1. Morphological Operators

2. More on Homogeneous Intensity Regions
Morphology
Refers to “form and structure”
Defined as set operations
Pixels that make up a region are the elements of the set
Pixels not in the region are the elements outside the set
Operate on binary images where connected components have been identified
Basic implementation is similar to that of convolution with a kernel

3. (connected components with boundaries)
Pixels as Sets 1 3 2 5 4
1 Set of pixels (white)
5 Sets of pixels
(connected components with boundaries)

4. Morphology Operations
Inputs
Binary image (connected components)
Structuring element (kernel)
Output
Processed regions
Region
Region
Morphological
Operation
Structuring
Element

5. Structuring Element Analogous to a convolution kernel
Acts as a “probe” of the binary image
Designate an origin of the element
May be the central pixel
May be some other pixel
Can be any shape or size

6. Structuring Element 1 1 1 1 1

7. Structuring Element Notice the empty spaces on the elements
These spaces are treated as “don’t cares”
Clearly we must represent them as something when writing code
You choose what the values should be and make sure you handle them consistently
More on this later

8. Basic Operations Five common Morphological Operations Dilation Erosion
Closing
Opening
Hit And Miss

9. Dilation Used to enlarge a region
Sweep the structuring element over the image (as you did with convolution)
Each time the origin lands on a pixel within the region…
Write a 1 wherever the structuring element is a 1
That is, replicate the structuring element for every region pixel
The effect will be to thicken lines, fill gaps and holes, and grow object boundaries

10. Dilation With a 3x3 structure element of all 1’s: INPUT OUTPUT
DIFFERENCE

11. Erosion Used to shrink a region
Sweep the structuring element over the image
When all structuring elements (the 1’s) cover region pixels
Place a 1 at the origin pixel of the output image
The effect will be to thin lines and shrink object boundaries

12. Erosion With a 3x3 structure element of all 1’s: INPUT OUTPUT
DIFFERENCE

13. Closing
Used to close internal holes and eliminate concavities from regions
Concatenation of dilation/erosion operations
Perform a dilation on the input image to produce an intermediate image
Perform an erosion on the intermediate image to produce the output image

14. Closing With a 3x3 structure element of all 1’s: INPUT OUTPUT
DIFFERENCE

15. Opening Used to remove “tails” from regions
Concatenation of erosion/dilation operations
Perform an erosion on the input image to produce an intermediate image
Perform an dilation on the intermediate image to produce the output image

16. Opening With a 3x3 structure element of all 1’s: INPUT OUTPUT
DIFFERENCE

17. Another Example INPUT DILATE ERODE OPEN CLOSE
With a 3x3 structure element of all 1’s:

18. Hit And Miss Searches for specific patterns in the image
Sweep the structuring element over the image
When all structuring elements match underlying image pixels exactly
Consider structure element 1’s (region pixels), 0’s (non-region pixels), and “don’t cares”
Logical OR the image pixel corresponding to the structure element origin into the output image (i.e. write a value to the output where the origin of the structuring element is)

19. Hit And Miss
Note that the Structuring Element contains three types of values
Foreground (region) indicators
Background (non-region) indicators
“don’t care” indicators
Foreground indicators must land on region pixels
Background indicators must land on non-region pixels
Don’t care indicators can land on anything

20. Hit And Miss 1 STRUCTURING OUTPUT ELEMENT INPUT
There is a white pixel here
(lower left corner of the square)

21. Compound Operations
Advanced operations can be created by combining the basic operations
Skeletonization (thinning)
Thin(i, j) = I(i, j) ∩ HitAndMiss(I(i, j))
This will take a “wide” object and produce a thin object

22. From Edges to Lines
The Hough Transform

23. From Edges To Lines Edges are a good start
They contain some important information and reduce the amount of raw data
We don’t have to deal with all the pixel intensities that may not be all that informative
But now we must do something with the edges
As we saw last week, computing texture was one of the things we can do

24. Binarized edgels (edge pixels), not a line!
Hough Transform
Hough Transform
Binarized edgels (edge pixels), not a line!

25. Binarized edgels (edge pixels), not a line!
Hough Transform
Hough Transform
Binarized edgels (edge pixels), not a line!

26. Binarized edgels (edge pixels), not a line!
Hough Transform
Hough Transform
Binarized edgels (edge pixels), not a line!

27. Binarized edgels (edge pixels), not a line!
Hough Transform
Hough Transform
Binarized edgels (edge pixels), not a line!

