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codebook with respect to the transformations induced by image motions has been clearly demonstrated: the results reported here are independent of which sequence is used to produce the codebook. A number of improvements can be made to the coders. The choice of wavelet filters affects the performance: longer symmetric filters perform better, both numerically and subjectively, albeit with increased coding and decoding delay, but we have yet to explore the use of biorthogonal filters. The temporal lowpass encoding would benefit from better quantization and a more effective VQ scheme, such as normalizing the vectors and sending a scale factor. Lastly, the use of a tree-based VQ algorithm would allow a wider variation in rate and distortion than the present scheme.
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Two-Dimensional Object Alignment Based on the Robust Oriented Hausdorff Similarity Measure Dong-Gyu Sim and Rae-Hong Park Abstract—This paper proposes an oriented Hausdorff similarity (OHS) measure for robust object alignment. The OHS measure is introduced by replacing the distance concept of conventional Hausdoff distance (HD) algorithms by the similarity concept of the Hough transform (HT). The proposed algorithm can be considered as the modified directed HT using the distance transform (DT). The orientation information at each pixel is also used to remove incorrect correspondences. Index Terms—Aerial image, distance transform, Hausdorff distance, Hough transform, matching, robust statistics, similarity measure.
REFERENCES [1] A. E. Jacquin, “A novel fractal block-coding technique for digital images,” in Proc. Int. Conf. Acoustics, Speech, Signal Processing ’90, 1990, pp. 18–30. [2] I. Levy and R. Wilson, “Predictive wavelet transform coding: Unifying fractal and transform coding,” in Proc. PCS ’96, Melbourne, Australia, 1996, pp. 347–352. [3] R. Watt, Visual Processing: Computational, Psychophysical and Cognitive Research. Hillsdale, NJ: Lawrence Erlbaum, 1988, ch. 6. [4] M. Balakrishnan, W. Pearlman, and L. Lu, “Variable-rate tree-structured vector quantizers,” IEEE Trans. Inform. Theory, vol. 41, pp. 917–930, 1995. [5] X. S. B.-J. Kim and W. A. Pearlman, “Very low bit-rate embedded video coding with 3D set partitioning in hierarchical trees,” IEEE Trans. Circuits Syst. Video Technol., vol. 8, pp. 243–250, 1998. [6] C. I. Podilchuk, N. S. Jayant, and N. Farvardin, “Three dimensional subband coding of video,” IEEE Trans. Image Processing, vol. 4, pp. 125–139, 1995. [7] I. Daubechies, Ten Lectures on Wavelets: SIAM, 1992, ch. 3. [8] Y. Linde, A. Buzo, and R. Gray, “An algorithm for vector quantizer design,” IEEE Trans. Commun., vol. 28, pp. 84–95, 1980. [9] A. Gersho, “On the structure of vector quantizers,” IEEE Trans. Inform. Theory, vol. 28, pp. 157–166, 1982. [10] L. Corte-Real and A. P. Alves, “A very low bit rate video coder based on vector quantization,” IEEE Trans. Image Processing, vol. 5, pp. 263–273, 1996.
