Image processing
Content
2011
1 Image representation 2 Image Filtering 3 Morphological transformations
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T. Chateau
Institut Pascal
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Image Representation
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Image Representation
The'image'model'used'here:
What'is'an'image?
f : [0, L
several'possible'defini/ons' • computer'point'of'view':'unsigned'char'table • Physicist:'observa/on'of'an'environment'by'an'op/cal'
1] ⇤ [0, M ]p I = f (x, y)
With':' • L:'number'of'lines'(height) • C:'number'of'columns'(width) ' 2 Np •M • p'='1'for'a'luminance'image'(grey'level) • p'='3'for'a'color'image'(RGB,'HSV,'...)
sensor'(2D'digi/zed'signal)
• Mathema/cian:'the'projec/on'of'a'3D'space'on'a' plane
• ...
T. Chateau
1] ⇥ [0, C
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Institut Pascal
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Image Representation
Image Representation
Distance between two pixels
Pixels, neighbourhood, and distances
Each pixel can be localised by its co-ordinates (x,y) into the image plane. Distances between pixels may be defined A distance measure must have the following properties:
d(P, Q) > 0 d(P, Q) = d(Q, P ) Triangle
Square
Hexagonal
d(P, Q) d(P, R) + d(R, Q)
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T. Chateau
Institut Pascal
Institut Pascal
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Image Representation
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Image Representation
Principal'distances • Manathan'distance'' d1 (P, Q) = |xp
• Euclidian'distance
d2 (P, Q) = (xp
Neighbourhood
xq | + |yp xq )2 + (yp
yq | yq )2
0.5
• Chessboard'distance d1 (P, Q) = max(|xp T. Chateau
xq |, |yp
Vk (p) = {Q : 0 < d(P, Q) k}
yq |)
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Institut Pascal
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Image Representation
Image Representation
Image'Scaling
Image'Scaling
bi-linear interpolation
Knn
In computer graphics, image scaling is the process of resizing a digital image 9
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Institut Pascal
Institut Pascal
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Image Representation
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Image Representation
Image'Scaling
Image'Scaling
bi-linear interpolation
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bi-linear interpolation 11
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Image Representation
Image Representation
Image'Scaling Without
With
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bi-cubic interpolation
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Image Representation
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Image Representation
Color'images
Addi/f'model'RGB'(Red,' Green,'Blue)
• Images'are'generaly'modelized'by'a'3'components' vector
• Many'color'spaces'exist'(RGB,'HSV,'Lab,'YCrCb,' YUV,'...)
• Only'two'of'them'are'presented'hereaXer' T. Chateau
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Institut Pascal
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Image Representation
Image Representation
Addi/f'model'RGB'(Red,' Green,'Blue)
T. Chateau
Addi/f'model'RGB'(Red,' Green,'Blue)
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Institut Pascal
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Image Representation
Image Representation
HSV'model:'[Hue,'Satura/on,'Value]
• • •
HSV'model:'[Hue,'Satura/on,'Value]
Value'is'the'grey'level' (luminance) Hue'(wavelength)'is'measured' by'the'angle'along'the'value' axis. Satura*on'is'the'module'of' the'normal'vector'along'the' value'axis
• T. Chateau
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chroma/c'cube'observed'from' the'white'color Lateral'view'of'HSV'hexagon 19
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Image Representation
Image Representation
HSV'model:'[Hue,'Satura/on,'Value]
HSV'model:'[Hue,'Satura/on,'Value]
Constant'brightness
Constant'satura/on
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Institut Pascal
Institut Pascal
