Publication 2000 Issue No. 3 - March Abstract - Consistent Gradient Operators
March 2000 (vol. 22 no. 3)
pp. 252-265
 ASCII Text x Shigeru Ando, "Consistent Gradient Operators," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 22, no. 3, pp. 252-265, March, 2000.
 BibTex x @article{ 10.1109/34.841757,author = {Shigeru Ando},title = {Consistent Gradient Operators},journal ={IEEE Transactions on Pattern Analysis and Machine Intelligence},volume = {22},number = {3},issn = {0162-8828},year = {2000},pages = {252-265},doi = {http://doi.ieeecomputersociety.org/10.1109/34.841757},publisher = {IEEE Computer Society},address = {Los Alamitos, CA, USA},}
 RefWorks Procite/RefMan/Endnote x TY - JOURJO - IEEE Transactions on Pattern Analysis and Machine IntelligenceTI - Consistent Gradient OperatorsIS - 3SN - 0162-8828SP252EP265EPD - 252-265A1 - Shigeru Ando, PY - 2000KW - Image processingKW - feature extractionKW - gradientKW - edgeKW - cornerKW - orientation.VL - 22JA - IEEE Transactions on Pattern Analysis and Machine IntelligenceER -

Abstract—Fine, accurate gradient information is required in many image-processing algorithms and systems including differential geometric methods, orientation analysis, and integrated vision sensors. In this paper, we propose optimal gradient operators based on a newly derived consistency criterion. This criterion is based on an orthogonal decomposition of the difference between a continuous gradient and discrete gradients into the intrinsic smoothing effect and the self-inconsistency involved in the operator. We show that consistency assures the exactness of gradient direction of a locally one-dimensional (1D) pattern in spite of its orientation, spectral composition, and subpixel translation. Stressing that inconsistency reduction is of primary importance, we derive an iterative algorithm which leads to accurate gradient operators of arbitrary size. We compute the optimum $3\times 3$, $4\times 4$, and $5\times 5$ operators, compare them with conventional operators and examine the performance for one synthetic and several real images. The results indicate that the proposed operators are superior with respect to accuracy, bandwidth, and isotropy.

