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15) The simulation as under Case 14 but with OS= 0.0, i.e.,using JM, reduced (Case 15) E1 and E2 significantly while E3increased. This reinforces the earlier observation that the un-normalized J tends to favor the equipopulation (which agreeswith the simulation).
16) An imperfect supervision using WJ' with samples of thetwo classes significantly mixed (Case 16) resulted in significantlyhigh E2 as the external labeling is considerably far from truthalthough 3 was set to 0.9. However, the algorithm was able toderive acceptable partition with fairly low E1 in spite of thisexternally imposed high wrong label bias.
17) A supervision similar to Case 16 using JM (Case 17)resulted in slightly lesser E1 and E2, but higher E3 as before.Thus, while the external imperfect labels do influence the re-
sults, the major factor influencing the process is still the basiccohesiveness of similarity in the data set as is desired of anyoptimal partitioning procedure. This is reflected in the fact thatin most cases (except for Cases 3 and 11) E1 is less than or equalto E2. In Cases 3 and 11 the external label bias matches theinherent bias due to distances and hence E2 is less than E1.(However, in these cases, the other measure E3 is equal to E2.Thus, in no case E2 is lower than both El and E3.)In summation, the simulation experience shows that the re-
sult obtained in a general case is the net effect of several ofthese biases such as equipopulation bias of JM, unequal popu-lation bias of JN', the label probability bias due to input ,B, therelative population bias 0, and the inherent distance based biasin the data set. Depending on which of these biases reflects thefactual state, the results measures (E1, E2, and E3) will be moreor less satisfactory. In general, in an imperfectly supervisedenvironment, it is deemed advisable to use .) as developed here.This leads to a result averaged over all these different biases,and to that extent, less sensitive to any one particular bias.Whenever external information is available confirming the va-lidity of one or the other of these biases, a correspondinglybiased JI(JM or JN') may be used.
VI. CONCLUDING REMARKS
The problem environment visualized here does not permitdeployment of traditional parametric techniques based on iden-tification of finite mixtures and the nonparametric approachesthat are presently available are computationally complex. Thistherefore requires development of some nonparametric group-ing or partitioning approach. Accordingly, the problem wasstructured as one of optimal linear/nonlinear partitioning ofthe given, imperfectly labeled training sample set. The emphasishas been to make the best use of available, albeit imperfect,labels in deriving the optimal partition. The algorithm, as con-structed, while giving some weightage to this label information,still retains the basic property of searching for the optimal par-tition based on the cohesiveness of the data. The integratedoptimality criterion proposed here provides the flexibility nec-essary for the user to influence the search if so desired or deemedadvisable on the basis of available external information in termsof relative cluster sizes and label probabilities. The reportedsimulation experience brings out the influence of the variousfactors contributing to the optimality of the partitions and en-hances the user's confidence in the effectiveness of this newtechnique. The simulation experience has also shown that theassociated computational loads are far less than under the alter-native approach of error-correcting procedures. This makesthe proposed methodology acceptable from a computationalpoint of view also. Accordingly, it is hoped that this will serveas a valuable tool for pattern analysis in imperfectly supervisedenvironments.
REFERENCES
[1] R. L. Kashyap, "Algorithms for pattern classification," in Adap-tive Learning and Pattern Recognition Systems, Fu and Mendel,Eds. New York: Academic, 1970, pp. 81-112.
[21 Y. C. Ho and R. L. Kashyap, "A class of iterative procedures forlinear inequalities," SIAMJ. Contr., vol. 4, pp. 112-115, 1966.
[3] R. E. Warmack and R. C. Gonzalez, "An algorithm for the opti-mal solution of linear inequalities and its application to patternrecognition," IEEE Trans. Comput., vol. C-22, pp. 1065-1075,Dec. 1973.
[4] B. V. Dasarathy, "DHARMA: Discriminant hyperplane abstract-ing residuals minimization algorithm for separating clusters withfuzzy boundaries," Proc. IEEE, vol. 64, pp. 823-824, May 1976.
[51 H. Teicher, "Identifiability of fiite mixtures," Ann. Math. Stat.vol. 32, pp. 1265-1269, Dec. 1963.
(6] K. Shanmugam and A. M. Breipohl, "An error correcting proce-dure for learning with an imperfect teacher," IEEE Trans. Syst.,Man, Cybern., vol. SMC-1, pp. 223-229, July 1971.
