394 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, MAY 1975

If the initial classification is background, the point is combined occluded objects and forward and backward motion seem to bewith the other previously processed background points. If the straight forward. Rotation (especially unrestricted, i.e., rotationmotion of an object is other than parallel to one of the axes, about any point), however, seems at this point to be beyond thisthere may be "shadow" areas, which are classified as type 6 approach to motion perception. It is hoped that a report on theregions. It is assumed that the pictures were taken close enough progress of the investigation of the aforementioned extensionsin time so that these areas are small. The points in these areas will be forthcoming shortly.will have to be reclassified on the basis of their position. Fig. 7illustrates these areas.

In this process, it is unlikely that a point that is initially The author wishes to express his gratitude to Dr. L. Uhr whomisclassified due to a peculiarity of the picture will be grouped provided many helpful suggestions and criticisms during thewith points from another region. Erroneous grouping can occur course of this research.only when the velocity values for the misclassified points arecoincidentally identical to a legitimate set of values. In this case, REFERENCESno action is taken and a false segmentation will occur. [1] H. B. Barlow, R. M. Hill, and W. R. Levick, "Retinal ganglion cells

responding selectively to direction and speed of image motion in therabbit," J. Physiol., vol. 173, pp. 377-407, 1964.

1 1 10 SUBSYSTEM [2] C. R. Brice and C. L. Fennema, "Scene analysis using regions,"Artificial Intelligence, vol. 1, pp. 205-226, 1970.

The preceding program can identify all the regions of a simple [3] C. L. Bristor, "The earth location of geostationary satellite imagery,"few of heordred ponts. Te 1110 Pattern Recognition, vol. 2, pp. 269-277, 1970.scene by processing just a few of the ordered points. The1110 [4] R. M. Brown, "On-line computer recognition of handprinted charac-

subsystem of which this program is only a part is aware of this ters," IEEE Trans. Comput.^, vol. C-13, pp. 750-752, 1964.[5] A. Guzman, "Decomposition of a visual scene into three dimensionalfact. Accordingly, it reorders the list of points as each point IS bodies," in 1968 Fall Joint Computer Conf., AFIPS Conf. Proc., vol. 33,

processed. In addition, it has a hierarchically organized scene 1968, pp. 291-304.[61 D. H. Hubel and T. N. Wrisel, "Receptive fields and functionaldescription component that attempts to identify the objects on architecture in two nonstriate visual areas of the cat," J. Neurophysiol.,

the basis of the data gathered while processing the points. vol. 28, pp. 229-289, 1965.[7] J. A. Leese, C. S. Novak, and V. R. Taylor, "The determination ofWhenever a scene has been satisfactorily described, the sub- cloud pattern motions from geosynchronous satellite image data,"

systemstopsand prints its results. [8Pattern Recognition, vol. 2, 1970.systemstops~~~ ~ ~ ~ ~ ~ ~ ~~~[]J. Y. Lettvin, 'H. R. Maturana, W. S. McCulloch and W. H. Pitts,"What the frog's eye tells the frog's brain," Proc. IRE, vol. 47, pp.

RESULTS 1940-1951, Nov. 1959.[9] H. R. Maturana and S. Frenk, "Directional movement and horizontal

The current system can theoretically handle any number of edge detection in the pigeon retina," Science, vol. 142, pp. 977-979,1900.rectangular objects in a scene. To this date, it has been success- [10] T. Marrill, A. K. Hartley, T. G. Evans, B. H. Bloom, D. M. R. Park,fully tested with three black rectangular objects moving in front T. P. Hart, and D. L. Darley, "CYCLOPS-1: A second-generation

recognition system," in 1963 Fall Joint Computer Conf., AFIPS Conf.of a white background. Each object moves with a different Proc., vol. 24, 1963, pp. 27-33.velocity. The scene is always successfully segmented into three [11] MiT and S. Papert, "S. Project MAC Progress Report IV,'MTPress, Cambridge, 1967.objects and background. In certain scenes with four or more [12] L. G. Roberts, "Machine perception of three dimensional solids," in

it.mayinitially misclassify points due to unique Optical and Electro-optical Information Processing, J. T. Tippet, et al.objects, it may initially misclassify points due to unique Eds. Cambridge, Mass.: M.I.T. Press, 1965.circumstances. Fig. 4 is an example of a scene where a back- [13] E. A. Smith and D. R. Phillips, "Automated cloud tracking using