[1] G. Robinson, “Edge Detection by Compass Gradient Masks,” Computer Graphic Image Processing, vol. 6, pp. 492-501, 1977.
[2] R. Nevatia and K.R. Babu, “Linear Feature Extraction and Description,” Computer Vision Graphic Image Processing, vol. 13, pp. 257-269, 1980.
[3] W. Frei, C.-C. Chen, “Fast Boundary Detection: A Generalized and a New Algorithm,” IEEE Trans. Computers, vol. 26, no. 10, pp. 988-998, 1977.
[4] I.J. Clark, "Authenticating Edges Produced by Zero-Crossing Algorithms," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, no. 1, pp. 43-57, Jan. 1989.
[5] D. Marr and E.C. Hildreth, “Theory of Edge Detection,” Proc. Royal Soc. London, B, vol. 207, pp. 187-217, 1980.
[6] J.J. Koenderink, “The Structure of Images,” Biological Cybernetics, vol. 50, pp. 363-370, 1984.
[7] L.M.J. Florack, B.M. ter Haar Romeny, J.J. Koenderink, and M.A. Viergever, “Scale and the Differential Structure of Images,” Image and Vision Computing, vol. 10, no. 6, pp. 376-388, 1992.
[8] R. Machuca, K. Phillips, “Applications of Vector Fields to Image Processing,” IEEE Trans. Pattern Analysis Machine Intelligence, vol. 5, no. 3, pp. 316-329, Mar. 1983.
[9] L.M.T. Florack, B.M. ter Harr Romeny, J.J. Koenderink, and M.A. Viergever, “General Intensity Transformation and Differential Invariants,” J. Mathematical Imaging and Vision, vol. 4, no. 2, pp. 171-187, 1994.
[10] J.B.A. Maintz, P.A. van den Elsen, and M.A. Viergever, “Evaluation of Ridge Seeking Operators for Multimodality Medical Image Matching,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 18, no. 4, pp. 353-365, Apr. 1996.
[11] J. Canny, “A Computational Approach to Edge Detection,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 8, no. 6, pp. 679-698, June 1986.
[12] S. Mallat and S. Zhong, “Characterization of Signals from Multiscale Edges,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 14, no. 7, pp. 710-732, July 1992.
[13] E.P. Lyvers and O.R. Mitchell, “Precision Edge Contrast and Orientation Estimation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 10, no. 6, pp. 927-937, 1988.
[14] R.M. Haralick, “Digital Step Edges from Zero-Crossings of Second Directional Derivatives,” IEEE Trans. Pattern Analysis Machine Intelligence, vol. 6, no. 1, pp. 58-68, Jan. 1984.
[15] O.A. Zuniga and R.M. Haralick, “Integrated Directional Derivative Gradient Operator,” IEEE Trans. Systems, Man, and Cybernetics, vol. 17, no. 3, pp. 508-517, 1987.
[16] E.P. Lyvers,O.R. Mitchell,M.L. Akey,, and A.P. Reeves,“Subpixel measurements using a moment-based edge operator,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, pp. 1,293-1,309, Dec. 1989.
[17] J. Koplowitz and V. Greco, “On the Edge Location Error for Local Maximum and Zero-Crossing Edge Detectors,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 16, no. 12, pp. 1,207-1,212, Dec. 1994.
[18] L. Ganesan and P. Bhattacharyya, “Edge Detection in Untextured and Textured Images—A Common Computational Framework,” IEEE Trans. Systems, Man, and Cybernetics, vol. 27, no. 5, pp. 823-834, 1997.
[19] M. Heath, S. Sarkar, T. Sanocki, and K. Bowyer, “Comparison of Edge Detectors: A Methodology and Initial Study,” Computer Vision, Graphics, and Image Understanding, vol. 69, no. 1, pp. 38-54, 1998.
[20] R. Nevatia, “Image Segmentation,” Handbook of Pattern Recognition and Image Processing, T.Y. Young and K.-S. Fu, ed., Academic Press, chapter 9, 1986.
[21] E.R. Davies, “Circularity—A New Principle Underlying the Design of Accurate Orientation Operators,” Image and Vision Computing, vol. 2, no. 3, pp. 134-142, 1984.
[22] R. Wilson and A.H. Bhalerao, “Kernel Designs for Efficient Multiresolution Edge Detection and Orientation Estimation,” IEEE Trans. Pattern Analysis Machine Intelligence, vol. 14, no. 3, pp. 384-390, Mar. 1992.
[23] J. Kittler, “On the Accuracy of the Sobel Edge Detector,” Image and Vision Computing, vol. 1, no. 1, pp. 37-42, 1983.
[24] R. Machuca and A.L. Gilbert, “Finding Edges in Noisy Scenes,” IEEE Trans. Pattern Analysis Machine Intelligence, vol. 3, no. 1, pp. 103-111, Jan. 1981.
[25] P.H. Gregson, “Using Angular Dispersion of Gradient Direction for Detecting Edge Ribbons,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 15, no. 7, pp. 682-696, 1993.
[26] J.-B. Martens, “Local Orientation Analysis in Images by Means of the Hermite Transform,” IEEE Trans. Image Processing, vol. 6, no. 8, pp. 1,103-1,116, 1997.
[27] A. Papoulis, Probability, Random Variables, and Stochastic Processes. Mc Graw Hill, chapter 7, 1965.
[28] L. Basano, B. Caprile, E.D. Micheli, A. Geminiani, and P. Ottonello, “Edge-Detection Schemes Highly Suitable for Hardware Implementation,” J. Optical Soc. Am. A, vol. 5, no. 7, pp. 1,170-1,175, 1988.
[29] R.O. Duda and P.E. Hart, "Use of the Hough transforms to detect lines and curves in pictures," Comm. ACM, vol. 15, no. 1, pp. 11-15, 1972
[30] E.R. Davies, Machine Vision: Theory, Algorithms, Practicalities. Academic Press, 1990.
[31] A. Korn, “Toward a Symbolic Representation of Intensity Changes in Images,” IEEE Trans. Pattern Analysis and Machine Intelligence, CS Press, 1988, pp. 610‐625.

Index Terms:
Image processing, feature extraction, gradient, edge, corner, orientation.
Citation:
Shigeru Ando, "Consistent Gradient Operators," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 22, no. 3, pp. 252-265, March 2000, doi:10.1109/34.841757