L24. More on Image File Processing

1
L24. More on Image File Processing

• Filtering Noise
• Edge Detection

2
Pictures as Arrays

• A black and white picture can be encoded
• as a 2D Array
• Typical
• 0 lt A(i,j) lt 255
• (black)
  (white)
• Values in between correspond to different
• levels of grayness.

3
Just a Bunch of Numbers
1458-by-2084
150 149 152 153 152 155 151 150
153 154 153 156 153 151 155 156
155 158 154 153 156 157 156 159
156 154 158 159 158 161 157 156
159 160 159 162
4
Dirt!
1458-by-2084
150 149 152 153 152 155 151 150
153 154 153 156 153 2 3 156
155 158 154 2 1 157 156 159
156 154 158 159 158 161 157 156
159 160 159 162
Note how the dirty pixels look out of place
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Can We Filter Out the Noise?
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Idea
1458-by-2084
150 149 152 153 152 155 151 150
153 154 153 156 153 ? ? 156
155 158 154 ? ? 157 156 159
156 154 158 159 158 161 157 156
159 160 159 162
Assign typical neighborhood gray values to
dirty pixels
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Getting Precise

• Typical neighborhood gray values

Could use Median Or Mean
radius 1
radius 3
Well look at Median Filtering first
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Median Filtering

• Visit each pixel.
• Replace its gray value by the median
• of the gray values in the neighborhood.

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Using a radius 1 Neighborhood
0 6 6 6 6 7 7 7 7
Before
After
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How to Visit Every Pixel
m 9
n 18
for i1m for j1n Compute new gray
value for pixel (i,j). end end
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Original
i 1
j 1
Filtered
Replace with the median of the values
under the window.
12
Original
i 1
j 2
Filtered
Replace with the median of the values
under the window.
13
Original
i 1
j 3
Filtered
Replace with the median of the values
under the window.
14
Original
i 1
j n
Filtered
Replace with the median of the values
under the window.
15
Original
i 2
j 1
Filtered
Replace with the median of the values
under the window.
16
Original
i 2
j 2
Filtered
Replace with the median of the values
under the window.
17
Original
i m
j n
Filtered
Replace with the median of the values
under the window.
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What We Need

• (1) A function that computes the median
• value in a 2-dimensional array C
• m medVal(C)
• (2) A function that builds the filtered
• image by using median values of radius r
• neighborhoods
• B medFilter(A,r)

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Computing Medians
x
x sort(x)
x
n length(x) n 7 m ceil(n/2) m
4 med x(m) med 36
If n is even, then use med ( x(m)
x(m1) )/2
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Median of a 2D Array

• function med medVal(C)
• p,q size(C)
• x
• for k1p
• x x C(k,)
• end
• Compute median of x and assign to med.

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Medians vs Means

• A
• 150 151 158 159 156
• 153 151 156 155 151
• 150 155 152 154 159
• 156 154 152 158 152
• 152 158 157 150 157
• Median 154 Mean 154.2

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Medians vs Means

• A
• 150 151 158 159 156
• 153 151 156 155 151
• 150 155 0 154 159
• 156 154 152 158 152
• 152 158 157 150 157
• Median 154 Mean 148.2

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Back to Filtering
m 9
n 18
for i1m for j1n Compute new gray
value for pixel (i,j). end end
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Window Inside
m 9
n 18
New gray value for pixel (7,4)
medVal( A(68,35) )
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Window Partly Outside
m 9
n 18
New gray value for pixel (7,1)
medVal( A(68,12) )
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Window Partly Outside
m 9
n 18
New gray value for pixel (9,18)
medVal( A(89,1718) )
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function B medFilter(A,r) B from A via
median filtering with radius r neighborhoods.
m,n size(A) B uint8(zeros(m,n))
for i1m for j1n C pixel
(i,j) neighborhood B(i,j) medVal(C)
end end
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The Pixel (i,j) Neighborhood

• iMin max(1,i-r)
• iMax min(m,ir)
• jMin max(1,j-r)
• jMax min(n,jr)
• C A(iMiniMax,jMinjMax)

m
A
r 1
r 2
n
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B medFilter(A)
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Original
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What About Using the Meaninstead of the Median?

• Replace each gray value with the
• average gray value in the radius r
• neighborhood.

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Mean Filter with r 3
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Mean Filter with r 10
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Why it Fails
150 149 152 153 152 155 151 150
153 154 153 156 153 2 3 156
155 158 154 2 1 157 156 159
156 154 158 159 158 161 157 156
159 160 159 162
The mean does not capture representative values.
85 86 87 88
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And Median Filters LeaveEdges (Pretty Much) Alone
200 200 200 200 200 200 200 200
200 200 200 100 200 200 200 200
100 100 200 200 200 100 100 100
200 200 100 100 100 100 200 100
100 100 100 100
Inside the box, the 200s stay at 200 and the
100s stay at 100.
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Finding Edges
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What is an Edge?

• Near an edge, grayness values change
• abruptly.

200 200 200 200 200 200 200 200
200 200 200 100 200 200 200 200
100 100 200 200 200 100 100 100
200 200 100 100 100 100 200 100
100 100 100 100
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The Rate-of-Change-Array

• Suppose A is an image array with integer
• values between 0 and 255
• B(i,j) be the maximum difference between
• A(i,j) and any of its eight neighbors.

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Example
81
Rate-of-change at middle pixel is 30
40

• function B Edges(P)
• P is a jpeg file
• B is the corresponding
• Rate-Of-Change array
• A rgb2gray(imread(P))
• m,n size(A)
• B uint8(zeros(m,n))
• for i2m-1
• for j 2n-1
• B(i,j) ?????
• end
• end

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Recipe for B(i,j)

• The 3-by-3 subarray that includes
• A(i,j) and its 8 neighbors
• Neighbors A(i-1i1,j-1j1)
• Subtract A(i,j) from each entry
• Diff Neighbors A(i,j))
• Take absolute value of each entry..
• posDiff abs(Diff)
• Compute largest value in each column
• colMax max(posDiff)
• Compute the max of the column maxs
• B(I,j) max(colMax)

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Rate-of-Change Array to Image

• B Edges('Tower.jpg')
• Compute 0-1 array that identifies
• those B entries bigger than 20
• importantPixels B gt 20
• Display those pixels with maximum
• brightness
• C uint8( 255importantPixels )
• imshow(C)

Bgt0 is a 0-1 array, the ones are located where
B(i,j)gt 20.
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Threshhold 40
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Threshhold 20
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Threshhold 30