Contains reports on two research projects.National Institutes of Health (Grant 5 PO1 GM14940 -05

Greenwood, R. E.

Stallings, W. W.

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IX. COGNITIVE INFORMATION PROCESSINGAcademic and Research StaffProf.Prof.Prof.Prof.Prof.Prof.Prof.Prof.Prof.R.J.B.R.J.A.A.W.D.L.M. EdenJ. AllenB. A. BlesserT. S. HuangF. F. LeeS. J. MasonW. F. SchreiberD. E. TroxelI. T. YoungR. BochnerE. BowieE. BoyleL. BrennerR. Ellis, Jr.E. FilipC. GoldsteinB. GrossmanW. HartmanP. A. HenckelsD. A. FayDeborah A. FinkelC. L. FontaineC. C. HsiehE. R. JensenD. E. RobinsonJ. M. SachsSandra A. SommersJ. S. VenturaDr. R. R. ArcherDr. G. H. GranlundDr. J. E. GreenDr. E. G. GuttmannDr. K. R. InghamDr. J. I. MakhoulDr. D. M. OzonoffDr. D. SankowskyDr. O. J. TretiakF. X. CarrollGraduate StudentsM. HubelbankP. D. HenshawT. KitamuraJ. W. KlovstadG. P. MarstonG. G. MatisonG. PoonenN. RashidR. SingerA. ESTIMATES OF THE HUMAN VISUAL LINE-SPREAD ANDPOINT-SPREAD FUNCTIONSThere has been much interest in the human visual spatial modulation transfer func-tion, line-spread function, and point-spread function. These three functions are mean-ingful only under conditions of vision when the human visual system is linear andspace -invariant. The modulation transfer function is the system function relating sen-sation (the output) to stimulus (the input). The line-spread function is the output attribu-table to a knife impulse input of unit area per unit length, and the point-spreadfunction is the output resulting from a point impulse input of unit volume. This reportshows the relationship among the three functions when circular symmetry is presentand uses experimental values of the easily measured modulation transfer functionI tocalculate values for the two spread functions.Cutrona 2 approximates the human visual spatial modulation transfer function, H(k),asH(k) = 470 (e-.1531k -e-.54k )kwhere k is spatial frequency in cpd (cycles per degree). The line-spread function,This work was supported principally by the National Institutes of Health(Grant 5 PO1 GM14940 -05).QPR No. 102D. G. SitlerA. A. SmithR. D. SolomonJ. H. StahlerW. W. StallingsW. D. StrattonK. P. WacksT. R. WillemainY. D. WillemsK. H. Yung149(IX. COGNITIVE INFORMATION PROCESSING)h f(x), is the one-dimensional impulse response of the visual system and is related toH(k).hp(x) =Sr-00H(k) cos (2lTkx) dk4 .153 .54(.153) 2 + ( 2 rrx) 2  (.54)2 + (2irx)where x is distance in degrees. The point-spread function is the two -dimensionalimpulse response. If the point-spread function, h (r), has circular symmetry, then ha (x)and h (r) are related by the Abel-like integral equationp002 2which is solved by substituting a = x + zh (r) -rand using Laplace transform techniques.dh (y) 1dy 2(y 2 )1/2(y -r)= 470 • 2Tr 153((.153)2 +(2Trr))3/Zwhere r is radial distance60,000in degrees. h and h arep shown in Fig. IX-1.3000 -2500 -2000 -1500 -40,000 -20,00020,000-20,0002000-0.02 0.06 0.18-2000L0.08 0.1;500Fig. IX-1. (a) Point-spread function vs angular distance in degrees.(b) Line-spread function vs angular distance in degrees.QPR No. 102_ .54(.54)2 + (2wr)2) 3/2150(IX. COGNITIVE INFORMATION PROCESSING)Although an experiment by Mitchell, Freeman, and Westheimer 4 indicates circularasymmetries in the visual system, the formula given for h is probably a fair approx-imation.It is interesting to note that the formula for h can be determined directly. h (r) isthe Fourier-Bessel transform of H(k) and is given byp 0where J denotes the zero-order Bessel function of the first kind. The two computations40agree.R. E. GreenwoodReferences1. The numerical analysis is based on Robson's work at 1 Hz. See J. G. Robson,J. Opt. Soc. Am. 56, 1141 (1966).2. L. Cutrona, Ph. D. Thesis, Department of Psychology, M. I. T., 1970.3. G. F. Carrier, M. Krook, and C. E. Pearson, Functions of a Complex Variable(McGraw-Hill Book Company, Inc., New York, 1966). I would like to thank DavidBenney for suggesting that the equation was similar to Abel's equation.4. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products(Academic Press, Inc., New York, 1965). Formulas 2.562. 