1. Fourier Transform: Real-World Images
February 22, 2018
Computer Vision Lecture 8: Edge Detection I

2. Fourier Transform: Real-World Images
February 22, 2018
Computer Vision Lecture 8: Edge Detection I

3. Fourier Transform: Real-World Images
February 22, 2018
Computer Vision Lecture 8: Edge Detection I

4. Fourier Transform: Low-Pass Filtering
February 22, 2018
Computer Vision Lecture 8: Edge Detection I

5. Fourier Transform: High-Pass Filtering
February 22, 2018
Computer Vision Lecture 8: Edge Detection I

6. Fourier Transform: Noise Removal
February 22, 2018
Computer Vision Lecture 8: Edge Detection I

7. Computer Vision Lecture 8: Edge Detection I
February 22, 2018
Computer Vision Lecture 8: Edge Detection I

8. Computer Vision Lecture 8: Edge Detection I
Why Edge Detection?
How can an algorithm extract relevant information from an image that enables further algorithms to recognize objects?
The most important information for the interpretation of an image (for both technical and biological systems) is the contour of objects.
Contours are indicated by abrupt changes in brightness.
We can use edge detection filters to extract contour information from an image.
February 22, 2018
Computer Vision Lecture 8: Edge Detection I

9. Types of Edges One-dimensional profiles of different edge types
February 22, 2018
Computer Vision Lecture 8: Edge Detection I

10. Computer Vision Lecture 8: Edge Detection I
Types of Edges
One-dimensional profile of actual edges
February 22, 2018
Computer Vision Lecture 8: Edge Detection I

11. Computer Vision Lecture 8: Edge Detection I
First we need some definitions:
An edge point is a point in an image with coordinates [i, j] at the location of a significant local intensity change.
An edge fragment corresponds to the i and j coordinates of an edge and the edge orientation , which is typically the gradient angle.
An edge detector is an algorithm that produces a set of edges (edge points or edge fragments) from an image.
February 22, 2018
Computer Vision Lecture 8: Edge Detection I

12. Computer Vision Lecture 8: Edge Detection I
A contour is a list of edges or the mathematical curve that models the list of edges.
Edge linking is the process of forming an ordered list of edges from an unordered list. By convention, edges are ordered by traversal in a clockwise direction.
Edge following is the process of searching the (filtered) image to determine contours.
February 22, 2018
Computer Vision Lecture 8: Edge Detection I

13. Computer Vision Lecture 8: Edge Detection I
Gradient
In the one-dimensional case, a step edge corresponds to a local peak in the first derivative of the intensity function.
In the two-dimensional case, we analyze the gradient instead of the first derivative.
Just like the first derivative, the gradient measures the change in a function.
For two-dimensional functions it is defined as
February 22, 2018
Computer Vision Lecture 8: Edge Detection I

14. Computer Vision Lecture 8: Edge Detection I
Gradient
Gradients of two-dimensional functions:
The two-dimensional function in the left diagram is represented by contour lines in the right diagram, where arrows indicate the gradient of the function at different locations. Obviously, the gradient is always pointing in the direction of the steepest increase of the function.
February 22, 2018
Computer Vision Lecture 8: Edge Detection I

15. Computer Vision Lecture 8: Edge Detection I
Gradient
In order to compute Gi and Gj in an image F at position [i, j], we need to consider the discrete case and get:
Gi = F[i+1, j] – F[i, j]
Gj = F[i, j+1] – F[i, j]
This can be done with convolution filters: 1 -1
Gi =
Gj = -1 1
To be precise in the assignment of gradients to pixels and to reduce noise, we usually apply 33 filters instead (next slide).
February 22, 2018
Computer Vision Lecture 8: Edge Detection I

16. Computer Vision Lecture 8: Edge Detection I
Sobel Filters
Sobel filters are the most common variant of edge detection filters.
Two small convolution filters are used successively on the same input image: -1 -2 1 2 Si Sj -1 1 -2 2
February 22, 2018
Computer Vision Lecture 8: Edge Detection I

17. Computer Vision Lecture 8: Edge Detection I
Sobel Filters
Sobel filters yield two interesting pieces of information:
The magnitude of the gradient (local change in brightness):
The angle of the gradient (tells us about the orientation of an edge):
February 22, 2018
Computer Vision Lecture 8: Edge Detection I

18. Gradient vs. Contour Orientation
Note: Edge (= gradient) and contour orientation are perpendicular to each other:
Here, the gradient orientation is horizontal (pointing to the right) and the contour orientation is vertical.
February 22, 2018
Computer Vision Lecture 8: Edge Detection I

19. Computer Vision Lecture 8: Edge Detection I
Sobel Filters
Calculating the magnitude of the brightness gradient with a Sobel filter. Left: original image; right: filtered image.
February 22, 2018
Computer Vision Lecture 8: Edge Detection I

20. Sobel Filters and Thresholding
February 22, 2018
Computer Vision Lecture 8: Edge Detection I