l environments, see Paliwal t451 where a "signal + noise" f o r m u l a t i o n is used to construct a noise suppressing filter. Implicit noise suppression is also carried o u t by curve fitting procedures t46) or those based o n t r u n c a t e d Fourier series expansions. Procedures for dealing with variations in a p p a r e n t size by "averaging" feature statistics over a range of sizes can also be used in this contex, tll a l t h o u g h they lead to an increase in variance. The "signal + noise" a p p r o a c h can be applied to o t h e r models including bivariate auto-regressions, 14~ a l t h o u g h its extension to non-linear models 116) is less straightforward. O u r a p p r o a c h can be modified to deal with coloured noise.
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Computer Vision, Seattle, Washington, pp.57-71 1 3 October (1990). 16. B. Kartikeyan and A. Sarkar, Shape description by time series, IEEE Trans. Pattern Analysis Mach. Intell. PAMI11, 977-984 (1989). 17. K. B. Eom and X. Chen, Maximum likelihood decision rules for recognizing noisy shapes, Proc. Int. Conf. on Acoustics, Speech, and Signal Process., New York, U.S.A. pp. 972-975 11-14 April (1988). 18. P. F. Singer and R. Chellappa, Machine perception of partially specified planar shapes, Proc. Conf. Vision and Pattern Recognition pp. 479-502 June (1985). 19. P. F. Singer and R. Chellappa, Classification of boundaries on the plane using stochastic models, Proc. IEEE Conf. Vision and Pattern Recognition, Washington DC, pp. 146-147 (1983). 20. R. L. Kashyap and R. Chellappa, Stochastic models for closed boundary analysis: representation and reconstruction, IEEE Trans. Inf. Theory IT-27, 627-637 (1981). 21. U. Grenander, A unified approach to pattern analysis, Advances in Computers, F. L. Alt and M. Rubinoff, eds, Vol. 50, pp. 175-215. Academic Press, New York (1970). 22. S. M. Kay, Noisy compensation for the autoregressive spectral estimates, IEEE Trans. Acoust. Speech Signal Process. ASSP-28, 292-303 (1980). 23. B. D. Ripley, Statistical Inference for Spatial Processes. Cambridge University Press, Cambridge (1988). 24. R. Molina and B. D. Ripley, Using spatial models as priors in astronomical image analysis, J. Appl. Statistics 16, 193-206 (1989). 25. M.V. Malakooti and K.A. Teague, CARMA model method of two dimensional shape classification: an eigen system approach vs the LP norm, Proc. Int. Conf. on Acoustics, Speech and Signal Processing, Dallas, Texas, U.S.A. pp. 583 586, April 6 9 (1987). 26. J.W. Goodman, Statistical Optics. Wiley, New York (1980). 27. Z. Chen and S.-Y. Ho, Computer vision for robust 3D aircraft recognition with fast library search, Pattern Recognition 24, 375-390 (1991). 28. S. Lakshmanan and H. Derin, Simultaneous parameter estimation and segmentation of Gibbs random fields using simulated annealing, IEEE Trans. Pattern Analysis Mach. lntell. PAMI-11, 799-813 (1989). 29. E. Parzen, Time series analysis for models of signal plus noise, Spectral Analysis of Time Series, B. Harris, ed., pp. 233-258. Wiley, New York (1967). 30. A. M. Walker, Some consequences of superimposed error in time series analysis, Biometrika 47, 33 43 (1960). 31. J. A. Cadzow, Spectral estimation: an over-determined rational model equation approach, Proc. IEEE 70, 907-939 (1982). 32. Y. T. Chan and R. P. Langford, Spectral estimation via the high-order Yule-Walker equations, IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 689-698 (1982). 33. D. F. Gingras, Asymptotic properties of high-order YuleWalker estimates of the AR parameters of an ARMA time series, I EEE Trans. Acoust. Speech Signal Process. ASSP33, 1095-1101 (1985). 34. D.F. Gingras and E. Masry, Autoregressive spectral estimation in additive noise, IEEE Trans. Acoust. Speech Signal Process. ASSP-36, 490-501 (t 988). 35. K.K. Paliwal, A noise-compensated long correlation matching method for AR spectral estimation of noisy signals, Signal Process. 15, 437-440 (1988). 36. L. Vergara-Dominguez, New insights into the high-order Yule-Walker equations, IEEE Trans. Acoust. Speech Signal Process. ASSP-38, 1649-1651 (1990). 37. S. P. Bruzzone and M. Kaveh, Information trade-offs in using the sample autocorrelation functions in ARMA parameter estimation, IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 701-715 (1984). 38. W. Gersch, Estimation of the autoregressive parameters
