Stan Z. Li, Han Wang, William Y.C. Soh. Fig. 1. Part of a ... plane will not have any e ect on the motion. It holds as an ...... H. Wang and M. Brady. \Corner ...
Journal of Mathematical Imaging and Vision, 8(2), March, 181{192 (1998)
c 1998 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.
Robust Estimation of Rotation Angles from Image Sequences Using the Annealing M Estimator STAN Z. LI
HAN WANG
WILLIAM Y.C. SOH
School of Electrical and Electronic Engineering
Nanyang Technological University, Singapore 639798
Received . Revised .
Abstract.
A robust method is presented for computing rotation angles of image sequences from a set of corresponding points containing outliers. Assuming known rotation axis, a least squares (LS) solution can be derived to compute the rotation angle from a clean data set of point correspondences. Since clean data is not guaranteed, we introduce a robust solution, based on the M-estimator, to deal with outliers. Then we present an enhanced robust algorithm, called the annealing M-estimator (AM-estimator), for reliable robust estimation. The AM-estimator has several attractive advantages over the traditional Mestimator: By de nition, the AM-estimator involves neither scale estimator nor free parameters and hence avoids instabilities therein. Algorithmically, it uses a deterministic annealing technique to approximate the global solution regardless of the initialization. Experimental results are presented to compare the performance of the LS, M- and AM-estimators for the angle estimation. Experiments show that in the presence of outliers, the M-estimator outperforms the LS estimator and the AM-estimator outperforms the M-estimator.
Keywords: Deterministic annealing, global optimization, M-estimator, motion analysis, robust statistics.
1. Introduction A problem in computer vision is to recover the motion from two dimensional projections. A rigid motion can be decomposed into two parts, the translation and the rotation. Of the two, the rotational motion is relatively dicult to compute. Ambiguity exists when the interpretation of the rotation axis is deep (far away) or shallow (close). This is called the bas-relief ambiguity [13]. Previously, we have shown that rotation angles can be computed uniquely from a set of two-view corresponding points, assuming that the direction of the rotation axis is known [32]. When more pairs are available, a least squares (LS) solution can also be obtained. The LS solution assumes that correspondences between frames are known and contains no mismatches or outliers. However, in practice, this assumption may be violated, especially under live environments. For example, due to acceleration
and deceleration, turning and occlusion, the measurements can change drastically. False matches, i.e. outliers, can occur. The LS estimate can get arbitrarily wrong when outliers are used in the estimation. In this paper, we present a robust method for the estimation of rotation angles in the presence of outliers. As the rst step, we de ne the angle estimate based on the robust M-estimator. The optimal estimate minimizes a cost function in which outliers are excluded from the calculation of the angle estimate. However, the M-estimators have the following problems: (i) it is not robust to the initial estimate; (ii) the de nition of the Mestimator involves some scale factors, such as the median of absolute deviation (MAD), the scale factors have to be estimated, and the estimation of the scale factors is itself a dicult problem. As the second step towards the robust estimation of rotation angles, we use an improved robust Mestimator referred to as the annealing M-estimator
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Fig. 1. Part of a sequence of images rotating at 10 degrees between adjacent frames. The image size is 256256.
(AM-estimator) [20], [21] to overcome the above problems. The AM-estimator has two parts: (i) the definition of the AM-estimator and (ii) the computational algorithm. The de nition involves neither scale estimate nor free parameter, avoiding problems with the instabilities therein. Therefore, order statistics, such as the median in the M-estimator, are not required and hence the job of sorting is not needed either. In order to approximate the global minimum, the algorithm incorporates a deterministic annealing schedule into the estimation process in a way similar to the graduated non-convexity (GNC) algorithm [4]. The functions used in the AM-estimator are continuous as opposed to piecewise continuous functions in the M-estimator. When the robust angle estimate is tracked from high to low temperatures during the annealing process, the convergence problem often encountered in the M-estimator is minimized, and so is the in uence of initialization on the nal estimate. Experiments are presented to compare LS, Mand AM- estimators in the estimation of rotation angles from image sequences. We show that the M-estimator performs better than the LS estimator, and that the AM-estimator further improves
the M-estimator to a signi cant extent. The AMestimator is quite stable and has an elegant behavior with respect to the percentage of outliers and noise variance, in contrast to the M-estimator. The paper is organized as follows: The least squares solution is derived in Section 2. In Section 3, a robust solution is de ned using the Mestimator; and the AM-estimator is presented. Experimental results are shown in Section 4, followed by conclusions in Section 5.