[7] A. L. Lakshminarasimhan, "A unified approach to feature selec-tion and learning in unfamiliar environments," Ph.D. dissertation,Indian Inst. of Sci., Bangalore, India, Nov. 1975, pp. 38-46.
[8] B. V. Dasarathy and A. L. Lakshminarasimhan, "Sequential learn-ing employing unfamiliar teacher hypothesis (SLEUTH) withconcurrent estimation of both the parameters and teacher charac-terisitcs,"Int. J. Comput. Inform. Sci., vol. 5, pp. 1-7, Mar. 1976.
[9] -, "Learning under a VEDIC teacher," Int. J Comput. Inform.Sci., vol. 8, no. 1., pp. 75-88, 1979.
[10] D. W. Peterson and R. L. Mattson, "A method for finding lineardiscriminant functions for a class of performance of criteria,"IEEE Trans. Inform. Theory, vol. IT-12, pp. 380-387, July 1966.
[11] K. Fukunaga and W. L. G. Koontz, "A criterion and an algorithmfor grouping data," IEEE Trans. Comput., vol. C-19, pp. 917-923,Oct. 1970.
[12] B. V. Sheela and B. V. Dasarathy, "OPAL: A new algorithm foroptimal partitioning and learning in non-parametric unsupervisedenvironments," Int. J. Comput. Inform. Sci., vol. 8, no. 3, pp.239-253, 1979.
[13] P. Ramamoorty and B. V. Sheela, "Modiflcation of the flexiblepolyhedron method for function minimization," Nat. Aeronaut.Lab., Bangalore, India, NAL AE-TM-4 -73, 1973.
[14] J. A. Nelder and R. Mead, "A simplex method for function mini-mization," Comput. J., vol. 7, pp. 308-313, 1965.
Some Notes on Cellular Logic Operators
KENDALL PRESTON, JR.
Abstract-Cellular logic machines used for feature extraction in pat-tern recognition have increased in speed to the point of making it pos-sible to execute programs equivalent to 1 billion general-purposecomputer instructions in 1 TV frame time. Unfortunately, most cellularlogic operators (CLO's) are designed ad hoc. It is important, therefore,to begin to systematize the generation of algorithms using CLO se-quences for pattern analysis. These notes systematically analyze someaspects of CLO's which are used in shape discrimination and idealiza-tion and in object counting and sizing. New extensions of subfieldnumbering schemes in the hexagonal tessellation are introduced.
Index Terms-Cellular logic, matched filtering, neighborhood logic,shape analysis.
Manuscript received January 2, 1980; revised July 25, 1980.The author is with the Department of Electrical Engineering, Carnegie-
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. PAMI-3, NO. 4, JULY 1981
I. INTRODUCTION
Cellular logic, as defined and developed in [ 11, is a Booleanimage algebra performed on arrays of logical numbers producedby thresholding the arrays of real numbers which constitutepictures or images. (Logical images are sometimes called"binary" images.) Each cellular logic operator (CLO) executesa multivariate logical function which is used to transform theset of all logical values which represent a logical image into anew set of logical values. The present value of each pictureelement in a logical array along with the values of its directlyadjacent picture elements are taken as the independent vari-ables. The dependent variable is the logical number whichconstitutes the new value of each picture element. Sequencesof CLO's may be written for the purpose of extracting rathercomplicated shape, size, context, and texture features frompictures.Two-dimensional cellular logic deals with logical functions of
at most nine variables, e.g., the variables represented by a 3 X 3picture element (pixel) cell. Thus, the output of each functionmay be computed by table lookup using a table of at most512 words. Cellular logic machines (CLM's) based on thistechnique and operating in several parallel channels on a single-instruction multiple-data-stream (SIMDS) basis have achievedoperational speeds equivalent to 1000 million general-purposecomputer instructions/s [2]. CLM's now under constructionwill reach 10-100 billion instructions/s for some cellular logiccomputations [3]. The possibility of conducting a programconsisting of 1 billion equivalent instructions in 1 TV frametime is clearly of interest in many applications. It is thispotential that makes cellular logic of particular interest.Unfortunately, although ingenious cellular logic programs
have been written for extracting rather complex features fromimages of, for example, human blood cells, there is no knownanalytic method for deriving these programs. Such programsconsist of sequences of cellular logic operators (CLO's). It isthe purpose of this note to begin to classify some commonlyused types of CLO's in a systematic manner. In the paragraphsbelow, CLO's for size and shape discrimination are discussed,followed by a short presentation on noise removal, and finally,a comparison of object counting techniques using the subfieldmethodology originally devised by Golay [41. It is hoped thatthis note will serve as a stimulus to others in methodizingcellular logic techniques in at least a semianalytic manner.
