1. Each pixel is 0 or 1, background or foreground Image processing to
Binary Images
Each pixel is 0 or 1, background or foreground
Image processing to
Enhance separation of objects of interest
Separate and count multiple objects
Understand the shapes of multiple objects

2. From image to binary image
Classification: dividing pixels into "foreground" and "background"
Thresholding
If a pixel has a value in range (min, max) it is foreground
Often min is 0 or max is maximum pixel value
Choice of range can be manual or automatic
(E.g. look for peaks / valleys in histogram)
More complex operations
Use information from neighboring pixels
Use properties besides pixel value (e.g. location) …

3. Neighborhoods can be defined for each pixel
Image Neighborhoods
Neighborhoods can be defined for each pixel
The two most common neighborhoods
4-neighborhood
8-neighborhood N W E S

4. Applying a Mask
Mask is a set of relative pixel positions. One is designated the origin (0,0) - usually at center
Each mask element is weighted
To apply the mask, put the origin pixel over the image pixel and multiply weights by the pixels under them, then add up all the values.
Usually this is repeated for every image in the pixel. Assumptions must be made for pixels near the edge of the image.

5. Mask application example
Mask = 1 1 1, origin at center
Apply to every pixel in image:
Result is

6. Shapes in a binary image1

7. Masks for object counting
External corners (origin = top left)
Internal corners (origin = top left)
Apply using exact match (result is 1 if every pixel in mask matches the image)
If any of the 4 external corner masks matches, the corner is external (& same for internal)

8. Counting 4-Connected Objects
The number of 4-connected objects in an image is
(E - I) / 4
where E is the number of external corners and I is the number of internal corners.
Assumptions:
Objects are 4-connected (2 diagonal pixels are 2 objects)
Objects do not contain "holes" within them
Go back and try it on example two slides back.
Note how the hole affects things.

9. "Blobs" are connected components
4-connected (diagonal neighbors don't count)
8-connected (diagonal neighbors are connected)
The following diagram has 3 4-connected components (red, blue, black) or 2 8-connected components (non-black, black)

10. Recursive Connected Components
Copy the image
While at least one 1-pixel exists in the existing image
Create a new label
REC_LABEL(label, pixel)
Label and remove the pixel
For each non-zero neighbor
REC_LABEL(label, neighbor)

11. Recursive Connected Components Example
First pixel
After 4 recursive calls, no 4-neighbors neighbors to color
Start again with a new color on an unmarked pixel

12. Two-Pass Connected Components
For every pixel in the image (left-right, top-bottom)
If the pixel is non-zero
If the pixel has no labeled neighbors (above/left)
Create a new label & label it
Else if all labeled neighbors are the same
Give the pixel the same label
Else if neighbors have 2 different labels
Give the pixel the largest label
Mark the smaller label as a parent of the larger one (parent [larger] = smaller)

13. Two-Pass Connected Components
Renumber all pixels using only one value for each "equivalence class"
This value is the root of the tree
While (parent[x] != 0) x = parent[x];

14. Two-pass Connected Component Example
2 lines scanned
3 lines scanned
4th line -- Conflict - set black = blue ?

15. Binary Image Morphology
Morphology = "Study of Shape"
Set of operations that are useful for processing connected components ("blobs") based on shape
Examples
Remove small holes or outcroppings
Remove objects below a given size
Smooth corners

16. Mask used for binary morphology
Structuring Element
Mask used for binary morphology
Like convolution masks, they slide over the image and operate on the pixels under them
Common elements:
Box (square or rectangle)
Disk (digital filled circle)
Bar (horizontal or vertical)

17. Morphology Operations
Dilation
Result is mask OR original
Erosion
Result pixel is 1 if origin pixel is 1 and all pixels covered by the mask are also 1

18. Morphology Operations
Closing = dilation followed by erosion
Opening = erosion followed by dilation

19. Applications of Morphology
General
Closing closes holes (up to size of element)
Opening opens spaces (up to size of element)
Specialized
Choose elements of size/shape based on your object
Eliminate objects that are too small / large
Isolate interesting features

20. Each pixel is distance to closest “1” in the original image.
Grassfire Transform
Each pixel is distance to closest “1” in the original image. 2 1 3 4 5 6