precisely aligned digital ATS pictures," IEEE Trans. Comput., vol.ground point could be initially misclassified as a shadow area C-21, pp. 715-729, July 1972.point.6 No effort is made in the current version of the program [14] W. Teitelman, "Real time recognition of hand-drawn characters,"in 1964 Fall Joint Computer Conf., AFIPS Conf. Proc., vol. 29, 1964,to reclassify misclassified points. As a result, under these pp. 559-575.circmstaces,fals sementtionwilloccr. I genral,the [15] L. Uhr, "The description of scenes over time and space," Univ.circumstances, false segmentation will occur. In general, the WicnnMaso,Th.Rp17,Fb193Wisconsin, Madison, Tech. Rep. 172, Feb. 1973.more objects moving in the scene, the poorer the performance. [16] L. Uhr and C. Vossler, "A pattern recognition program that generates,The current version is also incapable of handling occluded evaluates, and adjusts its own opertos, iPtrRotnL. Uhr Ed. New York, Wiley, 1966, pp. 349-364.objects, oblique motion parallel to an edge, rotation, or forward [17] G. L. Walls, The Vertebrate Eye and Its Adaptive Radiation. Newand backward motion. In addition, due to the simple imple-mentation of the discontinu.ity matching, the discontinuities mustbe quite pronounced.

CONCLUSION

This first attempt to use motion for segmentation was highly A Modification of a Clustering Methodsuccessful in its ability to separate moving rectangular objects IVAN TOMEKfrom stationary background. A more general ability of thisnature would be most useful in a real time situation where an Abstract-A modification of a simple clustering method is described,entire scene could not always be processed; it would enable the and its performance is improved considerably in a number of cases withanalytical selection of cohesive regions of a scene for more practically the same amount of computation. The method is based onextensive analysis. dimensionwise application of the original method. An example is shown.The program is, however, severely restricted by the types of

objects and motion it can handle. However, investigation is INTRODUCTIONcurrently under way with a cross-shaped template that promises In a recent paper on cluster analysis [1] D. J. Eigen et a!.to extend potential segmentation to any arbitrarily shaped proposed a simple clustering method based on approximation toobject undergoing any type of velocity mnovement. Extensions to marginal densities. The method is quite simple, but results of its

6 The conditions under which misclassifications can occur are quite Manuscript received July 26, 1974; revised December 2, 1974. This workvaried. They are not only a function of the!number of objects in motion was supported by the Canadian Heart Foundation under a Postdoctoralin a scene but also of the relationships between the direction of the motion Fellowship.and spatial position. The nature of these conditions has not been completely The author is with the Department of Medicine, University of Alberta,explored yet. Edmonton, Alta., Canada.

395

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X XX X X X X X X X X X

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X X Xxx x

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(C)Fig. 2. (a) Three two-dimensional clusters with normal distribution. (b) Clustering of three clusters obtained by original algorithm.

(c) Clustering obtained by modified algorithm.

396 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, MAY 1975

application are not always ideal (in the sense that in many cases can be avoided and the second stage clustering made easier atit generates clusters in which "natural" boundaries, although practically no additional computational cost.well defined on the given data set, are violated by the resulting One of the referees brought to the attention of the author thepartition). This, however, is not a serious obstacle since the fact that the proposed algorithm is related to that described bymethod is intended as the first stage in a two-stage clustering Fu [2].approach: global clustering followed by a local one performedon preliminary clusters obtained in the first stage. In an n- REFERENCESdimensional space containing N vectors Xi = (xi1, .,xi.) the [1] D. J. Eigen et al., "Cluster analysis based on dimensional informationwith applications to feature selection and classification," IEEE Trans.proposed algorithm works essentially as follows. Syst., Man, Cybern., vol. SMC-4, pp. 284-294, May 1974.