I. INTRODUCTION 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 on the properties of the features and the matching measure used. A distance transform (DT) and a Hausdorff distance (HD) have been widely investigated because they are simple and insensitive to changes of image characteristics [4]. The chamfer matching scheme and the HD matching method based on the minimum distance value between the point sets are efficient. Borgefors proposed a chamfer matching algorithm based on the chamfer DT, in which optimal polygon vertices were detected in the closed contour objects [5]. HD matching proposed by Huttenlocher et al. [4] does not require to establish correspondences, i.e., it does not need to find feature points such as polygon vertices, instead makes use of a set of points extracted by an edge operator. They proposed a partial HD measure based on the ranked order statistics to estimate the similarity between two sets of edge points extracted from the object model and the test image in the presence of occlusion. Also a censored HD (CHD) measure based on the ranked order statistics was proposed [6]. The CHD is efficient for luminance changes of gray level images contaminated by additive Gaussian noise. Dubuisson and Jain [7] analyzed the properties of the HD measures, and proposed the modified HD (MHD) based on the average distance to estimate the similarity between two objects. Sim et al. proposed the robust HD measures based on the M-estimator and least trimmed squares (LTS) [8]. These algorithms (M-HD and LTS-HD) can cope with a large percentage of noise because they are derived from the concept of robust statistics [9]. On the other hand, multiresolution approaches have been proposed to reduce the search time for locating an object. These approaches have shown that the optimum matching point can be detected with a small number of searches [10], [11]. This paper proposes an oriented Hausdorff similarity (OHS) measure for object alignment. By adopting the robust similarity function, it is robust against outliers, where outliers are defined by anomalous data Manuscript received February 17, 1999; revised October 17, 2000. This work was supported in part by the Agency for Defence Development and by Automatic Control Research Center, Seoul National University, Seoul, Korea. This paper was presented in part at the IAPR Workshop on Machine Vision Applications, Chiba, Japan, November 1998. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Josiane B. Zerubia. The authors are with the Department of Electronic Engineering, Sogang University, Seoul 100-611, Korea (e-mail:
[email protected]). Publisher Item Identifier S 1057-7149(01)01658-X.
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(a)
along with their DT maps, and the gradient images are necessary to compute the proposed OHS. The DT map is constructed using Borgefors’ algorithm [17] to reduce the computation time. Then, the gradient image is obtained by the Sobel edge operator and edge points are detected by thresholding the gradient image. The directed OHSs are computed with these gradient, edge, and DT images. The final OHS is determined by the minimum of two directed OHS’s. Note that the proposed OHS is based on a similarity, whereas the conventional HD algorithms are based on the distance between two images [4], [7], [8]. The proposed algorithm is introduced by adopting the similarity concept of the HT. The HT calculates the similarity by accumulating the matching points that are within the search interval. In the proposed algorithm, the similarity is calculated by accumulating the distance value of the DT map. As a result, the similarity can be considered as the combination of two directed Hausdorff similarities. The final Hausdorff similarity value is determined by using the directed OHS values that are calculated from two sets of edge points A and B obtained from gray level images and , and the two gradient images AG and BG . The proposed OHS is defined as
A
HOHS
= min(hOHS (AG ;
B
AE ; dB ); hOHS (BG ; BE ; dA ))
where hOHS (AG ; AE ; dB ) denotes the directed OHS, and AE and BE represents the edge images extracted from images and , respectively. dB (a) = minb2B ka 0 bk is the distance map value of the image at position a [17]. The HD algorithm can be accelerated by using the distance map. To consider the worst case, the proposed algorithm employs the minimum operator, whereas the conventional algorithm makes use of the maximum operator. The directed OHS is defined as
A
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hOHS (AG ; AE ; dB ) =
Fig. 1. Similarity function of the (a) proposed algorithm and (b) the HT. =
far from the assumed error distribution. While the conventional algorithms have been proposed by introducing the ranking operator based on robust statistics, the proposed algorithm is constructed based on the Hausdorff measure using the concept of the Hough transform (HT) [12], [13]. In robust statistics, the HT is known to be robust against a large percentage of outliers, assuming that outliers are distributed randomly. Thus, the proposed algorithm is derived by embedding the robust Hausdorff measure into accumulating operation of the HT. It becomes robust to severe noise/degradation. The orientation computed at each pixel is also used for eliminating wrong correspondences. The conventional algorithms [7], [14], [15] used a specific similarity function for the HD, not orientation information. Also the orientation information was employed in the partial HD to remove the false alarm [16]. This paper presents a hybrid matching method, in which the robust Hausdorff measure and orientation of a pixel are embedded into the structure of the HT. Performance comparison of several alignment methods is presented for various test images with varying noise level and noise type. The rest of the paper is structured as follows. In Section II, the proposed algorithm is presented. In Section III, experimental results are shown for various test images with varying noise level and noise type. Finally, conclusions are summarized in Section IV.