II. SIZE AND SHAPE DISCRIMINATION
Many CLO's or CLO sequences have the effect of boundarymodification. Fig. 1 provides an illustration. This figure showsthe result of using the cellular logic subroutine in the Carnegie-Mellon SUPRPIC image processing language. This subroutineassumes that the image array is in the square (Cartesian) tes-sellation. The subroutine uses as inputs each value of the logi-cal variable Xoj in the N X N logical image array in conjunc-tion with the logical values of its eight neighbors Xij (i = 1,2 ... 8). These constitute the independent variables. Thequantity N is the span of the N X N array of logical numberswhich describes the input function. The index j iterates overthe entire array. The index i defines the eight neighborhoodpositions in the 3 X 3 cell, starting with the value 1 in the up-per left corner and indexing clockwise.The value of the dependent or output variable X6j is given
by computing, for each of the N2 neighborhoods of the inputfunction, the following quantities:
oj Xij Xi Lxij - Xi + I, i
where bj is the number of logical 1's in the jth neighborhood
ol _* *-m-+-+--NULL(a)
_ * _@_ -- NULL
(b)
U..-.-*-4-*-+- -NULL
* *+N NULL *-.E.NULL*- +(d) (g) (h)
I_P-* - *-a-NULL 4mNULL(e) (i)
*---L- -NULL
(f)
I- NULL
Fig. 1. Illustrations of cellular logic sequences for shape discriminationusing the cellular logic command of the Carnegie-Mellon SUPRICimage processing system with the parameter 1 equal to 4 (a)-(c),equal to 5 (d)-(f), equal to 6 (g), and equal to 7 (h)-(i).
10
-J-j:Dz0
(/)
L-JC-)
5
0 5MAXIMUM CHORD
10
Fig. 2. Graph showing the number of cycles to null versus object maxi-mum chord for the square, octagon, and diamond using ¢ = 4 (dashedlines) and 'F = 5 (dotted lines).
and Xi is the number of 0-1 or 1-0 transitions in that neigh-borhood. Note that /j = 8 indicates an interior point in thecluster of contiguous l's; 4j = 0, an exterior point; Xi = 2, anedge point; yj = 4, a link in a chain of l's; etc.The value of the dependent variable is calculated by first
detemining the values of two intermediate variables Aj and Bjwhich are given by
AO if q5j.'T?1 otherwise
ro if X<Bj { i dj
1 otherwise
where 1 and are parameters of the subroutine. Finally,
X'oj =Xoj (Ai +Bj).The sequences shown in Fig. 1 have the diamond, the regular
octagon, and the square as input functions. Fig. 1 (a)-(c)shows the result of operating with 4 = 4 and any value ofgreater than 2. As can be seen, at each step of the sequence,the boundary of the input function moves inward until the setof l's in the input function is reduced to the empty set (null).In the first three cases shown, the number of steps or cycles
t-p-
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required for reduction to null is 6, 7, and 9, respectively. ForF = 5, the result is quite different, as shown in Fig. 1 (d)-(f).Here reduction to null occurs in four cycles for all input func-tions irrespective of shape. This illustrates that using (F = 4yields an operation which provides for shape discrimination,while (F = 5 yields sensitivity only to the length of the maxi-mum axial chord (maximum chord along the X or Y axis).
Fig. 1 also provides results for (F = 6 for the diamond (Fig.1(g)] which reduces to null in three cycles and 4( = 7 for thediamond [Fig. 1(h)] and the octagon [ Fig. 1 (i)], yielding twoand three cycles, respectively. The other cases for 4( = 6 and(F = 7 are not shown, since in these cases, the number of cyclesto null is the same as for (= 5. Fig. 2 tabulates cyclesto null for several other sizes of the same input functions forboth 4F = 4 and ( = 5. It appears from this figure that thecomparative utility of using (F = 4 for shape discrimination and(F = 5 for size discrimination is a general property, although ananalytic proof of this property has yet to be developed.
