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1.5. DIFFERENTIAL OPERATORS

Lecture



Detection and selection of contours (borders) in the image points with sharp differences in brightness, usually carried out by differentiating the function   1.5.  DIFFERENTIAL OPERATORS by spatial coordinates and comparison of the result of differentiation with a threshold. First derivatives with respect to spatial coordinates   1.5.  DIFFERENTIAL OPERATORS and   1.5.  DIFFERENTIAL OPERATORS defined as

  1.5.  DIFFERENTIAL OPERATORS (1.5.1a)

  1.5.  DIFFERENTIAL OPERATORS (1.5.1b)

Derivative in direction   1.5.  DIFFERENTIAL OPERATORS constituting an angle   1.5.  DIFFERENTIAL OPERATORS with the horizontal axis, by definition [3, p. 10] is equal to

  1.5.  DIFFERENTIAL OPERATORS (1.5.2)

The gradient module is

  1.5.  DIFFERENTIAL OPERATORS (1.5.3)

Second derivatives by spatial coordinates   1.5.  DIFFERENTIAL OPERATORS and   1.5.  DIFFERENTIAL OPERATORS by definition equal

  1.5.  DIFFERENTIAL OPERATORS (1.5.4a)

  1.5.  DIFFERENTIAL OPERATORS (1.5.4b)

The sum of derivatives (1.5.4) gives the Laplacian

  1.5.  DIFFERENTIAL OPERATORS . (1.5.5)

It should be noted that the Laplacian is a scalar quantity that does not depend on the direction on the plane   1.5.  DIFFERENTIAL OPERATORS while the gradient is a vector dependent vector.


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Digital image processing

Terms: Digital image processing