a2A a2A
OA (a) 1 OB (a) T (dB (a)) s(a)T (dB (a))
where an orientation vector OA (a) represents a unit gradient vector of at position a; s(a) = OA (a) 1 OB (a) denotes the dot product of two gradient vectors obtained from two images. The symmetric threshold function, T (x), is shown in Fig. 1(a). The proposed algorithm uses the distance value by considering the number of matching points as the HT does. As a result, it becomes robust to severe noise. The distance is given by the proximity and linearly weighted by the orientation, rather than given by the maximum operation between the distance difference and the orientation difference [17]. Note that the maximum operator may yield incorrect large distance by combining the distance and orientation information whereas in the proposed algorithm the distance is given by the proximity and weighted linearly by the orientation, reducing the distance value. If the directed Hausdorff similarity makes use of the threshold function as shown in Fig. 1(b), it is considered as a directed HT. Furthermore, because the orientation information is used, the point that has a small DT value with different orientations can be effectively removed. The conventional algorithms are based on the distance map alone, so their performance could be disturbed by edges having different orientations. The proposed algorithm is more robust than the LTS-HD. This distance measure can be used to detect a matching point between model and input images. The matching process can be described as follows.
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II. PROPOSED OHS The proposed OHS is composed of edge detection, the DT map construction, and the proposed similarity computation. The edge maps
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Fig. 2. Input and model images: (a) crowd; (b) taxi; (c) airport; (d) interchange; (e) model of (a); (f) model of (b); (g) model of (c); and (h) model of (d).
for for for for
where Sx and Sy represent the search area for detecting the matching point. The matching point is determined by finding a point having the maximum proximity, with the model image sliding within the search area.
The computational complexity of the proposed algorithm is similar to that of the partial HD, LTS-HD, MHD algorithms that have linearorder complexity in computing the HD measure at a point. In the cases of the partial HD, LTS-HD, and MHD, efficient search algorithms can be employed. For the proposed algorithm, fast search algorithms can be also developed by subsampling the edge images of a model and an input. In the coarser level, the matching point is detected in the Hough space generated by subsampled edge pixels. In a finer level search, other edge information is used to accumulate the Hough translational space, thus yielding an accurate matching point. III. EXPERIMENTAL RESULTS The effectiveness of the proposed OHS is shown by alignment experiments, in which an object is detected in an input image. The align-
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Fig. 3. Detection results for Gaussian noise cases (
ment point is detected by finding a point that yields a minimum translational Hausdorff measure H ( 8 t; ), where 8 denotes the translation operation and t represents a translational parameter. To evaluate the performance, several test images are employed, in which reference images are taken from the same test images. Experiments also show the alignment result for aerial and satellite images that were acquired by different sensors. All test images are quantized to eight bits except for the Indian remote sensing (IRS) satellite image which is quantized to six bits.
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A. Alignment Performance of the Conventional and Proposed Algorithms as a Function of the Noise Level for Gaussian and Impulse Noise Cases The performance of the proposed algorithm is compared with that of the original HD, MHD [7], partial HD [4], and LTS-HD [8], with a number of test images containing various noise types and levels.
