Abstract
Histograms are used to analyze and index images. They have been found experimentally to have low sensitivity to certain types of image morphisms, for example, viewpoint changes and object deformations. The precise effect of these image morphisms on the histogram, however, has not been studied. In this work we derive the complete class of local transformations that preserve or scale the magnitude of the histogram of all images. We also derive a more general class of local transformations that preserve the histogram relative to a particular image. To achieve this, the transformations are represented as solutions to families of vector fields acting on the image. The local effect of fixed points of the fields on the histograms is also analyzed. The analytical results are verified with several examples. We also discuss several applications and the significance of these transformations for histogram indexing.
Similar content being viewed by others
References
Abraham, R. and Marsden, J.E. 1978. Foundations of Mechanics. Benjamin/Cummings: New York.
Anastassiou, D. 1989. Error diffusion coding for A/D conversion. IEEE Transactions on Circuits and Systems, 36(9):1175-1186.
Andronov, A.A., Leontovich, E.A., Gordon, I.I., and Maier, A.G. 1973. Qualitative Theory of Second-Order Dynamic Systems. John Wiley and Sons: New York.
Arneodo, A., Decoster, N., and Roux, S.G. 1999. Intermittency, log-normal statistics, and multifractal cascade process in highresolution satellite images of cloud structure. Physical Review Letters, 83(6):1255-1258.
Arnold, V.I. 1989.Mathematical Methods of Classical Mechanics. Springer-Verlag: New York.
Bach, J.R., Fuler, C., Gupta, A., Hampapur, A., Horowitz, B., Humphrey, R., Jain, R., and Shu, C. 1996. TheVirage image search engine: An open framework for image management. In SPIE Conference on Storage and Retrieval for Image and Video Databases IV, March 1996. Vol. 2670, pp. 76-87.
Basri, R. 1996. Paraperspective=Affine. International Journal of Computer Vision, 19(2):169-179.
Bouzouba, K. and Radouane, L. 2000. Image identification and estimation using the maximum entropy principle. Pattern Recognition Letters, 21:691-700.
Cohen, S. and Guibas, L. 1999. The earth mover's distance under transformation sets. In Proc. of the 7th International Conference on Computer Vision, Vol. 2, Kerkyra, Greece. Sept. 1999, pp. 1076-1083.
Finlayson, G.D., Chatterjee, S.S., and Funt, B.V. 1996. Color angular indexing. In Proc. of the 4th European Conference in Computer Vision, Vol. 2, Berlin, Germany, 1996, pp. 16-27.
Foley, J.D., van Dam, A., Feiner, S.K., and Huqhes, J.F. 1996. Computer Graphics Principles and Practice. Addison Wesley: Reading, MA.
Ford, R.M., Strickland, R.N., and Thomas, B.A. 1994. Image models for 2-D flow visualization and compression. CVGIP: Graphical Models and Image Processing, 56(1):75-93.
Giachetti, A. and Torre, V. 1996. The use of optical flow for the analysis of non-rigid motions. International Journal of Computer Vision, 18(3):255-279.
Ginneken, B.V. and Romeny, B.M.H. 2000. Applications of locally orderless images. Journal of Visual Communication and image Representation, 11:196-208.
Glasbey, C.A. 1993. An analysis of histogram-based thresholding algorithms. Computer Vision, Graphics, and Image Processing, 55(6):532-537.
Griffin, L.D. 1997. Scale-imprecision space. Image and Vision Computing, 15:369-398.
Haaser, N.B. and Sullivan, J.A. 1971. Real Analysis. Dover Publications: New York.
Hadjidemetriou, E., Grossberg, M.D., and Nayar, S.K. 2000. Histogram preserving image transformations. In Proc. of the IEEE Conference on Computer Vision and Pattern Recognition, Vol. 1, South Carolina, June 2000, pp. 410-416.
Halsey, T.C., Jensen, M.H., Kadanoff, L.P., Procaccia, I., and Shraiman, B.I. 1986. Fractal measures and their singularities: The characterization of strange sets. Physical Review A, 33(2):1141-1151.
Huang, T.S. 1990. Modeling, analysis and visualization of nonrigid object motion. In Proc. of IEEE International Conference of Pattern Recognition, June 1990, pp. 361-364.
Jagersand, M. 1995. Saliency maps and attention selection in scale and spatial coordinates: An information theoretic approach. In Proc. of the 5th IEEE International Conference on Computer Vision, June 1995, pp. 195-202.
Kass, M. and Witkin, A. 1987. Analyzing oriented patterns.Computer Vision Graphics and Image Processing, 37:362-385.
Koenderink, J.J. and Van Doorn, A.J. 1999. The structure of locally orderless images. International Journal of Computer Vision, 31(2/3):159-168.
