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DTU Compute
Introduction to Medical Image
Analysis
Rasmus R. Paulsen
DTU Compute
[email protected]
http://www.compute.dtu.dk/courses/02511
http://www.compute.dtu.dk/courses/02512
Plenty of slides adapted from Thomas Moeslunds lectures
DTU Compute
Lecture 4 – Neighbourhood Processing
9.00 – 11.30
Lecture
Exercises
2
12.00 – 13.00
Lunch break
13.00 -
Exercises
DTU Compute, Technical University of Denmark
Introduction to Medical Image Analysis
25/2/2015
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What can you do after today?
 Describe the difference between point processing and neighbourhood
processing
 Compute a rank filtered image using the min, max, median, and
difference rank filters
 Compute a mean filtered image
 Decide if median or average filtering should be used for noise removal
 Choose the appropriate image border handling based on a given input
image
 Implement and apply template matching
 Compute the normalised cross correlation and explain why it should be
used
 Apply given image filter kernels to images
 Use edge filters on images
 Describe finite difference approximation of image gradients
 Describe the concept of edge detection
3
DTU Compute, Technical University of Denmark
Introduction to Medical Image Analysis
25/2/2015
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Point processing
Input
Output
1
2
0
1
3
2
1
4
2
2
1
0
1
0
1
1
2
1
0
2
2
5
3
1
2
12
9
• The value of the output pixel is only dependent on the value of one input pixel
• A global operation – changes all pixels
4
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Introduction to Medical Image Analysis
25/2/2015
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Point processing
 Grey level enhancement
– Process one pixel at a time independent of all other pixels
– For example used to correct Brightness and Contrast
Correct
5
Too high
brightness
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Too low
brightness
Too high
contrast
Too low
contrast
Introduction to Medical Image Analysis
25/2/2015
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Neighbourhood processing
Input
Output
1
2
0
1
3
2
1
4
2
2
1
0
1
0
1
1
2
1
0
2
2
5
3
1
2
12
9
• Several pixels in the input has an effect on the output
6
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Introduction to Medical Image Analysis
25/2/2015
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Use of filtering
Noise removal
Enhance edges
Smoothing
• Image processing
• Typically done before actual image analysis
7
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Introduction to Medical Image Analysis
25/2/2015
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Salt and pepper noise
 Pixel values that are very
different from their
neighbours
 Very bright or very dark
spots
 Scratches in X-rays
What is that?
8
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Introduction to Medical Image Analysis
25/2/2015
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Salt and pepper noise
 Fake example
– Let us take a closer look at
noise pixels
They are all 0 or 255
Should we just remove all the 0’s and 255’s from the image?
9
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Introduction to Medical Image Analysis
25/2/2015
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What is so special about noise?
 What is the value of the pixel
compared to the neighbours?
 Average of the neighbours
– 170
 Can we compare to the
average?
– Difficult – should we remove
all values bigger than
average+1 ?
 It is difficult to detect noise!
172, 169, 171, 168, 0, 169, 172, 173, 168
10
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Introduction to Medical Image Analysis
25/2/2015
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Noise – go away!
 We can not tell what pixels
are noise
 One solution
– Set all pixels to the average
of the neighbours (and the
pixel itself)
 Oh no!
170
149
– Problems!
– The noise “pollutes” the
good pixels
172, 169, 171, 168, 0, 169, 172, 173, 168
169, 168, 0, 170, 172,173, 170, 172, 170
11
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Introduction to Medical Image Analysis
25/2/2015
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What is the median Value?
A) 170
B) 173
C) 169
D) 171
E) 172
30
169, 168, 0, 170,172, 173, 170, 172, 170
0
A
12
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0
B
0
C
Introduction to Medical Image Analysis
D
2
E
25/2/2015
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The median value
 What is the median value of the values below?
 The values are sorted from low to high
 The middle number is picked
– The median value
169, 168, 0, 170,172, 173, 170, 172, 170
0,168, 169, 170, 170, 170,172, 172,173
Median
Noise has no influence on the median!
