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HandWiki. Image Derivatives. Encyclopedia. Available online: https://encyclopedia.pub/entry/31744 (accessed on 22 September 2026).
HandWiki. Image Derivatives. Encyclopedia. Available at: https://encyclopedia.pub/entry/31744. Accessed September 22, 2026.
HandWiki. "Image Derivatives" Encyclopedia, https://encyclopedia.pub/entry/31744 (accessed September 22, 2026).
HandWiki. (2022, October 28). Image Derivatives. In Encyclopedia. https://encyclopedia.pub/entry/31744
HandWiki. "Image Derivatives." Encyclopedia. Web. 28 October, 2022.
Image Derivatives
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Image derivatives can be computed by using small convolution filters of size 2 x 2 or 3 x 3, such as the Laplacian, Sobel, Roberts and Prewitt operators. However, a larger mask will generally give a better approximation of the derivative and examples of such filters are Gaussian derivatives and Gabor filters. Sometimes high frequency noise needs to be removed and this can be incorporated in the filter so that the Gaussian kernel will act as a band pass filter. The use of Gabor filters in image processing has been motivated by some of its similarities to the perception in the human visual system. The pixel value is computed as a convolution where [math]\displaystyle{ \mathbf{d} }[/math] is the derivative kernel and [math]\displaystyle{ G }[/math] is the pixel values in a region of the image and [math]\displaystyle{ \ast }[/math] is the operator that performs the convolution.

small convolution laplacian convolution

References

  1. H. Farid and E. P. Simoncelli, Differentiation of discrete multi-dimensional signals, IEEE Trans Image Processing, vol.13(4), pp. 496--508, Apr 2004. http://www.cns.nyu.edu/pub/lcv/farid03-reprint.pdf
  2. H. Farid and E. P. Simoncelli, Optimally Rotation-Equivariant Directional Derivative Kernels, Int'l Conf Computer Analysis of Images and Patterns, pp. 207--214, Sep 1997. http://www.cs.dartmouth.edu/~farid/downloads/publications/caip97.pdf
  3. A. Hast., "Simple filter design for first and second order derivatives by a double filtering approach", Pattern Recognition Letters, Vol. 42, no.1 June, pp. 65--71. 2014. http://www.sciencedirect.com/science/article/pii/S0167865514000282
  4. W.T. Freeman, E.H. Adelson, The design and use of steerable filters, IEEE Trans. Pattern Anal. Mach. Intell. 13 (1991) 891–906. http://people.csail.mit.edu/billf/www/papers/steerpaper91FreemanAdelson.pdf
  5. A. Savitzky, M.J.E. Golay, Smoothing and differentiation of data by simplified least squares procedures, Anal. Chem. 36 (1964) 1627–1639. https://pubs.acs.org/doi/full/10.1021/ac60214a047
  6. J. Luo, K. Ying, P. He, J. Bai, Properties of savitzky–golay digital differentiators, Digit. Signal Process. 15 (2005) 122–136.
  7. H. Scharr, Optimal second order derivative filter families for transparent motion estimation, in: M. Domanski, R. Stasinski, M. Bartkowiak (Eds.), EUSIPCO 2007. https://www.researchgate.net/profile/Hanno_Scharr/publication/228927501_Optimal_second_order_derivative_filter_families_for_transparent_motion_estimation/links/004635151972e4d50b000000.pdf
  8. Scharr, Hanno, 2000, Dissertation (in German), Optimal Operators in Digital Image Processing . http://nbn-resolving.de/urn/resolver.pl?urn=urn:nbn:de:bsz:16-opus-9622
  9. B. Jähne, H. Scharr, and S. Körkel. Principles of filter design. In Handbook of Computer Vision and Applications. Academic Press, 1999.
  10. B. Jähne, P. Geissler, H. Haussecker (Eds.), Handbook of Computer Vision and Applications with Cdrom, 1st ed., Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 1999, pp. 125–151 (Chapter 6).
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