2. Least Squares Solutions Consider an image sequence as in Fig.1 in which the car is rotated on a turn-table. Assume that corner points are detected as in Fig.2 and their correspondence is found between frames. The problem concerned in this paper is to compute the rotation angle between consecutive frames, using corresponding points in 2D. Harris [11] has presented an algorithm that uses the prior knowledge of known rotation axis. The argument is that under many circumstances, the rotation axis is obviously provided. For example, the car manoeuvring on a horizontal road can be regarded as rotation about a vertical axis. His algorithm gave dual solutions and requires an addi-
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Fig. 2. Corners detected. Due to the parallel projection assumption, all the corner coordinates are image coordinates with aspect ratio 1:1. Therefore the camera calibration is not required.
tional phase to determine which one is the wanted solution. Wang and Li [32] show that with the assumption of known rotation axis and orthographic 3D2D projection, the unique solution exists from only one pair of correspondence and the LS solution can be obtained using more pairs. Further detail can be found in [30] together with the error analysis that proves the method is statistically unbiased. The 2-view algorithm reported here can also be extended into 3-views where the prior knowledge of the rotation axis is not required. Assuming parallel projection, let Pi = (Xi ; Yi ; Zi ) denote a point on the object and this point is projected onto the image plane at pi = (xi ; yi ), hence (xi ; yi ) = (Xi ; Yi ). This implies that the distance (Zi ) from the object to the image plane will not have any e ect on the motion. It holds as an approximation for the full perspective and thereby turn it into a weak perspective. Further, we denote the motion by a rotation, speci ed by R (orthogonal matrix, 3 3, followed by a translation, speci ed by t = (tx ; ty ; tz )> , hence,
0 X0 1 0 r @ Yii0 A = @ r2111 Zi0 | r31
10
1 0 1
Xi tx r12 r13 r22 r23 A @ Yi A + @ ty A Zi tz r32 r33
{z R
}
(1)
where (Xi0 ; Yi0 ; Zi0 )> is the corresponding point of (Xi ; Yi ; Zi )> after motion. This equation contains three linear equations of which the rst two are used since the depth is not recoverable. Replacing (Xi0 ; Yi0 )> and (Xi ; Yi )> with (x0i ; yi0 )> and (yi ; yi )> , we have x0 = r x + r y + r Z + t 11 i 12 i 13 i x i yi0 = r21 xi + r22 yi + r23 Zi + ty : (2) The translational components tx and ty can be dropped if we do the following manipulation: nd a point (x1 ; y1 ) on the object, and set the coordinate origin to this point; after the motion, this point becomes (x01 ; y10 ) and we move it to the origin again. Making use of the identities of orthogonal matrix: r11 r23 ? r21 r13 = ?r32 (3) r12 r23 ? r22 r13 = r31 ; we now have
r23 x0 ? r13 y0 + r32 x ? r31 y = 0:
(4)
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This equation shows that under orthographic projection, there is an in nite number of solutions from 2 views regardless the number of correspondences (Huang and Lee [13]). Introducing the constraint of known axis, we show that a unique solution exists by using only one pair of correspondence. Let ` = (`x; `y ; `z )> denote the unit vector of the rotation axis, the rotation matrix R of angle around ` can be represented by 0 C + `2 V ` ` V ? ` S ` ` V + ` S 1 xy z xz y x R = @ `y `x V + `z S C + `2y V `y `z V ? `xS A `z `xV ? `y S `z `y V + `x S C + `2z V (5) where C = cos ; S = sin , and V = 1 ? cos (see [16] for details). Substitute the elements from equation (5) into equation (4) and rearrange it, we obtain,
a(1 ? cos ) + b sin = 0; where
to
(6)
a = `z (`y (x + x0 ) ? `x (y + y0 )) b = `x(x ? x0 ) + `y (y ? y0 ):
(7) Because tan 2 = 1?sincos , equation (6) evolves (a tan 2 + b) sin = 0:
(8)
In equation (8), sin 6= 0 because the assumption of known axis e ectively implies that 6= 0. Hence,
= 2 arctan( ?ab ):
(9)
The above equation establishes the relation of the rotation angle from two views using only one correspondence with the a priori knowledge of rotation axis. There are singular points which might violate the uniqueness, i.e. when a = 0. This violation is caused by the symmetric projection of the corresponding pairs: (1) when `z = 0, that is equivalent to when the rotation axis is frontal parallel to