III. NOISE REMOVALTwo-dimensional logical image arrays are generated by
thresholding a multivalued image function. Noise in theoriginal image function frequently produces a noncontiguousgroup of 1's in the logical image functions which correspondto a single object in the original function. An extreme exampleof such a result is shown in the left column of Fig. 3. Noisemay be removed by the inverse transform, i.e., where the 1'sand O's are interchanged, and where (F = 4 and has any valuegreater than 2. Fig. 3(a) shows how, after four cycles of theinverse transform, the noisy image merges into an "idealization"of the correct object shape. In this case, the result is a convexoctagon. This pattern is stable under further cycles of thetransform, and its boundary has previously been referred to asthe "convex hull" and has been found useful in biomedicalimage analysis for locating the arm tips of chromosomes [5],[6].Since the hexagonal tessellation is frequently used in cellular
logic, Fig. 3 (b), (c) shows noise removal using the Golayneighborhood [41:
N6=Xij ieGe
N6=Xij ieGo
where N6 is the six-neighborhood, Ge is the set of integers(1, 2, 4, 6, 7, 8), and Go is the set of integers (2, 3, 4, 5, 6, 8).[See Fig. 3(f).] Cellular logic transforms using the Golaytechnique use Ge for even lines in the image and Go for oddlines. Fig. 3(b) assumes that the first line of the object is aneven line, and Fig. 3(c), odd.
Fig. 3(d), (e) uses two new neighborhoods developed duringthe compilation of this note which have the advantage thatodd and even line neighborhood alternation is not required.These neighborhoods are given by
N6=Xij iESl
N6=Xij iES2
where SI is the set of integers (2, 3, 4, 6, 7, 8) and S2 is theset of integers (1, 2, 4, 5, 6, 8). Fig. 3(d) uses the neighbor-hood S1, and Fig. 3(e), S 2. In each of the cases shown inFig. 3 (b)-(e), the output function is a figure which, whenremapped onto the hexagonal tessellation, is a convex hexagon.Note the difference in the number of pixels in each final result,namely, 21, 19, 23, 19, respectively. Clearly, there is less varia-bility in the number of pixels in the output function whenusing the Golay transform (±5 percent) than when using thenew technique (±10 percent). However, the new technique issimpler to implement in a table lookup computer in that thesame table is used for each line.
(a) t ~ 9 + * * ^
(b) IGHBORHOD CEL
(d) _ 0
EIGHT-NEIGHBORHOOD CELL
(fG)
GOLAY EVEN CELL GOLAY ODD CELL
1~~~~~~FSI CELL S2 CELL
Fig. 3. Illustration of shape idealization using the cellular logic subrou-tine (a), the Golay hexagonal transformation (b)-(c), and a newmodification of the hexagonal transformation (d)-(e). The corre-sponding neighborhood configurations are shown in (f).
Finally, Fig. 3 shows how the estimate of the number ofpixels in the actual object varies according to both theneighborhood and the tessellation used in noise removal. Notealso that, in all but one case, five cycles are required to reachan output function which is stable upon further cycles of thetransform.
IV. COUNTING AND SIZING
In addition to shape discrimination and noise removal, oneof the major uses of CLO sequences is in object counting andsizing. This operation requires the retention of residues. (Aresidue is defined as a pixel whose logical value is 1 and all ofwhose neighbors have the value 0.) A count of the residues inthe array of logical numbers at the end of each cycle leads tothe residue histogram which has frequently been used as a fea-ture vector in pattern recognition [ 7]. This residue histogramis a cumulative histogram of the number of objects in the origi-nal logical image as a function of their size.The retention of residues requires the use of subfields as
originally introduced by Golay [41. If subfields are not used,the group of contiguous 1's corresponding to a specific objectmay be reduced to the empty set [Fig. 4(a)] or may be frag-mented into a multiplicity of residues [Fig. 4(b)]. Both ofthese problems may be avoided while using the appropriatevalues of the quantities (F and and operating upon pixels inone subfield at a time. Fig. 4(c) shows an example of reduc-tion to a single residue using the SUPRPIC subfield map andF = 6 with 4. A final problem which may occur when toolarge a value of (F is used is shown in Fig. 4(d) for (F = 7 and
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(a) -- 9-
(b) 34213 4
341 --q-a. -I
(d) ---- ~ * @@ ~ ~ >
Fig. 4. Examples of some problems encountered when reducing to residues. The object transforms to the empty set ifsubfields are not used (a). Transforming in subfields with = 6 and : greater than 4 leading to fragmentation (b). Cor-rect results when cD = 6 and = 4 are utilized (c). Generation of closed polygons (instead of residues) when using xP = 7anda=4 (d).
pling latice) for the Golay hexagonal transformation, the SUPRICsquare transformation, and two modifications of the hexagonaltransformation.