Marsden, J.E. and Tromba, A.J. 1988. Vector Calculus. W.H. Freeman and Company: New York.
Moghaddam, B. and Pentland, A. 1995. Maximum likelihood detection of faces and hands. In International Workshop on automatic Face-and-Gesture Recognition, 1995, pp. 122-128.
Murase, H. and Nayar, S. 1995. Visual learning and recognition of 3D objects from appearance. International Journal of Computer Vision, 14:5-24.
Niblack, W. 1993. The QBIC project: Querying images by content using color, texture, and shape. In SPIE Conference on Storage and Retrieval for Image and Video Databases, April 1993, Vol. 1908, pp. 173-187.
Pass, G., Zabih, R., and Miller, J. 1996. Comparing images using color coherence vectors. In Proc. of ACM Multimedia, 1996, pp. 65-73.
Rao, A.R. and Jain, R.C. 1992. Computer flow field analysis: Oriented texture fields. IEEE Transactions on Pattern Analysis and Machine Intelligence, 14(7):693-706.
Rose, J.S. 1994. A Course on Group Theory. Dover: New York.
Royden, H.L. 1968. Real Analysis. MacMillan: New York.
Sander, P.T. and Zucker, S.W. 1992. Singularities of principal direction fields from 3-D images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 14(3):309-317.
Schiele, B. and Crowley, J.L. 2000. Recognition without correspondence using multidimensional receptive field histograms. International Journal of Computer Vision, 36(1):31-50.
Sirovich, L. and Kirby, M. 1987. Low-dimensional procedure for the characterization of human faces. Journal of the Optical Society of America, 4(3):519-524.
Smith, J. and Chang, S.F. 1996. Tools and techniques for color image retrieval. In Proc. of SPIE, Feb.1996, Vol. 2670, pp. 1630-1639.
Spivak, M. 1965. Calculus on Manifolds. Benjamin/Cummings: New York.
Sporring, J. and Weickert, J. 1999. Information measures in scalespaces. IEEE Transactions on Information Theory, 45(3):1051-1058.
Stricker, M. and Orengo, M. 1995. Similarity of color images. In Proc. of SPIE Conference on Storage and Retrieval for Image and Video Databases III, Feb. 1995, Vol. 2420, pp. 381-392.
Swain, M.J. and Ballard, D.H. 1991. Color indexing. International Journal of Computer Vision, 7(1):11-32.
Swaminathan, R. and Nayar, S.K. 1999. Non-metric calibration of wide angle lenses and polycameras. In Proc. of the IEEE Conference on Computer Vision and Pattern Recognition, 1999, Vol. II, pp. 413-419.
Tsallis, C. 1988. Possible generalization of Boltzmann-Gibbs statistics. Journal of Statistical Physics, 52(1/2):479-487.
Turk, M. and Pentland, A. 1991. Eigenfaces for recognition. Cognitive Neuroscience, 3(1):71-86.
Ulichney, R.A. 1988. Dithering with blue noise. Proceedings of the IEEE, 76:56-79.
Vehel, J.L., Mignot, P., and Berroir, J.P. 1992. Multifractals, texture, and image analysis. In Proc. of the IEEE Conference on Computer Vision and Pattern Recognition, Champaign, Illinois, June 1992, pp. 661-664.
Verri, A., Girosi, F., and Torre, V. 1989. Mathematical properties of the two-dimensional motion field: From singular points to motion parameters. Journal of the Optical Society of America A, 6(5):698-712.
Verri, A. and Poggio, T. 1989. Motion field and optical flow: Qualitative properties. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(5):490-497.
Weng, J., Cohen, P., and Herniou, M. 1990. Calibration of stereo cameras using a non-linear distortion model. In Proc. of the IEEE Conference on Computer Vision and Pattern Recognition, 1990, Vol. I, pp. 246-253.
Wu, H.S. and Barba, J. 1998. Minimum entropy restoration of star field images. IEEE Transactions on Systems, Man, and Cybernetics-Part B, 28(2):227-231.
Zhang, H.J., Kankanhali, A., and Smoliar. S.W. 1993. Automatic partitionang of full-motion video. Multimedia Systems, 1:10-28.
Zhang, H.J., Low, C.Y., Smoliar W., and Wu, J.H. 1995. Video parsing, retrieval and browsing: An integrated and content-based solution. ACM Multimedia, pp. 15-24.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hadjidemetriou, E., Grossberg, M.D. & Nayar, S.K. Histogram Preserving Image Transformations. International Journal of Computer Vision 45, 5–23 (2001). https://doi.org/10.1023/A:1012356022268
Issue Date:
DOI: https://doi.org/10.1023/A:1012356022268