1. Introduction to Image Processing
Edge Detection and Sharpening
Grass
Sky
Tree ?

2. Edge Detection and Sharpening
The Basic Theory
Properties of 1st and 2nd Derivatives
Edge Detection
gradient operators based on 1st derivatives
Laplacian as the 2nd derivative operator
Edge Sharpening
composite Laplacian
unsharp masking

3. Origin of Edges Edges are caused by a variety of factors
depth discontinuity
surface color discontinuity
illumination discontinuity
surface normal discontinuity
Edges are caused by a variety of factors
Source: Steve Seitz

4. Edge Detection
Edge is marked by points at which image properties change sharply
To detect changes in a digital 2D image intensity function, f(x, y), a continuous image can be reconstructed, followed by the differentiation operator OR
We take the discrete derivative (finite difference) shown below:

5. The Basic Theory
An intensity jump (edge) generates a peak in the 1st derivative, and the peak generates a zero-crossing in the 2nd derivative (when the first derivative is at a maximum, the second derivative is zero)
Gradient methods detect edges by looking for the maximum and minimum in the first derivative of the image
Laplacian methods search for zero crossings in the second derivative of the image to find edges

6. Alternative 1st and 2nd Derivatives
Sharpening filters are based on computing spatial derivatives of an image.
The first-order derivative of a one-dimensional function f(x) is
The second-order derivative of a one-dimensional function f(x) is

7. Properties of the 1st and 2nd Derivatives

8. Profile for 1st Derivative5 4 3 2 1 6 7 -1 6 -6 1 2 -2 7 8

9. Profile for 2nd Derivative5 4 3 2 1 6 7 -1 1 6
-12 -4 7 -7 9

10. Computing 1st Derivativex -1 1 -2 2
Derivative of function I at 1:
I’(1) = (I(2) - I(0))/2
Rearranging it we have:
2*I’(1) = -1*I(0) + 0*I(1) + 1*I(2)
Equivalent to local filter operation using -1 -2 1 2 1 2 -1 -2
The standard definition of the Sobel operator omits the 1/8 term
doesn’t make a difference for edge detection
the 1/8 term is needed to get the right gradient value, however -1 1

11. 1st Derivative Gradient Operators
Roberts cross-gradient operators
Prewitt operators
Sobel operators

12. 1st Derivative Gradient Operators
Prewitt masks for
detecting diagonal edges
Sobel masks for

13. Properties of Image Gradient
First-order derivatives:
The gradient of an image Gxy at location (x,y) is defined as the vector:
The gradient points in the direction of most rapid increase in intensity
The edge strength is given by the gradient magnituge
O OR approx.
The gradient direction is given by:
How does this relate to the edge direction?

14. Gradients in 2D
For an image function, I(x,y), the gradient direction, (x,y), gives the direction of steepest image gradient: (x,y)  atan(Gy/Gx)
This gives the direction of a line perpendicular to
the edge
Gxy Gy  Gx

15. Sobel Operator
Original Sobel

16. Gradient Operators: Examples

17. Gradient Operators: Examples

18. Gradient Operators: Examples

19. Important Observation
Note that Prewitt operator is a box filter convolved with a derivative operator
Also note a Sobel operator is a [1 2 1] filter convolved with a derivative operator

20. Computing 2nd Derivativesx
The theory can be carried over to 2D as long as there is a way to approximate the derivative of a 2D image
I’’(1) = (I’(1.5) - I’(0.5))/1
I’(0.5) = (I(1) - I(0))/1 and
I’(1.5) = (I(2) - I(1))/1
I’’(1) = 1*I(0) – 2*I(1) + 1*I(2)
Equivalent to local filter operation with 1 -2

21. Laplacian Operator Development of the Laplacian method
The two dimensional Laplacian operator for continuous functions:
The Laplacian is a linear operator.

22. Laplacian Operator

23. Laplacian Operator
Applying the Laplacian to an image we get a new image that highlights edges and other discontinuities
Original Image
Laplacian Filtered Image
Laplacian Filtered Image Scaled for Display 23

24. But That Is Not Very Enhanced!
Laplacian Filtered Image Scaled for Display
The result of a Laplacian filtering is not an enhanced image
We have to do more work in order to get our final image
Subtract the Laplacian result from the original image to generate our final sharpened enhanced image 24

25. Laplacian Image Enhancement- =
Original Image
Laplacian Filtered Image
Sharpened Image
In the final sharpened image edges and fine detail are much more obvious 25

26. Simplified Image Enhancement
The entire enhancement can be combined into a single filtering operation 26

27. Composite Laplacian Mask
This gives us a new filter which does the whole job for us in one step -1 5 27

28. Composite Laplacian Mask

29. Unsharp Masking & Highboost Filtering
to generate the mask: Subtract a blurred version of the image from itself
add the mask to the original
Highboost:
k>1
gmask(x,y) = f(x,y) - f(x,y)
g(x,y) = f(x,y) + k*gmask(x,y)

30. 1st & 2nd Derivatives Comparisons
Observations:
1st order derivatives generally produce thicker edges
2nd order derivatives have a stronger response to fine detail e.g. thin lines
2nd order derivatives produce a double response at step changes in grey level
The 2nd derivative is more useful for image enhancement than the 1st derivative
Stronger response to fine detail
Simpler implementation
Because these kernels approximate a second derivative measurement on the image, they are very sensitive to noise. To counter this, the image is often Gaussian smoothed before applying the Laplacian filter 30

31. Acknowlegements
Slides are modified based on the original slide set from Dr Li Bai, The University of Nottingham, Jubilee Campus plus the following sources:
Digital Image Processing, by Gonzalez and Woods