= 20): (a) crowd, (b) taxi, (c) airport, and (d) interchange. Fig. 2(a) and (e) show a test input (crowd, 256 2 256) and its model image, respectively. Fig. 2(b) and (f) show a test input (taxi, 256 2 230) and its model image, respectively. Fig. 2(c) and (g) show a test input (airport, 256 2 256) and its model image, respectively. Fig. 2(d) and (h) also show a test input (interchange, 320 2 240) and its model image, respectively. Note that the 64 2 64 model images are taken from the test inputs. Detection results by the proposed algorithm are shown in Fig. 3, for the four test images contaminated by Gaussian noise ( = 20), where denotes the standard deviation of the Gaussian noise added. Edges of corrupted input and model images are obtained by the Sobel operator, in which the threshold T h for edge detection is experimentally set to 100. Note that in experiments, noise is added in the image space, thus a single outlier in the image space may generate a number of outlier observations. Edge points used in computation of the HD measure are extracted from the gray level image, and the observations with the edge pixels are used for computing the distance measure. The Gaussian noise added in the image space can be considered as outliers in terms of HD
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Detection ratios as a function of the Gaussian noise level : (a) crowd, (b) taxi, (c) airport, and (d) interchange.
estimators based on edge points. So, in spite that the LTS-HD can cope with 50% outliers in terms of observations of estimator, the estimator could fail in the case in which the fraction of outliers in the image space is much less than 50%. Fig. 3 shows the alignment result by superimposing the edge map of each model image on the corresponding edge map of the input image. Even though the input image is severely distorted by noise, the proposed algorithm yields a correct alignment result. Fig. 4 shows the detection ratio of six methods as a function of the Gaussian noise level , where the detection ratio is obtained by 100 experiments with different random realizations for each noise level. Fig. 4 shows that the proposed algorithm yields the best alignment results on the whole. The MHD algorithm yields good results for Gaussian noise cases. The proposed algorithm can well detect the desired object for severe Gaussian noise of which the standard deviation is up to 30 for the crowd, taxi, and interchange images. But for the airport image, the correct corresponding point is detected for Gaussian noise of which the standard deviation is less than 60. Because the gray level in corresponding target area shows a large contrast, its edge is unlikely to be deteriorated by the Gaussian noise. The LTS-HD algorithm yields
good alignment results, comparable with the proposed algorithm. The proposed algorithm is more robust than the LTS-HD for severe noise cases. Fig. 4 shows that the performance of the proposed algorithm with orientation information is better than that of the algorithm without the orientation information (denoted by “no orientation” in Figs. 4 and 5). Note that because of the robustness of the proposed algorithm, the proposed algorithm without the orientation information yields better results than the conventional ones. Fig. 5 illustrates the detection ratio of six methods as a function of the impulse noise level, which is defined by the percentage of the number of flipped pixels to the total number of edge pixels. The original HD and MHD algorithms are not based on robust statistics, thus, their performance is likely to be deteriorated by the severe noise or distortion. Because the partial HD algorithm is based on the ranked order statistics, it can not deal with the large percentage of outliers. The LTS-HD yields the best alignment result next to the proposed algorithm, because the LTS-HD algorithm is also based on the robust statistics. Note that the proposed algorithm is based on the HT that is known to be robust to large percentage of outliers, so it can correctly detect objects for severer noise cases.
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Fig. 5.
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Detection ratios as a function of the impulse noise level: (a) crowd, (b) taxi, (c) airport, and (d) interchange.
B. Alignment of an Aerial Image with a Satellite Image To show the effectiveness of the proposed algorithm, the aligment of an aerial image with a satellite image is performed. Fig. 6(a) and (b) show the aerial image (320 2 240) and the IRS satellite image (256 2 256), respectively. This satellite image is quantized to six bits with 5m ground resolution. Note that two images were acquired by different sensors and that the aerial image (IRS image) corresponds to the model (test) image. The input aerial image is compensated to match the IRS satellite image according to the attitude of the aircraft, and the compensated image is shown in Fig. 6(c). The edges of the aerial and satellite images are obtained by the Sobel edge operator with the threshold T h set to 100 and 40, respectively. Fig. 6(d) shows an alignment result by the proposed algorithm, in which in solid box, the compensated aerial edge image is superimposed on the IRS satellite edge image. In the case of the aerial image alignment, the search area is set to 200 pixels 2 200 pixels. The same alignment result is obtained by the four conventional algorithms as well as the proposed algorithm. Fig. 7(a) and (b) show another aerial test image (320 2 240) and the image compensated by the attitude change of an aircraft, respectively, to match a corresponding IRS satellite image (1000 2 1000). The edges of the aerial and satellite images are obtained by the Sobel edge operator with the threshold T h set to 100 and 40, respectively.
