Image Edge Detection - First Order Differential Operator Roberts, Sobel, Prewitt, Kirsch, Robinson

Edge detection refers to the process of detecting edge points and edge segments from an image and describing the edge direction.This paper records the first-order differential operators commonly used in edge detection.

Image edges

The edge of the image generally refers to the position where the grayscale change rate of the image is the largest.

The main reasons are as follows:

1. The grayscale of the image changes discontinuously in the surface normal direction;
2. The spatial depth of objects in the image is inconsistent;
3. Inconsistent colors on smooth surfaces;
4. Lighting and shadowing of objects in the image

Edge Detection

• Edge detection refers to the process of detecting edge points and edge segments from an image and describing the edge direction.The image can be regarded as a binary function f(x, y), (x, y) is the position of the pixel, and f(x, y) is the gray value there, so the image can be imagined as a surface.After viewing it as a surface, it can be processed mathematically.The edge is the most drastic position on the surface, and this position is also the position of the local extreme point of the surface. Everyone knows that the extreme value can be calculated by the derivative.

First-order differential operator detects edges

• Gradient field of the image (difference between adjacent pixels):

\nabla \mathrm{f}(\mathrm{x}, \mathrm{y})=\left(\frac{\partial \mathrm{f}}{\partial x}, \frac{\partial \mathrm{f}}{\partial y}\right)

• The magnitude of the gradient is:

|\nabla \mathrm{f}(\mathrm{x}, \mathrm{y})|=\sqrt{\left(\frac{\partial \mathrm{f}}{\partial x}\right)^{2}+\left(\frac{\partial \mathrm{f}}{\partial y}\right)^{2}}

• The gradient direction is:

\angle \nabla \mathrm{f}(\mathrm{x}, \mathrm{y})=\arctan \left(\frac{\partial \mathrm{f}}{\partial x}\right)/\left(\frac{\partial \mathrm{f}}{\partial x}\right)

From this, we get that the fastest changing direction of a point (x, y) in the image is the direction of the gradient, and the slowest changing direction is the direction perpendicular to the gradient.

First-order operator

Roberts operator

• Differencing at (\mathrm{i}+1 / 2, j+1 / 2)

|\nabla \mathrm{f}(1, j)|=|f(\mathrm{i}, j)-\mathrm{f}(\mathrm{i}+1, j+1)|+|f(1+1, j)-\mathrm{f}(\mathrm{i}, \mathrm{j}+1)|

• Converting to a template is

For all differential operators, if only threshold judgment is performed for |Rxf|, the edge perpendicular to the x direction is selected; if only |Ryf| for threshold judgment, the edge perpendicular to the y direction is selected; if |Rxf|+|Ryf| is selected for judgment, the edge in the general direction is obtained.

Sobel operator

• Central difference, for the four points of the horizontal and vertical lines, the weight is high, the template is:

More in practice

Prewitt operator

• Template is

Kirsch operator

8 templates, corresponding to 8 directions, the maximum value is selected

Robinson operator

• The rules are the same as above, but also 8 templates

References

• https://blog.csdn.net/u014485485/article/details/78339420