Object Matching Algorithms Using Robust Hausdorff ... - CiteSeerX

IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 8, NO. 3, MARCH 1999

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[2] D. Stauffer and A. Aharony, Introduction to Percolation Theory, 2nd ed., rev. London, U.K.: Taylor & Francis, 1994. [3] D. C. Rapaport, “Cluster size distribution at criticality,” J. Stat. Phys., vol. 66, pp. 679–682, 1992. [4] J. Hoshen, M. Berry, and K. S. Minser, “Percolation and cluster structure parameters. I. The enhanced Hoshen-Kopelman algorithm,” Phys. Rev. E, vol. 56, pp. 1455–1460, 1997. [5] A. Rosenfeld and J. L. Pfalz, “Sequential operations in digital picture processing,” J. Assoc. Comput. Machin., vol. 13, pp. 471–494, 1966. [6] R. Lumia, L. Shapiro, and O. Zuniga, “A new connected components algorithm for virtual memory computers,” Comput. Vis., Graph, Image Process., vol. 22, pp. 287–300, 1983. [7] J. L. C. Sanz and D. Petkovic, “Machine vision algorithm for automated inspection of thin-film disk head,” IEEE Trans. Pattern Anal. Machine Intell., vol. 10, pp. 830–848, 1988. [8] R. M. Haralick, “Some neighborhood operations,” in Real Time/Parallel Computing Image Analysis, M. Onoe, K. Preston, and A. Rosenfeld, Eds. New York: Plenum, 1981, pp. 11–37. [9] R. Cypher and J. L. C. Sanz, “SIMD architecture and algorithms for image processing and computer vision,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 37, pp. 2158–2174, 1989. [10] D. M. Perry, P. J. Moran, and G. M. Robinson, “Three dimensional surface metrology of magnetic recording materials through direct-phasedetecting microscopic interferometry,” J. Inst. Electron. Radio Eng., vol. 55, pp. 145–150, 1985. [11] W. E. King et al, “X-Ray tomographic microscopy investigation of the ductile rupture of an aluminum foil bonded between Sapphire blocks,” Scripta Metall. Mater., vol. 33, pp. 1941–1946, 1995. [12] J. Hoshen, “Percolation and cluster structure parameters: The radius of gyration,” J. Phys. A., vol. 30, pp. 8459–8469, 1977.

on the type of the features used, matching measure criterion, and so on. Low-level matching algorithms, i.e., algorithms using a distance transform (DT) [4]–[6] and a Hausdorff distance (HD) [7]–[9] have been investigated because they are simple and insensitive to changes of image characteristics. In this paper, we apply the robust statistics of regression analysis [10], [11] to the computation of the HD measures for object matching, resulting in two robust HD measures: M-HD based on M-estimation and least trimmed square-HD (LTS-HD) based on LTS. The two proposed robust approaches yield the correct results, even though the input data are severely corrupted. The rest of the paper is structured as follows. Section II reviews the conventional HD measures and related robust statistics. In Section III, two proposed robust object matching algorithms based on HD measures are presented [12]–[14]. Experimental results for images with occlusions and noisy images are shown in Section IV, and conclusions are given in Section V. II. OBJECT MATCHING ALGORITHMS USING THE CONVENTIONAL HD MEASURES The HD measure computes a distance value between two sets of edge points extracted from the object model and a test image. The classical HD measure [7] between two point sets A = fa1 ; 1 1 1 ; aN g and B = fb1 ; 1 1 1 ; bN g of sizes NA and NB , respectively, is defined as H (A; B )

Object Matching Algorithms Using Robust Hausdorff Distance Measures Dong-Gyu Sim, Oh-Kyu Kwon, and Rae-Hong Park

= max(h(A;

Index Terms—Hausdorff distance, least trimmed square, M-estimation, object matching.

