of points that exist in only one set increases, the likelihood of missing the correct match will also increase. If point features of different types from the images can ...
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of points that exist in only one set increases, the likelihood of missing the correct match will also increase. If point features of different types from the images can be used, then the number of pairings of edges can be reduced as done by Stockman [10]. Each point might have a label such as "center of gravity" or "T intersection", which would greatly reduce the number of points with which it could correspond. The algorithm can also make good use of a priori bounds on the transformation parameters. Execution of the expensive step 5 of the algorithms could be bypassed whenever out-of-bounds parameter estimates are reached. REFERENCES [1] [2] [3] Fig. 13. The resampled MSS image.
[4]
[5] [6] [7] [8] [9] [10] [11] [12] [13] Fig. 14. The overlaid MSS and T M images.
[14]
D. Marr and E. Hildreth, "Theory of edge detection," Proc. Royal Soc. London, vol. B-207, 1980, pp 187-217. S. Yam and L. S. Davis, "Image registration using generalized Hough transform," in Proc. Pattern Recognition and Image Processing, 1981, pp. 525-533. L. N . Kanal, B. A. Lambird, D. Levine, and G. C. Stockman, "Digital registration of images from similar and dissimilar sensors," in Proc. Int. Conf. Cybern. Society, 1981, pp. 347-351. A. Goshtasby, " A symbolically-assisted approach to digital image registration with application in computer vision," Tech. Rep. 83-013, Dept. of Computer Science, Michigan State University, East Lansing, MI, 1983. W. A. Davis and S. K. Kenue, "Automatic selection of control points for the registration of digital images," in Proc. 4th Int. Joint Conf. Pattern Recognition, 1978, pp. 171-193. E. L. Hall, D. L. Davis, and M. E. Casey, " T h e selection of critical subsets for signal, image, and scene matching," IEEE Trans. Patt. Anal. Mach. IntelL, pp. 313-322, 1980. S. Ranade and A. Rosenfeld, "Point pattern matching by relaxation," Pattern Recognition, vol. 12, 1980, pp. 269-275. C. Wong, H. Sun, S. Yamada, and A. Rosenfeld, "Some experiments in relaxation image matching using corner features," Pattern Recognition, vol. 16, 1983, pp. 167-182. C. T. Zahn, " A n algorithm for noisy template matching," in IFIP Cong. 1974, pp. 698-701. G. C. Stockman, S. Kopstein, and S. Benett, "Matching images to models for registration and object detection via clustering," IEEE Trans. Patt. Anal. Mach. IntelL, vol. 4, no. 3, 1982, pp. 229-241. M. A. Fischler and R. C. Bolles, "Random sample consensus: A paradigm for model fitting with application to image analysis and automated cartography," Commun. ACM, vol. 24, no. 6, June 1981, pp 381-395. D. C. S. Allison and M. T. Noga, "Some performance tests of convex hull algorithms," BIT, vol. 24, 1984, pp. 2 - 1 3 . H. Samet and A. Rosenfeld, "Quadtree structures for image processing," Proc. 5th Int. Conf. Pattern Recognition, 1980, pp. 815-818. R. A. Jarvis, " O n the identification of the convex hull of a finite set of points in the plane," Inform. Processing Lett., vol. 2, 1973 pp 18-21.
