Detection of partial symmetry using correlation with rotated-reflected images

Pattern Recognition, Vol. 26, No. 8, pp. 1245 1253, 1993 Printed in Great Britain

0031 3203/9356.00+.00 Pergamon Press Ltd © 1993 Pattern Recognition Society

DETECTION OF PARTIAL SYMMETRY USING CORRELATION WITH ROTATED-REFLECTED IMAGES TAKESHI MASUDA, KAZUHIKO YAMAMOTOand HIROMITSU YAMADA Electrotechnical Laboratory, l-1-4 Umezono, Tsukuba, Ibaraki 305, Japan (Received 14 July 1992; in revisedform 19 January 1993; receivedfor publication 8 February 1993) Abstract--Symmetry is one of the important structural properties of a figure both in visual psychology and in computer vision. Symmetrical properties of planar two-dimensional figures are considered and a new method for the extraction of rotational symmetry and reflectional symmetry is presented. A figure is said to have symmetry if it is invariant with some congruent transformation which consists of translation, rotation and reflection. The approach is to obtain the congruent transformations which make the transformed image a good match with the original image. The directional correlation of edge features is used to evaluate how well the original and transformed images match. This method does not require any segmentation nor knowledge of the centroid position, and the symmetrical properties of partially symmetric figures can also be detected. The approach is described and this method is applied to synthetic and real images. Rotational symmetry Reflectional symmetry Glide symmetry matching Correlation Directional pattern matching

I. INTRODUCTION Symmetry is one of the important structural properties of a figure both in visual psychology and in computer vision. In this paper, we consider the extraction of symmetrical properties of a planar two-dimensional figure, and present a new approach based on congruent transformations. We say a figure has symmetry if it is invariant under some congruent transformation, which consists of translation, rotation and reflection. The figure which has rotational symmetry is the same as the original one if it is rotated. The rotation takes place about the center of rotational symmetry. The number of folds of rotational symmetry signifies how many times the rotated figure matches the original one in one cycle. The figure which has reflectional symmetry is unchanged if it is reflected. The reflection takes place about the axis of reflectional symmetry. The composite transformation of reflection and translation is glide. We shall only concentrate on those symmetries given by congruent transformations, which do not include skewed symmetry. Our approach is to obtain the congruent transformations which make the transformed image a good match with the original image. We use the correlation of directional edge features to evaluate how well the original and transformed images match. We apply a congruent transformation and then perform a twodimensional correlation between the transformed image and the original image for all possible amounts of parallel translation. If we obtain a high value for the correlation, then we know that this corresponds to some symmetrical property of the image. By examining the transformation which gives a high correlation, we can determine which symmetrical properties have been detected.

Partial symmetry

Pattern

To extract rotational symmetry, we take the original image and then apply all possible rotations to it and correlate each result with the original image. The transformation is carried out using each pixel as the center of rotation in turn. A good match signifies rotational symmetry. To extract reflectional symmetry, we reflect the original image and then apply all possible rotations and translations and correlate each result with the original image. Again a high degree of matching signifies reflectional symmetry. Symmetry has been investigated by several researchers. A common approach is based upon determining the centroid of the figure. For both reflected and rotated symmetry, the centroid either lies on the axis of symmetry or forms the center of rotational symmetry. Atallah C1)detected the axis of reflectional symmetry by first determining the centroid position and then using a string pattern matching technique, which considers all possible lines passing through the centroid. Marola 12) detected multiple axes of symmetry with respect to the centroid, and also considered non-symmetric images. He determined axes of symmetry under the assumption that the axis of symmetry of an almost symmetrical figure passes near the centroid. Bolles~a)detected rotational and reflectional symmetry using a structural description of the properties of a figure, which again requires the position of the centroid. These global approaches have difficulties in extracting symmetric properties from figures which are not symmetric globally, but contain some symmetric parts. The determination of local symmetry involves detection of symmetry in only a small part of a figure. Blum 14)explored planar figures by fitting circular disks inside the figures and noting the locus of their centers, which he called the sym-ax or symmetric axis transform. Brooks (s) and Brady and Asada 161 studied local sym-