the viewer. Hence all the rotation appears to be translation; (2) when `y (x + x0 ) ? `x(y + y0 ) = 0, this is equivalent to y + y0 = `y x + x0 `x which means that the rotation causes the matching points cross the rotation axis in exactly symmetric manner. This singular point can be eliminated when there is more than one matching pairs. When noise is present, equation (8) will not hold exactly. Hence an estimation method is needed. A simple formulation is the least squares by which the following is minimized
E () =
m X i=1
fAi tan(=2) + Bi g2
(10)
where m is the number of observations (corresponding pairs). The optimal solution is obtained by di erentiating E () and setting @E@() = 0, resulting in
= 2 arctan(? where
PA B P Ai 2 i ) i
(11)
Ai = `z (`y (xi + x0i ) ? `x (yi + yi0 )) (12) Bi = `x(xi ? x0i ) + `y (yi ? yi0 ):
This approach gives a unique solution which is statistically unbiased; the error is in the range of 10?7 to 10?6 in units of degrees [30].
3. Robust Solutions
The analysis and derivation made in the previous section is based on an assumption that all the pairs f(xi ; yi ); (x0i ; yi0 )g are correct correspondence. In this section, we present a robust solution to the angle estimation problem when there are false matches, i.e. outliers, in the set of correspondences. Robust methods [29], [14], [25] are tools for statistical regression problems with outliers. They have been applied in image processing and vision problems in recent years. Kashyap and Eom [17] developed a robust algorithm for estimating parameters in an autoregressive image model where the noise is assumed to be a mixture of a Gaussian
Robust Estimation of Rotation Angles from Image Sequences Using the Annealing M Estimator and an outlier process. Shulman and Herve [26] proposed to use Huber's robust M-estimator [14] to compute optical ow involving discontinuities. Stevenson and Delp [28] used the same estimator for curve tting. Besl et al. [1] proposed a robust M window operator to prevent smoothing across discontinuities. Haralick et al. [10], Kumar and Hanson [19] and Zhuang et al. [34] used robust estimators to nd pose parameters. Jolion and Meer [15] identi ed clusters in feature space based on the robust minimum volume ellipsoid estimator. More recently, Boyer et al. [5] presented a procedure for surface parameterization using a robust M estimator. Other recent works in this area can be found in the Proceedings of International Conference on Robust Computer Vision [9], [6]. Black et al. [3], [2] applied a robust operator not only to the smoothness term but also to the data term. Li showed close relationships between robust Mestimators and discontinuity-adaptive MRF priors [20], [21]. 3.1. M-Estimator
The essential form of the M-estimation problem is the following: Given a set of m data samples r = fri j 1 i mg where ri = f + i , the problem is to estimate the location parameter f . The distribution of noise i is not assumed to be known exactly. The only underlying assumption is that 1 ; : : : ; m obey a symmetric, independent, identical distribution (symmetric i.i.d.). A robust estimator has to deal with departures from this assumption. Let the residual errors be
i = ri ? f
(13)
and each error is penalized by a potential function g(i ). The M-estimate f is de ned as the minimum of a global energy function
f = arg min E (f ) f where
E (f ) =
X i
g(ri ? f )
(14)
(15)
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To minimize the above, it is necessary to solve the following equation X @g(ri ? f ) =0 (16) i
@f
for the minimum f = f . When g(i ) is only a function of i2 , as always taken in this paper, its rst derivative has the following form
g0 (i ) = 2i h(i ) (17) where h() is an even function. In this case, the estimate f can be expressed as the following weighted sum of the data samples P f = Pi hh((i) )ri i i
(18)
where i = ri ? f are the residual errors, and h, referred to as the interaction function, acts as the weights. The above represents a xed point equation. The choice of the interaction function is important. Ideally, the function h should provide adaptive interaction; data points with larger errors have smaller weights and those with in nitely large errors have the zero weight. In robust Mestimation, the h takes a truncated shape. For example, Tukey's biweight function [29] is de ned as
( ? i 2 2 if ji j < cS h(i ) = 1 ? cS 0
otherwise
(19)
where S is an estimate of spread, c is a constant parameter and cS is the scale estimate. Possible choices of c and S are S = medianfi g with c set to 6 or 9; or S = medianfji ? medianfi gjg (median of absolute deviation (MAD)) with c = 1:4826 chosen for the best consistency with the Gaussian distribution. The M-estimator requires some scale estimates parameters and its performance relies crucially on them [29], [14], [25]. Classical scale estimates such as the median and MAD are not robust themselves. How to make scale estimates robust is an important topic in robust statistics (see also