13
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Introduction to Medical Image Analysis
25/2/2015
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Noise away – the median filter
 All pixels are set to the
median of its neighbourhood
 Noise pixels do not pollute
good pixels
170
170
172, 169, 171, 168, 0, 169, 172, 173, 168
169, 168, 0, 170, 172,173, 170, 172, 170
14
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Introduction to Medical Image Analysis
25/2/2015
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Noise removal – average filter
Scanned X-ray with salt and pepper
noise
15
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Average filter (3x3)
Introduction to Medical Image Analysis
25/2/2015
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Noise removal – median filter
Scanned X-ray with salt and pepper
noise
16
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Median filter (3x3)
Introduction to Medical Image Analysis
25/2/2015
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Image Filtering
 Creates a new filtered image
 Output pixel is computed based
on a neighbourhood in the input
image
 3 x 3 neighbourhood
– Filter size 3 x 3
– Kernel size 3 x 3
– Mask size 3 x 3
 Larger filters often used
– Size?
 7x7
– Number of elements?
 49
17
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Introduction to Medical Image Analysis
25/2/2015
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Median filter
A) 25
B) 90
C) 198
D) 86
E) 103
25
0
0
A
18
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B
0
C
Introduction to Medical Image Analysis
D
2
E
25/2/2015
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Rank filters
 Based on sorting the pixel
values in the neighbouring
region
 Minimum rank filter
– Darker image. Noise problems.
 Maximum rank filter
– Lighter image. Noise problems.
 Difference filter
– Enhances changes (edges)
0,168, 169, 170, 170, 170,172, 172,173
20
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Introduction to Medical Image Analysis
25/2/2015
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Median filter
A) 3
B) 84
C) 112
D) 73
E) 202
30
0
0
A
21
DTU Compute, Technical University of Denmark
B
0
C
D
Introduction to Medical Image Analysis
1
E
25/2/2015
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Correlation
Math Skills
 What is it?
 Two measurements
Low
Bench Press
Muscles
High
– Low correlation
– High correlation
 High correlation means that
there is a relation between
the values
 They look the same
 Correlation is a measure of
similarity
Muscles
23
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Introduction to Medical Image Analysis
25/2/2015
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Why do we need similarity?
 Image analysis is also about recognition of patterns
 Often an example pattern is used
 We need something to tell us if there is a high match
between our pattern and a part of the image
Find matches
Example pattern
Image
24
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Introduction to Medical Image Analysis
25/2/2015
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Correlation (1D)
Kernel
1
4
Signal (1D image)
1
2
1
1
1
2
2
1
1
2
2
1
1
1*1+2*1+1*2=5
Normalise! 1
4
5
4
25
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Introduction to Medical Image Analysis
25/2/2015
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Correlation (1D)
1
26
1
4
1
2
1
1
1
2
2
1
1
2
2
1
5
4
7
4
7
4
5
4
5
4
7
4
7
4
5
4
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Introduction to Medical Image Analysis
25/2/2015
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Normalisation
• The sum of the kernel elements is used
• Keep the values in the same range as the input image
1
4
1
2
1
Sum is 4
1
1
2
2
1
1
2
2
1
1
1*1+2*1+1*2=5
Normalise! 1
4
5
4
27
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Introduction to Medical Image Analysis
25/2/2015
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Normalisation
 Normalisation factor
– Sum of kernel coefficients
h(x)
1
2
1
ℎ 𝑥 =1+2+1
𝑥
28
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Introduction to Medical Image Analysis
25/2/2015
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Correlation on images
 The filter is now 2D
Kernel coefficients
1
9
Input
1
29
2
0
Output
1
3
1
1
1
1
1
1
1
1
1
Kernel
1
2
1
4
2
2
2
1
0
1
0
1
3
1
2
1
0
2
4
2
5
3
1
2
2
2
1
3
1
6
3
DTU Compute, Technical University of Denmark
12
9
Introduction to Medical Image Analysis
25/2/2015
DTU Compute
Correlation on images
1
9
1
1
1
1
1
1
1
1
1
Input
1
30
2
0
Output
1
3
1
2
1
4
2
2
2
1
0
1
0
1
3
1
2
1
0
2
4
2
5
3
1
2
2
2
1
3
1
6
3
DTU Compute, Technical University of Denmark
12
9
11
9
Introduction to Medical Image Analysis
25/2/2015
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Correlation on images
The mask is moved row by row
No values at the border!