21. Another Example

22. Two-Pass Algorithm for Grassfire
Set all boundary pixels to Max
First pass: top left to bottom right
If original pixel was 1, pixel is 0
Else pixel = min(above + 1, left + 1);
Second pass: bottom right to top left
Pixel = min(pixel, below+1,right+1);

23. Distance Transforms and Medial Axis
Grassfire as described here measures 4-connected distance to the region of interest
Variations
Use a different distance transform, e.g. Euclidean
Use 0 instead of a large value at the boundaries
Medial axis is set of points furthest from a boundary
Set of pixels with maximal values in grassfire
Very sensitive to changes in boundary

24. Examples
Figure 3.9 (f = dilated, g=grassfire, h=components)

25. Other Useful Binary Image Operations
Pixelwise AND, OR
Pixelwise Subtraction
0 if first image is 0, or both images are 1
Translation
Every pixel is copied (dx, dy) away from its original position

26. Shape = list of features
Shape Representation
Shape = list of features
Boundary points, segments
Region features (center, convex hull, etc.)
“Interesting” points, e.g. corners
Shape invariants
Shape = mathematical description
Function type & parameters (e.g. circle, radius 5, centered at (0,0))
Mathematical function approximation of boundary, e.g. B-spline
Shape Classes

27. Shape Features Region features Boundary features
Simple properties (area, Euler’s #, projections, bounding box, eccentricity, elongatedness, direction, compactness)
Statistical moments
Convex Hull
Region decompositions (hierarchical representation)
Skeleton
Boundary features
Boundary points (e.g. chain code)
Geometric properties (perimeter, curvature, chords, etc.)

28. Compactness = (Perimeter & Perimeter) / Area
Simple Properties
Area
Number of pixels in the region
Perimeter
Number of boundary pixels
Compactness = (Perimeter & Perimeter) / Area
Maximal for circle, minimal for thin, long rectangle
Centroid
Average x value, average y value

29. Statistical Moments
In a binary image, a moment is the sum of relevant x’s and y’s.
(0,0) moment = area
(1,0) moment / area = average x coordinate
(0,1) moment / area = average y coordinate

30. Variations on Statistical Moments
Central moment
First translate the shape so that its center is (0,0), i.e. subtract the (average x, average y) values from all pixel locations
Scaled central moment
Divide the central moment by a power of the scale factor and the area
Unscaled central moment
Divide the central moment by a power of the area only

31. 2nd Order Moments and Ellipses
Central (2,0) moment relates to the horizontal axis of an ellipse approximating the shape
Central (0, 2) moment relates to the vertical axis of an ellipse approximating the shape
Central (1,1) moment relates to the rotation of an ellipse approximating the shape

32. More Region Properties
Bounding box
Min, max x-value, min-max y-value
Extremal points (on the bounding box)
Topmost left, topmost right, leftmost top, leftmost bottom, etc.
Lengths of extremal axes (e.g. top left -> bottom right)
These approximate the Convex Hull of the object
Convex hull is the shape you get by pounding nails into the black pixels and then wrapping them with a rubber band.

33. Boundary Pixels and Perimeter
4-connected object (black)
Boundary pixels have at least one white 8-neighbor
8-connected object (black)
Boundary pixels have at least one white 4-neighbor
If the object is 4-connected, the background is 8-connected and vice versa
Perimeter is the number of boundary pixels
Chain code: start at uppermost, leftmost boundary pixel - list DIRECTION to next pixel until the first one is reached again

34. Centroid, center of mass Center of contour
Where is the center?
Centroid, center of mass
(average x over all pixels, average y over all pixels)
Center of contour
(average x, y over boundary pixels only)
Example (note sensitivity to contour!)

35. Basis: thinning (many algorithms!)
Region Skeletons
Basis: thinning (many algorithms!)
Maximum values of grassfire transform (medial axis)
Last pixels to disappear with repeated erosion

36. Hierarchical or Graph Shape Description
Define a set of primitive shapes
Define a set of operations (concatenation, union, intersection, etc.)
Define a shape as a network of primitive shapes (parts) connected by operations
Recognize a shape by recognizing its parts and the relationships between them.

37. Region Adjacency Graph
Primitive shape = "connected set of pixels"
Operations = "adjacent to"
Element of region 1 is in the neighborhood of element of region 2
In binary images, all regions with no holes are adjacent to the single background region
All holes are adjacent to the objects that contain them