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of a mixed moving average time series, IEEE Trans. Automat. Contr. AC-15, 583-588 (1970). M. Pagano, Estimation of autoregressive signals plus white noise, Ann. Statist. 2, 97-108 0974). L. B. Jackson, J. Huang, K. P. Richards and H. Chen, AR, ARMA and AR in noise modeling by fitting windowed correlation data, IEEE Trans. Acoust. Speech Signal Process ASSP-37, 1608-1612 (1989). G. N. Bebis and G. M. Papadourakis, Object recognition using invariant object boundary representations and neural network models, Pattern Recognition 25, 25-44 (1992). B.A. Wichmann and I. D. Hill, Algorithm AS 183. An efficient and portable pseudo-random number generator, J. R. Statist. Soc. Set. C 31, 188-190 (1982). R. Hansen and R. Chellappa, Empirical robust estimates
for 2-D non-causal autoregressive models, Proc. Int. Conf. on Acoustics, Speech and Signal Processing, Albuquerque, New Mexico, U.S.A. pp. 2005-2008, 3-6 April (1990). 44. J. R. Casar Corredera and G. de Miguel Vela, A CFAR AR-based method for radar detection in clutter, Signal Processing V. Theory and Application, L. Torres, E. Masgrau and M. A. Lagunas, eds, pp. 2019-2022. Elsevier, Amsterdam (1990). 45. K. K. Paliwal, Estimation of noise variance for the noisy AR signal and its application in speech enhancement, 1EEE Trans. Acoust. Speech Signal Process. ASSP-36, 292-294 (1988). 46. I. Weiss, Shape recognition on a varying mesh, IEEE Trans. Pattern Analysis Mach. Intell. PAMI-12, 345-362 (1990).
About the Author--RICHARD H. GLENDINNING received a B.Sc. from Salford University in mathematics in 1975 and a M.Sc. and Ph.D. in statistics from Newcastle and London Universities in 1976 and 1986, respectively. He has held Post-doctoral positions at the Universities at Bath and Newcastle and was a lecturer in Statistics at University College Cork, Ireland. His research interests include time series analysis, image analysis, random sets, stochastic processes, central limit theorems and transport in heterogeneous media. He has published a number of papers in his specialities.
Pattern Recoonition, Vol. 27, No. 6, pp. 777 784, 1994 Elsevier Science Ltd Pattern Recognition Society Printed in Great Britain
Pergamon
0031-3203(94)E0001-2
SIGNAL PLUS NOISE MODELS IN SHAPE CLASSIFICATION R. H. GLENDINNING Defence Research Agency(DRA), Malvern, WR14 3PS, U.K. (Received 6 January 1993; in revised form 15 December 1993; received for publication 6 January 1994)
Abstract--In this study we show how a signalplus noiseformulationcan be used to improvethe performanceof auto-regressive shape classifiersin environments with time varying noise characteristics. Variations of this type may be due to changes in atmospheric conditions or the relative effect of discretization errors. This approach is compared to existing methodologies using a number of noisy templates in a cluttered environment. Typical applications are to aircraft identification and automatic signature validation. Shape
Classification Signalplus noise
Auto-regressiveprocess
1. INTRODUCTION Various time series models have been used to classify three-dimensional objects using the shape of their silhouette. There is ample evidence to demonstrate the practical value of this approach in classifying objects into a pre-determined number of classes. However, their performance can be significantly degraded by changes in the statistical characteristics of the imaging noise(i) in cluttered environments. Such variations may be due to changes in atmospheric conditions or the relative effect of discretization errors. In this study we show how the important class of auto-regressive shape classifiers can be modified to give improved performance in environments with time varying noise characteristics. Typical applications are to aircraft identification,(z) the classification of biological species, {3) automatic inspection of industrial processes ('*)and the automatic recognition of handwritten characters. (5~ We observe a sequence of points (Y~)sampled from the noise corrupted boundary of an object and decompose it into a signal component (Xi) and a noise component (qi) where Yi=Xi"['qi,
i = 1 ..... n.