Input
31
Output
1
2
0
1
3
1
2
1
4
2
2
2
1
0
1
0
1
3
1
2
1
0
2
4
2
5
3
1
2
2
2
1
3
1
6
3
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Introduction to Medical Image Analysis
25/2/2015
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Correlation on images – no normalisation
A) 4
B) 7
C) 16
D) 23
E) 25
32
1
2
1
1
3
1
1
2
1
1
2
0
1
3
1
2
1
4
2
2
2
1
0
1
0
1
3
1
2
1
0
2
4
2
5
3
1
2
2
2
1
3
1
6
3
DTU Compute, Technical University of Denmark
27
?
2
0
A
0
B
C
D
2
E
Introduction to Medical Image Analysis
25/2/2015
DTU Compute
Correlation on images – no normalisation 2
A)
B)
C)
D)
E)
33
0,10
3, 3
6, 2
10, 15
11, 32
-1 -2 -1
0
0
0
1
2
1
1
2
0
1
3
1
2
1
4
2
2
2
1
0
1
0
1
3
1
2
1
0
2
4
2
5
3
1
2
2
2
1
3
1
6
3
DTU Compute, Technical University of Denmark
26
0
0
0
?
A
B
C
D
Introduction to Medical Image Analysis
?
0
E
25/2/2015
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Mathematics of 2D Correlation
𝑔 𝑥, 𝑦 = 𝑓 𝑥, 𝑦 ∘ ℎ(𝑥, 𝑦)
Correlation operator
1
2
0
1
3
1
2
1
4
2
2
2
1
0
1
0
1
3
1
2
1
0
2
2
5
3
1
2
1
3
1
1
2
1
4
1
3
1
2
2
1
2
1
6
3
h
f
37
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Introduction to Medical Image Analysis
25/2/2015
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Mathematics of 2D Correlation
𝑅
𝑅
𝑔 𝑥, 𝑦 =
ℎ(𝑖, 𝑗) ∙ 𝑓(𝑥 + 𝑖, 𝑦 + 𝑗)
𝑗=−𝑅 𝑖=−𝑅
Example g(2,1)
1
2
0
1
3
1
2
1
4
2
2
2
1
0
1
0
1
3
1
2
1
0
2
2
5
3
1
2
1
3
1
i = 1, j = 0
1
2
1
4
1
3
1
2
2
1
2
1
6
3
f
38
i = -1, j = -1
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h
h(0,0)
f(2,1)
Introduction to Medical Image Analysis
25/2/2015
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Mathematics of 2D Correlation
𝑅
𝑅
𝑔 𝑥, 𝑦 =
ℎ(𝑖, 𝑗) ∙ 𝑓(𝑥 + 𝑖, 𝑦 + 𝑗)
𝑗=−𝑅 𝑖=−𝑅
𝑔(𝑥, 𝑦 = 1 ∙ 2 + 2 ∙ 0 + 1 ∙ 1 + 1 ∙ 1 + 3 ∙ 4 + 1 ∙ 2 + 1 ∙ 0 + 2 ∙ 1 + 1 ∙ 0
1
2
0
1
3
1
2
1
4
2
2
2
1
0
1
0
1
3
1
2
1
0
2
2
5
3
1
2
1
3
1
1
2
1
4
1
3
1
2
2
1
2
1
6
3
h
f
39
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Introduction to Medical Image Analysis
25/2/2015
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𝑅
𝑅
𝑔 𝑥, 𝑦 =
ℎ(𝑖, 𝑗) ∙ 𝑓(𝑥 + 𝑖, 𝑦 + 𝑗)
𝑗=−𝑅 𝑖=−𝑅
𝑔
ℎ
ℎ
ℎ
40
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2,2 =
−1, −1 ⋅ 𝑓 1,1 + ℎ 0, −1 ⋅ 𝑓 2,1 + ℎ 1, −1 ⋅ 𝑓 3,1 +
−1,0 ⋅ 𝑓 1,2 + ℎ 0,0 ⋅ 𝑓 2,2 + ℎ 1,0 ⋅ 𝑓 3,2 +
−1,1 ⋅ 𝑓 1,3 + ℎ 0,1 ⋅ 𝑓 2,3 + ℎ 1,1 ⋅ 𝑓 3,3
Introduction to Medical Image Analysis
25/2/2015
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2D Kernel Normalisation
Normalisation factor:
ℎ(𝑥, 𝑦)
𝑥
𝑦
1 + 2 + 1 + 1 + 3 + 1 + 1 + 2 + 1 = 13
1
2
1
1
3
1
1
2
1
h
41
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Introduction to Medical Image Analysis
25/2/2015
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Correlation
A) 50122
B) 123001
C) 11233
D) 2550
E) 90454
26
0
0
A
42
DTU Compute, Technical University of Denmark
1
0
B
C
D
Introduction to Medical Image Analysis
E
25/2/2015
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Smoothing filters
 Also know as
– Smoothing kernel, Mean filter, Low pass filter, blurring
 The simplest filter:
– Spatial low pass filter
– Removes high frequencies
1
9
 Another mask:
– Gaussian filter
Why Gaussian?
44
DTU Compute, Technical University of Denmark
1
16
1
1
1
1
1
1
1
1
1
1
2
1
2
4
2
1
2
1
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25/2/2015