Object matching in two-dimensional (2-D) images has been an important topic in computer vision, object recognition, and image analysis [1]–[3]. The performance of the matching method depends Manuscript received June 26, 1997; revised May 27, 1998. This paper was presented in part at the IEEE International Conference on Image Processing, Lausanne, Switzerland, September 1996. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Dmitry B. Goldgof. D.-G. Sim and R.-H. Park are with the Department of Electronic Engineering, Sogang University, Seoul 100-611, Korea (e-mail: [email protected]). O.-K. Kwon was with the Department of Electronic Engineering, Sogang University, Seoul 100-611, Korea. He is now with MJL Korea, Ltd., Seoul 150–010, Korea. Publisher Item Identifier S 1057-7149(99)10560-8.

=

th

Ka

2A dB (a)

(2)

where dB (a) represents the minimum distance value at point a to the point set B , and Kath2A denotes the K th ranked value of dB (a). This HD measure needs one parameter, f = K=NA , whose range is from 0.0 to 1.0. Depending on the fractional value of f , its performance widely varies. Experimentally, when f is about 0.6, good matching results are obtained. By modifying the HD based on the ranked order statistics, Azencott et al. proposed the CHD measure in comparing binary images [8]. The directed distance of the CHD is defined as hK; L (A; B )

I. INTRODUCTION

(1)

where h(A; B ) and h(B; A) represent the directed distances between two sets A and B . Huttenlocher et al. proposed the partial HD measure in comparing partial portions of images containing severe occlusions or degradation [7]. The directed distance of the partial HD is defined as hK (A; B )

Abstract—A Hausdorff distance (HD) is one of commonly used measures for object matching. This work analyzes the conventional HD measures and proposes two robust HD measures based on M-estimation and least trimmed square (LTS) which are more efficient than the conventional HD measures. By computer simulation, the matching performance of the conventional and proposed HD measures is compared with synthetic and real images.

B ); h(B; A))

=

th

th

2

2

jj 0 jj

(3)

dB (a):

(4)

Pa A Qb B

a

b

th ranked value of Qbth2B jja 0 bjj, with 2 representing the Qth ranked value of the Euclidean distance set. The CHD measure requires two parameters: p = P =NA and q = Q=NE , where NE denotes the size of the Euclidean distance set. The range of both parameters is from 0.0 to 1.0. Experimentally, when p is from 0.8 to 0.9 and q is from 0.01 to 0.05, good matching results are obtained. Dubuisson and Jain proposed the MHD based on the average distance value in comparing the synthetic images contaminated by four types of noise [9]. The directed distance of the MHD is defined as where

th Qb B

th

2 denotes the

Pa A

1057–7149/99$10.00  1999 IEEE

P

hMHD (A; B )

=

1 NA

2A

a

426


In the proposed LTS-HD based on the LTS scheme [11], [12], the directed distance hLTS (A; B ) is defined by a linear combination of order statistics

hLTS (A; B ) =

(a)

(b)

(c)

2

Fig. 1. Matching result of a synthetic test image. (a) Object model (72 72). (b) Test image (256 256). (c) Matching result of (a) and (b) by the proposed LTS-HD (h = 0:6).

2

The MHD measure does not require a parameter, however its matching performance is not good, compared with that of the partial HD and the CHD, because it employs the summation operator over all distances some of which might be computed from outliers. III. PROPOSED OBJECT MATCHING ALGORITHMS USING ROBUST HD MEASURES This work proposes two robust HD measures based on the robust statistics such as M-estimation and LTS. The directed distance hM (A; B ) of the proposed M-HD based on M-estimation is realized by replacing the Euclidean distance by the cost function that can eliminate outliers [11], [14], [15]. M-estimation minimizes the summation of a cost function . The directed distance hM (A; B ) is defined as

hM (A; B ) =

1

NA a2A

(dB (a))

where the cost function is convex and symmetric and has a unique minimum value at zero. In our experiments, we use the cost function defined by

