Binary Image Analysis

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Binary Image Analysis

• Thresholding
• Convolution
• Morphology
• http//www.mmorph.com/mxmorph
• 4. Region Properties
• 5. Edge Properties

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1. Region Properties2. Edge properties
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Region Properties
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Object Counting

• How many objects are there in an image?

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Define external and Internal Corners

• Slide the masks over image and find
  external_match (L,P) and
• internal_match (L,P)

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Object Counting
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Document analysis

• How many letters , words, paragraphs
• in a page?

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Connected Components Labeling
Once you have a binary image, you can identify
and then analyze each connected set of
pixels. The connected components operation takes
in a binary image and produces a labeled image
in which each pixel has the integer label of
either the background (0) or a component.
connected components
binary image after morphology
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Methods for CC Analysis

1. Recursive Tracking
2. Parallel Growing (needs parallel hardware)
3. Row-by-Row

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Recursive Labeling
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Row-by-row labeling algorithm

• First pass Propagate a pixels label to its
  neighbors to the right and below it. Whenever
  two different labels can propagate to the same
  pixel, these labels are recorded as an
  equivalence class.
• Second pass Perform a translation, assigning to
  each pixel the label of its equivalence class.
• A union-find data structure is used for
  efficient construction and manipulation of
  equivalencenclasses represented by tree
  structures.

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Equivalent Labels
Original Binary Image
0 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 1 1
1 0 0 0 1 1 1 1 0 0 0 1 1 0 0 0 1 1 1 1 1 0 0 1 1
1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 0 0 1 1
1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 0 0 0 1 1
1 1 1 1 1 1 1 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 0 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1
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Equivalent Labels
The Labeling Process
0 0 0 1 1 1 0 0 0 0 2 2 2 2 0 0 0 0 3 0 0 0 1 1 1
1 0 0 0 2 2 2 2 0 0 0 3 3 0 0 0 1 1 1 1 1 0 0 2 2
2 2 0 0 3 3 3 0 0 0 1 1 1 1 1 1 0 2 2 2 2 0 0 3 3
3 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 3 3 3 0 0 0 1 1
1 1 1 1 1 1 1 1 1 0 0 3 3 3 0 0 0 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 0 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1
1 ? 2 1 ? 3
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Run-Length Data Structure
0 1 2 3 4
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4
row scol ecol label
Binary Image
U N U S E D 0 0 0 1 0 0 3 4 0 1 0 1 0 1
4 4 0 2 0 2 0 2 4 4 0 4 1 4 0
0 1 2 3 4 5 6 7
Rstart Rend

• 2
• 4
• 6
• 0 0
• 7 7

0 1 2 3 4
Row Index
Runs
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Run-Length Algorithm
Procedure run_length_classical initialize
Run-Length and Union-Find data structures
count lt- 0 / Pass 1 (by rows) / for each
current row and its previous row
move pointer P along the runs of current row
move pointer Q along the runs of previous row
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Case 1 No Overlap
Q
Q
///// /////
////
/// /// /////
P
P
/ new label / count lt- count 1 label(P) lt-
count P lt- P 1
/ check Qs next run / Q lt- Q 1
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Case 2 Overlap
Subcase 2 Ps run has a label that is different
from Qs run
Subcase 1 Ps run has no label yet
Q
Q
/////// ///// /////////////
/////// ///// /////////////
P
P
label(P) lt- label(Q) move pointer(s)
union(label(P),label(Q)) move pointer(s)

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Pass 2 (by runs)
/ Relabel each run with the name of the
equivalence class of its label / For each run M
label(M) lt- find(label(M))
where union and find refer to the operations of
the Union-Find data structure, which keeps track
of sets of equivalent labels.
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Thresholding and connected compmonent labeling
connected components of 1s from thresholded image
Gray level Image
Binary Image
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Region Properties-Features
Properties of the regions can be used to
recognize objects.

