Today:

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EE545 Image Processing

• Today
• Image Formation Physics, image tesselation
• Binary Image Processing
• Transformations on image coordinates

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EE545 Image Processing

• 2D Image I (r,c) size N x M
• Digital Image Processing Improving and
  transforming pictorial information
• Computer Vision Processing of Images to extract
  information about the world

c
I
r
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Examples of Image Processing
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Examples of Image Processing
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Image formation

• How is the image obtained ?
• A simple (but unrealistic) model
• Orthographic (parallel) projection

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Image formation

• A more realistic model
• Perspective projection (using pinhole model)

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Perspective Images of Renaissance Painters
The Scholar of Athens, Raffaello Sanzio
(Rafael), 1518
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Pinhole
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Image formation Brightness
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Image formation Lenses
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Image formation Sampling Image Space

• The Image pixels can be tiled with three regular
  polygons

Rectangular
Triangular
Hexagonal
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Binary Image Processing(Two levels for pixel
values)

• Gray Level Image (8 bit/pixel) Histogram

of pixels
0 (white)
255 (black)
TThreshold if I(r,c)gtT I(r,c)255 else
I(r,c)0
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Adjacency in rectangular sampling
4-adjacent pixels
8-adjacent pixels
p
p
8 path
4 path
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Foreground/Background paradoxes for binary images
8-8 connectivity for fg-bg
4-4 connectivity for fg-bg
Foreground Closed loop Background No hole
(problem)
Foreground No closed loop Background Hole
(problem)
Solution Use 8-4 or 4-8 connectivity for fg-bg
No such problems for hexagonal tessellation !
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Distance
D r
D c
Euclidian distance Manhattan distance Chess-boar
d distance
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First Binary Image Processing Algorithm
Connected Component (CC) Labeing

• 8 connected component Maximal set of 1's for
  which all pairs of 1's in S are 8 connected by an
  8 path in S.
• Connected component labeling (Iterative
  algorithm)
• P gets its label from one of its neighbors
• Top/down Bottom/up passes

p
p
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Binary Image Processing

• Connected component labeling (using memory)
• Examples

N
j
a
b
c
abcd0 then L(p)iNj
p
d
i
M
1
0
1
1
0
1
0
1
0
0
1
0
0
1
1
1
1
1
1
0
L(p)L(a) L(c)L(a)
L(p)L(a) L(d)L(c)L(a)
L(d)L(b)
Assignments
L(p)L(b) L(d)L(b)
Equivalences
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Representation of Binary Images

• Matrices I(r,c)
• Chains
• Run length coding

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5
6
p
0
4
2
3
1
021244567
Boundary following algorithm Start from the
first pixel in the scan order
Look for a border pixel at box 0,1,2,3
Do the same for the new pixel until
the first pixel is reached
0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 1 0 1
The code is ((11144)(214)(52355))
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Representation of Binary Images

• Hierarchical data structures
• Pyramids
• Quadtrees

Level 0
Level 1
Level 2
Pixel at Level k Average of pixels at Level
(k1)
11
10
0
0
2
3
13
10
11
13
2
3
120
121
122
123
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Binary Image ProcessingRegion Properties of
CCs

• Area
• Centroid
• Eulers number (genus)

,
C of connected components H of holes
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Binary Image ProcessingRegion Properties of
CCs

• Perimeter (P) of boundary pixels
• Compactness (A circle is the
  most compact figure)
• Moments
• Central moments

Where the centroid can be found as
,
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Binary Image ProcessingRegion Properties of CCs

• Direction
• Rectangularity Area of bounding rectangle

Area of the object
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Binary Image Processing

• Objects (or their parts) are represented by
  isolated connected components
• Main applications Document processing, Character
  recognition, Cell counting,
• Several descriptors of an object can be extracted
  from an image (size, centroid position, of
  holes, perimeter, compactness, dominant
  direction, horizontal/vertical/diagonal
  projections )

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1D Projections on Binary Images
Vertical
Horizontal
Diagonal
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Binary Image Processing

• In practice binary
• images (obtained by
• thresholding) need
• conditioning
• The tools necessary
• for conditioning are
• provided by
• Mathematical Morphology

Noisy
Conditioned
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Mathematical Morphology

• mathematical framework used for
• pre-processing
• noise filtering, shape simplification, ...
• enhancing object structure
• skeletonization, convex hull...
• segmentation
• watershed,
• quantitative description
• area, perimeter, ...

