Why is the laplace transform not popular for image processing convolution? Most textbooks only conver the Fourier transforms.

  • $\begingroup$ Can you give some reasons for and against? As it stands, the question seems to be not very constructive. $\endgroup$
    – Raphael
    Apr 11 '13 at 7:36

The Laplacian is indeed used in image processing routinely but, possibly not as much as Fourier transforms. Reasons (other than just the difference in span of uses, narrow vs wider) may be: Fourier transforms have been highly optimized due to their wide application, and are possibly less complicated theoretically than the Laplacian. sometimes the Laplacian of the Gaussian is taken for "blob detection".[2][3]

From the book Digital signal processing fundamentals By Ashfaq A. Khan p105:

Convolution is the primarily tool in image processing while Laplace Transform is used mainly in signal processing, such as speech and controls systems.

[1] Laplace filter in image processing (with edge detection and motion estimation applications)

[2] Laplacian in blob detection intuition (mathoverflow)

[3] blob detection

  • $\begingroup$ "Fourier transforms have been highly optimized" is that true for the normal Fourier transform (not the Fast Fourier Transform) as well? Do you know how much faster? Do you have other examples with mathmatical description and source code? $\endgroup$ Apr 12 '13 at 23:36
  • $\begingroup$ was alluding to FFT in the answer. other examples of what? the wikipedia article compares FFT with other Fourier transform algorithms. $\endgroup$
    – vzn
    Apr 12 '13 at 23:40
  • $\begingroup$ DFT vs laplace transform, benchmark and sourcecode? $\endgroup$ Apr 14 '13 at 9:22
  • $\begingroup$ the two transforms are not really used for the same specific purposes therefore it seems unlikely/uncommon for authors to compare them directly with each other. $\endgroup$
    – vzn
    Apr 14 '13 at 15:00
  • $\begingroup$ does the popularity of the fourier transform have something to do with the 'you can't work with a la place transform and a delta function' thing and convergence... $\endgroup$ Apr 21 '13 at 20:48

A Laplace transform is (in principle) a one-sided Fourier transform with expontial attenuation term. This makes it suitable for many problems with a starting condition (e.g. starting a circuit's voltage supply). For image analysis a plain Fourier transform seems to be all one needs. The Laplacian expresses the second derivate. It has nothing to do with Laplace transform.


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