This Article 
 Bibliographic References 
 Add to: 
Scale-Space Properties of Quadratic Feature Detectors
October 1996 (vol. 18 no. 10)
pp. 987-999

Abstract—Feature detectors using a quadratic nonlinearity in the filtering stage are known to have some advantages over linear detectors; here, we consider their scale-space properties. In particular, we investigate whether, like linear detectors, quadratic feature detectors permit a scale selection scheme with the "causality property," which guarantees that features are never created as scale is coarsened. We concentrate on the design most common in practice, i.e., one dimensional detectors with two constituent filters, with scale selection implemented as convolution with a scaling function. We consider two special cases of interest: constituent filter pairs related by the Hilbert transform, and by the first spatial derivative. We show that, under reasonable assumptions, Hilbert-pair quadratic detectors cannot have the causality property. In the case of derivative-pair detectors, we describe a family of scaling functions related to fractional derivatives of the Gaussian that are necessary and sufficient for causality. In addition, we report experiments that show the effects of these properties in practice. Thus we show that at least one class of quadratic feature detectors has the same desirable scaling property as the more familiar detectors based on linear filtering.

[1] A. Rosenfeld and M. Thurston, "Edge and Curve Detection for Visual Scene Analysis," IEEE Trans. Computers, vol. 20, pp. 562-569, May 1971.
[2] D. Marr and E. Hildreth, "Theory of Edge Detection," Proc. Royal Soc. London, vol. B207, pp. 187-217, 1980.
[3] A.P. Witkin, "Scale-Space Filtering," Int'l Joint Conf. Artificial Intelligence, 1983, pp. 1,019-1,021, Karlsruhe.
[4] J.J. Koenderink, "The Structure of Images," Biol. Cybern., vol. 50, pp. 363-370, 1984.
[5] J. Babaud, A. Witkin, M. Baudin, and R. Duda, "Uniqueness of the Gaussian Kernel for Scale-Space Filtering," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 8, pp. 26-33, Jan. 1986.
[6] J.J. Clark, "Singularity Theory and Phantom Edges in Scale Space," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 10, no. 5, pp. 720-727, 1988.
[7] A. Yuille and T. Poggio, "Scaling Theorems for Zero Crossings," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 8, pp. 15-26, Jan. 1986.
[8] R.A. Hummel and A.R. Moniot, "Reconstructions From Zero Crossings in Scale-Space," IEEE Trans. Acoust., Speech, Signal Processing, Dec. 1989.
[9] L. Wu and Z. Xie, "Scaling Theorems for Zero-Crossings," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 12, pp. 46-54, Jan. 1991.
[10] V. Anh, J.Y. Shi, and H.T. Tsui, "Scaling Theorems for Zero Crossings of Bandlimited Signals," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 18, no. 3, pp. 309-320, Mar. 1996.
[11] P. Perona and J. Malik, "Scale-Space and Edge Detection Using Anisotropic Diffusion," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 12, no. 7, pp. 629639, July 1990.
[12] Geometry-Driven Diffusion in Computer Vision, B.M. ter Haar Romeny, ed. Kluwer, 1994.
[13] G. H. Granlund, "In Search of a General Picture Processing Operator," Computer Graphics and Image Processing, vol. 8, pp. 155-173, 1978.
[14] E. Adelson and J. Bergen, "Spatiotemporal Energy Models for the Perception of Motion," J. Optical Soc. Am., vol. 2, no. 2, pp. 284-299, 1985.
[15] J. Daugman, "Uncertainty Relation for Resolution in Space, Spatial Frequency, and Orientation Optimized by Two Dimensional Visual Cortical Filters," J. Optical Soc. Am., vol. 2A, no. 7, pp. 1,160-1,169, 1985.
[16] M.C. Morrone and R.A. Owens, "Feature Detection From Local Energy," Pattern Recognition Letters, vol. 6, pp. 303-313, 1987.
[17] R. Owens, S. Venkatesh, and J. Ross, "Edge Detection is a Projection," Pattern Recognition Letters, vol. 9, pp. 233-244, 1989.
[18] W.T. Freeman and E.H. Adelson, "The Design and Use of Steerable Filters," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 13, pp. 891-906, 1991.
[19] P. Perona and J. Malik, "Detecting and Localizing Edges Composed of Steps, Peaks and Roofs," Proc. Third Int'l Conf. Computer Vision,Osaka, Japan, pp. 52-57, 1990.
[20] P. Perona and J. Malik, "Boundary Detection Using Quadratic Filters: Performance Criteria and Experimental Assessment," Proc. SPIE Conf. on Applications of Artificial Intelligence, SPIE vol. 1,708, Orlando, Apr. 1992.
[21] P. Kube, "Properties of Energy Edge Detectors," Proc. IEEE Computer Soc. Conf. Computer Vision and Pattern Recognition, 1992.
[22] P. Perona, "Deformable Kernels for Early Vision," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 17, no. 5, pp. 488-499, May 1995.
[23] G.H. Granlund and H. Knuttson, Signal Processing for Computer Vision. Kluwer, 1995.
[24] T. Poston and I. Stewart, Catastrophe Theory and its Applications.London: Pitman, 1978.
[25] V.I. Arnold, Singularity Theory. Cambridge Univ. Press, 1981.
[26] R.N. Bracewell, The Fourier Transform and Its Applications. McGraw Hill, 1978.
[27] M.C. Morrone and D.C. Burr, "Feature Detection in Human Vision: A Phase Dependent Energy Model," Proc. Royal Soc. of London, vol. B235, pp. 221-245, 1988.
[28] C. Ronse, "On Idempotence and Related Requirements in Edge Detection," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 15, pp. 484-491, 1993.
[29] A.I. Barvinok, "Feasibility Testing for Systems of Real Quadratic Equations," STOC, 1992.
[30] P. Kube and P. Perona, "Scale-Space Properties of Quadratic Edge Detectors," Proc. Third European Conf. Computer Vision, J.-O. Eklundh, ed., LNCS-Series, vol. 800-801, pp. I-115-122.Stockholm: Springer-Verlag, May 1994.

Index Terms:
Feature detection, edge detection, scale space, nonlinear filtering, energy filters, quadratic filters, causality.
Paul Kube, Pietro Perona, "Scale-Space Properties of Quadratic Feature Detectors," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 18, no. 10, pp. 987-999, Oct. 1996, doi:10.1109/34.541408
Usage of this product signifies your acceptance of the Terms of Use.