Derivatives & Edges

Question 1

How do you calculate the 1st derivative?

Answer 1

First derivative can be calculated by taking the 2nd value - 1st value in the raw data e.g. 5 is the 1st, 4 is the 2nd, so 1st derivative would be -1.
Can also be approximated by (2nd value - 1st value)*2

Question 2

How do you calculate the 2nd derivative?

Answer 2

(Value before number + value after number) - 2(current value)
Also can be derived by estimating the 1st derivative at x + 0.5 and x-0.5 and computing the derivative of the resulting area.

Question 3

Give some examples of 1st derivative filters?

Answer 3

Roberts’ Cross Operators
Sobel Operators

Question 4

What are the 4 different steps when enhancing edges?

Answer 4

Take the original image
Gaussian smooth it
Subtract the smoothed version from the original to make an unsharp mask
Add the mask to the original to make the edge appear more obvious

Question 5

How can you enhance edges using derivative filters, and which one is better at it?

Answer 5

2nd Derivative is more useful for image enhancement than 1st derivative - stronger response to fine detail, and simpler implementation
Using derivatives, you can use a Laplacian

Question 6

What are the benefits of using a Laplacian?

Answer 6

Isotropic
One of the simplest image sharpening filters
Straightforward digital implementation via convolution

Question 7

What is the Laplacian’s grid made up of?

Answer 7

3x3
Top Row - 0, 1, 0
Middle Row - 1, -4, 1
Bottom Row - 0, 1, 0

Question 8

What is the aim when using a Laplacian?

Answer 8

Highlights edges and other discontinuities

Question 9

What can you do with a Laplacian to produce a sharpened image?

Answer 9

Subtract the Laplacian result from the original image to generate the final sharpened enhanced image

Question 10

What does single operator mean with regards to image enhancement?

Answer 10

Using a single operator with convolution performs image sharpening in a single step

Question 11

What is the goal of edge detection?

Answer 11

Mark points at which image intensity changes sharply, which are edges of objects.

Question 12

How do you detect edges (theory)?

Answer 12

To detect edges, you find peaks in the 1st derivative of intensity or zero-crossings in the 2nd derivative

Question 13

How can you combine edge detection with thresholding?

Answer 13

Significant peaks in magnitude of 1st derivative are high
Apply a threshold, all peaks higher than the threshold value are significant, all others are ignored.

Question 14

What are the properties of Roberts’ Cross Operator?

Answer 14

Very quick to compute - 4 pixels, only subtractions and additions, but is very sensitive to noise and only gives a strong response to very sharp edges

Question 15

What are the differences between Roberts’ Cross and Sobel?

Answer 15

Both use a user-supplied threshold, however, Sobel is still in use, but Roberts’ is less common.
Larger Sobel operators are also more stable in noise

Question 16

What does Marr-Hildreth imply?

Answer 16

Gaussian smooth, compute Laplacian
OR
Convolve with the Laplacian of a Gaussian

Question 17

What does LoG and DoG mean?

Answer 17

LoG - Laplacian of Gaussian
DoG - Difference of Gaussians

Question 18

What are the differences and similarities between 1st Derivative and 2nd Derivative methods?

Answer 18

Peaks in 1st derivative:
Strong response at edges, but also responds to noise
Peak detection and threshold selection need care
Zero crossing in 2nd Derivative:
Well-defined, easy to detect
Must form smooth, connected contours
Tend to round off corners

1st Derivative methods are much more common in practical applications

Question 19

What does an optimal edge detector require to be optimal?

Answer 19

Good Detection - it should mark all the edges and only all the edges
Good localisation - the points marked should be as close to the real edge as possible
Minimal response - each edge should be reported only once

Question 20

What was the Canny operator made up of?

Answer 20

A sum of 4 exponential terms, but is very closely approximated by the 1st Derivative of a Gaussian i.e. 1st derivative of a Gaussian smoothed image
Gives a cleaner response to a noisy edge than square operators

Question 21

What is Non-Maximal Suppression?

Answer 21

Check if pixel is a local maximum along the gradient direction
Select a single maximum across the width of the edge

Question 22

What does the industry standard thresholding method contain?

Answer 22

Allows a band of variation, but assumes continuous edges
User still selects parameters, but its easier and less precise

Question 23

What is thresholding with hysteresis, and what does it aim to do?

Answer 23

The effect is to keep weak edges if they connect strong edges, as long as the strong edges are really strong, and the weak edges aren’t really weak.

Question 24

What was it that Canny had done?

Answer 24

Showed that 1st Derivative of a Gaussian smoothed image is the optimal way to detect step edges in noise - explained why 1st Derivates are a good idea
Designed the industry standard thresholding method - Non-Maximal suppression, and thresholding with Hysteresis
Effectively solved the edge detection problem