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Image Representation
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Image Representation
HSV'decomposi/on
HSV'model:'[Hue,'Satura/on,'Value] Constant'hue
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Institut Pascal
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Image Representation
Content
Addi/onal'informa/on'on'color' images 1 Image representation 2 Image filtering 3 Morphological transformations
The Colour Image Processing Handbook Sangwine, Stephen J.; Horne, Robin E.N. (Eds.) 1998, 456 p. 1
Hardcover, ISBN 978-0-412-80620-9
P. Bonton et al. Lasmea ISBN : 2-10-006843-1 in French
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Image Filtering
Image Filtering: Global Methods
The'same'func/on'is'applied'on'all'the' pixels H(x) = Card{p : I(p) = x}
methods'are'divided'into'two'main' categories
•Global'methods'(the'same'func/on'
Histogram:'a' basic'tool'for' global'filtering
is'applied'on'all'the'pixels)
•Local'methods'(the'func/on'applied' to'one'pixel'is'related'to'it' neighbourhood) T. Chateau
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Image Filtering: Global Methods
Image Filtering: Global Methods
Histogram:'some'examples
Global'transforma/ons filtered image 255
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255 Input image
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Institut Pascal
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Image Filtering: Global Methods
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Image Filtering: Global Methods
histogram'stretching'
histogram'stretching' filtered image 255
0 min
max 255 input image
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Institut Pascal
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Image Filtering: Global Methods
Image Filtering: Global Methods
histogram'equaliza/on'
histogram'equaliza/on'
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T. Chateau
Institut Pascal
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Image Filtering: Global Methods binary image
binariza/on
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Image Filtering: Global Methods
And'many'other'transforma/ons:
255
✓stretching 0
threshold
✓equaliza/on
255 Input image
✓Area'extrac/on' ✓Inverted'image' ✓Gamma'correc/on ✓…
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✓
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Image Filtering: Spatial Methods
Image Filtering: Spatial Methods
Spatial Methods
Spatial Methods
Def:'modify'the'pixels'in'an'image' based'on'some'func/on'of'a'local' neighborhood'of'the'pixels'
Def:'modify'the'pixels'in'an'image' based'on'some'func/on'of'a'local' neighborhood'of'the'pixels'
I2 (x) = f (I1 (x), VI1 (x)) T. Chateau
I2 (x) = f (I1 (x), VI1 (x)) 37
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Image Filtering: Spatial Methods
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Image Filtering: Spatial Methods
Linear'based'filters:
Spatial Methods
Two'categories:
✓The'simplest ✓Replace'each'pixel'by'a'linear'combina/on'of'its'
neighbors.'
✓linear'based'filters,
✓'The'prescrip/on'for'the'linear'combina/on'is'
called'the'“convolu/on'kernel”
✓non^linear'based'filters.
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Institut Pascal
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Image Filtering: Spatial Methods
Image Filtering: Spatial Methods
Some'classic'kernels'(average'operators) 0 1 1 1 1 1 W1 = @1 1 1A 9 0 1 1 1 1 1 1 1 1 @1 2 1A W1 = 10 0 1 1 1 1 1 2 1 1 @2 4 2A W1 = 16 1 2 1
Let'W'be'the'kernel'(matrix)'of'size' [^n,n]x[^m,m]' X I2 (x) = W (u)I1 (x + u) u2W
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Institut Pascal
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Image Filtering: Spatial Methods
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Image Filtering: Spatial Methods
Some'classic'kernels'(average'operators)
Classic'kernels'(Gaussian'filter)
Average (neighbourhood) 5x5
W1 (i, j) = C exp
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Institut Pascal
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i +j 2 2 2
2
!