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Use of smoothing
3x3
45
7x7
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11x11
15x15
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25/2/2015
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Use of smoothing
 Large kernels smooth more
 Removes high frequency information
 Good at enhancing big structures
3x3
46
15x15
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Introduction to Medical Image Analysis
25/2/2015
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Mean filter
A) 166
B) 113
C) 12
D) 51
E) 245
22
2
0
A
47
DTU Compute, Technical University of Denmark
B
2
C
D
Introduction to Medical Image Analysis
4
E
25/2/2015
DTU Compute
Border handling
No values at the border!
Input
49
Output
1
2
0
1
3
1
2
1
4
2
2
2
1
0
1
0
1
3
1
2
1
0
2
4
2
5
3
1
2
2
2
1
3
1
6
3
DTU Compute, Technical University of Denmark
Introduction to Medical Image Analysis
25/2/2015
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Border handling – extend the input
Input
• Zero padding – what happens?
0 0 0 0 0 0
0 1 2 0 1 3
0 2 1 4 2 2
0 1 0 1 0 1
0 1 2 1 0 2
50
• Zero is black – creates dark border around the image
1
2
3
4
2
5
3
1
2
2
2
1
3
1
6
3
DTU Compute, Technical University of Denmark
Introduction to Medical Image Analysis
25/2/2015
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Correlation with zero padding
A)
B)
C)
D)
E)
51
5, 8
12, 16
13, 3
15, 21
17, 12
1
2
1
1
3
1
1
2
1
29
1
2
0
1
3
1
2
1
4
2
2
2
1
0
1
0
1
3
1
2
1
0
2
4
?
2
5
3
1
2
2
2
1
3
1
6
3
A
DTU Compute, Technical University of Denmark
?
0
1
0
B
C
D
Introduction to Medical Image Analysis
3
E
25/2/2015
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Border handling – extend the input
Input
• Reflection
1 1 2 0 1 3
1 1 2 0 1 3
2 2 1 4 2 2
1 1 0 1 0 1
1 1 2 1 0 2
53
• Normally better than zero padding
1
2
3
4
2
5
3
1
2
2
2
1
3
1
6
3
DTU Compute, Technical University of Denmark
Introduction to Medical Image Analysis
25/2/2015
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Template Matching
 Template
– Skabelon på dansk
 Locates objects in images
Template
Input
54
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Output: Correlation image
Introduction to Medical Image Analysis
25/2/2015
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Template Matching
 The correlation between the template and the input
image is computed for each pixel
Template
Input
55
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Output: Correlation image
Introduction to Medical Image Analysis
25/2/2015
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Template Matching
 The pixel with the highest value is found in the
output image
– Here is the highest correlation
Template
Input
56
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Output: Correlation image
Introduction to Medical Image Analysis
25/2/2015
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Template Matching
 This corresponds to the found pattern in the input
image
Template
Input
57
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Output: Correlation image
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25/2/2015
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Problematic Correlation
 Correlation matching has problem with light areas –
why?
