medical imaging and virtual medicine - ggg

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Dirk Bartz, Visual Computing for [email protected]

Lecture 2:

Histograms, Classification,Segmentation

Medical Imaging and Virtual Medicine

[email protected] 2005

Literature (1)

Paper:• G. Kindlmann, D. Durkin: Semi-Automatic Generation of

Transfer Functions for Direct Volume Rendering, Proc. IEEE Symposium on Volume Visualization, 1998.

• V. Pekar, R. Wiemker, D. Hempel: Fast Detection of Meaningful Isosurfaces for Volume Data Visualization, IEEE Visualization, 2001.

• J. Kniss, G. Kindlmann, C. Hansen: Interactive Volume Rendering Using Multi-Dimensional Transfer Functions and Direct Manipulation Widgets, IEEE Visualization, 2001.

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Outline

Histograms and Classification

Windowing

Histogram Filter

Segmentation


Histogram and Classification (1)

• Recall:Volume Data are typically 8-12bit deep

• Value distribution of voxels of dataset is called histogram:• With 8bit: value distribution of 256

possible data values• With 12bit: 4096 possible data values

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MRI Dataset, T2 weighted (fluids)

Linear histogram



Logarithmic histogram

MRI Dataset, T2 weighted (fluid)

(filtered brighter)

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Scaled linear histogram (filtered brighter)





5









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• Histogram are important for data analysis, but show only materials

• Quest for material interfaces• But: frequency of voxel value is often not

sufficient

• Gradient magnitude exposes material boundary:• High gradient magnitude

large differences• Small gradient magnitude

homogeneous area



Artificial Dataset: Low-Pass filtered Sphere

(enlarged)Linear histogram

Not very informative!

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Artificial Dataset: Low-Pass filtered Sphere

(enlarged)

Gradient histogram (f´(v))Voxel value f(v)

∇v

|f´(v)|

Frequency Intensity

[Kindlmann, Durkin, VolVis 1998.]



More Information:First directional derivative of voxel values f´(x) along gradient:

Maximum indicates material interface

Second directional derivative of voxel values f´´(x) along gradient:

Zero-crossing indicates material interface

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Voxel value f(v), f´(v), f´´(v) at material interface

From:[Kindlmann, Durkin, VolVis 1998.]

f(v)f´(v)f´´(v)

v


• First derivative approximated from gradient magn‘:

D^ f = ||∇f||

• Second derivative not easy to construct

• Approximation by gradient magnitude ofgradient field


∇f

Direction

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Position x, voxel value f(v), and gradient f´(v)

v

v

f(v)

f(v)

f´(v)

f´(v)




Position x, voxel value f(v), and f´´(v)

f(v)

f(v)

f´´(v)

f´´(v)

v

v From:[Kindlmann, Durkin, VolVis 1998.]

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Histogram Volume:Voxel value f(v), gradient f´´(v), and f´´(v)

f(v)

f(v)

f´´(v)

f´´(v)

f´(v)f´(v)




• Don’t display full value range for f´(v) and f´´(v); otherwise material interfaces can be missed

• Histogram representation is projected from histogram volume in viewing direction

f´(v) f´´(v)

f(v) f(v)

Projected along: f´´(v), f´(v)

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Cylinder and embedded Cylinder

Dataset

One material interface(one maximum/zero-crossing)

Three material interfaces(three maximazero-crossings)

f´(v)

f´(v) f´´(v)

f´´(v)

f(v) f(v)

From: [Kindlmann, Durkin, VolVis 1998.]



More Examples

From: [Kindlmann, Durkin, VolVis 1998.]

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Variation of Method:[Pekar et al. IEEE Visualization, 2001.]

• Examines further feature criterions:• Total Gradient Feature Curve• Average gradient• Surface of isosurface• Volume of isosurface (count voxels)

• Combines criterion into decision function



Total Gradient Feature Curve:• Calculates Laplace-Operator for every voxel:

2. derivative considered Zero-crossing at material interfaces)

• Accumulates results in voxel value (like histogram)

• Results in Laplace-weighted histogram

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Total Gradient Feature Curve:• Features are emphasized by Maxima

Image slice Total Gradient Feature Curve 3D Rendering

From: [Pekar et al., IEEE Vis 2001.]



Mean Gradient Feature Curve:• Divides Total Gradient Feature Curve

by measured surface along isosurface

• More independent from actual size

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Mean Gradient Feature Curve:• Peak size more independent of actual feature size

Image slice Mean Gradient Feature Curve 3D Rendering

From: [Pekar et al., IEEE Vis 2001 ]



More useful metrics for Data Analysis:• Statistics of 1. order:

• Local, average gray value

mk(p(x,y)) = 1 / N ∑∑v(x+i,y+j)

• Local varianceσ2

k(p(x,y)) = 1 / (N-1) ∑∑(v(x+i,y+j) – mk(p))2

i j

k k

i j

k k

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More useful ...• Measure for smoothness R:

If σ2 gets small, R approaches 0 and indicates homogeneous/smooth region.