(1)
Here (~/i)describes imaging noise which may be caused by discretization errors or changes in atmospheric conditions. In this paper we use the equal angle algorithm described by Dubois and Glanz{6) to collect this sequence of points, although there are several competing methods. (~) In the spirit of Grenander and K e e n a n F ) we express our knowledge about possible variations in the shape of (X~) by a circular auto-regressive process. Such variation may be due to changes in orientation or sampling effects. Our aim is to generate shape descriptors from the parameters of this stochastic process. The © Crown Copyright 1994.
Robustness
advantage of this approach is that noise characteristics can be estimated separately from the shape of the underlying object boundary. This "signal + noise" structure differs from earlier work where the process (Y/) is modelled directly. (4'6's 20) With one exception,(~6) low order univariate, bivariate or complex-valued auto-regressive processes are used. This means that the shape descriptors derived from the parameters of the model depend on the variance of the imaging noise which may vary over time. We have already seen that the latter may have a profound effect on the performance of such classifiers in environments with time varying noise characteristics. ") Noise corrupted auto-regressive models are well known and have been applied to a variety of problems.(21-24) Circular ARMA models which include noise corrupted auto-regressive processes as a special case have been used to describe noise corrupted object boundaries,(25) although the emphasis is on improved spectral estimation rather than classification. There appears to be no work on the use of such models in classifying objects in an environment with time varying noise characteristics, although the general approach is well known in other areas. A "signal + noise" structure also provides a natural means of generating nonGaussian models as we may use a Gaussian model for (Xi) and drawn (ql) from a long tailed distribution to describe the effects of atmospheric turbulence.(26) The classification of a three-dimensional object may be improved by an estimate of pose. One approach is to match a given silhouette to a library of admissible shapes which represent an object at varying orientations. This search typically uses Fourier descriptors, (27) although the shape descriptors used in this study can be used. 2. ESTIMATING THE SHAPE DESCRIPTORS
We observe a sequence (Yi) of points sampled from the noisy boundary of an object. We use a model of 777
778
R.H. GLENDINNING
the form Yi=Xi+tli,
i=1 ..... n
(2)
where 0/3 is a sequence of r a n d o m variables with zero mean and variance f2. Here (Xi) and (r/z) are mutually independent processes. In this study we assume that (X3 is drawn from the circular auto-regressive process P
Xi=~"~- ~
OjXti_sjl+Wi,
i=1 ..... n
(3)
j=t
2.2. Hi#her order Yule-Walker equations
where 1 < s~ < ... < sp < n is a sequence of integers, (w3 a sequence of independent normally distributed random variables with variance f 2 and I-x] is x interpreted periodically on the integers, 1,2,...,n. Put 0 = (01 . . . . . Op) and write X ~ CAR(O, f2,/~). A variety of shape descriptors t2°) can be defined using any rotation and scale invariant function of (0, f2,/z). These include (0), (0, f l ), f~ = #f~ ~and (0, f2 ), f2 = fix(1 - Y~_~~0j)# - ~, where a x2 is the variance of X~ and #(1 - 5:~= 10j) - ~ its mean. Note that a 2 can be written in terms of 0 and f2. Replacing (0, a 2, #) by appropriate estimates gives the corresponding "shape statistic".