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metry based on a similar approach to that of Blum but using the locus of a midpoint of a symmetric pair of edge points. Ponce 17) showed these symmetry axes do not agree with each other except for special cases. All of these methods define the local symmetry axis as the locus of a point determined by parts of the figure. In practice, for general figures, the symmetry axes are obtained as a set of discretely scattered points. Thus, some postprocessing is necessary to integrate this scattered information. Bigiin~s) defined symmetry in a different manner which is based on the use of harmonic functions, and detected several kinds of symmetry using gray scale images. Many other methods have been proposed such as the use of a point plotting method/91 In some cases, these methods use local features which are delicate to small local variance of the figure. The method described in this paper differs from previous global approaches in that we do not assume that the axis of reflectional symmetry passes through the centroid, nor do we assume that the centroid forms the center of rotational symmetry. Thus, we can detect the symmetric properties of partially symmetric figures. Our method also uses the directional correlation to evaluate the matching between the original and the transformed images. It does not produce scattered points in the usual fashion of the methods that extract local symmetry, but can directly determine the geometry of a symmetric property. At the same time, we can extract not only reflectional symmetry, but also glide symmetry which is derived from glide transformation. In Section 2 we show how to extract rotational symmetry using correlation with a simple example and in Section 3 the reflectional symmetry case is presented. Details of experiments and the results are shown in Section 4.

f(x) and the rotated image f(u) as aim(f; ~, 0) = ~ f ( u ) f ( x + ~)dx.

(3)

Now, we define the directional correlation for the parallel translation ~ = (¢1, ~2) as

aair(f; ~) =

I J g r a d j ( x ) ( g r a d ~ f ( x + ~))~ dx

= ~~[ ~ t f (xl,x2), ~;2f (x',x2) ]

dx 1 dx 2.

•

(4)

L~--~2f(xI + ~I'X2 + ~2)_.J This is the correlation of the inner products of the gradients of the images. When we apply this correlation with the rotated image, we have to transform not only the position but also the gradient. In the Euclidean space, gradients are transformed in the same way as positions. The gradient is then transformed by a rotation as

L ,
151

or

gradu = gradx R(O)~ where R(O)T is the transposed matrix of R(O). We can represent the gradient of the rotated image as

(grad,,f (R (O)x)) R (O)T

(6)

and finally we extend the directional correlation with the rotational image as adir(f; ~, 0) = S~ g r a d u f ( u ) ( g r a d j ( x + ~))T dx

= ~S(gradxf(R(O)x))R(O)T(gradxf(x + ~))Tdx. (7) 2. EXTRACTION OF ROTATIONAL SYMMETRY

2.1. Correlation between an image and its rotated image In this section we describe our definition of the directional correlation, which we use to evaluate the match between the original and the rotated images. We represent the original image as the intensity function f ( x ) = f ( x l , x 2 ) on the image plane whose coordinates are x = (xl, x2). The intensity autocorrelation of the image f(x) for the parallel translation ~ = (~1, ¢2) is defined as aint(f; ~) = ~ f ( x ) f ( x + ~) dx.

(1)

The rotated image about the origin by the angle 0 is represented as f(u)

=f(R(O)x)

(2)

where =[

cos0

R(O) L - s i n 0

sin01 cos0 "

We define the correlation between the original image

We calculate the values of adir(f; ~, 0) for the set of all possible values of the parameters ~ and 0. We take the original image and rotate it by each sampled 0, and the outputs from the rotations are correlated with the original image for possible sampled ~. This produces adir in the same manner as the ordinary correlation. In practice we obtain the gradient images of the smoothed figure by convolving the original image with the gradient of the Gaussian filter. Convolving with the gradient of the Gaussian filter satisfies the equation of the relation between the gradient and the rotated gradient of equation (5). This approach falls within the group of pattern matching methods called directional pattern matching. This approach has a reduced cost of computation and there is some biological support in that there are edge orientation receptive fields in the human visual pathway. This approach is similar to already existing directional pattern matching methods such as the generalized Hough transform of Ballard ~1o) and the M A P method of Yamadatl 1) if we add some kind of non-linear operation such as quantization and saturation into adir.

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These methods have advantages over ours because they reduce the cost of computation, increase the sharpness of the directional properties, and ensure selection of only well-matched parts of the correlation function, etc. However, they require heuristic parameters and the outputs are relatively noisy. Our approach is more theoretical and general than these methods. 2.2. R o t a t i o n a l correlation We have defined the directional correlation adi, between the rotated image f(u) and the original image f(x) in equation (7). However, the rotation considered in adir takes place about the origin. The combination of a translation ~ and a rotation about the origin R(O) can be represented by a rotation about the point to which satisfies the equation = to - RtO)Tto.

