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image data processing
image filtering and segmentation in 2D and 3D
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aims and objectives
• visualization • data optimization • data segmentation • feature recognition • measuring • object reconstruction • ...
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filte
ring
image data processing and visualization: workflow
patient/object data
image data convolution
feature extraction
data set registration
2D (image) visualization
3D (volume) visualization
measuring
fused imaging
2.5D (surface) visualization
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what is an image?
- a distribution of a variable I in space (x,y) or (x,y,z) - a scalar field (= an array/matrix/volume of numbers)
image image “landscape” = imagescape
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what is an image filter?
input image I output image O
a mathematical function that transforms an imput image into an output image: I(x,y,z) à O(x,y,z)
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extract/enhance “relevant” information remove/reduce“irrelevant” information
why image filters?
input image I output image O
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1. Visualizing 2D biomedical image data
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windowing function di
spla
y un
its
(8 b
it/pi
xel)
input
output 255
0
à see neural response function
example: visualizing a CT cross section
-1000 (air)
3095 (maximum)
-0 (water)
window width
window center
Hounsfield density units (12bit/pixel)
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windowing: CT data example
low contrast window
soft tissue window “inverted” window
bone threshold
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example of a false-color scale
R
G
C M
B
Y
path through color space
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example of a false color scale
input
output
255
1791
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false color application in 3D
bone thickness map"
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discrete false-color scale
→ see LUT, multiple thresholding, data segmentation, enhanced reality
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2. Image processing with 2D and 3D filters
using data from many input elements (pixels/voxels) to calculate
the value of one output elements (pixel/voxel)
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what is an image filter?
input image I output image O
filter function = kernel filtering = convolution ⊗
mathematical function with a spatial structure (“kernel”) that transforms an input image I into an output image O
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principle of filtering
input output
input image output image convolution: apply a function (an operator)
to all image elements (pixels or voxels)
kernel function (size n x n)
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principle of filtering
convolution
=
⊗
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some examples
• the maths of simple filters • mean and Laplace filter • statistical filters: Gauss, median, histogram • edge detection
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the maths of simple filters
Ix-1,y+1 Ix,y+1 Ix+1,y+1
Ix-1,y Ix,y Ix+1,y
Ix-1,y-1 Ix,y Ix+1,y-1
input image elements: 3x3 region around pixel (x,y)
3x3 filter kernel (weight matrix)
Ox,y = Ix+i,y+ j !wi, j( )j="1
1
#i="1
1
#
Ox,y
output image element: new value at pixel (x,y)
⊗ = w-1,1 w0,1 w1,1
w-1,0 w0,0 w1,0
w-1,-1 w0,-1 w1,-1
…
…
…
…
…
…
… … …
… … …
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mean filter („low-pass“)
1/9 1/9 1/9
1/9 1/9 1/9
1/9 1/9 1/9
3x3 kernel
1/25 1/25 1/25 1/25 1/25
1/25 1/25 1/25 1/25 1/25
1/25 1/25 1/25 1/25 1/25
1/25 1/25 1/25 1/25 1/25
1/25 1/25 1/25 1/25 1/25
output images
input image
5x5 kernel
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Laplace filter („high-pass“)
0 -1/4 0
-1/4 1 -1/4
0 -1/4 0
see: http://www.wiley.com/go/virtualreconstruction
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statistical filter 1: noise reduction with Gauss filter
random noise from image data acquisition device
Gauss filter
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statistical filter 2: impulse noise reduction with median filter
median filter (a statistical filter): evaluate median value in nxn neighborhood of each pixel
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statistical filter 3: histogram filter
output intensity is a function of the frequency of input intenstiy
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edge detection
• identify object boundaries
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edge detection in the retina
principle of edge detection: • center: positive weight • periphery: negative weight
+- - - -
-
-
+-
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edge detection: computer
Laplace filter kernel: 2nd spatial derivative of the image
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spatial derivatives
I=f(x) Ii-1 Ii Ii+1
xi xi+1 xi-1
(1⋅Ii-1-2⋅Ii+1⋅Ii+1)
Ii- Ii-1 Ii+1-Ii ΔI/Δx
Δ2I/Δx2
1 -2 1
function
1st deriv.
2nd deriv.
Laplace filter
position x
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example image: I = f(x,y)
x
y
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example image: I = f(x,y)
x
y
I
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example image: first spatial derivatives
Ix’ =dI/dx
Iy’=dI/dy
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example image: gradient [Ix’ Iy’]
vector components are first derivatives along x and y
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example image: Laplacian Ix’’ + Iy’’
sum of second derivatives along x and y
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example image: Laplacian Ix’’ + Iy’’
sum of second derivatives along x and y
edges
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example image: I = f(x,y)
x
y
I edges
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edge detection with Laplace filter
input output
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concatenating filters
Gauss: noise reduction
Laplace: edge detection
input 1 output 1 = input 2 output 2
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3. image data segmentation
filters for data classification
transformation of bitmap graphics into vector graphics
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data segmentation
principal aim: classification
• attribute specific properties (labels) to each voxel of the data set
• create explicit object representations • methods:
– automated: thresholding – interactive: feature extraction with gradient-based
and morphological filters
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example 1: thresholding attribute one output value to a range of input values
thresholding multiple thresholding (à lookup tables)
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example 2: morphological operators (filters)
• extract pre-defined features from 2D/3D images
• reconstruct explicit contours, surfaces, volumes in the form of geometric descriptions
à transform bitmap graphics into vector
graphics
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geometric „primitives“
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contour reconstruction in 2D with line primitives
marching squares: morphological 2D filter kernel
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contour data
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surface reconstruction in 3D
marching cubes: morphological 3D filter kernel
etc... 28=256 variants)
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surface data
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free-form surface representation
1 cm
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example 3: deformable templates
• define a-priori forms (templates) based on anatomical knowledge
• adapt (deform) templates to 2D/3D image data
• extract 2D/3D features • measure difference between actual feature
and template
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freeform 2D-deformable templates
• use an a-priori model as a template to classify image data
• deform the template until it optimally fits the image data
• label the image data accordingly
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freeform 3D-deformable templates: brain extraction tools Ventrice (PhD thesis, 2011)
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cylindrical template (cylindrical surface parameterization)
D E
l a m p l
0⁰ 90⁰ 180⁰ 270⁰ 360⁰
a l
m
0
1
c = 1
ma
z
F
0⁰ 90⁰
180⁰ 270⁰
Morimoto et al. (2011)
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spherical template (spherical surface parameterization)
chimpanzee human
Specht et al. (2007)