𝑅
𝑅
𝑔 𝑥, 𝑦 =
ℎ(𝑖, 𝑗) ∙ 𝑓(𝑥 + 𝑖, 𝑦 + 𝑗)
Very High!
𝑗=−𝑅 𝑖=−𝑅
Real max
Template (h)
Fake max
Output: Correlation image
Input (f)
58
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Normalised Cross Correlation
NCC x, y =
Correlation
Length of image patch ⋅ Length of template
Real max
Template (h)
Output: Correlation image
Input (f)
59
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Length of template
 Vector length
– Put all pixel values into a vector
– Compute the length of this vector
 Describes the intensity of the template
– Bright template has a large length
– Dark template has a small length
𝑅
𝑅
Length of template =
ℎ(𝑖, 𝑗) ∙ ℎ(𝑖, 𝑗)
𝑗=−𝑅 𝑖=−𝑅
Template (h)
60
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Length of image patch
 Vector length based on pixel values in image patch
 Describes the intensity of the image patch
Template (h)
Input (f) with patch
61
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Normalised Cross Correlation
 The length of the image patch and the length of
template normalise the NCC
 If the image is very bright the NCC will be “pulled
down”
Correlation
NCC x, y =
Length of image patch ⋅ Length of template
62
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Normalised Cross Correlation
 NCC will be between
– 0 : No similarity between template and image patch
– 1 : Template and image patch are identical
NCC x, y =
Correlation
Length of image patch ⋅ Length of template
Real max
Template (h)
Output: Correlation image
Input (f)
63
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Normalised Cross Correlation
Der udføres template matching med og normalized cross
correlation (NCC) beregnes. Hvad bliver NNC i den markerede
pixel?
A) 0.10
B) 0.33
C) 0.83
D) 0.62
E) 0.98
23
1
0
A
64
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1
B
1
C
D
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Edges
Gray level profile
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 An edge is where there is a
high change in gray level
values
 Objects are often separated
from the background by
edges
Introduction to Medical Image Analysis
25/2/2015
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Edges
′
𝑓 (𝑑)
 The profile as a function f(d)
 What value is high when there
is an edge?
– The slope of f
– The slope of the tangent at d
f (d)
67
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Finite Difference
 Definition of slope
 Approximation
𝑓′
𝑓 𝑑 + ℎ − 𝑓(𝑑)
𝑑 = lim
ℎ→0
ℎ
𝑓′
𝑓 𝑑 + ℎ − 𝑓(𝑑)
𝑑 ≈
ℎ
 Simpler approximation 𝑓 ′ 𝑑 ≈ 𝑓 𝑑 + 1 − 𝑓(𝑑)
h= 1
68
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DTU Compute
Edges
 Discrete approximation of f’(d)
 Can be implemented as a filter
f (d)
𝑓 ′ 𝑑 ≈ 𝑓 𝑑 + 1 − 𝑓(𝑑)
-1 0
f(d-2) f(d-1) f(d)
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DTU Compute, Technical University of Denmark
1
f(d+1)
f(d+2) f(d+3)
Introduction to Medical Image Analysis
25/2/2015
DTU Compute
Edges in 2D
 Changes in gray level values
–
–
–
–
Image gradient
Gradient is the 2D derivative of a 2D function f(x,y)
Equal to the slope of the image
A steep slope is equal to an edge
𝛻𝑓 𝑥, 𝑦 = 𝐺 (𝑔𝑥 , 𝑔𝑦 )
70
DTU Compute, Technical University of Denmark
Introduction to Medical Image Analysis
25/2/2015
DTU Compute
Edge filter kernel
-1
0
1
-1
0
1
-1
0
1
 The Prewitt filter is a typical
edge filter
 Output image has high
values where there are
edges
Vertical Prewitt filter
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DTU Compute, Technical University of Denmark
Introduction to Medical Image Analysis
25/2/2015
DTU Compute
Prewitt filter
Original
Prewitt
Prewitt
Hot colormap
72
DTU Compute, Technical University of Denmark
Smooth
15x15
Introduction to Medical Image Analysis
Smooth 15x15
Prewitt
25/2/2015
DTU Compute
Edge detection
 Edge filter
– Prewitt for example
 Thresholding
– Separate edges from nonedges
 Output is binary image
– Edges are white
73
DTU Compute, Technical University of Denmark
Introduction to Medical Image Analysis
25/2/2015
DTU Compute
Edge filtering
Der udføres en filtrering (correlation) med et filter på billedet. Der bruges ikke
coefficient normalization. Den filtrerede værdi i den markerede pixel bliver -119.