= 4 where the final stable output function consists of twoclosed polygons rather than a residue. This problem does notpresent itself when using the neighborhoods Ge, Go, S 1, andS2.Four subfield maps are shown in Fig. 5: 1) the SUPRPIC
subfields-of-four, 2) Golay's original subfields-of-three, and3) the S 1 and S2 neighborhoods. This is illustrated in Fig. 6.For all cases shown in Fig. 6, the value of is set equal to 4 in
order to prevent fragmentation into multiple residues. Fig.6(a), (b) shows the results of using 4D = 5 and 1D = 6 using theSUPRPIC subfields with the subfield order 1-3-4-2, respec-tively. Fig. (c), (d) and Fig. 6 (e), (f) show the results of usingthe Golay transform assuming that the first line of the objectis odd and even, respectively. The first of these pairs of figuresuse 4 = 4 and the second 4' = 5. Fig. 6 (g), (h) and Fig. 6 (i),(j) show similar results for the neighborhoods S2 and S 1,respectively. For Fig. 6 (c)-(j), the subfield order in all casesin 1-3-2.
Fig. 6 illustrates primarily that, unlike operating withoutsubfields (Section II), altering the value of does not allowone to shift between shape discrimination and size discrimina-tion. In all cases in Fig. 6, the same original logical function isused as the input and, with only one exception [Fig. 6(i)],exactly two major cycles are required to produce the residue.(A major cycle is completed once each time all subfields are
processed.)
Finally, Fig. 7 has been prepared to illustrate the effect ofshifting the original input function with respect to the subfieldmap and of changing the subfield order. Fig. 7 (a), (b) showstransforming to a residue with 4f = 5 and = 4 for the fourpossible positions of the original input function with respectto the four subfields. As can be seen, there is a negligible dif-ference in the number of subfield cycles required to achievethe residue. This implies that objects of a given size and shapewill produce residues in essentially the same bin of the residuehistogram relatively independent of where the object to whichthe input function corresponds is found in the original inputimage.
Fig. 7 (e), (g) shows the effect of changing the subfield orderfrom 1-3-4-2 to 3-4-2-1, 4-2-1-3, and 2-1-3-4, respectively. Ascan be seen from these figures and Fig. 7(a), the number ofsubfield cycles to achieve the residues is 8, 7, 9, and 7, respec-tively, representing a variation of ±15 percent. Therefore, ingeneral, it can be said that it would appear reasonable to havethe residue histogram quantized into approximately ten binswhen analyzing the size distribution of objects in the sizerange dealt with in this note.
V. CONCLUSION
This note presents a systematic comparison of some CLO se-quences for shape discrimination and idealization and for ob-ject counting and sizing. Also presented are two new subfieldnumbering schemes for the six-neighborhood which are easierto implement in table lookup computers than the originalscheme of Golay. The question remains, however, of how toassess the merits of one sequence or scheme versus another.Previously, Deutsch [8] addressed this problem by presentinga comparison of CLO sequences for thinning (skeletonizing)logical input functions generated by thresholding images ofhandprinted characters. The merits of the various sequenceswere then compared by testing the resultant character recogni-tion success rate for each scheme using Highleyman's data [9].This author has performed similar tests of various PCA se-quences for shape idealization by measuring identification suc-cess rates for the shapes created by each sequence [ 10].Further analytic studies along these lines are clearly required,and it is hoped that this note will serve to establish a precedent.Such an approach is already being used by such workers asAbdou and Pratt as regards algorithms for boundary detection[11]. The need for such studies is evident due to the increas-
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(a) 4 _--- I --O _ --o s
3 412
(c) 3 --l _ ---
2 312
(d) I1 _-~~1
(e) _*
A soR--_
-_ gm m---"
_l.,-*,_.U
(i) _ | M l s _ _
0j) 32_ 1_12
Fig. 6. A given object is transformed to its residue using E = 4 and tendifferent cellular logic sequences. The cellular logic command for =
5 and = 6 (a)-(b). The Golay hexagonal transformation using 4 =
4 and = 5 with the first line of the object resting on both even andodd lines (c)-(f). The modified hexagonal transformation using =
4 and = 5 for each of the subfield numbering schemes shown in Fig.5 (g)-(j). For the cellular logic command in SUPRIC, the subfieldorder is 1-3-4-2 and for the hexagonal transformations the subfieldorder is 1-3-2. Note that the six-neighborhood hexagonal transforma-tions do not form closed polygons.