2, 2.563.2, and 6.621.1were useful.5. D. E. Mitchell, R. D. Freeman, and G. Westheimer, J. Opt. Soc. Am. 57, 246 (1967).B. TRANSFORMATIONS ON DIGITIZED PICTURES USING LOCALPARALLEL OPERATORS1. IntroductionA vital area of concern in automatic pattern recognition and image-processing appli-cations is that of preprocessing. Usually, a picture presented for analysis to a compu-ter is in digitized form. This report is concerned with the (common) case in whicha picture is represented as a rectangular array of 1 's and 0's with 1 corresponding toblack and 0 corresponding to white. Preprocessing can then be defined to be any trans-formation on the array such that some points are changed from 1 to 0 (erased), and someare changed from 0 to 1 (filled in).A family of such transformations can .be defined with the useful property thatthe connectivity of the picture is not altered. The following properties can also beimposed.Local - Each point in the new pattern is a function of the corresponding point in theold pattern and its eight neighbors (see Fig. IX-2).QPR No. 102P4 P3 P2P5 P P1P6 P7 P8Fig. IX-2. A point P and its eight neighbors.1001P 01001 1 1OP 10 0 10101P1011Fig. IX-3. Changing P will not alter connectivity.111OP 01 1 11 0 11P 00 1 11011P 0O1 0 1Fig. IX-4. Changing P will alter connectivity.01100PQ 00 1 1 00 1 1 01 1 0 00PQ 0001100001 1P 11 1Q 10 0 0 00 0 1 1OP 1 010001 0 0 0111000Fig. IX-5. Removal of only P or Q will not disconnect the set ofblack points in these pictures. Removal of both will.QPR No. 102 152(IX. COGNITIVE INFORMATION PROCESSING)Parallel -A new copy of the picture is generated separately from the old one, sothat each newly generated point is a function only of points on the original pattern.A transformation, then, can be viewed as an array of local operators, one for eachpoint, operating in parallel. In practice this is implemented sequentially, with one oper-ator generating a new picture point-by-point from the original.Any transformation can therefore be completely specified by specifying the condi-tions under which a single point may be changed, either from 1 to 0 or 0 to 1. One ofthese conditions is that changing the point does not alter connectivity. This conditionis discussed next, followed by a description of two particular transformations.2. ConnectivityA subset of a pattern is connected if for any two points p and q of the subset thereexists a sequence of points in the subset:p = poP P1' ... Pn-,Pn = qsuch that p.i is a neighbor of Pil , 1 < i - n. As in Fig. IX-2, each point is consideredto have 8 neighbors.Consider Fig. IX-3. In each of the three pictures, changing P either from blackto white or from white to black will not alter the connectivity of the black points. Thisis because the set of l's (blacks) around P is already connected. In each of the picturesin Fig. IX-4, the set of 1 's will be connected only if P is black. Changing P either wayalters connectivity.From this, we can state a rule that must be satisfied to change a point: A point maynot be changed, either from black to white or white to black unless the set comprisingits 8 neighbors contains at most one connected subset of black points.A simple means of calculating the number of connected black components in theneighborhood of P (excluding P) has been reported by Hilditch. Define the crossingnumber of P with respect to the subset containing its black neighbors as the number oftimes a "bug" taking a walk around P by way of its neighbors would have to cross fromoutside to inside the subset. The bug is permitted to take diagonal steps to avoid leavingthe black area. If the crossing number is 0, the number of connected black componentsis 0 or 1. Otherwise, the number of components is equal to the crossing number. InFig. IX-3, for example, the first two pictures have crossing number 1 and the last pic-ture has crossing number 0. In Fig. IX-4, the first two pictures have crossingnumber 2 and the last picture has crossing number 3.The crossing number X(P) is calculated as follows:4X(P) = b,i=l 1QPR No. 102 153(IX. COGNITIVE INFORMATION PROCESSING)where b. = 1 if P = 0, and either P2i or P = 1, and b. = 0 otherwise.3. Hole FillingConsider the transformation specified by the following set of rules. A point P willbe filled in, that is, changed from white to black, if and only if all of the following aretrue:1. P = 0. The point must be white initially.82. I P. > 4. Most of its neighbors are black.i=l 13. X(P) = 0 or 1.All other points remained unchanged.This transformation changes some points from white to black. A point is changedif most of its 8 neighbors are black and if connectivity is preserved. Such a transforma-tion can be used to eliminate