- Email: [email protected]
On the Wechsler de Souza discussion
Pattern Reco~t~ition Vol. 16, No. 2, pp. 269 270, 1983. Printed in Great Britain.
0031 320383,020269 02 NI3.000 Pergamon Press Ltd. Pattern Recognition Society
COMMENT ON THE WECHSLER-DE
SOUZA DISCUSSION*
J. K. PERCUS Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012, U.S.A. (Received for publication 29 November 1982)
abilities p~ that a walker starting at the center pixel first leave the internal square along one of the four outer edges, p~ are of course compatable functions of N ( x , y ) and constitute the desired descriptive parameters of the image in the window. A single composite parameter, the squared gradient gz = (p~ _ p~)2 + (PA -- PR)2, is also set up, where L, R, A, B refer to left, right, top, bottom. Thus far, there is no controversy. But then Wechsler and Citron, instead of computing pj directly, a somewhat time-consuming task, adopt a Monte Carlo strategy ; they literally run a sequence of r a n d o m walks and note the numbers of arrivals at the four edges. In their first application, Wechsler and Citron test the hypothesis that the image in the window is uniform by performing a ~2 test against the expected arrival frequencies on the four sides on the assumption of {AN = N(x,y) - N(x',y')/(x - x') 2 + ( y - y,)2 = 1}, uniformity of intensity, de Souza ~2~now argues that the statistical input to this procedure is nil ; p~ are fixed by although the set {AN} is highly linearly dependent. known N(x, y) and so only the fluctuations or noise in There are now three steps to be taken: 1. choice of the r a n d o m walk statistics are being tested. Indeed he appropriate combinations of AN as characteristic says, and Wechsler is led to agree, the random walk parameters ; 2. numerical construction of these parasequence is being used as a non-deterministic defimeters ; 3. use of the parameters for classifying texture. nition of uniformity, and one that depends upon the An ingenious suggestion has been made by Wechsler number of walks chosen and the level of confidence and coworkers (see Wechsler and Citron Ill and reused in the Zz test. In fact, one can best describe the ferences contained therein) that AN be used to define a procedure by saying that it corresponds to a pararandom walk in which the relative transition probmetric set of fuzzy tests, and then assume that some ability from a pixel to a neighbor is precisely [AN 1. pair of parameters corresponds to an empirical conThe r a n d o m walk samples texture by preferentially sensus as to the division into uniform and non-uniform wandering along gradients in the intensity map of the distributions, with the non-uniqueness of the criterion image. To characterize the r a n d o m walk, Wechsler et mirroring the variety of b o u n d a r y cases effectively al. isolate the n - 2 x n - 2 internal square with its including psychological input. It should be regarded as 2(n - 1 )(n - 2) connections and choose the four proba tentative approach which may then serve heuristically in the development of a quantifiable procedure. In their second more extensive application, Wechsler and Citron introduce the squared gradient alluded *Supported in part by DOE, Contract DE-AC02to above to characterize each window associated with 76ER03077 269
! would like to comment on the discussion that has developed concerning a test for texture classification. Suppose that the portion of a black and white image covered by a square window of n units by n units is to be classified and that each pixel, or unit subsquare centered at location (x, y), x, y = 1,..., n, is represented by its mean grey level N(x, y). At this level of resolution, all information is contained in the set {N(x, y)} of n 2 elements. The basic problem is to select combinations of N(x, y) which categorize the texture of the image, or as a subcase, which distinguish it from one which is essentially structureless or uniform. This is clearly a global problem in which neighbor relations are important, so that a permutation of {N(x, y)} will certainly destroy the type of texture. As a start, then, it is more suitable to consider the 2 n ( n - 1) nearest neighbor differences
270
Comment
a grid placed over the image in question, and then characterize the full image by the sequence of squared gradients. Two images are to be regarded as equivalent in texture if their sequences are sufficiently close in a suitable metric, or one may use the metric to partially order the sequences. This is a test in which the exact squared gradients could in principle be used, although the authors choose the presumably more efficient random walk estimate instead. In summary, Wechsler et al. present a possible, and plausible, set of parameters to describe texture, the structure beyond uniformity. In the form they emphasize, with random walks starting from a central pixel alone, this may be rather a test of anisotropy, but it is certainly an entering wedge. Their computation of
these parameters is put in a statistical context which taken literally is wholly inappropriate, as de Souza points out in detail (reinforced by a simulation study which, however, does not include the relations among AN). Nevertheless, I would expect that this procedure will ultimately lead to one, utilizing related underlying concepts, in which the test parameters are sufficiently simple that recourse to Monte Carlo estimation at high resolution will prove unnecessary. REFERENCES
1. H. Wechslerand T. Citron, Feature extraction for texture classification, Pattern Recognition 12, 301 (1980). 2. P. de Souza, A note on a random walk model for texture analysis, Pattern Recognition 16, 219 (1983).