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a recent review [23]). The AM-estimator is partly aimed to avoid instability therein. Now turn to the rotation angle estimation problem. Denote the set of m corresponding point pairs by r = f(pi ; p0i ) j i = 1; : : : ; mg (20) where pi = (xi ; yi ); p0i = (x0i ; yi0 ) are point pairs between two images. In rotation angle estimation, the errors are represented by i = Ai tan(f=2) + Bi . The M-estimator minimizes, instead of (10), the following
E (f ) =
m X i=1
g(Ai tan(f=2) + Bi )
(21)
By setting ddEf = 0 and using g0 () = 2h(), one obtains m X i=1
rithms can get stuck at an unfavorable local solution. The M-estimator has the theoretical breakpoint of K1+1 where K is the number of unknown parameters to be estimated; but in practice, the breakpoint can be well below this value because of the problem of local minima. Secondly, the de nition of the M-estimator involves some scale factors, such as the median of absolute deviation (MAD), which must be estimated. The robustness of such estimates is questionable and deserves a devoted study. So far, they have been sources of sensitivity and instability. The work by Zhang et al. [34] is one of few works in vision aimed to enhance robust estimation. They adopt a di erent approach by modeling contaminated Gaussian noise. The resulting algorithm contains a few heuristically determined thresholds and is dependent of initialization. Here we present another treatment.
f[Ai tan(f=2) + Bi ] Ai h(Ai tan(f=2) + Bi )g = 0 3.2. The AM-Estimator
(22) Rearranging the above gives the following xedpoint equation
Pm h A B i i i f = 2 arctan(? Pi=1 m h A2 ) i=1 i i
(23)
where hi = h(Ai tan(f=2)+ Bi ). The M-estimator produces more reliable estimates than the LS estimator, as will be shown in the section of experiments. Existing M-estimators have the following problems which a ect its performance: Firstly, it is not robust to the initial estimate. This is a problem common to nonlinear regression procedures [24]. The convergence of the algorithm depends on the initialization. Even if the problem of convergence is avoided, the need for good initial estimate cannot be ignored for convergence to the global estimate. It is also pointed out by vision researchers [10], [23], [34], [27] that the choice of the initial estimate has signi cant in uence on the quality of the M-estimate. This is because most M-estimators are de ned as the global minimum of a generally non-convex energy function and hence the commonly used gradient based algo-
We present an improved robust M-estimator referred to as the annealing M-estimator (AMestimator) to overcome the above problems. The AM-estimator takes the same form as the Mestimator in (18). It di ers from the M-estimator in the following way: (i) The scale estimate in the h() is replaced by a parameter > 0. Now, the energy function depends on the parameter, E (f ) = E (f ), and so does its minimum P h ( ) r f = arg min E (f ) = Pi h (i ) i (24) f i i f is reduced to the least squares solution when h () constant for all . (ii) Given an initial f , the nal solution is obtained by tracking f for from a large value to a very small one (! 0+), f = lim f (25)
!0+
Mathematically, it is required that g () or h () is chosen so that for a large enough value, g () is convex on an interval and the global minimum of E (f ) can be found easily. Computationally, is initially set to a high enough value. Once a locally minimal solution
Robust Estimation of Rotation Angles from Image Sequences Using the Annealing M Estimator (which is the global one for large values) is found at current , is decreased toward 0+ . This is the annealing process. In this way, a sequence of solutions ff g is generated for the decreasing and f is the last one in the sequence. The convex approximation and the annealing process together constitute a GNC [4] like algorithm. Given the form of the AM-estimator (25), it is the weighting function h and the annealing schedule that determine the nal AM-estimate. Corresponding to Tukey's biweight function, we have the following
8 22 < i if ji j < h (i ) = : 1 ? 0 otherwise
(26)
Given a xed interval B = [min; max ], one can always nd a suciently large such that g 00 () is positive de nite and g () is strictly convex for any 2 B . This is a characteristic we want to use. The de nitions of h and g are not limited to the ones given by the existing M-estimators. We have the following general de nitions for these functions.