(8)

We define the rotational correlation arot(f; to, 0) by representing ad~r(.f;~,0) as the function of to instead of

Fig. 1. Star image. using equation (8). If we let x' = x + ~, we obtain the rotational correlation as arot(f; to, 0) = ~ g r a d x , f ( R ( O ) ( x ' -

to) + to)

R ( O ) T ( g r a d x , f ( x ' ) ) T dx'.

m m mmm mmmm mmm (~) 0 = 0

(b) 0 = 7 r / 1 6

(c) 0 = 2 7 r / 1 6

(d) 0 = 3 7 r / 1 6

(e) O = 4 7 r / 1 6

(f) O = 5 7 r / 1 6

(g) O = 6 7 r / 1 6

(h) O = 7 ~ r / 1 6

(i) 0 = 8zr/16

(j) 0 = 9~-/16 (k) 0 = 10zr/16(1) 0 = 11zr/16

(111) 0 = 12~r/16(n) 8 = 13~r/16(o) 8 = 1 4 7 r / 1 6 ( p ) 8 = 15~r/16 Fig. 2. Result of arot(Star, oJ, 0).

(9)

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Thus, for each to and O,afot(f;to, O) is equal to the rewritten by letting x" = R(0)x instead ofx' in the followcorrelation which makes each position x' correspond ing fashion: to the rotated position about to, R(0)(x' - to) + to. By arot(f; co, 0) = SSgradx.f(x") R( - 0)x using the rotational correlation, we can obtain correlation values invariant with the coordinate system (gradx,,f(R(-O)(x" - to) + to)Tdx". (10) because each point can be the center of the rotation. This is equal to the correlation arot(f; co, - 0 ) , then Though the direct calculation of the rotational corwe can conclude that relation arot is very complicated, we can evaluate this with the combination of two basic operations, first we arot(f; to, 0) = arot(f; CO, -- 0). (11) evaluate adir(f; g, 0) for all possible values of(g, 0), then we transform the parameter space from (g, 0) to (to, 0) using equation (8). Finally we obtain the result stored 2.3. Extraction of rotational symmetry from the correin a three-dimensional array (to1, to2, 0). lation result Let us consider an example. Figure 1 shows a binary If a figure has rotational symmetry about a point, image whose size is 128 x 128 containing a star-shaped the rotational correlation of the point peaks periodifigure with 5-fold rotational symmetry. Applying equacally as the rotation angle varies. We can extract the tion (7) to the shape of Fig. 1, then transforming it by equation (8), we obtain the correlation result of Fig. 2, number of folds of rotational symmetry by investigating arot(star;to, 0), where light pixels indicate strong cor- the periodicity of the rotational correlation. In the relation, and dark pixels indicate negative correlation. following, for simplicity, we shall represent arot(f; co, 0) The zero value is indicated as gray. The local maximum as arot(0) unless otherwise stated. To evaluate the n-fold rotational symmetrical proppoint in Fig. 2(g) shows that we obtain an image that is a good match to the original image if we rotate the erty (n _> 2), we use original image by 6n/16 (~2n/5). There is another p(f;to, n)= 1 ,--1 { 2nk'~ n_ik~ arot~f;to,~) (12) local maximum in Fig. 2(n) near 0 = 4n/5. Note that the correlation value for 0 = 0 is constant for all over co. This signifies that nothing changes as which resamples the correlation function at the expeclong as no rotation takes place about each position. ted angles which correspond to n-fold symmetry. Our This correlation value is the maximum value of arot. definition ofp(n) does not include a,ot(0), because arot(0) We show the exact value of arot with normalization by is constant at all image points. When n > IN/2], where N is the number of angular sampling, we represent p(n) this maximum value in the following fashion. The point g has a circular locus of movement which as p(~). The value o f p ( ~ ) is the mean of the rotational passes through the origin and whose center is at to. As correlation of all sampled angles. Thus p(oo) returns a 0 changes through angles of 0, n, 2n, so (~1, ~2) moves high value at the center of circular figures. We term from (0, 0) via (2~o1, ~o2) back to (0, 0), respectively. If this p(n) the spectrum of rotational symmetry, and we we introduce 0 as a third coordinate axis, then (~1, ~2, 0) can obtain p(n) at each pixel by applying equation (12) sweeps out the surface of a cylinder. Note that the to the correlation results. We find the local maximum three-dimensional spiral swept out by (~1, ~2, 0) as 0 of the spectrum of rotational symmetry of each fold. changes from 0 to 2n simply is mapped to a straight From this we determine where the centers are located, how many folds there are, and how reliable is the line (co1, 092, 0) (see Fig. 3). The rotational correlation of equation (9) can be estimation of the rotational symmetry. Each fold of the rotational symmetry of the original image gives a local maximum in the spectrum of rotational symmetry. However, for the folds which are 0 multiples and divisors of the exact folds, the spectrum of rotational symmetry may form local maxima. The spectra for folds of higher multiples are lower than the spectrum for the exact fold because they evaluate the means including poor correlation values for extra folds. The spectra for folds of lower divisors have similar values to that for the exact fold, for example, for a 6-fold rotational symmetry we get a strong local (c01,c0z,0) maxima not only at the 6-fold but also at the 2-fold or the 3-fold. However, this is reasonable because a 6-fold rotationally symmetric figure is also a 2-fold or a 3-fold rotationally symmetric figure. It is usual to employ the Fourier series expansion to investigate the periodicity of a function defined within Fig. 3. Locus of (¢1,~z,0) mapped to constant (o91,o92) for a finite interval. However, each part of the rotational O
\