0031 320383,020269 02 NI3.000 Pergamon Press Ltd. Pattern Recognition Society
COMMENT ON THE WECHSLER-DE
SOUZA DISCUSSION*
J. K. PERCUS Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012, U.S.A. (Received for publication 29 November 1982)
abilities p~ that a walker starting at the center pixel first leave the internal square along one of the four outer edges, p~ are of course compatable functions of N ( x , y ) and constitute the desired descriptive parameters of the image in the window. A single composite parameter, the squared gradient gz = (p~ _ p~)2 + (PA -- PR)2, is also set up, where L, R, A, B refer to left, right, top, bottom. Thus far, there is no controversy. But then Wechsler and Citron, instead of computing pj directly, a somewhat time-consuming task, adopt a Monte Carlo strategy ; they literally run a sequence of r a n d o m walks and note the numbers of arrivals at the four edges. In their first application, Wechsler and Citron test the hypothesis that the image in the window is uniform by performing a ~2 test against the expected arrival frequencies on the four sides on the assumption of {AN = N(x,y) - N(x',y')/(x - x') 2 + ( y - y,)2 = 1}, uniformity of intensity, de Souza ~2~now argues that the statistical input to this procedure is nil ; p~ are fixed by although the set {AN} is highly linearly dependent. known N(x, y) and so only the fluctuations or noise in There are now three steps to be taken: 1. choice of the r a n d o m walk statistics are being tested. Indeed he appropriate combinations of AN as characteristic says, and Wechsler is led to agree, the random walk parameters ; 2. numerical construction of these parasequence is being used as a non-deterministic defimeters ; 3. use of the parameters for classifying texture. nition of uniformity, and one that depends upon the An ingenious suggestion has been made by Wechsler number of walks chosen and the level of confidence and coworkers (see Wechsler and Citron Ill and reused in the Zz test. In fact, one can best describe the ferences contained therein) that AN be used to define a procedure by saying that it corresponds to a pararandom walk in which the relative transition probmetric set of fuzzy tests, and then assume that some ability from a pixel to a neighbor is precisely [AN 1. pair of parameters corresponds to an empirical conThe r a n d o m walk samples texture by preferentially sensus as to the division into uniform and non-uniform wandering along gradients in the intensity map of the distributions, with the non-uniqueness of the criterion image. To characterize the r a n d o m walk, Wechsler et mirroring the variety of b o u n d a r y cases effectively al. isolate the n - 2 x n - 2 internal square with its including psychological input. It should be regarded as 2(n - 1 )(n - 2) connections and choose the four proba tentative approach which may then serve heuristically in the development of a quantifiable procedure. In their second more extensive application, Wechsler and Citron introduce the squared gradient alluded *Supported in part by DOE, Contract DE-AC02to above to characterize each window associated with 76ER03077 269
! would like to comment on the discussion that has developed concerning a test for texture classification. Suppose that the portion of a black and white image covered by a square window of n units by n units is to be classified and that each pixel, or unit subsquare centered at location (x, y), x, y = 1,..., n, is represented by its mean grey level N(x, y). At this level of resolution, all information is contained in the set {N(x, y)} of n 2 elements. The basic problem is to select combinations of N(x, y) which categorize the texture of the image, or as a subcase, which distinguish it from one which is essentially structureless or uniform. This is clearly a global problem in which neighbor relations are important, so that a permutation of {N(x, y)} will certainly destroy the type of texture. As a start, then, it is more suitable to consider the 2 n ( n - 1) nearest neighbor differences
270
Comment
a grid placed over the image in question, and then characterize the full image by the sequence of squared gradients. Two images are to be regarded as equivalent in texture if their sequences are sufficiently close in a suitable metric, or one may use the metric to partially order the sequences. This is a test in which the exact squared gradients could in principle be used, although the authors choose the presumably more efficient random walk estimate instead. In summary, Wechsler et al. present a possible, and plausible, set of parameters to describe texture, the structure beyond uniformity. In the form they emphasize, with random walks starting from a central pixel alone, this may be rather a test of anisotropy, but it is certainly an entering wedge. Their computation of
these parameters is put in a statistical context which taken literally is wholly inappropriate, as de Souza points out in detail (reinforced by a simulation study which, however, does not include the relations among AN). Nevertheless, I would expect that this procedure will ultimately lead to one, utilizing related underlying concepts, in which the test parameters are sufficiently simple that recourse to Monte Carlo estimation at high resolution will prove unnecessary. REFERENCES
1. H. Wechslerand T. Citron, Feature extraction for texture classification, Pattern Recognition 12, 301 (1980). 2. P. de Souza, A note on a random walk model for texture analysis, Pattern Recognition 16, 219 (1983).