Image Enhancement
(Spatial Domain Filters)
Spatial Domain
• Spatial Filtering
– (As opposed to frequency domain filter)
– For linear filters
• Foundation based on the convolution theorem
– g(x,y) = h(x,y)*f(x,y)
– G(u,v) = H(u,v)F(u,v)
• Goal is to either remove, or isolate
frequencies in the image
• Types
– Linear
– Non-linear
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Response of a linear mask
R = w1z1 + w2z2 + w3z3 ... + w9z9
Where wi are mask coefficients and zi are pixel
intensities.
Spatial Filtering using a Mask
• Neighborhood operators
– Convolution
z1 z2 z3
w1 w2 w3
z4 z5 z6
w4 w5 w6
z7 z8 z9
w7 w8 w9
Response of a linear mask, R,
R = ( w1z1 + w2z2 + w3z3 ... + w9z9) =
9
∑ wizi
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f(x,y) is centered around “z5”
so g(x,y) = R =filter_about*f(x,y)
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Smoothing Filters
• Smoothing Filters
– blurring
• “pre-processing”
– removal of small details before object extraction
– noise reduction
• removal of noise in an image
• Often called “Low-Pass” Filters
– Filter lets low-frequencies pass
– Stops high-frequencies
Low-pass spatial filtering
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Frequency Domain
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Spatial Domain
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Low-pass spatial filtering
•We can build a gaussian filter
w1 w2 w3
w4 w5 w6
w7 w8 w9
•But, only requirement for
a low-pass filter is that wi be positive
•Note, that the result can be
larger than the valid output range(L-1)
•Can pre-scale the filter
scale_factor =
(Σ wi )-1
Low-pass spatial filter
1/9 *
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Average Filter
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Extends to larger filters
1 1 1
1/9 * 1 1 1
1 1 1
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Example
original
n=15 (nxn mask)
n=5 (nxn mask)
n=25 (nxn mask)
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Other arrangements
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Gaussian
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Weighted
Remember that the resulting gray-levels
may be out of the range of the original image.
Isotropic Gaussian Filter
G ( x, y ) =
1
2πσ 2
−
e
x2 + y 2
2σ 2
2D Gaussian distribution
with mean (0,0) and σ=1
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Computing a discrete gaussian mask
Center the Mask at (0,0). So, a 5x5
mask would compute values by at:
-2
-1
0
1
2
-2
G ( x, y ) =
-1
0
1
2πσ 2
−
e
x2 + y 2
2σ 2
Use G(x,y) to compute
values for mask.
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Often scaled by Max-Gray
Level; ie, 255*G(x,y)
Computing a discrete gaussian mask
G ( x, y ) =
1
2πσ 2
−
e
x2 + y 2
2σ 2
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Linear vs. Non-linear
• “Convolution”/Linear Filters
– Linear operation
– Have corresponding frequency domain filter
• Non-linear Filters
– Mask used to determine the proper substitution
of a “good” pixel value
– Examine neighbors using various orderings
• Often use Rank or Order Statistics
– Harder to interpret effect in frequency domain
Selected Smoothing
• (non-linear filter)
• kth nearest neighbors
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Find the kth nearest neighbors
of (x,y) in a neighborhood nxn
k=6
nearest neighbors = {2,3,3,4,4,2}
(x,y) = (2+3+3+4+4+2} * (1/6) = 3
Note: Straight Averaging would have equaled 5
pixel values about (x,y)
window 3x3
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Max Homogeneity Smoothing
• non-linear
• Use 5 different windows with f(x,y)
included in all of them
Max Homogeneity Smoothing
• Compute the gradient of each of these
“windows”
• f(x,y) is the average of the window
with the smallest “variation”
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Max Homogeneity Smoothing
• This filter is trying to find the
window that (x,y) is the most similar
too.
?
Median Filter
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•
•
•
Very popular non-linear filter
Find the median of the window
Preserves edges
Removes impulse noise, avoids excessive smoothing
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neighbor sort = {2,2,3,3,4,4,8,9,10}
f(x,y) = median
pixel values about (x,y)
window 3x3
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Example
Noisy Image
Original
Median Filter
Low-Pass Filter
Median Filter
• Often referred to as “de-speckle
operation”
• It converges
– That is, if you perform it over and over,
eventually the image will not change.
