1.5. DIFFERENTIAL OPERATORS

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

(1.5.1a)

(1.5.1b)

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

(1.5.2)

The gradient module is

(1.5.3)

Second derivatives by spatial coordinates and by definition equal

(1.5.4a)

(1.5.4b)

The sum of derivatives (1.5.4) gives the Laplacian

. (1.5.5)

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