Consider the following model which describes the boundary of our templates.
Xi=l.~-[-O1X[i_t]+O2Xti_2]+wi,
i= 1. . . . . n.
(9)
We estimate 0 using (6) and the following set of 12 over-determined H O Y W equations based on (9) x x x Ck--OlCk_l+O2Ck_ 2,
k - - 3 . . . . . 14.
(10)
Then we estimate p by
where 0j is an estimate of 0r Replacing c~ by the usual estimate gives
2.1. The Yule-Walker equations Typically ( 0 , f 2 , f 2 , # ) are u n k n o w n and must be estimated from (Y/). Here we use the Yule-Walker equations as maximum likelihood leads to a difficult optimization problem, t2a) First we introduce some notation. Let c~ = cov(Ytq, Yti+jl)J = 0 . . . . . n -- 1
(4)
and r~ = c~/c~. The corresponding quantities for (X 3 and (r//) are defined analogously. Then c~ = c~ + c], r~ = (r; + flrT)(1 + fl)- 1, fl = c[(c~)-i (5) where (1 + fl)- ~ is the signal accuracyJ 29) When (r/z) is a sequence of independent r a n d o m variables we have c ~ = c ~ ) + c ~ , c ~ = c ~ and r ~ = r ~ ( l + f l ) - l , j > l . (6) This simple relationship can be used to estimate (0, f~, f2,/~) through an appropriate set of estimating equations. These are derived from (3) by multiplying by Xtk ~, [k] < i and taking expectations. This gives P
C[x-k] = E OjctX-s~-M"4-EX[klWi,
this study we use higher order Yule-Walker equations (HOYW) which are a set of equations not containing c~. This approach has an extensive literature, t22'2s,30-36) Substantial improvements in performance have been reported using more than p H O Y W equations (the over-determined case), although the degree of improvement ~33'34~appears to be problem dependent. Further improvements can be expected in the circular case. t2 sj
Ik-I < i
(7)
j=l
where we assume that EXi = 0 without loss of generality. It is easy to see that the term containing wi in (7) tends to zero as n tends to infinity as the circular b o u n d a r y conditions have less effect as n increases. The first p equations are given by P
ct~k]= ~ OjCI~k_~A, k = l . . . . . p
(8)
j=l
and are known as the Yule-Walker equations (YW). A wide variety of techniques have been used to estimate (0, f~, f2, #) using equations of this form. In
f^2 x = (cY2_ 0tc~)02-1
(12)
2 a^2, = c~ - #x.
(13)
and
The residual variance a~ is estimated by t34) 2
dw-dx-
~
(14)
j=l
The extension of this approach to higher order models is immediate. Several other techniques have been proposed in the literature. These include variance deflation where the diagonal elements of the YW equations t22~are corrected by an estimate of the variance of ~/i (as c~ = c~ + c~). The limitations of the H O Y W equations and diagonal deflation are well known, t22'35'37~ An alternative approach describes a noise corrupted autoregressive process as an A R M A ( p , p) process. Then 0 is estimated using standard techniques, t38~ Such estimates may be statistically inefficient and improved estimates have been derived using non-linear regression techniques, t39~ Other techniques include matching the sample and theoretical auto-correlation function by least squares, t4°~ 3. THE CLASSIFICATION SCHEME
The "signal + noise" formulation is used to allocate noisy silhouettes to a n u m b e r of categories using the following scheme. (1) Model selection. The first step in our is to extract a n u m b e r of points from the of each template using an appropriate scheme.t t 3. t 4.41) Auto-regressive models of
algorithm boundary sampling increasing
Signal plus noise models in shape classification order are fitted to this sequence using the YW equations and the process repeated for a number of random rotations of each template. We choose an appropriate model order by examining the change in the average residual variance as parameters are added. (2) Training. We use a training set to estimate the mean and covariance matrix of the shape statistics for various a,2 covering a realistic range of values. The training set is made up of a number of realizations derived from each representative template. These may be real world (noisy) images, randomly rotated idealized templates corrupted by additive random noise or realizations from a CAR(O, tr2,1~) corrupted by additive random noise. The number of realizations used will depend on the variability of the shape statistics and is problem dependent. (3) Classification. In this study we classify noisy silhouettes using the shape statistic 0r. The expected value of this statistic is g which takes the unique value gk for the kth template. For cluttered environments we use a classification rule of the form: Allocate (Xi) to process k if