4.-I-
(f)
( 12(g)2
(h) 23123
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IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. PAMI-3, NO. 4, JULY 1981
(a) 4 2 3 43 341223
_ *_+p-. * - * -em.. -e I-_ *I-_-_ _ _-e. -*.
(b) 4 , -123'41 --4 2 3 ---ql*--qw~ -.0 % -s I --ft I --
2 3 41
11 234
41
(d) ;2343 4 1232 3 4
(f) _ -~ ~ ~ - 41 - o W --. 1% -- I --a
(g) w- _ ~+_II " _E _~u_ -
Fig. 7. Illustrations showing the effect of using the SUPRIC cellularlogic command to generate the residue of an object with the objectin all four possible locations on the square latice (a)-(d). The sub-field order is 1-3-4-2. The result of using other subfield orders areillustrated, namely, 3-4-2-1, 4-2-1-3, 2-1-3-4 (e)-(g).
ing use of cellular logic machines and cellular automata invarious image processing applications in communications [12],medicine [13], and military reconnaisance [14].
REFERENCES
[(1 K. Preston, Jr., M.J.B. Duff, S. Levialdi, P. E. Norgren, and J-i.Toriwaki, "Basics of cellular logic with some application inmedical image processing," Proc. IEEE, vol. 67, pp. 826-856,May 1979.
(2] Perkin-Elmer diff3 System Users Manual, Norwalk, CT, Nov.1976.
[31 S. R. Sternberg, "An architecture for real-time biomedical imageprocessing," in Proc. COMPSAC 79, Chicago, IL, Nov. 1979.
(5] J. Sklansky, "Recognition of convex blobs," Pattern Recognition,vol. 2, pp. 3-10, 1970.
(6] K. Preston, Jr., "Applications of cellular logic to biomedical imageprocessing," in Computer Techniques in Biomedical Engineering.New York: Auerbach, 1973, p. 295.
[7] M. Ingram, P. E. Norgren, and K. Preston, Jr., "Automatic differ-
entiation of white blood cells," in Image Processing in BiologicalScience. Berkeley, CA: Univ. California Press, 1968, pp. 97-136.
[8] E. S. Deustch, "On some pre-processing techniques for characterrecognition," in Computer Processing in Communications.Brooklyn, NY: Polytechnic Press, 1969, pp. 221-234.
[9] W. H. Highleyman, "Linear decision functions with application topattern recognition," in Optical Character Recognition, G. L.Fischler et al., Eds. Washington, DC: Spartan, 1962, pp. 249-285.
[10] K. Preston, Jr., "Tissue section analysis: Feature selection andimage processing," Pattern Recognition, vol. 13, pp. 17-36, 1981.
(11] I. E. Abdou and W. K. Pratt, "Quantitative design and evaluationof enhancement/thresholding edge detectors," Proc. IEEE, vol.67, pp. 753-763, May 1979.
(12] T. Usubuchi, S. Mizuno, and K. linuma, "Entropy reduction offacsimile pictures by a thinning process," in Abstracts ofPresen-tation-1977 Picture Coding Symp., Tokyo, Japan, Aug. 1977,pp. 55-56.
(13] J.-I. Toriwaki, Y. Suenaga, T. Negoro, and T. Fukumura, "Patternrecognition of chest X-ray images," Comput. Graph. Image Pro-cessing, vol. 2, no. 3/4, pp. 252-271, 1973.
[14] J. C. Simon and A. Rosenfeld, Eds., Digital Image Processing andAnalysis (NATO Advanced Study Institute, Bonas, 1975).Noordhoff: Leyden, 1977.
-*400 -iw -s4w - fi --w I -o I -U
481
---Op
----w
---4w
---qw ---Ow I ---w ---Qp --.O- m --owm
(e) ---400 ---Oo 40 ----w 0 ---4w ---o-= ---w N ---e- a ----w n