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The point is located at the same position as the center of the star shape and so our method has extract 5-fold rotational symmetry of the star shape. None of the other spectrums for n-fold rotational symmetry except that for n = 5 shows any significant point.

0.1 0.6 0.4 0.2

3. EXTRACTIONOF REFLECTIONALSYMMETRY 5

5

5

5

Fig. 4. Rotational correlation at the center of the star image. Correlation valuesare normalizedby the constant value arot(0).

To extract reflectional symmetry, we define the reflectional correlation in a similar way to the correlation with the rotated image, except that we use M(O) instead of R(O) as the transformation as follows: aref(f; ~, 0) = SS(gradxf(M(O)x))M(O) r

1

(gradx f ( x + g))Tdx

0.8 where

0.6

M(0)=[-c°s0

0.4

-sin0

0.2 --~5-

lO

I5

20

25

30 ~*

Fig. 5. Spectrum of rotational symmetry,at the center of the star image. Spectrum values are normalized by the constant value arot(0 ).

Fig. 6. Spectrum of 5-fold rotational symmetry of star.

the target figure and parameters used in the experiment (see Section 4). This dramatically affects the Fourier series expansion. For all the figures which have rotational symmetry the rotational correlation value is periodical. Therefore, we conclude that the Fourier series expansion is not suitable for extracting the rotational symmetry. This is the reason why we use p(n) in equation (12) instead of the Fourier series expansion. We show an example for the original image of Fig. 1. Figure 4 is the rotational correlation for the center of the star shape. Figure 5 is the spectrum of the rotational symmetry at that point, which is calculated according to equation (12). Figure 6 shows the spectrum of 5-fold rotational symmetry for all image pixels. Only a single point is visible. This indicates that the center of 5-fold rotational symmetry is most likely located at that point. PR 26:8-E

(13)

-sin0] cos0J

The matrix M(O) represents the reflection and at=f(f; ~, 0) is the correlation between the original image and the image reflected by M(O). This correlation is calculated in a similar way to the correlation with the rotated image (Section 2). Now we need to consider what kind of transformation corresponds to each parameter set (~,0). The slant of the axis of reflection M(O) is 0/2. The translation can be considered the composite of two translations, one is orthogonal to the axis of reflection, and the other is parallel to it. The reflection and the translation which is orthogonal to the axis form a reflection, whose axis has the slant of 0/2 and passes through ~/2. The reflection and the translation along the axis cause the glide transformation. We can obtain the parameters of the axes of reflectional symmetry and the amount of glide translation along them directly from the local maxima of the reflectional correlations. We show the result of the extraction of reflectional symmetry for the heart shape of Fig. 7, which is a 128 × 128 gray scale image obtained by an image scanner. The variation in intensity of this image is so small that the image is almost a binary image. This figure has an axis of reflectional symmetry which can be clearly seen.

Fig. 7. Heart image.