(~
-
gk)'Xk(~.~)
- 1(0~ -
gk)
< (0y -- gi)tZj(~r~)- 1(0y -- 9j), jv~k
(15)
(0r -
(16)
and Ok)'Zk(~)-
l(0k
-- Ok) <
Zp+l(q) 2
where the variance of r/i is a,z. Here the covariance matrix Ek(a,z) of 0y depends on a,.z To implement this rule we estimate ~ using the over-determined YW equations and find the nearest value in the training set. The corresponding "sample" means and covariance matrices are used to replace Okand Ek for all templates in the training set. The former is included as there are small variations in Ok as a~ varies. This algorithm can be modified~6) to account for large changes in apparent size. Each template being approximated to an appropriate degree of detail in each case, 4. T H E E X P E R I M E N T
In this section we compare the effectiveness of the "signal + noise" approach with the conventional one described in Dubois and Glanz. tr~ Our aim is to show that the "signal + noise" approach is less sensitive to variations in the variance of the imaging noise. We also study the robustness of the "signal + noise" approach to changes in the distributional shape of the imaging noise. Our comparisons are based on a cluttered environment classification rule as this gives a demanding trial for the relatively well separated templates used in this study. We use the shape descriptor (01, 02, f2) with the "signal + noise" formulation as it proved difficult to estimate f l for some templates using the overdetermined YW equations. This problem becomes progressively worse as the variance of the imaging noise increases and is due to negative estimates of the residual variance.
779
Our comments are based on the ability of each scheme to classify 1000 noisy versions of the hand sketched templates presented in Fig. 1. We generate test and training data by extracting a number of points from the boundary of each randomly rotated template using the equal angles unwrapping algorithm and adding random noise to the radial components. We call this sequence a "centroid profile". This approach was used for comparability with earlier work/6) In our experiments we implement the "signal + noise" approach using 160 angles and the conventional approach with 64. This gives an even handed comparison as the robustness of the conventional approach appears to deteriorate as the number of angles increases where as model fit improves which favours the "signal + noise" approach. All random numbers used in this study were generated by the algorithm described in Wichmann and Hill~42~ using the repeatable seed (40273, 579, 3271). Normal deviates were generated using the Box-Muller method. 4.1. Training Each template was randomly rotated about the centre of gravity of its vertices and its boundary sampled using the equal angles unwrapping algorithm. (6) Autoregressive models of increasing order were fitted to the centroid profile of 1000 randomly rotated versions of each template using the YW equations. By examining the decrease in the residual sum of squares as more complex models are fitted we decide that a model with p = 2 provides a reasonable fit to most templates. The "signal + noise" classifier and the conventional approach were trained on 1000 randomly rotated versions of each template. The resulting centroid profiles were corrupted by random normally distributed noise with zero mean and variance a,.2 This process was repeated for tr2 taking the values 1, 2. . . . . 15. The mean and variance of the shape statistics are presented in Table 1 for or,-2 _ 1. The "signal + noise" estimates are derived from a set of 12 over-determined YW equations. The corresponding means and co-variance matrices were collected and used in our classification scheme. On the basis of the templates used in this study we saw that there can be substantial differences between the HOYW and YW estimates of 01 and 02 even if they are calculated from the same centroid profile. This is a consequence of the approximate nature of our model. However the use of a large number of estimating equations generally leads to a substantial reduction in the variance of the parameter estimates. 4.2. A comparison with varying conditions In this section we examine the robustness of the "signal + noise" approach and the corresponding autoregressive scheme to changes in the distributional properties of the noise (~/i).Such changes occur in imaging systems whose noise characteristics vary over time. This may be due to changes in atmospheric conditions
780
R.H. GLENDINNING
1
3
2
3
aircraft shapes
4
machine parts Fig. 1. The templates.
