Chapter 3 cont’d.Binary Image Analysis
Binary image morphology
(nonlinear image processing)
CLOSING
Definition: closing BBABA
CC BABA
simply dilation followed by erosion when –B = B
comp((comp(A) opened by B))
APPLICATION TO MEDICAL IMAGES
APPLICATION TO GEAR INSPECTION (MISSING TEETH)
Extracting shape primitives
►translation invariance (E,D,O,C)
Morphological filter properties
xBABxA
►increasing (E,D,O,C)
Morphological filter properties
BABAA 2121 A t h e n i f
►commutative (D)
►associative (D)
Morphological filter properties
ABBA
CBACBA
This property can lead to a more efficient implementation. How?
►antiextensive (O)
Morphological filter properties
ABA
►extensive (C)
Morphological filter properties
BAA
►idempotent (O,C)►“Idempotence is the property of certain
operations in mathematics and computer science, that they can be applied multiple times without changing the result beyond the initial application.” - wikipedia
Morphological filter properties
AA
►The origin is important for E and D, but not O and C.
Morphological filter properties
BOUNDARIES
Boundary detection► Let B be a disk centered at the origin.
Then we can detect 3 types of boundaries:
1. External boundary
2. Internal boundary
3. Both (morphological gradient)
ABA
BAA
BABA
CHGHG :Recall
Boundary example
More examplesinput dilation
erosion external
internal morphologicalgradient
More examples
Dilation & distance maps/transforms
►Dilation can be used to create distance maps.
►Di is the set of all pixels at distance i from object.
►The operators denote the ith and i-1th applications of dilation.
BABAD iii 1
CONDITIONAL DILATION
Conditional dilation►(Two somewhat conflicting) problems:
1. Dilation may merge separate objects.
2. Erosion may remove parts of an otherwise complete object that we are trying to target.
Conditional dilation (Dougherty et al. method) addressing problem #1 (i.e., dilation may merge separate objects)► If B contains the origin, expands A
without bound. Recall dilation:
►So we define conditional dilation:
- the conditional dilation of A by B relative to C
BA
CBA
AaCaBBA C
|
(hitting) | e)commutativ (since |
addition) (Minkowski |
AxBxAaaBBbbA
BABA C
Conditional dilation (Dougherty) cont’d.►But the previous definition may cause
dilation to join disconnected objects so another form may be used:
and | CaBAaaBBA
Conditional dilation (Dougherty) example
C
B
Conditional dilation (Dougherty) example
Conditional dilation (Shapiro and Stockman) addressing problem #2
►Recall problem #2: “We want the entire component(s), not just what remains of them after erosion.”
► “Given an original binary image B, a processed image C (e.g., C=B(-)V for some structuring element V), and a structuring element S, let
The conditional dilation of C by S w.r.t. B is defined bywhere the index m is the smallest index satisfying Cm = Cm-1.”
mB CSC |
. a nd 10 BSCCCC nn
Example of Shapiro and Stockman conditional dilation
Note: C is (typically) a processed version of B.
From previous slide:“Given an original binary image B, a processed image C, and a structuring element S, let
The conditional dilation of C by S w.r.t. B is defined by
where the index m is the smallest index satisfying Cm = Cm-1.”
Basically, we grow (dilate) object seeds in C until we completely recover the object in B corresponding to those seeds.
S
S
. a nd 10 BSCCCC nn
mB CSC |
Conditional dilation example► Below are the input image (left), C, and the result of
its dilation by an euclidean disk of diameter 7 (right), C0<i<m.
► Below are the conditional image (left), B, and the resultant image (right), Cm.
SKELETONIZATION
Skeletonization►a common thinning
procedure
►Medial Axis Transform (MAT)
►Connected sets can have disconnected skeletons!
Skeletonization
SkeletonizationDefinition: Given a Euclidean (continuous)
binary image, consider a point within an object. Next, consider that point to be the center of the largest disk centered at that point s.t. the edge of the disk touches the border of the object. Two possibilities:
1. This disk is contained in a larger disk (centered somewhere else)
or2. This disk is NOT contained in a larger disk.
If (2) holds, the disk is maximal and the center is on the medial axis of the object.
Skeletonization
Skeletonization►A and B are maximal; C is not.
Skeletonization
Skeletonizationexample
Note: I’m not sure why these three aren’t exactly the same!
Skeletonization example
Skeletonization►Define skeletal subset Skel(S; n) to be the
set of all pixels x in S s.t. x is the center of a maximal disk nB (the n-fold dilation) where n=0,1,… and
(n in nB is, in effect, the size.)►Define skeleton to be the union of all skeletal
subsets:
n
BBBnB ...
,...2,1,0,
;,...2,1,0),;(
nBnBSnBSSSkelBnBSnBSnSSkel
nnSSkelSSkel
Morphological operations on gray images
►Believe or not, these operations can be (have been) extended to gray image data!
►This ends our discussion of binary morphological operations.
► References:►E. Dougherty, “An Introduction to Morphological Image
Processing,” SPIE Press, 1992.►E. R. Dougherty and J. Astola, “An Introduction to Nonlinear
Image Processing,” SPIE Press, 1994.►C.R. Giardina and E.R. Dougherty, “Morphological Methods
in Image and Signal Processing,” Prentice Hall, 1988.►L.G. Shapiro and G.C. Stockman, “Computer Vision,”
Prentice Hall, 2001.