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W // //

(a) tT=O

(b) O=Tr/8 (c) 0=27r/8 (d)/9=37r/8

(e) 0 : 4 r / 8

(f)/9=5~r/8 (g)/9=67r/8 (h) 0= 77r/8

(i) O=8r/S

(j)/7=OTr/S (k) O= 107r/8(1) 0= llr/S

// // // ( m ) 0 = 121r/8(n) 0 = 13~r/S(o) 0 = 1 4 f / 8 ( p )

0 -- 15~r/8

Fig. 8. Result of aref(heart; ~, 0).

with intensities according to each axis correlation value. The result in Fig. 9 demonstrates that our method has extracted the right axis of symmetry. Note that other local maximum axes of reflectional symmetry can also be distinguished, which represent reflectional symmetry of local parts of the figure.

4. EXPERIMENTSAND RESULTS 4.1. Implementation

Fig. 9. Axes of reflectional symmetry of heart.

First we calculated the reflectional correlation of this figure a,¢f(heart; ~, 0), the result is shown in Fig. 8. Then we extracted local maximum points of the reflectional correlation, and obtained the axes of symmetry corresponding to each of them. We drew these axes

Our actual implementation differs a little from the theoretical approach described in Sections 2 and 3. To evaluate the rotational correlation or the reflectional correlation, we model the image to be rotated or to be reflected using the position and gradient on the outline of the figure, which we correlate with the gradient of another whole image. This reduces the computational cost of the correlation in comparison to the method of correlations using two entire images, and also reduces the computational cost needed for rotation and reflec-

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tion. This is sufficient for gray scale images which are almost binary. We convolve with the gradient of the Gaussian filter instead of the gradient (Section 2), and use tr = 1.0 as the standard deviation of the Gaussian filter in the experiments. We sample the 0 of R(O) or M(O) up to N = 64. We implement our method on the S U P E R 3 2000 whose C P U is R3000, which is manufactured by Japan

C o m p u t e r Corporation. We show the times needed to compute the example in the case of the star image. We used 145 points on the outline, and spent 6 min to evaluate adi r (star, ~, 0), 8 s to transform it to arot(star; to, 0), and 2 min to obtain p(star; to, n).

Fig. 10. Star and triangle image.

Fig. 12. Leaces image.

4.2. Results We have already shown the result of extraction of rotational symmetry for the star image. We then per-

(b)

(c)

(d)

Fig. l 1. Result of detection of rotational symmetry of star and triangle. The first, second, third and fourth extracted rotational symmetries are indicated in (a), (b), (c) and (d), whose arot values are 0.31,0.30,0.23, and 0.19, respectively. Each " + " symbol represents the center of corresponding rotational symmetry, and the original image and rotated images are superimposed. Edges of the original image and the rotated images which are well matched are shown brightly.

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(a)

~

!~

igl/

(b)

,i ~

(c)

(d)

Fig. 13. Result of detection of reflectional symmetry of leaves. The first, second, third and fourth extracted reflectional symmetries (including glide symmetries) are indicated in (a), (b), (c) and (d), whose ar,t values are 0.25, 0.23, 0.20 and 0.18, respectively. Each light straight line represents the axis of corresponding reflectional symmetry, and the original image and reflected images are superimposed. Edges of the original image and reflected image which are well matched are shown brightly.

formed experiments using two figures located adjacent to each other; star and triangle image (Fig. 10) including a star shape and a triangle shape. The star shape has 5-fold rotational symmetry and the triangle has 3-fold rotational symmetry. Note that these symmetrical properties do not lie at the centroid of the whole composite figure. The results of extracted rotational symmetry of the four strongest local maxima are shown in Fig. 11. The extracted 5-fold rotational symmetry of the star shape is shown in Fig. 1l(a). Figure 1l(b) shows the second extracted 3-fold rotational symmetry of the triangle shape. Figures 1l(c) and (d) signify that there are some other local maximum points. These 2-fold rotational symmetries near the center of each image are caused by the fact that two of the edges of the star shape are nearly parallel to one of the edges of the triangle. We also show another result of the extraction of reflectional or glide symmetry from a real image of Fig. 12. This is a 128 × 128 gray scale image of alternate leaves obtained by an image scanner. This image contains glide symmetry about the stem and each leaf has reflectional symmetry. We illustrate the extracted symmetry from this image in a similar way to the rotational symmetry in Fig. 13.