Fig. 8(a) shows the edge image of the satellite image and Fig. 8(b) shows the alignment results superimposed onto the enlarged area of the satellite image, obtained by the proposed and four conventional algorithms. The search area is set to as large as 1000 pixels 2 1000 pixels. Estimation errors of the proposed algorithm and the LTS-HD are (01, 01) and (4, 5), respectively, where the estimation error is defined by the difference between the actual position and the estimated position. The LTS-HD yields the corresponding result comparable to that of the proposed algorithm. Whereas the other conventional algorithms yield the wrong alignment result, because the satellite image shows characteristics different from those of the input aerial image. Actually, estimation errors of the MHD, the partial HD, and the original HD algorithms are (233, 051), (249, 024), and (427, 069), respectively. Even though the search area is very large, the proposed algorithm yields the correct alignment because it is based on the estimator showing a high robustness. Whereas the conventional algorithms can not successfully detect the object in the aerial image, because edges of the satellite image are severely contaminated. IV. CONCLUSIONS In this paper, a new HD similarity measure is introduced based on the HT that is robust to severe noise. Simulation results for various test
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Alignment result with aerial and satellite images. (a) Input aerial image, (b) IRS satellite image, (c) compensated image of (a), and (d) alignment result.
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Aerial image and its compensated image. (a) Input aerial image and (b) compensated image of (a).
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Alignment result: (a) superimposed matching results by the conventional and proposed algorithms and (b) enlarged matching results.
images with varying noise level and noise type show that the proposed algorithm can cope well with the large percentage of outliers contained in an input image. Further research will focus on the development of the fast algorithm and the extension of the proposed algorithm to scale and rotation changes. 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] D. P. Huttenlocher, G. A. Klanderman, and W. J. Rucklidge, “Comparing images using the Hausdorff distance,” IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-15, pp. 850–863, Sept. 1993. [5] G. Borgefors, “Hierarchical chamfer matching: A parametric edge matching algorithm,” IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-10, pp. 849–865, Nov. 1988.
[6] R. Azencott, F. Durbin, and J. Paumard, “Multiscale identification of buildings in compressed large aerial scenes,” in Proc. 13th Int. Conf. Pattern Recognition, vol. 2, Vienna, Austria, Aug. 1996, pp. 974–978. [7] M.-P. Dubuisson and A. K. Jain, “A modified Hausdorff distance for object matching,” in Proc. 12th Int. Conf. Pattern Recognition, Jerusalem, Israel, Oct. 1994, pp. 566–568. [8] D.-G. Sim, O.-K. Kwon, and R.-H. Park, “Object matching algorithm using robust Hausdorff distance measures,” IEEE Trans. Image Processing, vol. IP-8, pp. 425–429, Mar. 1999. [9] P. J. Rousseeuw and A. M. Leroy, Robust Regression and Outlier Detection. New York: Wiley, 1987. [10] D. P. Huttenlocher and W. J. Rucklidge, “A multi-resolution technique for comparing images using the Hausdorff distance,” Dept. Comput. Sci., Cornell Univ., Ithaca, NY, Tech. Rep. 92-1321, Dec. 1992. [11] W. J. Rucklidge, “Efficiently locating objects using the Hausdorff distance,” Int. J. Comput. Vis., vol. 24, pp. 251–270, Sept. 1997. [12] D. H. Ballard, “Generalizing the Hough transform to detect arbitrary shapes,” Pattern Recognit., vol. 13, pp. 111–122, Mar. 1981. [13] V. F. Leavers, “The dynamic generalized Hough transform: Its relationship to the probabilistic Hough transforms and an application to the concurrent detection of circles and ellipse,” Comput. Vis., Graph., Image Process., vol. 56, pp. 381–398, Nov. 1992.
