Transcript lmv5- lecture

Machine Vision
ENT 273
Image Filters
Hema C.R.
Lecture 5
Why Filters are needed?
• Image processing converts an
input image into an enhanced
image from which information
about the image can be retrieved.
• To enhance images any unwanted
information or distortions called
noise has to be removed.
• Filtering is the process which
removes noise from an image
[which also includes lightening
darker regions to enhance quality
of the image or suppresses
unwanted information /region]
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Original Image
Image with Noise
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Image Noise
• Images are formed by light
falling on a sensor
• Noises are introduced due to
– Quantization – which reduces
the light levels to 256
– Imperfect sensors
– Imperfect lighting conditions
during acquisition
– Compression formats
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Features of Filter
• Reduce the size of the data on which we
want to build
• our inferences
• Reduction should retain most informative
pieces
– Edges and Corners
– Texture
• Mathematically/computationally
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Types of Noise
Original Image
• Salt and Pepper
– In salt and pepper noise pixels in the image are vastly different
in color from their surrounding pixels.
– The color of a noisy pixel bears no relation to the color of
surrounding pixels.
– Generally this type of noise will only affect a small number of
image pixels.
– When viewed, the image contains dark and white dots, hence
the term salt and pepper noise.
.
Salt and Pepper
Noise
• Gaussian Noise
– In Gaussian noise (dependent noise), an amount of noise is
added to every part of the picture.
– Each pixel in the image will be changed from its original value
by a (usually) small amount.
– Taking a plot of the amount of distortion of a pixel against the
frequency with which it occurs produces a Gaussian distribution
of noise
• Uniform Noise
Gaussian Noise
Uniform Noise
– Pixel values are usually close to their true values.
– Average value is equal to the real one
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Noise Removal
• Most Noise removal processes
are called filters
– Applied to each point in an image
[convolution]
– Use information in the small local
windows of a pixel
• Noise removal Filters
– Linear Filters
– Non-Linear Filters
• Linear Filters
– Gaussian filters
– Mean Filter
• Non-Linear Filters
– Median Filter
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Linear Systems
• Space Invariant System
– A system whose response remains the
same irrespective of the position of the
input pulse
Input impulse
( x  x0 , y  y0 )
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Linear
Space
Invariant
System
Output impulse
response
g( x  x0 , y  y0 )
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Linear Space Invariant Systems
In LSI systems the output h(x,y) is a convolution
of f(x,y) with impulse response g(x,y)
h( x, y)  f ( x, y)  g ( x, y)
Input Image
f ( x, y )
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Linear
Space
Invariant
System
g(x,y)
Output Image
h( x , y )
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Convolution
• Convolution
– A neighborhood operation in which each
output pixel is a weighted sum of
neighboring input pixels. The weights are
defined by the convolution kernel. Image
processing operations implemented with
convolution include smoothing, sharpening,
and edge enhancement.
– Convolution is a spatially invariant operation
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Convolution (cont.)
• One of the most important processes on
images.
• Actually a class of operations allowing
much variety.
• Used in almost all Computer Vision
programs as a starting point.
• Used in many Computer Graphics
processes.
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Example: horizontal differencing as
convolution
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a combination of multiplication
and addition
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Example: Image after convolution
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Convolution Mask
P1 P2 P3
P4 P5 P6
A
B
C
D
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F
G
H
I
P7 P8 P9
h [i,j]
h[ i , j ]  AP1  BP 2  CP3  DP 4  EP5  FP6  GP7  HP8  IP 9
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Mask
• The mask can contain any numbers. E.g. the
mask below amounts to
averaging adjacent pixels and so produces a
smoothing effect on the image.
• The mask can be vertically oriented instead of
horizontally. The
• differencing mask will then be sensitive to
horizontal edges.
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• The mask can be 2-dimensional—in fact,
just like a small piece of image.
• The example below also combines
smoothing and differencing, and is
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• The mask can be 2-dimensional—in fact,
just like a small piece of image. The
example below also combines smoothing
and differencing, and is used in practical
applications. (It is referred to as a Sobel
operator.)
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Example of 2-D convolution
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Some special masks
• Horizontal differences
• Vertical differences
• Diagonal differences
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• Differencing masks may be more complex e.g.
a horizontal bar mask
• An arbitrary mask can be given this property by
computing the mean of its weights, and
subtracting the mean from each weight in turn to
get a newweight.
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A simple smoothing mask
• This takes the average of each 3x3 region
of the image.
• The simple smoothing mask is not
particularly good, because its point-spread
function has sharp edges.
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The Gaussian smoothing mask
• A more effective smoothing mask falls off
gently at the edges. One of the bestis the
Gaussian mask. Its width is described by a
parameter called  (sigma).
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Example on the effect of progressively increasing
values are 1, 2 and 4 in the 3 smoothed images.
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. Its