An Application of the c-Varieties Clustering 100-percent overlap. In these cases the best match configuration Algorithms to Polygonal Curve Fitting could be missed. Also, when only a small set of feature points are available, any subsetting procedure is suceptible to error. JAMES C. BEZDEK, MEMBER, IEEE, AND IAN M. ANDERSON Although our programs were not written with total attention on speed, the timing results show that stage 2 (control-point Abstract—An algorithm is described that fits boundary data of planar correspondence) of the registration process can be performed in less time than stage 1 (control-point selection). Since, automatic shapes in either rectangular coordinate or chain-coded format with a set of control point selection is a more difficult stage, the deliberate straight line segments. The algorithm combines a new vertex detection search for an optimal match with imperfect data seems justifiable method, which locates initial vertices and segments in the data, with the c-elliptotype clustering algorithm, which iteratively adjusts the location of for stage 2. With feature points spread somewhat uniformly in the plane, these initial segments, thereby obtaining a best polygonal fit for the data in the computational savings due to use of the convex hull will the mean-squared error sense. Several numerical examples are given to become larger as the number of points increases. The chance of exemplify the implementation and utility of this new approach. missing the correct match becomes smaller provided that by I. INTRODUCTION increasing the points in a set, the number of points on the Let B = { b x , b 2 ,..., b N } be TV points in the plane ordered boundary of its convex hull increases also. If the two sets of points have only translational, rotational, and scaling differences, along the boundary of some planar shape. For such a data set, we there is no chance of missing the correct match by any version of Manuscript received October 6, 1984; revised May 13, 1985. This work was the matching algorithm because at some step in the iteration, two supported in part by the National Science Foundation under grants IST-84correctly corresponding edges from the convex hulls must be 07860 and MCS-80-02328. J.C. Bezdek is with the Department of Computer Science, University of paired and subsequently all N point correspondences verified. As South Carolina, Columbia, SC 29208, USA. I. M. Anderson is with the the amount of noise in either or both sets increases or the number Department of Mathematics, Utah State University, Logan, U T 84322, USA. 0018-9472/85/0900-0637$01.00 ©1985 IEEE
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present a new algorithm that finds a set of e Une segments, which collectively provide a good polygonal fit to the points in B. This problem, variously referred to in the literature as segmentation of plane curves, curve fitting, linear splining or polygonal approximation, plays an important role in a wide variety of imageprocessing applications. Excellent surveys of many existing polygonal approximation routines, together with their many applications may be found in Duda and Hart [4], Hall [5], Pavlidis [7], and Rosenfeld and Kac [8]. Generally, polygonal approximation routines consist of three steps. The first step is one of initialization—vertices are tentatively located in the data that provide an initial polygonal approximation. In the second step, this initial polygon is iteratively adjusted, either by adding additional edges or by changing the slopes and centers of existing edges. This adjustment continues until some specific goodness of fit criteria is met. Finally, in step three the polygonal approximation may be "tidied up"—edges which are very nearly the same may be merged or concatenated, edges may be extended to form a connected polygonal approximation or, alternatively, edges supported by too few data points may be deleted altogether. The algorithm described here, henceforth called boundary-fit, uses in step 1 the vertex detection method described in detail in Anderson and Bezdek [1]. This method is based upon a new geometric property of sample variance-covariance matrices; viz., that the matrix commutator of two such matrices constructed from successive arcs in the boundary data provides an analytical measure of the tangential deflection between these arcs. This leads to a highly reliable placement of the initial vertices (each vertex is a point in the data set B) and results in quicker convergence of the iterative process. To adjust this initial polygonal approximation, the Boundaryfit program uses the fuzzy c-elliptotype (FCE) algorithm developed by Bezdek and Coray et. al. [3]. This algorithm is a sophisticated extension of the well-known "scatter matrix, eigenvalue-eigenvector" line-fitting method (see Duda and Hart [4] pp. 332-334), and it is designed specifically to accommodate data sets with many linear structures. It can be described briefly as follows. First, the FCE algorithm assigns, during each iteration, to each point bj, j' = 1,2, · · · N of the boundary, a membership value wZ7 in each line segment Lz, i = 1,2, · · · c. These membership values are constrained by the conditions