De nition 1. An adaptive interaction function (AIF) h () is one which satis es the following properties (i) h () is at least C 0 continuous and (possibly piecewise) di erentiable in , (ii) h () = h (?), (iii) h () > 0, (iv) h0 () < 0 (8 > 0), (v) lim!1 jh ()j = C < 1, and (vi) h () = h1 (=p ). The class of AIFs is de ned as the collection of all such h and is denoted by IH . 2 Tukey's biweight function (26) is an example of AIF; see Table 1 for more examples. The continuity of (i) means that the interaction varies continuously with respect to . The C 0 continuity in is required because the presence of discontinuities can cause the global minimum of E (f ) not to vary continuously with respect to the data { a potential instability [22]. The evenness of (ii) makes the interaction relative to the magnitude
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of the di erence i = ri ? f only, regardless of its sign. The positive de niteness of (iii) keeps the interaction positive. The monotony of (iv) leads to decreasing interaction as the error magnitude increases. In (v), C 0 is a constant whose value is the asymptote of jh ()j. To satisfy this property, it is necessary that lim!1 h () = 0. This is essential for the adaptation to discontinuities and robustness to outliers; it suppresses the interaction for data points having in nitely large . When these conditions are satis ed, data items ri for which jri ? f j are large (relative to ) will be weighted by small h (ri ? f ) values. This makes ri with larger jri ? f j less in uential and is why h is called an adaptive interaction function. The property of (vi) de nes the scaling e ect of the parameter . From (vi) and (i), it follows that h () is continuous also in . The above characterizes the properties that an AIF h should possess rather than instantiates some particular forms; therefore the de nition is rather broad and o ers many suitable choices. The de nition of the AIF has more implications than the robust estimation: It also describes how neighboring pixels of an image should interact for discontinuity adaptive MRF prior potentials [22]. In fact, underlying mechanisms in adaptive MRF prior models (such as line process [8] and mean eld approximation [7])) and robust M-estimation are the same [21].
De nition 2. The adaptive potential function (APF) corresponding to an h 2 IH is de ned R by g () = 2 0 0 h (0 )d0 . 2 Basically, g () is at least C 1 continuous; it is even: g () = g (?); its derivative function is odd g 0 () = ?g 0 (?). However, it is not necessary for g (1) to be bounded. Furthermore, g is strictly monotonically increasing as the error jj increases because g () = g (jj) and g 0 () = h () > 0 for > 0. This means larger jj leads to larger potential (penalty) g (). It conforms to the original spirit of the quadratic potential function gq () = 2 . Most existing error potential functions in M-estimation do not have such a property: the potential there does not increase as the error jj increases beyond certain value.
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Stan Z. Li, Han Wang, William Y.C. Soh Table 1. Three possible choices of h (), the corresponding g () and the bands.