Table 1. The mean and variance of the shape statistics using the "signal + noise" formulation (HOYW) with 160 angles and the Dubois and Glanz approach (YW) using 64. Each template is corrupted by normally distributed noise with variance tr~ = 1.0. The sample variance x 106 is given in parentheses HOYW
YW
Template
01
02
1 Shuttle
1.7998 (17) 1.8925 (8) 1.6782 (3469) 1.7912 (16)
-0.8200 (15) --0.9158 (6) --0.7116 (2762) -0.8195 (15)
1.7625 (198) 1.8635 (344) 1.8341 (59) 1.8850 (4)
-0.7723 (187) -0.8762 (303) -0.8521 (53) -0.9465 (5)
2 F 16 3 B1 4
DC 10
1 part 2 part 3 part 4 part
f2 Aircraft 0.4522 (3) 0.3217 (2) 0.4477 (98) 0.4802 (4) Machine parts 0.6426 (5) 0.7501 (7) 0.4656 (7) 0.3714 (6)
01
02
fl
1.6391 (2180) 1.3867 (2221) 1.1591 (4006) 1.4904 (1246)
-0.7370 (2142) --0.5684 (2377) -0.3312 (3322) -0.6433 (1232)
0.9774 (7313) 1.4848 (11,059) 0.8309 (13,931) 0.9919 (2579)
1.5305 (22,085) 1.5486 (4677) 1.3798 (16,901) 0.9936 (976)
-0.5774 (20,526) -0.6186 (3998) -0.4910 (13,850) -0.5166 (520)
0.3826 (2894) 0.4089 (1208) 0.7359 (4612) 2.1961 (3828)
Signal plus noise models in shape classification
781
Table 2. The percentage of correctly classified aircraft using a cluttered environment classifier with q = 0.01. Each template is corrupted by normally distributed noise with variance a.. 2 The conventional scheme is trained on templates corrupted by normally distributed noise with variance = 1 Template
Gn2
Model
1.0
Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz
2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
1 97.5 98.5 96.3 86.7 94.5 63.9 92.8 39.4 89.9 23.8 88.5 11.5 88.7 6.2 88.1 2.6 86.8 1.1 87.3 1.1
2
3
4
Average
98.3 99.7 93.0 96.6 86.5 90.4 75.1 84.3 67.6 75.2 61.3 66.9 59.8 62.3 62.7 49.1 67.7 45.0 71.2 37.5
99.7 96.8 100.0 96.6 99.9 94.8 99.7 95.4 99.7 94.0 99.8 92.3 99.7 89.8 99.6 89.7 99.3 87.8 99.2 83.1
97.6 99.0 96.0 96.0 91.3 92.5 90.2 84.8 91.3 77.0 91.3 67.7 91.7 59.9 92.1 52.2 91.8 43.5 92.7 36.7
98.3 98.5 96.3 94.0 93.1 85.4 89.4 76.0 87.1 67.5 85.2 59.6 85.0 54.6 85.6 48.4 86.4 44.3 87.6 39.6
Table 3. The percentage of correctly classified machine parts using a cluttered environment classifier with q = 0.01. Each template is corrupted by normally distributed noise with variance a 2. The conventional scheme is trained on templates corrupted by normally distributed noise with variance = 1 Template 2
Model
1
2
3
4
Average
1.0
Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz Signal + noise Dubois and Glanz
89.1 97.5 94.7 94.8 97.1 91.5 98.5 85.7 98.4 80.3 97.3 69.8 98.0 62.0 98.4 53.7 97.9 45.9 98.4 37.1
100.0 99.7 99.7 97.5 99.2 94.9 98.4 90.7 99.7 85.3 99.7 77.7 99.3 75.5 99.5 66.7 99.4 60.6 99.7 58.8
98.9 96.8 97.2 98.6 94.2 94.4 94.2 90.1 94.4 86.4 95.6 76.2 94.8 71.2 97.5 64.6 95.3 56.0 95.7 53.2
99.2 99.3 94.1 96.7 89.9 91.9 80.5 84.0 76.0 75.8 77.8 67.2 74.6 56.9 73.8 56.4 76.1 52.0 73.5 43.0
96.8 98.3 96.4 96.9 95.1 93.2 92.9 87.6 92.1 81.9 92.6 72.7 91.7 66.4 92.3 60.3 92.2 53.6 91.8 48.0
2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
or the relative effect ofdiscretization errors. In Tables 2 and 3 we examine the effect of incorrectly specifying the noise variance a~2 on the performance of each scheme. Imaging noise is represented by a sequence of indep e n d e n t normally distributed r a n d o m variables with zero mean and variance tr.. 2 The conventional autoregressive classifier is trained on 1000 noisy templates