(x) =

jxj; ;

xj xj >

1

H

H i=1

dB (a)(i)

where H denotes h 2 NA , as in the partial HD case, and dB (x)(i) represents the ith distance value in the sorted sequence (dB (x)(1) dB (x)(2) 1 1 1 dB (x)(N ) ). The measure hLTS (A; B ) is minimized by remaining distance values after large distance values are eliminated. So, even if the object is occluded or degraded by noise, this matching scheme yields good results. An optimal fraction h, whose range is from zero to one, depends on the amount of occlusion. If h is equal to one, this HD measure corresponds to the conventional MHD. In the proposed algorithms, a full search is adopted for finding the optimal matching location, that is, the point that yields a minimum distance is selected because the proposed algorithm are based on a distance measure. The computation time can be reduced by introducing efficient search algorithms such as hierarchical or pyramid schemes. The object matching algorithms based on two robust HD measures are insensitive to outliers and occlusions, because of employment of the robust estimation in computing the HD measures. Also they yield an efficiency larger than that of the conventional ones, because of the average operation embedded into them, resulting in algorithms relatively insensitive to the change of parameters, e.g., of the M-HD and h of the LTS-HD. The M-HD measure requires comparison and summation operations, whereas the LTS-HD measure requires sorting and summation operations. The partial HD requires a sorting operation as in LTSHD measure computation, where sorting operations can be computed with linear time and real-time algorithms [16], [17]. As a result, the computational complexity of the two proposed HD measures is almost the same as that of the conventional HD measures such as the partial HD and the MHD. Note that the DT map can be used for fast computation of various HD measures and that the computational complexity of the CHD is much higher than that of other HD measures. IV. EXPERIMENTAL RESULTS

AND

DISCUSSIONS

To show the matching performance of the proposed HD measures, we compare the performance of the conventional and proposed HD measures, in terms of the matching position, for synthetic and several real images with various levels of noise, distortion, and occlusions. The matching position is defined by the position, detected by each HD measure, of the object model with respect to the test image. Because the object matching by the CHD is not attractive in terms of the computational load and its matching performance greatly depends on two parameters p and q , we just select the partial HD and MHD as the conventional HD measures in performance comparison. They have almost the same computational complexity and the same number of parameters as those of the proposed HD measures.

j j

where is a threshold to eliminate outliers, so the outliers yielding large errors are discarded. In M-HD, because the matching performance depends on the parameter , it is important to determine appropriately. If is set to infinity, this HD measure is equivalent to the conventional MHD. Because the cost function is associated with the distance value dB (a), the threshold value between three and five is selected experimentally.

A. Performance Comparison of HD Measures for Matching of Occluded and Degraded Objects Fig. 1 shows the matching result of a synthetic test image containing occlusion. The 72 2 72 object model, a square, is shown in Fig. 1(a), and a 256 2 256 test image composed of two quadrangles, a circle, and a triangle is shown in Fig. 1(b). Also the matching result of the proposed LTS-HD, with h = 0:6, is shown in Fig. 1(c), where the object model is superimposed on the test image. Note that the


MATCHING POSITION RESULTS

(a)

BY

TABLE I CONVENTIONAL

(b)

Fig. 2. Matching result of a real test image. (a) Object model (72 by the two proposed HD measures.

427

AND

PROPOSED HD MEASURES

(c)

2 72). (b) Test image (256 2 256). (c) Identical matching result of (a) and (b)

proposed HD yields the correct position, even if the object model is partially occluded. Table I shows the comparison of the matching performance in terms of the matching position detected by the conventional and proposed HD measures, for the synthetic image shown in Fig. 1(b), where the correct position is (74, 62). The proposed HD measures yield the position error smaller than the partial HD and MHD. In experiments with the synthetic image, the proposed HD measures give more accurate position than the partial HD, because of the average operation embedded in the proposed HD measures. Also, the proposed robust HD measures produce better results than the MHD by effectively removing outliers caused by occlusions.

As a real test image, we use the 256 2 256 “road” image contaminated by Gaussian noise (standard deviation = 30) [8]. Also about 35% pixels of the target portion of an input image is deleted. The 72 2 72 object model and the test image are shown in Fig. 2(a) and (b), respectively. The matching result obtained by the two proposed HD measures is shown in Fig. 2(c), where the object model is superimposed on the test image. Table I shows that the proposed HD measures yield better results than the conventional ones for the real image. The matching position estimated by the two proposed HD measures (M-HD with = 3; 4; 5 and LTS-HD with h = 0:60; 0:70; 0:80), is correct, thus the superimposed matching results are identical, as shown in Fig. 2(c). On the other hand, the matching position detected by

428


(a)

(b)

(c) Fig. 3. Matching results by the conventional and proposed HD measures with the various amount of uniform noise (U ) and fixed line noise (V = 10): (a) partial HD; (b) M-HD; (c) LTS-HD.

two conventional HD matching (with f = 0:6 for the partial HD, matching results are almost correct) is not correct. According to Table I, a false alarm (123, 158) is detected. This point yields a small distance with a large similarity, whenever the correct matching point is detected.