AIF
APF
h1 () = exp(?2 = )
g1 () = ? exp(?2 = )
h2 () = 1=[1 + 2 = ]2
= ? =[1 + 2 = ] g3 () = log(1 + 2 = )
h3 () = 1=[1 + 2 = ]
g2 ()
It is usually not necessary to de ne the potential function g for the purpose of robust estimation. What is the most important for the AMestimator, and M-estimator in general, is the AIF h . Nonetheless, knowing g is helpful for use to study the convexity property of the corresponding AM-estimator. For a given g (), there exists a region of within which the function is convex: B = [bL ; bH ] = f j g 00 () 0g (27) We refer to this region B as the band. The lower and upper bounds bL; bH correspond to the two extrema of g 0 (), which can be obtained by letting g 00 () = 0, and we have bl = ?bH when g is even. When bL < < bH , g 00 () > 0 and thus g () is strictly convex. For example, supposing we are using h3 () = 1=[1 + 2 = ] for which
Band
B1 B2
p =2; p =2) p p = (? =3; =3)
= (?
B3
= (?p ; p )
B3 = (?p ; p ), we can always choose a value high enough so that ?p < i = ri ? f < ?p for all ri and f . When this is satis ed, E (f ) is
convex because the sum of convex functions is a convex function. Table 1 instantiates three possible choices of AIFs, the corresponding APFs and the bands. Fig.3 shows the qualitative shapes of the three APFs g () and their derivative functions g 0 () = 2h (). The nal estimates f with di erent h shapes vary but insigni cantly. If the continuity of h () is relaxed to be piecewise, then our de nition also includes the priors used by Geman and Geman [8] and Blake and Zisserman [4]. Although piecewise h () can also do the job, they present some theoretical instability [22] as mentioned earlier.
Fig. 3. The qualitative shapes of the three APFs (bottom) and and their derivative functions (top).
Robust Estimation of Rotation Angles from Image Sequences Using the Annealing M Estimator 1. The AM-Estimator 2. set t = 1, f (0) = f LS ; choose initial ; 3. do f 4. t t + 1; 5. compute errors i = ri ? f (t?1)P ; 8i; ( t ) 6. compute weighted sum f = Pi hh
((i )ir)i ;
7. if (jf (t) ? f (t?1) j < ) 8.
lower( ); 9. g until ( < ) 10. f f (t); 11. End pseudo-code.
Fig. 4
i
The AM-estimation algorithm.
The AM-estimator has been de ned mathematically. It is completed with a computational algorithm. When g () is non-convex, as can be with small , the direct method using the xed point iteration
P h (r ? f (t)) r Pi h (ir ? f (t)) i i i
(28)
can get stuck at a local minimum because the above equation is derived based on gradient descent. This problem is particularly serious for small . To solve this problem, we incorporate the GNC technique [4] into the solution nding process. GNC is a deterministic algorithm for approximating the global optimum of a non-convex function. It uses the continuation strategy like [12], [18], [33]; but it approximates the energy function to be minimized from a convex condition. It can signi cantly improve the quality of the estimate. In the AM-estimator, the continuation is performed by gradually decreasing the parameter value during the iteration. Initially, the parameter is set to a suciently large value (0) such that the APF g () is strictly convex. With such (0), it is easy to nd the unique minimum of the global error function E (f ) using the gradient descent method, regardless of the initial value f . The minmum is then used as the initial value for the next phase of (0)
/* converged */ /* frozen */
3.3. Annealing Procedure
f (t+1) =
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minimization under a lower to obtain the next minimum. As is lowered, g () may no longer be convex and local minima may appear. However, if we track the global minima from high to
! 0+ , we can approximate the global minimum f under ! 0+ . The AM-estimator algorithm is summarized in Fig.4. Obviously, whatever the initialization, the rst iteration always gives a value equal to the LS estimate m X 1 LS f = m ri i=1
(29)
This is because when ! +1, all weights are the same as h (i ) = 1; the strict convexity is guaranteed in this situation. The initial is chosen to satisfy the following
ji j = jri ? f LS j < bH ( )
(30)
where bH ( ) (= ?bL( )) is the upper bound of the band in (27). This guarantees g 00 (i ) > 0 and hence the strict convexity of g . The parameter is lowered according to a schedule in the function lower( ). The parameters and in the convergence conditions are some small numbers. An alternative way is to decrease according to a xed schedule regardless of whether f (t) is converged at current , which is equivalent to setting a big value for , and let the algorithm be frozen after dozens of iterations. This quick annealing is used in the experiment.