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Iterative Smoothing
• Local Averaging
– Converges to an image with constant intensity
• Max-homogeneity and K-nearest
– converges to a constant intensity or a piecewise
constant intensity
• Median Filter
– Converges to an image invariant to the filter
Sharpening
• Objective of sharpening
– Highlight fine detail in an image
– Enhance detail that has been blurred
– We can think of this as high-pass
filtering
• Letting High Frequencies Pass
• Removing Low Frequencies
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High-pass spatial filtering
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Frequency Domain
Spatial Domain
Basic hi-pass spatial filter
•Positive coefficients near its center
1/9 *
-1
-1
-1
-1
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-1
-1
-1
-1
•Negative coefficients near the outer
periphery
•Note, the sum of the coefficients is
zero
•Thus, when the mask is over an
area of constant intensity, the result
is zero
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Hi-pass filter
• The results may be negative
• You’ll need to scale and/or clip so that the
gray levels of the result span [0, L-1]
• Don’t take absolute value of the response
– Bad idea
Example
Original
High-Pass
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High Pass filter
• Another way to think about the high
pass filter:
– Highpass = Original – Lowpass
• Should be able to prove this using the
linear ‘R’ response equality and examples
of high-pass and low-pass filters
High Boost Filter
• High Boost = (A)(Original) – Lowpass
= (A-1)(Original) + Original - Lowpass
= (A-1)Original + Highpass
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Setting ‘A’ controls how much of the original image you would like to have
show up in the result
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A=1 gives you the high-pass filter
A > 1 allows some of the low-frequencies back in
–
Result, an image that looks more like the original with accents on the high-frequencies
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Constructing a High Boost
Filter
1/9 *
-1
-1
-1
-1
w
-1
-1
-1
-1
w = 9A-1
Examples
Original
A=1.1
A=1.15
A=1.2
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Derivative Filters
• Averaging pixels
– blur
– analogous to integration
• Differentiation
– has the opposite effect of “blurring”
– sharpens an image
First Order Derivatives
• Gradient
• Function of 2 variables x, y
⎡ ∂f ⎤
⎢ ∂x ⎥
⎢ ⎥
∇f = ⎢ ⎥
⎢ ∂f ⎥
⎢ ∂y ⎥
⎣ ⎦
x
y
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First Order Derivatives
• For each (x,y) you are storing two values:
⎡ ∂f ⎤
⎢ ∂x ⎥
⎢ ⎥
∇f = ⎢ ⎥
⎢ ∂f ⎥
⎢ ∂y ⎥
⎣ ⎦
• Often have two images to represent this
• X-Gradient and Y-Gradient
– Can be computed independently
First Order Derivatives
• X-Gradient and Y-Gradient
Original
df/dx
df/dy
f
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Gradient
• Gradient Magnitude
⎡⎛ ∂f ⎞ 2 ⎛ ∂f ⎞ 2 ⎤
∇f = mag(∇f) = ⎢⎜ ⎟ + ⎜⎜ ⎟⎟ ⎥
⎢⎣⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎥⎦
1/ 2
Gradient
• Gradient Angle/Direction
⎛ ∂f/∂y ⎞
Ψ (∇f ) = tan −1 ⎜
⎟
⎝ ∂f/∂x ⎠
Note: the angle is with respect to the “x” image axis
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Basic Derivative
• Consider the pixels
z1 z2 z3
z4 z5 z6
z7 z8 z9
•
f at z5 can be computed
∇f ≈ [( z5 − z6 ) 2 + ( z5 − z8 ) 2 ]1/ 2
Basic Derivative
• Consider the pixels
z1 z2 z3
z4 z5 z6
z7 z8 z9
•
f at z5 can be computed quicker
∇f ≈| z5 − z6 | + | z5 − z8 |
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Basic Derivative
z1 z2 z3
z4 z5 z6
z7 z8 z9
•
f sometimes is computed using the
“cross” difference
∇f ≈| z6 − z8 | + | z5 − z9 |
Robert Gradient
• Roberts cross-gradient operators
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1
0
-1 0
0
-1
0
Even masks are awkward to implement
Results in a phase-shift
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Odd Sized Masks
∇f ≈| ( z7 + z8 + z9 ) − ( z1 + z 2 + z3 ) | +
| ( z3 + z6 + z9 ) − ( z1 + z 4 + z7 ) |
• Difference between first and third row
(df/dy)
• Difference between first and third column
(df/dx)
Odd sized mask
• Prewitt operator
-1 0
1
-1 -1 -1
-1 0
1
0
0
0
-1 0
1
1
1
1
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Odd sized mask
• Sobel operator
-1 0
1
-1 -2 -1
-2 0
2
0
0
0
-1 0
1
1
2
1
• Weights closer neighbor a little more
Requirements for Derivative
Mask
• Measuring Change
– df/dx = A*f(x,y) – B* f(x+1,y)
– df/dy = A*f(x,y) – B*f(x,y+1)
• Derivative
– Sum of the coefficients has to be zero
• This makes sure the derivative is zero in a
region of constant intensity
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Example
Second Order Derivative
• Laplacian of a 2-D function f(x,y)
• is a second-order derivative defined
as:
∂ 2f ∂ 2f
∇f= 2 + 2
∂ x ∂ y
2
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Second Order Derivative
• For a 3x3 region, the form most
encountered is:
∇ 2 f ≈ 4 z 5 − ( z 2 + z 4 + z 6 + z8 )
• Requirements
– Center pixel coefficient be positive
– Outer coefficients be negative
– Its a derivative, so sum of coefficient be zero
Example of a Laplacian
• Typical Laplacian Operator
(not used in practice)
0 -1 0
-1 4 -1
0 -1 0
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Example
This is actually an example of a mask
which is a Laplacian of a Gaussian (LoG Filter)
See pp 582 of GW
Spatial Filters
• Spatial Convolution
– We are convolving a function about each (x,y)
– approximation of filters in the frequency domain
(at least for the linear filters . . non-linear is hard to analyze)
– types
• Blurring, Smoothing, Sharpening, Derivatives
• and the non-linear (Max Homogeneity, Kth Nearest, Median)
• Input gray-levels may be different than output levels
– very common
– May need to scale your image for visualization
• Filter coefficients do not have to be integers
– Results are non-integer
– use a float image
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