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Structuring element (SE)

• small set to probe the image under study
• for each SE, define origo
• SE in point p origo coincides with p
• shape and size must be adapted to geometric
  properties for the objects

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Basic idea

• in parallel for each pixel in binary image
• check if SE is satisfied
• output pixel is set to 0 or 1 depending on used
  operation

pixels in output image if check is SE fits
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Basic morphological operations
shrink

• erosion
• dilation
• combine to
• opening
• closening

grow

• keep general shape but smooth with respect to
• object
• background

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Mathematical Morphology

• An algebraic system of operators
• Complex shapes can be decomposed into meaningful
  parts
• Set theory notation (AImage points
  BOperator points)
• Dilation where
• Erosion
• Opening Closing

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Mathematical MorphologyDilation

• Example 1
• Example 2

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Dilation (example 3)
SE
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Dilation (example 4)
SE
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Mathematical MorphologyErosion

• Example

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Erosion
SE
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Erosion
SE
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Mathematical MorphologyDilation and Erosion
Dilation and Erosion by
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• Erosion is useful for
• removal of structures of certain shape and size,
  given by SE
• Dilation is useful for
• filling of holes of certain shape and size,
• given by SE

Combining erosion and dilation
WANTED remove structures / fill holes without
affecting remaining parts
SOLUTION combine erosion and dilation (using
same SE)
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Mathematical MorphologyOpening and Closing
Opening and Closing by Opening Closing
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Opening roll ball(SE) inside object
see B as a rolling ball
boundary of AB points in B that reaches
farthest into A when B is rolled inside A
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Closing roll ball(SE) outside object
see B as a rolling ball
boundary of A?B points in B that reaches
farthest into A when B is rolled outside A
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Mathematical MorphologyApplication (An example)

• Computing the genus ( of CCs - of holes)

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An application example (of MM) Isolating IC pins
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MM ApplicationConditioning fingerprint images

• erode
• A?B

2. dilate (A?B)?B AB
4. erode ((AB)?B)?B (AB)?B
3. dilate (AB)?B
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Imaging Geometry

• The motion of a camera can be modeled by
  geometric transformations

?
Affine Transformations
Perspective Tr.
(Orthographic camera model)
(Perspective camera model)
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Projective models for Camera Motion

• 2D Affine Transformation

Where and are old
and new coordinates
of a pixel
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Affine Transformation

• is the translation component
• A?

Scale Rotation
Vertical Shear
Horizontal Shear
Altered Aspect Ratio
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Affine and Perspective Transformations in matrix
form (in Homogenous Coordinates)

• Affine Transformation
• Perspective Transformation

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Perspective Transformations

• Another form Homographies

Note 6 unknowns for affine transformations
8 unknowns for perspective transformations
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Forward vs Backward mapping

• Forward
• Backward

Scan the input image, find new coordinates,
assign pixel value
Scan the output image, find old coordinates,
assign pixel value
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Interpolation (in gray level images)

• Forward mapping produces holes
• Backward mapping of a pixel produces

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• Nearest Neighbor Interpolation
• p gets its value from the nearest pixel
• Bilinear Interpolation
• p gets its value from the plane on top of 4
• neighboring pixels

u
220
150
v
1
p
220
150
p
100
60
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NNI p150 BI pp1(1-v)p2(v)
where p1 220(1-u)150(u)
p2100(1-u)60(u)
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