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Feature of Linear Filter
• About modifying pixels based on
neighborhood.
• Linear means linear combination of
neighbors.
– Linear methods simplest
– Can combine linear methods in any order to
achieve same result
– Maybe easier to undo
– “Filter Banks”
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Mean Filter
• Implemented by a local
averaging operation
where the value of each
pixel is replaced by the
average of all the values
in the local neighborhood:
where M is the total
number of pixels in the
neighborhood N
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h[i, j ] 
M
 f [k , l ]
( k ,l )N
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• 3x3 neighborhood about [i,j]
1 i 1 j 1
h[i, j ]    f [k , l ]
9 k i 1k  j 1
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Mean Filter
• The mean filter is a simple
sliding-window spatial filter
that replaces the center
value in the window with
the average (mean) of all
the pixel values in the
window.
• The window, or kernel, is
usually square but can be
any shape.
• An example of mean
filtering of a single 3x3
window of values is shown
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2 6 4
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Example of mean filter using 3x3 neighborhood
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• Another example of mean filter
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Median Filter
• The median filter is also a sliding-window
spatial filter, but it replaces the center value 3
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in the window with the median of all the
pixel values in the window.
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• As for the mean filter, the kernel is usually
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square but can be any shape.
• An example of median filtering of a single
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3x3 window of values is shown
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• Median filter remove 'impulse' noise
(outlying values, either high or low).
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• The median filter is also widely claimed to
be 'edge-preserving' since it theoretically
preserves step edges without blurring.
* * *
• However, in the presence of noise it does
blur edges in images slightly.
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Example of median filter using 3x3 neighborhood
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Gaussian Filter
• Gaussian filters removes noise by
smoothing but also blurs the image,
• The degree of smoothing is determined by
the standard deviation of the Gaussian.
(Larger standard deviation Gaussians, of
course, require larger convolution masks
in order to be accurately represented.)
• The Gaussian outputs a `weighted
average' of each pixel's neighborhood,
with the average weighted more towards
the value of the central pixels.
• This is in contrast to the mean filter's
uniformly weighted average.
• Gaussian provides gentler smoothing and
preserves edges better than a similarly
sized mean filter
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x2
 2
1
Gx  
e 2
2
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• 2D Gaussian function
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Gaussian Filter
• Can be constructed
using binomial
expansion
• Example
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n n n
n
(1  x) n       x    x 2  ....    x n
 0  1  2
n
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• We need to get the
Gaussian mask,
where c is a
normalizing constant
• By rewriting and
choosing a value of
the variance  2 , we
can find the nxn mask
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G( x, y)  ce

G ( x, y )
e
c
( x2  y2 )

22
( x2  y2 )
22
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2

 2; n  7
Example of Gaussian distribution for
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By choosingthe bottom right corner as the value of one
G (3,3)
e
k
k  91

( 32  32 )
2( 2)2
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 0.011
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• To normalize the
mask to ensure
uniform intensity
• Thus the convolution
is
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3
  G(i, j)  1115
i  3 j  3
H (i, j ) 
1
F (i, j )G (i, j )
1115
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Histogram Equalization
• Histogram equalization is a common technique
for enhancing the appearance of images.
Suppose we have an image which is
predominantly dark. Then its histogram would be
skewed towards the lower end of the grey scale
and all the image detail is compressed into the
dark end of the histogram. If we could `stretch
out' the grey levels at the dark end to produce a
more uniformly distributed histogram then the
image would become much clearer.
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Histogram Modification
• Histogram equalization
– Is a method for stretching contrast of unevenly
distributed gray values by uniformly redistributing
the gray values
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References
• Computer Vision – Linda G Shapiro &
George Stockman
• http://en.wikipedia.org/wiki/Image_noise
• Mat lab reference notes
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Machine Vision
End of Lecture 5