B. Performance Comparison of HD Measures for Noisy Binary Images Fig. 3 shows the RMS matching position error for the noisy road image by the conventional and proposed HD measures, as a function of the corresponding parameter of each measure, with the amount of uniform noise level U and fixed line noise (V = 10) varying. As the uniform noise U increases, the RMS position error increases. Fig. 3(a) shows the RMS matching position error by the conventional partial HD, when the partial fraction f is from 0.5 to 1.0. Because the outliers are not effectively removed with respect to ranked order statistics, when f is larger than 0.6, the matching performance is not good. In Fig. 3(b), because there is no data which can measure the similarity between objects, when is equal to 0.0, the matching result is not good. Also for large , because the outliers are not effectively removed by the cost function , the RMS position error is large. Fig. 3(c) shows that if the parameter h of the LTS-HD is from 0.50 to 0.90, the RMS position error is 0, i.e., the matching results are perfect. If h is larger than 0.90, the LTS-HD yields incorrect results, because the outliers are not effectively removed. The parameters of the proposed M-HD and LTS-HD can not be accurately determined. Though the parameters of the proposed robust HD measures depend on the amount of noise and various kinds of objects/noise, appropriate and h of M-HD and LTS-HD are about 5 and 0.7, respectively, which are selected experimentally.

V. CONCLUSIONS This correspondence proposes two efficient HD measures that are robust to outliers and occlusions. They are realized by applying the robust estimators such as M-estimation and LTS methods to the conventional HD algorithm. The effectiveness of the proposed HD measures are tested with various images and noise environments. Further research will focus on the application of the hierarchical structure to two proposed HD measures to reduce the computation time. Also application of other robust estimators to computation of the HD measures is to be investigated. REFERENCES [1] E. Persoon and K. S. Fu, “Shape discrimination using Fourier descriptors,” IEEE Trans. Syst., Man, Cybern., vol. SMC-7, pp. 170–179, Mar. 1977. [2] N. Ayache and O. D. Faugeras, “Hyper: A new approach for the recognition and positioning of two-dimensional objects,” IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-8, pp. 44–54, Jan. 1986. [3] B. Bhanu, “Shape matching of two-dimensional objects,” IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-6, pp. 137–155, Mar. 1984. [4] G. Borgefors, “Hierarchical chamfer matching: A parametric edge matching algorithm,” IEEE Trans. Pattern Anal. Machine Intell., vol. 10, pp. 849–865, Nov. 1988. [5] H.-C. Liu and M. D. Srinath, “Partial shape classification using contour matching in distance transform,” IEEE Trans. Pattern Anal. Machine Intell., vol. 12, pp. 1072–1079, Nov. 1990. [6] J. You, E. Pissaloux, W. P. Zhu, and H. A. Cohen, “Efficient image matching: A hierarchical chamfer matching scheme via distributed system,” Real-Time Imag., vol. 1, pp. 245–259, Oct. 1995. [7] D. P. Huttenlocher, G. A. Klanderman, and W. J. Rucklidge, “Comparing images using the Hausdorff distance,” IEEE Trans. Pattern Anal. Machine Intell., vol. 15, pp. 850–863, Sept. 1993. [8] R. Azencott, F. Durbin, and J. Paumard, “Multiscale identification of building in compressed large aerial scenes,” in Proc. 13th Int. Conf. Pattern Recognition, Vienna, Austria, Aug. 1996, vol. 2, pp. 974–978. [9] M.-P. Dubuisson and A. K. Jain, “A modified Hausdorff distance