Hvilket et 3x3 filter blev benyttet?
A) Mean
B) Horizontal Sobel
C) Vertical Prewitt
D) Horisontal Prewitt
E) Median
18
4
3
1
0
A
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DTU Compute, Technical University of Denmark
B
C
Introduction to Medical Image Analysis
D
E
25/2/2015
DTU Compute
Niveau af dagens emne / forelæsing
A) Meget let
B) Let
C) Passende
D) Svært
E) Meget svært
25
4
1
A
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DTU Compute, Technical University of Denmark
2
B
C
Introduction to Medical Image Analysis
D
0
E
25/2/2015
DTU Compute
Mit eget udbytte af dagen
A) Jeg har lært meget lidt
B) Lidt læring
C) Passende
D) God læring
E) Jeg har lært meget
14
7
4
3
0
A
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DTU Compute, Technical University of Denmark
B
C
Introduction to Medical Image Analysis
D
E
25/2/2015
DTU Compute
Sidste uges øvelser (pixelwise operations)
A) Meget lette
B) Lette
C) Passende
D) Svære
E) Meget svære
10
2
1
A
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DTU Compute, Technical University of Denmark
10
2
B
C
Introduction to Medical Image Analysis
D
E
25/2/2015
DTU Compute
Denne uges brug af clickers
A) click, click, click – jeg bliver
sindssyg!
B) For meget click
C) Passende
D) Jeg vil gerne have lidt mere
E) Jeg kan ik’ få nok
1
1
A
79
DTU Compute, Technical University of Denmark
15
14
1
B
C
Introduction to Medical Image Analysis
D
E
25/2/2015
DTU Compute
What can you do after today?
 Describe the difference between point processing and neighbourhood
processing
 Compute a rank filtered image using the min, max, median, and
difference rank filters
 Compute a mean filtered image
 Decide if median or average filtering should be used for noise removal
 Choose the appropriate image border handling based on a given input
image
 Implement and apply template matching
 Compute the normalised cross correlation and explain why it should be
used
 Apply given image filter kernels to images
 Use edge filters on images
 Describe finite difference approximation of image gradients
 Describe the concept of edge detection
80
DTU Compute, Technical University of Denmark
Introduction to Medical Image Analysis
25/2/2015
DTU Compute
Øvelsesrapport over øvelse 4
 Øvelser tæller cirka 30% af karakteren
 Skal afleveres senest den 11. marts kl. 23.59
 Kan afleveres af 1 eller 2 personer
– I skal skrive jeres egen øvelsesrapport
– Brug mindst et af jeres egne billeder
 Afleveres som PDF
 Lad være med at kopiere opgaveteksten ind i jeres
opgave
 Put Matlab kode i et Appendix og henvis fra teksten
81
DTU Compute, Technical University of Denmark
Introduction to Medical Image Analysis
25/2/2015
DTU Compute
Exercises
?
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DTU Compute, Technical University of Denmark
Introduction to Medical Image Analysis
25/2/2015