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Image Filtering: Spatial Methods
Image Filtering: Spatial Methods
Classic'kernels'(Gaussian'filter)
Classic'kernels'(Shen^Castan)
Gaussian (sig = 3 and support = 5x5)
W1 (i, j) = C exp
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Institut Pascal
! |i| + |j| b
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Image Filtering: Spatial Methods
Image Filtering: Spatial Methods
Gradient'approxima/on'kernels'(Sobel' filter)
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Horizontal
1 @ 0 W1 = 1 T. Chateau
2 0 2
1 1 0A 1
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Gradient'approxima/on'kernels'(Sobel' filter)
Vertical
0
1 W2 = @ 2 1
1 1 2A 1
0 0 0
W2 = W1T
Horizontal 47
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Vertical 48 Institut Pascal
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Image Filtering: Spatial Methods
Image Filtering: Spatial Methods
Gradient'approxima/on'kernels' (Laplacian'filter)
Gradient'approxima/on'kernels' (Laplacian'filter)
@2I @2I L(x, y) = + @x2 @y 2 Approximated'by: 0
0 @ 1 W1 = 0 T. Chateau
1 4 1
1 0 1A 0
0
1 @ 1 W1 = 1
1 1 1A 1
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Image Filtering: Spatial Methods
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Image Filtering: Spatial Methods
Linear'filters:'Some'proper/es'(separable' filters)
Linear'filters:'Some'proper/es'(separable' filters)
I2 (x, y) = I1 (x, y) ⇤ Hxy (x, y) If Hx,y (x, y) = H1x (x) ⇤ H2y (y)
a b
Then I2 (x, y) = [I1 (x, y) ⇤ H1x (x)] ⇤ H2y (y) T. Chateau
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0 1 0 2 a a a ⇤ @ b A = @ab a a2
ab b2 ab
1 a2 abA a2 52
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Image Filtering: Spatial Methods
Image Filtering: Spatial Methods
Linear'filters:'Some'proper/es'(separable' filters)
Non4linear4filters
Exercise:'show'that'the'following'filters' are'separable
^'Calcula/on'within'the'kernel'is'defined'by'non^ linear'mathema/cal'and''sta/s/cal'opera/ons'
^'Mathema/cal'Neighborhood'Operators
^'Sobel'filter'
•'Minimum''
^'Gaussian'filter
•'Median
•'Maximum'' •'Range •'Majority'' •'Standard'devia/on,'...
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Image Filtering: Spatial Methods
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Image Filtering: Spatial Methods
Median4filter Input
Mean
Median4filter ^'Robust'Filter
Can be computed using a quick sort algorithm
^'Non^linear'opera/on ^'Each'pixel'is'modified'according'to'the' median'value'of'it'neighbourhood'
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Median
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Image Filtering: Spatial Methods
Content
Median4filter
1 Image representation 2 Image filtering 3 Morphological transformations
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Binary image filtering: Mathematical morphology
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Binary image filtering: Mathematical morphology What can we do with MM ?
MM'was'originally'developed'for'binary' images,'and'was'later'extended'to' grayscale'func/ons'and'images born'in'1964'from'the'collabora/ve'work' of'Georges4Matheron'and'Jean4Serra,'at' the'École&des&Mines'de'Paris,'France
Remove noise separate shapes compare shapes T. Chateau
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Binary image filtering: Mathematical morphology
Binary image filtering: Mathematical morphology
Main'idea:''probe'an'image'with'a'simple,'pre^ defined'shape,'drawing'conclusions'on'how'this' shape'fits'or'misses'the'shapes'in'the'image.'
Some structuring elements
This'simple'"probe"'is'called'structuring'element,' and'is'itself'a'binary'image'(i.e.,'a'subset'of'the' space'or'grid).
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Binary image filtering: Mathematical morphology
Binary image filtering: Mathematical morphology
Basic'operators:'erosion
Basic'operators:'dila/on
Erosion of the binary image A by the structuring element B:
A
Before
SE
After
Dilation of the binary image A by the structuring element B:
B = {z 2 E|Bz ✓ A}
A
translation of B by z
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SE
Before
After
B = {z 2 E|(B s )z \ A 6= ?} symmetric of B
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Binary image filtering: Mathematical morphology
Binary image filtering: Mathematical morphology
Example
Example
Initial image
eroded 1 time
Initial image
dilated 1 time
eroded 2 times
eroded 3 times
dilated 2 times
dilated 3 times
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Binary image filtering: Mathematical morphology
Binary image filtering: Mathematical morphology
Basic'operators:'opening
Basic'operators:'closing
Before
A B = (A
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B)
After
Before
A • B = (A
B
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B)
After
B
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Binary image filtering: Mathematical morphology
Binary image filtering: Mathematical morphology
Basic'operators:'opening Initial image
Basic'operators:'closing opening
Initial image
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closing
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Binary image filtering: Mathematical morphology
Binary image filtering: Mathematical morphology
Some'proper/es
Basic'operators:'opening+closing Initial image
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A
result
B⇢A⇢A
B
A B ⇢A⇢A•B
A (A B) = A B A • (A • B) = A • B
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Binary image filtering: Mathematical morphology
Image Processing
Exercices
References
histogram stretching (0