Figures 13(a) and (b) represent the twin pair of local maxima which is derived from one glide symmetry of the leaves. These local maxima derived from the reflection about the common axis and the translation of the same amount but only whose direction is opposite. Figures 13(c) and (d) are the results of extraction of the reflectional symmetry which each leaf has. This result shows that our method can extract glide symmetry, which has not been treated yet at the same time with the reflectional symmetry. 5. C O N C L U S I O N S

We have described a new method of extracting symmetrical properties of planar two-dimensional figures which performs correlation with the rotated and reflected images. We have chosen to use a directional pattern matching technique to determine correlation values for symmetry, and the results of experiments with several images are shown. Our method has the advantages that we do not need any knowledge of the centroid and we can also detect the symmetrical properties of partially symmetric figures. One reason for these advantages is that we opted for directional pattern matching of edge features. Our

Detection of partial symmetry m e t h o d can extract glide s y m m e t r y at the same time as reflectional symmetry. The principle d i s a d v a n t a g e s are associated with the high c o m p u t a t i o n a l cost a n d the m e m o r y requirements. O u r m e t h o d can be used with gray scale images a n d it is also suitable for parallel implementation. Symmetry is one of the i m p o r t a n t structural properties of a figure, so we think that symmetry is effective to examine the s e g m e n t a t i o n of figures. A c k n o w l e d g m e n t s - - T h e authors would like to thank the Image Understanding Section staff at Electrotechnical Laboratory, especially R. Baldwin and G. Roth for reading the drafts of this paper and providing many comments.

3. 4. 5. 6. 7. 8. 9.

REFERENCES

I. M. J. Atallah, On symmetry detection, IEEE Trans. Cornput. C-34, 663-666 0985). 2. G. Marola, On the detection of the axes of symmetry of symmetric and almost symmetric planar images, IEEE

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Trans. Pattern Analysis Mach. Intell. PAMI-ll, 104 198 (1989). R. C. Bolles, Symmetry analysis of two-dimensional patterns for computer vision, Proc. Int. Joint Conf. Artif. Intell., pp. 70 72, August (1979). H. Blum, Biological shape and visual science, J. Theoret. Biol. 38, 205-287 0973). R.A. Brooks, Symbolic reasoning among 3-D models and 2-D images, Artif. Intell. 17, 285 348 (1981). M. Brady and H. Asada, Smoothed local symmetries and their implementation, Int. J. Robotic Res. 3(3), 36-60 (1984). J. Ponce, Ribbons, symmetries, and skewed symmetries, Proc. DARPA Image Understanding Workshop, pp. 1074 1979, April (1988). J. Big/in, Recognition of local symmetries in gray value images by harmonic functions, Proc. 9th Int. Conf. on Pattern Recognition, pp. 345 347, November (1988). S. Y. Yuen, Shape from contour using symmetries, Proc. I st Eur. Conf. on Comput. Vision, pp. 437-453, April (1990). D. H. Ballard, Generalizing the Hough transform to detect arbitrary shapes, Pattern Recognition 13, I II 122 (1981). H. Yamada, MAP matching--elastic shape matching by Multi-Angled Parallelism, Trans. IEICE Japan J73-DI1(4), 553-561 (1990) fin Japanese).

About the Author TAKESHIMASUDA received the Bachelor of Engineering degree from the University of Tokyo in 1989. Since then, he has been with the Image Understanding Division, Electrotechnical Laboratory (ETL) of the Ministry of International Trade and Industry. He is a member of the Institute for Electronics, Information and Communication Engineers (IEICE) of Japan and the Information Processing Society (IPS) of Japan.

About the Author--KAZUHIKO YAMAMOTOreceived the Bachelor, Master, and Ph.D. degrees in engineering from the Tokyo Denki University in 1969, 1971, and 1983, respectively. Since 1971, he has been with the Electrotechnical Laboratory (ETL), studying handprinted character recognition. He participated in the development of OCR in the Pattern Information Processing System project from 1971 to 1979. From 1979 to 1980 he was a Visiting Researcher in the Computer Vision Laboratory, University of Maryland where he worked on computer vision. He is a chief of the Image Understanding Section of ETL. His research interests are pattern recognition and artificial intelligence. Dr Yamamoto is a member of IEEE, IEICE Japan, and IPS Japan.

About the Author--HIROMITSU YAMADA received the Bachelor, Master and Ph.D. degrees in electrical engineering from the Waseda University in 1970, 1972, and 1985, respectively. From 1972 he has been with the Electrotechnical Laboratory (ETL). He is now a senior research scientist in the Image Understanding Section, Machine Understanding Division of ETL. He is currently working in handprinted character recognition, flexible shape matching by dynamic programming, and parallel operations for feature extraction and shape matching. Dr Yamada is a member of IEICE Japan, and IPS Japan