In each dimension a window size wd is determined and histo- [2] K. S. Fu, Adaptive, Learning and Pattern Recognition Systems Theory andApplications, J. M. Mendel, K. S. Fu, Eds. New York: Academic,grams calculated. A threshold td is then used to determine valid 1970, pp. 72-75.

clusters in each dimension (via comparison of histograms andthresholds). In this way each dimension is subdivided intosi, i 1, n sections (we will assume for simplicity that si > 0,for i = 1,n). A two-dimensional illustration of the obtained gridis shown in Fig. 1. It is clear that this method will create at most Optimal Maintenance Policies for Machines Subject toC = ]J.1 si, i = 1, n clusters. In many cases the number of Deterioration and Intermittent Breakdownsclusters will be smaller since some of the obtained hypercubes V. V. S. SARMA AND MANSOOR ALAMwill be empty, nevertheless, the number of obtained clustersgrows very fast (essentially exponentially with the number ofdimensions), and obviously many of the obtained clusters will Asrc-pia rvniemineac oiis o ahndimensions) and obviously many of the obtained clusters will subject to deterioration with age and intermittent breakdowns andbe artificial and will only hamper local clustering in the second repairs, are derived using optimal control theory. The optimal policiesstage. are shown to be of bang-bang nature. The extension to the case whenAn example of this is shown in Fig. 2 where three clusters of there are a large number of identical machines and several repairmen in

normally distributed samples were generated in a two-dimensional the system is considered next. This model takes into account the waitingspace and the preceding algorithm resulted in the generation of line formed at the repair facility and establishes a link between thisnine clusters most of which were artificial. We are going to problem and the classical "repairmen problem."propose a simple modification of [1] that improves its perfor-mance considerably in a number of cases without decreasing its I. INTRODUCTIONcomputational efficiency. Optimal control theory has been applied recently by several

investigators to obtain optimal maintenance and replacementpolicies for equipment. Thompson [1], Arora and Lele [2], and

A look at Fig. 2 suggests that if clustering were performed Sethi [3 ] consider the degradation of a machine's capability withdimensionwise (coordinate by coordinate) rather than globally, age and assume that the deterioration can be partly offset viathe resulting clustering could be much more acceptable in a preventive maintena ce. Alam and Sarma [4] consider the effectnumber of cases. We will define the modification more formally as of random catastrophic failure on the maintenance policies.follows: 1) m = n,M = 1, and S, = (Xi,.-..,XN); 2) apply This correspondence considers the intermittent breakdownsalgorithm [1] to coordinates xim of vectors Xi e Sk,k = 1,. M, and associated repairs of the equipment and the stochasticobtaining M clusters Si, i = 1, * - -,M, with a new value of M nature of these processes. The approach leads to unification of(determined by the nature of data); 3) m = m - 1; 4) if m = 0, this work with the now classical "repairmen problem" [5 ], whenthen the algorithm ends, otherwise go to step 2). Here m is an there are large number of machines in the system.iteration count and M is the number of clusters.The performance of this algorithm could be further improved II. PROBLEM FORMULATION-SINGLE MACHINE CASE

if coordinates were not taken sequentially but rather in the order A. Thompson's Modelfor Machine Deteriorationof quality of generated clusters (e.g., in Fig. 2(a) the secondcoordinate would be better to start with). Ordering could be In order to define the model the following notation is used:made in a number of ways. A natural method using information T sale date of the machine,obtained during the calculation of histograms is to assign each V(T) discounted profit during the life of the machine plusdimension a number qi, which measures the prominence of the discounted salvage value at time T, in dollars,generated clusters, S(t) salvage value of the machine at time t, in dollars,

qi = E g(h), along the histogram (1) r rate of interest,u(t) number of dollars spent on preventive maintenance at

with g(h) given, for example, by time t satisfying the constraint 0 < u(t) . U (preven-

g(h) = h2, for h > ti, g(h) = 0, otherwise. (2) tive maintenance here means money spent over andabove the minimum spent on necessary repairs because

CONCLUSION ~~~~~~~ofintermittent breakdowns),CONCLUSION ~~~~~~f(f)maintenance effectiveness function at time t, in dollarsThe proposed modification improves the global clustering in a added to S per dollar spent on maintenance,

number of cases such as that in Fig. 2 where three clusters exist,the algorithm [1] generates nine, but the proposed modifiedalgorithm generates the correct three clusters. The presented Manuscript received Aphril17,D197a4;revsteedfNoEmberia 19, 1974. i°example is artificial and favorable to the proposed method. Engineering, Indian Institute of Science, Bangalore, India.

. . . . . . ~~~~~~~~~~~~~M.Alam iS with the School of Automation, Indian Institute of Science,It l lustrates, however, thnat in a number of cases artificil results Bangalore, India.