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for object matching,” in Proc. 12th Int. Conf. Pattern Recognition, Jerusalem, Israel, Oct. 1994, pp. 566–568. P. Meer, D. Mintz, D. Y. Kim, and A. Rosenfeld, “Robust regression methods for computer vision: A review,” Int. J. Comput. Vis., vol. 6, pp. 59–70, Apr. 1991. P. J. Rousseeuw and A. M. Leroy, Robust Regression and Outlier Detection. New York: Wiley, 1987. I. Pitas and A. N. Venetsanopoulos, Nonlinear Digital Filters. Boston, MA: Kluwer, 1990. P. J. Besl, J. B. Birch, and L. T. Watson, “Robust window operators,” in Proc. 2nd Int. Conf. Computer Vision, Tampa, FL, Dec. 1988, pp. 591–600. C. V. Stewart, “A new robust operator for computer vision: Theoretical analysis,” in Proc. IEEE Conf. Computer Vision and Pattern Recognition, Seattle, WA, June 1994, pp. 1–8. D.-G. Sim et al., “Navigation parameter estimation from sequential aerial images,” in Proc. Int. Conf. Image Processing, Lausanne, Switzerland, Sept. 1996, vol. 2, pp. 629–632. D. S. Richards, “VLSI median filters,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 38, pp. 145–153, Jan. 1990. J. P. Fitch, “Software and VLSI algorithms for generalized ranked order filtering,” IEEE Trans. Circuits Syst., vol. CAS-34, pp. 553–559, May 1987.

Hybrid Estimation of Navigation Parameters from Aerial Image Sequence Dong-Gyu Sim, Sang-Yong Jeong, Doh-Hyeong Lee, Rae-Hong Park, Rin-Chul Kim, Sang Uk Lee, and In Chul Kim Fig. 1.

Proposed integrated position estimation system.

Abstract— This work presents a hybrid method for navigation parameter estimation using sequential aerial images, where navigation parameters represent the position and velocity information of an aircraft for autonomous navigation. The proposed hybrid system is composed of two parts: relative position estimation and absolute position estimation. Computer simulation with two different sets of real aerial image sequences shows the effectiveness of the proposed hybrid parameter estimation algorithm. Index Terms—Aerial image, digital elevation model, image matching, navigation, recovered elevation model.

I. INTRODUCTION Estimation of navigation parameters is important for autonomous navigation and many approaches have been presented [1]–[5]. This paper investigates the estimation of navigation parameters for an Manuscript received August 13, 1997; revised April 27, 1998. This work was supported in part by the Agency for Defense Development and by ACRC (Automatic Control Research Center), Seoul National University. An earlier version of this paper was presented at the IEEE International Conference on Image Processing, Lausanne, Switzerland, September 1996. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Dmitry B. Goldgof. D.-G. Sim and R.-H. Park are with the Department of Electronic Engineering, Sogang University, Seoul 100-611, Korea. S.-Y. Jeong was with the Department of Electronic Engineering, Sogang University, Seoul 100-611, Korea. He is now with the Kia Motors Corporation, Kwangmyung City, 423–050 Korea. D.-H. Lee was with the Department of Electronic Engineering, Sogang University, Seoul 100-611, Korea. He is now with Samsung Electronics Company, Ltd., Suwon City, 442–742, Korea. R.-C. Kim is with the School of Information and Computer Engineering, Hansung University, Seoul 136-792, Korea. S. U. Lee is with the School of Electrical Engineering, Seoul National University, Seoul 151-742, Korea. I. C. Kim is with the Agency for Defense Development, Daejon 305-600, Korea. Publisher Item Identifier S 1057-7149(99)01557-2.

Fig. 2. Relative position estimation.

aircraft using sequential aerial images. Because only the aerial image sequence is used as an input in our navigation system, the navigation system has advantages in that it is not detected by enemies nor guided by external signals, compared with other active approaches. Also, it can be attached to an aircraft without any special apparatus for compensation of an attitude change. Two test aerial sequences used in this work were acquired from a camera fixed on a light airplane and a helicopter, in which the optical axis of the camera varies according to the aircraft attitude. This work presents an integrated system for navigation parameter estimation using aerial sequence images. The main objective of the paper is to develop an effective algorithm for real-time implementation. The proposed system is composed of two parts: relative position estimation and absolute position estimation. The former is based on stereo modeling of two successive image frames, whereas the latter is accomplished by image matching with reference images or by using digital elevation model (DEM) information.

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