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4. Experimental Results The following experiments compare the performances of the LS estimator, the M-estimator and the AM-estimator, using the image sequence data shown in Fig.1. Corners are detected from these images using Wang-Brady detector [31] and was shown in Fig.2. True corner correspondences in the sequence are established. After that a certain percentage of random correspondences are added to simulated outliers. The input to the algorithm consists of the true correspondences and the outlier ones. For the AM-estimator, the AIF is chosen to be h () = h3 () = 1:0=(1 + 2= ). The schedule in lower( ) is: 1:5100=t ? 1 where t is the iteration number starting from 1. Two hundred iterations are performed and so the nal temperature is 1:5100=200 ? 1. For the M-estimator, Tukey's biweight function in Equ.(19) is used with c = 6:0 and S being the median. Figures 5 and 6 compare between the LS and the M- estimators and between the M- and the AM-estimators, respectively. In these two gures, the abscissa represents frame number and the ordinate rotation angle. The solid line represents the mean angle and the vertical dotted lines indicates the standard deviation computed for each frame. Large deviations indicate breakdowns due to outliers. From Fig.5 we see that the LS estimator has broken down while the M-estimator still works quite well. In fact, the breakdown point of the LS estimator is less than 5%. Fig.6 illustrates that the M- and AM-estimators are comparable when 2
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the data contains less than 20% of outliers. Above this percentage, the AM-estimator demonstrates its enhanced stability: The AM-estimator continues to work well when the M-estimate is broken by outliers and has only one breakdown at 60% of outliers. This illustrates that the AM-estimator has a considerably higher actual breakpoint than the M-estimator. The comparison of the experimental results are made by using exactly the same data for the compared methods. Further, each statistic is made based on 1000 random tests and the data set. Therefore, the results should be suciently reliable.
5. Conclusion The existing M-estimator performs better than the LS estimator. The AM-estimator further improves the M-estimator to a signi cant extent. It nds a good approximation to the global solution and has additional advantages of getting rid of scale estimates and being insensitive to parameter selection. Divergence is minimal in the AM-estimator because the initial estimate for the current value is the convergence point obtained with the previous value.
References 1. P. J. Besl, J. B. Birch, and L. T. Watson. \Robust window operators". In Proceedings of Second International Conference on Computer Vision, pages 591{ 600, Florida, December 1988.
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Fig. 6. Results computed from correspondence data containing 20% (row 1), 40% (row 2), 50% (row 3) and 60% (row 4) of outliers using the M-estimator (left) and the AM-estimator (right).
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Stan Z. Li, Han Wang, William Y.C. Soh
Biography Stan Z. Li received the B.Eng degree from Hu-
nan University, China, in 1982, M.Eng degree from the National University of Defense Technology, China, in 1985 and Ph.D degree from the University of Surrey, UK, in 1991. All degrees are in EEE. His research interests include computer vision, pattern recognition, image processing and optimization methods. He authored the book Markov Random Field Modeling in Computer Vision (Springer-Verlag, 1995).
Han Wang received his B.Eng in computers and
its applications in 1982 from Northeast Heavy Machinery Institute, China and his Ph.D in Parallel Image Processing in 1990 from Leeds University, UK. He has been a visiting scientist (1988) in Carnegie Mellon University for developing a parallel image processing language for the Warp machine and for Transputers. From 1989 to 1992, he has been a research ocer in the Robotics Research Group and the Computing Laboratory, Oxford University. The research involved in real
time implementation of 3D image processing algorithms for mobile robot navigation. Dr. Wang is currently lecturing in the Nanyang Technological University and his research interests are in object based real-time tracking and determination of hand pose from image sequences that can be used for control of a 5 degree-of-freedom robot head.
William Y.C. Soh received his B.Eng in electri-
cal and electronic engineering from the University of Canterbury, New Zealand, in 1983, and his Ph.D degree in electrical engineering from the University of Newcastle, Australia in 1987. From 1986 to 1987, he was a research assistant in the Department of Electrical and Computer Engineering, University of Newcastle. He joined the Nanyang Technological University, Singapore, in 1987 where he is currently a senior lecturer and the Head of the Control and Instrumentation Division, School of Electrical and Electronic Engineering. His current research interests are in the area of robust system theory, robotics, estimation and ltering, and signal processing. He has published over 70 research articles.
