Chapter 3 Binary Image Analysis

Types of images ► Digital image = I[r][c] is discrete for I, r, and c.  B[r][c] = binary image - range of I is in {0,1}  B’[r][c] – complement or inverse of B = 1 if B[r][c]=0 = 0 if B[r][c]=1  Gray-scale image = range of I is in {min, min+1,…,max-1,max}  Multispectral image = vector-valued ► Ex. Color image = range is in { | (r,g,b) in {0,1,…,max} }

How do we create binary images? ► By selecting (foreground) pixels of interest and thereby separating them from background pixels = segmentation  Ex. Thresholding = selecting ranges of gray (or color) values ► Ex.  1 if value>100  0 otherwise  0 = background  1 = foreground

Pixel and its neighbors ► 4-neighbors/4-adjacent/4-connected  {,,, }  Or as displacements: ► {,,, } ► How about 8-adjacent?  Distances?

Masks

► Mask = set of pixel positions and corresponding values called weights ► Mask origin = usually center ► How? 1.Calculate sum of products ► Boundary 1.Replicate nearest pixel value 2.Use 0 2.Normalize (or an amplitude shift will occur) Applying masks to images

► Derived from convolution: ► Discrete form is cross correlation: ► where f is the input image, h is the mask/filter kernel, and g is the output image result

Convolution ► See for some nice animations.

1*40+2*40+1*80+ 2*40+4*40+2*80+ 1*40+2*40+1*80 =800

1*40+2*80+1*80+ 2*40+4*80+2*80+ 1*40+2*80+1*80 =1120

But what about borders? ► When we are missing data at the edges, we typically do one of the following: 1.Copy to the missing value, the nearest neighboring value. 2.Use 0 for the missing value(s). (Regardless, it really doesn’t matter, but we need to consistently do something.)

1*40+2*40+1*40+ 2*40+4*40+2*40+ 1*40+2*40+1*40 =640 Method 1: copy nearest

The importance of normalizing the result. ► Otherwise, the values might get larger and larger (brighter and brighter).

=16 640/16

Counting objects (instead of holes) ► If we complement our masks from the hole counting algorithm, we can count objects! ► (Or if we complement our image, we can count objects!)

I don’t care for this format!

Object counting assumptions (complement of hole counting assumptions) 1. All image border pixels must be 0s. 2. Each region of 1s (objects) must be 4- connected. 3. Objects must also be simply connected (not contain any holes).

Implement object counting algorithm ► Check assumptions 1 and 2.  (Skip 3 for now.) ► Report number of objects. ► Exercise 3.1 p Efficiency of counting objects.  What is the maximum number of times that the function count_objects examines each pixel of the image?  How can functions external_match and internal_match be coded to be as efficient as possibly?

More object analysis ► So far, we can count objects!  But how big are they?  What shape are they?

Paths There exists a path from (x 1,y 1 ) to (x n,y n ) iff there exists a sequence (x 1,y 1 ), (x 2,y 2 ), …, (x n,y n ) s.t. all (x i,y i ) and (x i+1,y i+1 ) are (4- or 8-) connected and B(x i,y i )=B(x i+1,y i+1 ).

Connected component A connected component of value v is a set of pixels C with: 1.each having value v 2.and for every c and d in C, there exists a path from c to d.

Objects are 1s (and white) here.

Labels ► 0 = background (not part of any object) ► 1 = first labeled object ► 2 = second labeled object ► … ► -1 = part of an unlabeled object (ready for labeling)

I don’t care for this format! Recursive connected components algorithm: 3 functions

I don’t care for this format! Recursive connected components algorithm // 0 --> 0; 1 --> -1 // incremented before use so first will actually be 1

I don’t care for this format! Recursive connected components algorithm (Why not use r and c instead of L and P, respectively?)

I don’t care for this format! Recursive connected components algorithm <-- What does this mean/do? <-- recursion! This is an example of a flood fill algorithm (see for nice animations).

Begin skip other algorithms in this section.

End skip.