with tr.2 1.0. N o t e that the variance of the imaging noise is different in the test and training sets when a 2 > 1.0. The results presented in this section are based on a cluttered e n v i r o n m e n t classification rule with q = 0.01. We see from Tables 2 and 3 that the use of a "signal + noise" formulation substantially improves the performance of an auto-regressive classifier when there are =
782
R.H. GLENDINNING
large changes in the variance of the imaging noise a 2. The conventional approach may give better results when the true value of tr 2 is known. This is due to the fact that tr2 is estimated in our approach and is therefore subject to random errors and bias. The results for small tr2 are generally untypical of the "signal + noise" approach as they are influenced by our treatment of estimates of tr 2 lying near zero which are allocated to %2 = 1.0. Significant improvements were obtained using the synthetic templates used in Glendinning. ") In the later experiment we used 128 angles for the "signal + noise" approach and 32 for the conventional classifier. 4.3. Robustness Next we examine the robustness of the "signal + noise" approach to changes in the distributional form of the imaging noise. In Tables 4 and 5 we consider the case where the "signal + noise" classifier is trained on templates corrupted by independent normally distributed noise with variance in the range 1-15. These are used to classify templates corrupted by
Table 4. The percentage of correctly classified aircraft using a cluttered environment classifier with q =0.01 and the "signal + noise" formulation. Imaging noise is drawn from the density f(x)= (1 -p)~b(0, 0.2)+ p~b(0,60.2), with p = 0.05 and variance 0.2 Template 0"72
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
Average
1
2
3
4
p = 0.05
p = 0.0
97.7 96.9 95.1 92.7 90.6 90.2 87.3 88.6 85.1 85.3
98.3 93.1 85.1 74.1 68.0 62.2 61.7 62.7 67.2 69.1
99.8 99.9 99.9 99.5 99.8 99.6 99.7 99.5 99.3 99.4
97.4 96.3 91.8 90.5 90.3 91.2 91.3 91.7 92.3 92.1
98.3 96.6 93.0 89.2 87.2 85.8 85.0 85.6 86.0 86.5
98.3 96.3 93.1 89.4 87.1 85.2 85.0 85.6 86.4 87.6
Table 5. The percentage of correctly classified machine parts using a cluttered environment classifier with q = 0.01 and the "signal + noise" formulation. Imaging noise is drawn from the density with variance 0.~2 and p = 0.05 given by f(x) = (1 -- p)t~ (0, 0"2) + pq~ (0, 60. 2 )
Template 0.2 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
1
2
3
88.1 100.0 98.8 92.9 9 9 . 8 97.6 96.8 9 9 . 3 94.7 98.1 98.9 94.3 98.6 9 9 . 3 94.2 97.1 99.7 95.3 97.9 9 9 . 3 94.9 98.6 9 9 . 5 97.2 98.5 9 9 . 5 95.4 98.0 99.6 95.9
Average 4
p = 0.05
p = 0.0
99.4 94.6 89.7 80.8 76.8 76.8 74.4 73.3 76.5 74.7
96.6 96.2 95.1 93.0 92.2 92.2 91.6 92.1 92.5 92.2
96.8 96.4 95.1 92.9 92.1 92.6 91.7 92.3 92.2 91.8
non-normal noise from the following distribution: "}
f(x) = (1 - p) ~b(0, tr 2) + p~b(0, ka 2), k = 6, p = 0.05 (17) where ~b(/~,~r2) is the normal density with mean # and variance a 2. This density is symmetric about zero and has "fatter" tails than a normal distribution with matched mean and variance. In our experiments we use values of a~ in the range 1-10. In each case a2 =
~ ( 1 + (k - 1)p)- 1. F r o m Tables 4 and 5 we see that the "signal + noise" approach is relatively insensitive to the distributional shape of the imaging noise. This is consistent with the behaviour of the conventional approach described in Glendinning.I ~) 5. S U M M A R Y