isolated small sets of 0's, that is, to fill in small holes inblack regions. To achieve this effect, the transformation is applied repeatedly, alwaysusing the most recent version of the picture, until no more points are changed.4. ThinningAnother algorithm that can be specified is thinning. Thinning is the process ofreducing a picture to a stick figure or skeleton. A linelike black region is reduced to ablack line which is only one point thick. Again, this effect is achieved by repeated appli-cation of a transformation. The transformation to be applied removes boundary pointsin the picture. A boundary point is a black point, at least one of whose axially adjacentneighbors (P1, P3, P5, P7) is white. Each application peels away the outer layer ofthe black regions. This is continued until all black regions have been shrunk to mini-mum thickness without erasing them.This algorithm is somewhat complicated to explain, because of an effect producedby the parallel nature of the computation. The effect can be observed by noting the pic-tures of Fig. IX-5. In each case, the removal of either P or Q (for example, changingP or Q from black to white) does not alter connectivity. The removal of both theseboundary points does, however.One way out of this difficulty is to perform the thinning operation sequentially,changing points directly on the original picture. Unfortunately, as points are removedfrom the boundary of an object, subsequent nearby points become eligible for removal.The result is that the skeleton will tend to be biased toward one side or the other of theoriginal set of black points, depending on the order in which the points are taken.To maintain an essentially parallel type of computation, the following procedure isQPR No. 102 154(IX. COGNITIVE INFORMATION PROCESSING)adopted: each point P is tested with its 8 original neighbors. If the point is eligiblefor removal, it must then satisfy the further condition that its removal in conjunctionwith any of its neighbors that has been previously removed does not alter connectivity.Since the sequence in which the points are being tested is known, it is only necessaryto test for this condition for those neighbors that precede P. If the picture is examinedstarting at the top and going left to right, then it is only necessary to consider P2, P3,P4, and P5.Now notice that P2 may be ignored. The reasoning is as follows:The only case of interest is that in which P is black, and P2 has been changed fromblack to white, and the crossing number X(P) with P2 black is one. This is the case inwhich we wish to change P and must check that this change plus the change in P2 doesnot alter connectivity.Such a change in connectivity would take place only if X(P) with P2 white were greaterthan one.Suppose that P1 and P3 are white. If P is P2's only black neighbor, then, as weshall show, P2 would not have been erased. On the other hand, if P2 has other blackneighbors, then X(P2) > 1 and P2 still would not have been erased.It follows that at least one of P1 and P3 is black. Therefore, removal of P2 cannotchange the value of X(P), and so P can be erased without altering connectivity.The same line of reasoning applies to P4. To guarantee that the parallel removalof two adjacent points does not alter connectivity, it is sufficient to say that P may notbe erased if either P3 or P5 has been.The thinning transformation can now be specified. A point P will be erased, thatis, changed from black to white, if and only if all of the following are true:1. P = 1. The point must be black initially.2. P1 + P3 + P5 + P7 < 4. The point must be a boundary point.83. P. 1. If the point is on the tip of a thin line, it should not be erased,i=lthereby shortening the line.4. X(P) = 1.5. Neither P3 nor P5 has been erased.All other points remained unchanged.5. ConclusionTwo examples of connectivity-preserving transformations on digitized pictures havebeen given. A family of such transformations can easily be specified. Examples ofother functions that could be performed are thickening black regions, erasing isolatedQPR No. 102 155(IX. COGNITIVE INFORMATION PROCESSING)small sets of black points, and reducing each connected black region to a singlepoint.W. W. StallingsReferences1. C. J. Hilditch, "Linear Skeletons from Square Cupboards," Proc. Machine Intel-ligence Workshop 4 (1969).QPR No. 102156

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