• geometric properties
• gray-tone properties
• color properties
• texture properties
• shape properties
• motion properties
• relationship properties

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Geometric and Shape Properties

• area
• centroid
• perimeter
• perimeter length
• circularity 1

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Distances
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Moments
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Bounding Box and Extremal points

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ExEllipse, square, rectangle
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Labeling shown as Pseudo-Color
connected components of 1s from thresholded image
connected components of cluster labels
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Region Adjacency Graph
A region adjacency graph (RAG) is a graph in
which each node represents a region of the image
and an edge connects two nodes if the regions are
adjacent.
1
1
2
2
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4
3
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2. Edge Operators

• a kind of filtering that leads to useful features

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Lines and ArcsSegmentation
In some image sets, lines, curves, and circular
arcs are more useful than regions or helpful in
addition to regions.

• Lines and arcs are often used in
• object recognition
• stereo matching
• document analysis

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Edge Detection
Basic idea look for a neighborhood with strong
signs of change.
81 82 26 24 82 33 25 25 81 82 26
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• Problems
• neighborhood size
• how to detect change

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Differential Operators

• Differential operators
• attempt to approximate the gradient at a pixel
  via masks
• threshold the gradient to select the edge pixels

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Example Sobel Operator
-1 0 1
1 2 1 Sx -2 0 2
Sy 0 0 0 -1 0 1
-1 -2 -1

• On a pixel of the image I
• let gx be the response to Sx
• let gy be the response to Sy

Then the gradient is ?I gx gy
T
2
2
1/2
And g (gx gy ) is the
gradient magnitude. ? atan2(gy,gx)
is the gradient direction.
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Sobel Operator on the Blocks Image
original image gradient
thresholded
magnitude gradient

magnitude
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Common Masks for Computing Gradient

• Sobel
• Prewitt
• Roberts

-1 0 1 -2 0 2 -1 0 1

• 2 1
• 0 0 0
• -1 -2 -1

-1 0 1 -1 0 1 -1 0 1

• 1 1
• 0 0 0
• -1 -1 -1

0 1 -1 0

• 0
• 0 -1

Sx
Sy
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Zero Crossing Operators
Motivation The zero crossings of the second
derivative of the image
function are more precise than
the peaks of the first derivative.
step edge
smoothed
1st derivative
zero crossing
2nd derivative
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How do we estimate the Second Derivative?

• Laplacian Filter ? f ? f / ?x ? f / ?y

2
2
2
2
2
0 1 0 1 -4 1 0 1 0

• Standard mask implementation
• Derivation In 1D, the first derivative
• can be computed with mask -1 0 1
• The 1D second derivative is 1 -2 1
• The Laplacian mask estimates the
• 2D second derivative.

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Properties of Derivative Masks

• Coordinates of derivative masks have opposite
  signs in order to obtain a high response in
  regions of high contrast.
• The sum of coordinates of derivative masks is
  zero, so that a zero response is obtained on
  constant regions.
• First derivative masks produce high absolute
  values at points of high contrast.
• Second derivative masks produce zero-crossings at
  points of high contrast.

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Marr/Hildreth Operator

• First smooth the image via a Gaussian
  convolution.
• Apply a Laplacian filter (estimate 2nd
  derivative).
• Find zero crossings of the Laplacian of the
  Gaussian.
• This can be done at multiple resolutions.

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Haralick Operator

• Fit the gray-tone intensity surface to a
  piecewise
• cubic polynomial approximation.
• Use the approximation to find zero crossings of
  the
• second directional derivative in the direction
  that
• maximizes the first directional derivative.
• The derivatives here are calculated from direct
• mathematical expressions wrt the cubic polynomial.

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Canny Edge Detector

• Smooth the image with a Gaussian filter with
  spread ?.
• Compute gradient magnitude and direction at each
  pixel of
• the smoothed image.
• Zero out any pixel response ? the two
  neighboring pixels
• on either side of it, along the direction of
  the gradient.
• Track high-magnitude contours.
• Keep only pixels along these contours, so weak
  little
• segments go away.

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Canny Examples
Canny ?4
Canny ?1
Canny ?1
Roberts 2X2
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Canny on Kidney
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Canny Characteristics

• The Canny operator gives single-pixel-wide images
  with good continuation between adjacent pixels
• It is the most widely used edge operator today
  no one has done better since it came out in the
  late 80s. Many implementations are available.
• It is very sensitive to its parameters, which
  need to be adjusted for different application
  domains.