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[14] D. P. Huttenlocher, R. H. Lilien, and C. F. Olson, “Object recognition using subspace methods,” in Proc. Eur. Conf. Comput. Vis., vol. 1, Cambridge, U.K., Apr. 1996, pp. 536–545. [15] C. F. Olson, “A probabilistic formulation for Hausdorff matching,” in Proc. IEEE Conf. Computer Vision Pattern Recognition, Santa Barbara, CA, June 1998, pp. 150–156. [16] C. F. Olson and D. P. Huttenlocher, “Automatic target recogntion by matching oriented edge pixels,” IEEE Trans. Image Processing, vol. 6, pp. 103–113, Jan. 1997. [17] G. Borgefors, “Distance transformations in digital images,” Comput. Vis., Graph., Image Processing, vol. 34, pp. 344–371, June 1986.
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Fig. 1. Two eight-connected 10 general method.
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Hierarchical Block Matching Method for Fast Rotation of Binary Images Sung-Il Chien and Yung-Mok Baek
Abstract—In this paper, we develop a block matching method for fast rotation of binary images. We define coarse and fine blocks to extract bit patterns of an original image and calculate their predrawn mapping patterns (PMPs) using the given rotation angle. The traditional calculation for rotation of images can be thus replaced with simple matching of bit patterns of blocks and drawing of their two-dimensional (2-D) PMPs at the output plane. A scheme of overlapping blocks is also used to solve problems of hole generation and topology variation usually occurring in rotation. Experimental results demonstrate that our proposed method performs best in terms of rotation speed compared to other algorithms. Index Terms—Binary image rotation, document image processing, skew correction.
Fig. 2. Block diagram of proposed hierarchical block matching method. The coarse block (CB) is made of 9 9 pixels and the fine blocks (FBs) inside a CB are made of 3 3 pixels. CBs are useful in rotating homogenous black regions of a large size while FBs in rotating specific bit patterns arising from characters or boundaries of an object.
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I. INTRODUCTION A fast rotation algorithm for a binary image is essential in various image processing applications. In particular, skew angle detection and correction are important stages in document image processing and analysis systems, since it is quite probable that the document may be misaligned during the scanning process. Such skew may cause problems in subsequent procedures of layout analysis and character recognition. The first step to skew correction is skew angle detection and for this, several methods have been developed by researchers. However, the detailed description of them is beyond the scope of our study, so we here briefly summarize the typical approaches that attract our attention. Hashizume et al. [1] have detected the skew angle by nearest neighbor clustering of the connected components. Hough transform based methods [2] have been also proposed by relating the Hough plane peaks to the estimated skew angle. Avanindra et al. [3] have proposed robust detection of skew by using interline cross-correlation in the scanned image. There are also other methods for detecting document skew based on Fourier transform [4] and morphological transform [5].
Manuscript received June 17, 1999; revised September 27, 2000. This work was supported by the BK’21 Program, Ministry of Education, Korea. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Brian L. Evans. S.-I. Chien is with the School of Electronic and Electrical Engineering, Kyungpook National University, Taegu, Korea (e-mail:
[email protected]). Y.-M. Baek is with the Agency for Defense Development, Taejon, Korea (e-mail:
[email protected]). Publisher Item Identifier S 1057-7149(01)00814-4.
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Fig. 3. PMP examples (45 rotation): (a) original 3 3 bit patterns of , , and , (b) their rotated bit patterns without midpoint filling, and (c) final bit patterns of , , and in which eight-connectedness is preserved.
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After finding a skew angle by a skew detection algorithm, each pixel in a skewed image will be rotated. Since we often have to deal with large size images, a very fast rotation algorithm is desired to make the document processing system more practical and valuable. A binary image has two pixel values, i.e., f (x; y ) = 0 or 1 (0 for a white pixel and
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