28. Binarized edgels (edge pixels), not a line!
Hough Transform
Hough Transform
Binarized edgels (edge pixels), not a line!

29. Binarized edgels (edge pixels), not a line!
Hough Transform
Hough Transform
Binarized edgels (edge pixels), not a line!

30. Binarized edgels (edge pixels), not a line!
Hough Transform
Hough Transform
Binarized edgels (edge pixels), not a line!

31. Binarized edgels (edge pixels), not a line!
Hough Transform
Hough Transform
Binarized edgels (edge pixels), not a line!

32. Hough Transform What exactly is going on?
The Hough Transform is a technique for finding straight lines in a mass of disconnected points
It’s a “model-based” technique
It utilizes a mathematical equation (a model) of a line and then tries to fit points to the equation

33. Hough Transform
Input image is [typically] a binarized edge magnitude map
You could define it to use edge magnitudes in a manner similar to the Canny edge detector
The process maps image space edge points to Hough Space coordinates
The output image is the location of lines in image space
What I showed was a rendering of Hough space which is not really the desired output

34. Hough Space A line in image space maps to a point in Hough Space
A point in image space maps to multiple points in Hough Space
Each point (ρ, θ) in Hough space represents line in image space
ρ is the perpendicular distance from the line to the origin
θ is the orientation of the line relative to a specified axis

35. Hough Transform
A single edge point in image space can be part of an infinite number of lines in image space
Since “infinite” is a bit difficult to deal with in the digital computer world we quantize the orientation of the lines into convenient bins
To compute the Hough Transform of a single point in image space we measure the distance from the origin to every possible line through the point

36. Hough Transform Specifically, we compute
for all values of θ. For example: θ ρ θ ρ
Xp, Yp
Xp, Yp

37. Hough Space The result is a sinusoidal waveform in Hough space
There is a single dot in here

38. Implementation Basic data structure is the accumulator matrixθ ρ
Similar to intensity histogram creation
Initialize each location to 0
Set θ
Calculate ρ
Increment the (ρ, θ) bin

39. Mechanization
When multiple dots contribute to the same line (are colinear) in image space the sin waves will intersect in a single point in Hough Space
The coordinates of that point represent the parameters of the line

40. One Dot

41. Two Collinear Dots

42. Three Collinear Dots

43. Four Collinear Dots

44. Implementation Considerations
Must settle on an origin
Upper left image corner
Image center
Lower left image corner
Whatever
Just choose one and document it
Must select a quantization of the orientation
Too fine of a quantization and the sinusoidal waves won’t intersect in a single point
Too coarse of a quantization and near parallel lines are accumulated into a single bin

45. Implementation Considerations
The image points won’t always be perfectly collinear and therefore the Hough Space sinusoidal waves won’t always intersect in a single point
Angle quantization
Non-maximal peak suppression on the Hough accumulator array may be employed to reduce clustering effects (similar to the process used by Canny)

46. Implementation Considerations
The answer is in here somewhere

47. Implementation Considerations
After non-maximal peak suppression you may want to binarize (threshold) the accumulator array leaving only the strongest peaks
Those represent locations where the most sinusoids intersected
This is where the most number of points contributed to a given line (high degree of colinearity)

48. Implementation Considerations
The basic Hough Transform does not keep track of which points contribute to a line
Make each accumulator bin a linked-list (or array) of image space point locations if you want to do this
Edge orientation information is completely ignored
May want to store this or use it to weight a particular edges contribution to the accumulator

49. “Gotchas”
Make sure your accumulator array is initialized to 0 each time you compute a Hough Transform
Make sure your accumulator array is big enough to hold all possible distances
Distances can be as far as the diagonal length of the image
Distances can be positive or negative dependent on how you choose your origin

50. Generalized Hough Transform
The only thing specific to a straight line is the parameterization – the process is general
Basically, any shape that can be quantitatively described can be detected with the Hough Transform
Many approaches to detecting circles have been published – especially fixed radius circles

51. Remainder of Class Period
Open for questions regarding current or past programming assignments
Write code to perform Morphology operations:
Erosion
Dilation
Open
Close
Hit and Miss

52. Things To Do Write code to perform Reading for Next week
The five basic morphological operations
The Hough Transform
Due 2 weeks from now
Reading for Next week
Chapter 6 – Color
Why now? – because it is used to do segmentation, region description, and other mid-level vision stuff (as well as low-level operations)
We’ll look at various color representations
I especially like this topic 