Fig. 1. Geometry of the FCE functional J. A is any positive definite (2 X 2) weight matrix. D^ is the OG distance from A, to L ; with respect to the inner product norm induced by matrix A and dfj = ||b,· — vjß = (b,· — ν,·)^(ο,- — v,)·
Here a, 0 < a < 1, is viewed as a mixing coefficient. Loosely speaking, for example, with a = 0.8 (the value used in the subsequent examples), 80 percent of J measures deviations from linear shape within each cluster or edge, while 20 percent of J assesses the central tendencies of these clusters. In ideal situations the memberships utj converge to zero or one, a may be assigned a value close to one, and J approximates the sum of the squared distances from the data points to their assigned Unes.1 Any inner product induced norm can be used for dtJ and /),·■; in the sequel we take A = I, corresponding to choosing the Euclidean norm on R2. Another characteristic of the FCE algorithm is reflected by the presence of the user specified parameter m, 1 < m < oo. Spurious boundary points will typically have lower memberships in many lines so that with m large, their contribution to the functional / is minimal. Thus implicit in the FCE routine is a simple yet very effective means of suppressing noisy data points, which might otherwise result in unwanted or inaccurate edges. Front-end filtering or smoothing of the boundary is therefore unnecessary. The value m = 2 was used in the numerical examples that follow. To minimize / , the edge centers { vÉ}, their directions { dt} and the membership matrix [ul ] are iteratively adjusted as follows. Initially, the {vt} and {*/, j are taken to be the midpoints and directions of the zth edge as constructed from the vertex detection routine. The initial membership matrix is a so-called "hard" membership matrix with
I
0 ^ K,., < 1 and
Σ «„■ = i. /-I
If the point b} is very close to edge Li9 bj has a membership value in L, that is close to one; otherwise the membership value is small. Points near vertices in the polygonal approximation are assigned memberships of nearly 1/2 in the adjacent edges. This is in contrast to other line-fitting routines, where each data point must be associated with one and only one edge in the polygonal approximation. The information contained in the c X N membership matrix [Uj ■] is used to define the goodness of fit criteria. Specifically, suppose the edges are described parametrically by Li = {v( + /d ; |û z < t < bt) where vi is a point on the edge and dx is a unit vector in the direction of L,. Let Z>/y be the othogonal distance from the point bj to the edge Lz and let */,· ■ be the distance from bj to Vj (Fig. 1). The FCE algorithm attempts to find a local minimum in the variables {(ί/^,ν,,α,)} of the functional
J=t
E[^]w[^2'7+(i-«)^y]-
/-ly-l
(i)
I,
if b, is between the ith
and (i + l)-th vertex 0, otherwise. These variables are updated in accordance with the theory developed in [3]. First the new points i?inew) are computed from
v, H( Y) iîH(X)*H(Y).
I H( X\Y)/H( X), \H(Y\X)/H(Y),
Note that this definition is legitimate since H(X) = H(Y) implies H(X\Y) = H(J\X) from (1). Theorem (A) (B) (C)
Entropy and Correlation YASUICHI HORIBE Abstract—A simple and intuitively interprétable correlation coefficient between two discrete random variables is defined. Its metric property is then proved.
Consider the discrete random variables X, Y, Z, · · ·, such that 0 < H(X)< oo, 0 < i / ( 7 ) < o o , 0 < H(Z)< o c , · · · , where H(X) denotes the uncertainty (entropy) of X in information theory. The conditional entropy H(X\Y), which measures the uncertainty of X remaining after knowing Y, is also well known. Hence f/(X, Y) = 1 - d'(X, 7), d'(X, Y) = H(X\Y)/H(X), serves as a correlation coefficient, and d'(X, Y) is a correlation distance between X and Y. It is not, however, symmetric. This is immediately seen from the equality H(X)
- H(X\Y)
= H(Y) -
H(Y\X)
(D) (E)
0
H(X\Y) H(X)
H(Z\Y) H(Z)
, '
H(X\Y) H(X)
H(Y\Z) H(Z)
H(X\Y) H(X)
H(Y\Z) H(X)
>
H(X\Z) H(X)
_ , ( γ 7 Λ -d^X'Z>
where the first inequality is due to (1) and the third to the following: H(X\Y)
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+ H(Y\Z)
>
H(X\Z).
(2)