In this study we compared the performance of a "signal + noise" formulation with an auto-regressive shape classifier. ~6) This work was motivated by the study presented in Glendinning ") which describes the limitations of the conventional approach in environments with time varying noise characteristics. We have shown that a "signal + noise" formulation gives a substantial improvement in performance when there are large variations in the variance of the imaging noise tr2. The conventional approach may be preferable when the true value of tr 2 is known. Both approaches are equally insensitive to changes in the distributional form of the noise. F r o m Table 1 we see that the use of an over-determined set of H O Y W equations leads to a significant reduction in the variance of the parameter estimates, although the "signal + noise" formulation is generally more sensitive to model inadequacies which may result in biased estimates. F o r example, the approximate bias in our estimates of tr~2 are about (0.9,2.3, - 0 . 9 , - 5 . 9 ) for the machine parts and (1.2, - 6 . 5 , 1.5, - 2 . 4 ) for the aircraft. This clearly limits the applicability of this approach. N o difficulties were encountered in the numerical solution of the over-determined Y W equations, although their use generally resulted in an additional computational burden. For the aircraft templates we estimated an increase of 37% over the conventional approach when tr~2 = 1.0. We have seen that this approach is relatively insensitive to variations in the distributional shape of the imaging noise. When this is not the case, robust estimators "a'15) may be used, although they require extensive optimization in this case. Computationally less demanding estimates may be obtained by replacing the covariances in the H O Y W equations by an appropriate robust estimator, t43) The problem of object classification in time varying environments is often encountered in the study of radar signals. In this case we estimate the structure of the noise separately using background clutter344) Other approaches to this problem include suppressing
Signal plus noise models in shape classification
noise at the pre-processing stage by spatial s m o o t h i n g or direct filtering of the centroid profile using a n app r o p r i a t e filter. Similar ideas are used to e n h a n c e noisy speech in n a t u r a l environments, see Paliwal t451 where a "signal + noise" f o r m u l a t i o n is used to construct a noise suppressing filter. Implicit noise suppression is also carried o u t by curve fitting procedures t46) or those based o n t r u n c a t e d Fourier series expansions. Procedures for dealing with variations in a p p a r e n t size by "averaging" feature statistics over a range of sizes can also be used in this contex, tll a l t h o u g h they lead to an increase in variance. The "signal + noise" a p p r o a c h can be applied to o t h e r models including bivariate auto-regressions, 14~ a l t h o u g h its extension to non-linear models 116) is less straightforward. O u r a p p r o a c h can be modified to deal with coloured noise.
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About the Author--RICHARD H. GLENDINNING received a B.Sc. from Salford University in mathematics in 1975 and a M.Sc. and Ph.D. in statistics from Newcastle and London Universities in 1976 and 1986, respectively. He has held Post-doctoral positions at the Universities at Bath and Newcastle and was a lecturer in Statistics at University College Cork, Ireland. His research interests include time series analysis, image analysis, random sets, stochastic processes, central limit theorems and transport in heterogeneous media. He has published a number of papers in his specialities.