11 + σ2(p)R = 1 -


Outline


Windowing

Histogram Filter

Segmentation

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Windowing (1)

Why Windowing?• Volume Data are usually 12-16bit/voxel deep

• Output devices (screen) only have 8bit/color depth, or only 8bit gray value depth

• Value range of dataset must be mapped into 8bit

Windowing


Windowing (2)

12bit Volume Dataset, mapped into positive range

0 4095

Logarithmic histogram

2047 3100

8bit window

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Windowing (3)

Mapping of 12bit value range into 8bit:• Selected value range is named window

• Set:• Window width• Window position (center)

• Window width controls contrast:• More wide, the smaller the contrast• More narrow, the larger the contrast


Windowing (4)

Window Width:• Enlarging the window range with downsampling:

eg. 4bits are mapped into 1bit,or the range of 0..15 into 0..1• Loss of accuracy

• Values below/above of window are mapped on window limits: 0 or 255

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Windowing (5)

Window Position:• Position marks features

• Features outside of window cannot be differentiated any more


Windowing (6)

0 40952047

Window width

Window position

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Windowing (7)

0 40952047

Window width withreduced contrast,but enlargedrange

Window position marks feature


Windowing (8)

Different Window Ranges

Position: 1023, Width: 4096 Position: 0, Width: 400

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Windowing (9)

• Maps linearly into window range

• Size of different features can be different

• Non-linear windowing would be good(TechReport, using ToneMapping)


Outline


Windowing

Histogram Filter

Segmentation

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Histogram Filter (3)

Histogram Stretching:• Not all volume-/image data exploit whole

value range

• Like Window-to-Viewport-mapping (CG 101)

• New lower limit implicitly 0

Is(x,y) = gmax

Ie(x,y) – min(Ie)

max(Ie) – min(Ie)



Histogram Stretching:• Not all volume-/image data exploit whole

value range

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Histogram Stretching:

Original Image Image after Stretching



Histogram Stretching:

Original Image Image after Stretching

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Selective Histogram Stretching :• Partial range is mapped into larger range

• Useful to compensate for noise(not used histogram range)

• Piecewise linear

• Logarithmical



Selective Histogram Stretching:• Partial range is mapped into larger range

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Selective Histogram Stretching:• Piecewise linear scaling:

• Regular stretching of partial range• Other ranges are kept identical (might be

translated), or culled



Selective Histogram Stretching:• Logarithmic scaling:

• Certain ranges are enhanced• More like human vision

gs(g + c3) = gmax

log(c1+ g )– log(c1)

log(c2) – log(c1)

c2 - c1

gmax

with c3 as translation in histogram, c1,c2 controlsteepness, and g as histogram.

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Selektive Histogram Stretching:• Logarithmic

scaling:

with

c1= 5, c2 = 15,

c3= 0,

gmax = 50.



Histogram Equalization:• Not all volume dataset exploit range equally

• Uses relative accumulation frequency as metric

• Usually not invertible

• No perfect results, due to discrete nature of data

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Histogram Equalization:Histogram



Histogram Equalization:Accumulation frequency

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Histogram Equalization:




Original Image Image after Equalization

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Original Image Image after Equalization



Image processing applets at

• http://www.gris.uni-tuebingen.de/projects/grdev/doc/html/Overview.htmlKursbuch Applet-Index Bildverarbeitung

Histogramm-Manipulation

(in German only, sorry)

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Outline


Windowing

Histogram Filter

Segmentation


Segmentation (1)

What is Segmentation?• Segmentation is identification of

structures/objects (eg. organs) in image / volume data.

• Assign a semantic into data

• Unfortunately: Many structures are easy to detect with eyes, but difficult with algorithms

• Merging criterion can be local or regional.

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Segmentation (2)

ComputedTomography:

Bone

SegmentedMastoid


Segmentation (3)

Classification of Segmentation Approaches:

Model-based

Level-Sets

Region-basedRegion Growing

Fuzzy Connectedness

Markoff Random Field

Scale Space

Contur-basedEdge Detectors

Snakes

LifeWire

Watershed-transformation

LocalIntensity

Texture Analysis

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Segmentation (5)

Segmentation:• Good data acquisition saves plenty of

segmentation work

• Sources of issues:• Partial volume effect• Insufficient resolution• Insufficient contrast• Optimal: High Intensity, Neighborhood low

intensity


Segmentation (6)

Examples for good Contrast:

MRI TSE:Fluidfilled cavities

MRI TOF:Blood vessels

MRI 3D CISS:Fluid filled cavities

Rot. AngiographyContrast enhanced cavities

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Segmentation (7)

Ventricles / background

Corpus callosum / brain tissue

MRI Flash/T1

Examples of poor Contrast


Segmentation (8)

CT Angiography:• Good bone contrast

• Good blood vessel contrast

• Poor ventricle contrast (noise)

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Segmentation (8)

Bone

Blood vessel

Ventricle


Segmentation (9)

[Image of Hastreiter et al.,Univ. Erlangen-Nürnberg]

CT Angiography:

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Segmentation (10)

• Binary Segmentation often leads to blocky appearance due to interpolation artifacts.(similar to staircase artifacts)

Add boundary layer


Segmentation (11)

Region-based:• Identifies Objects in 2D/3D regions

• Considers voxel neighborhood

• Needs merging criterion

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Segmentation (12)

Merging Criterions:• Distance metrics are measures for the

similarity of two regions (voxel sets), which might be merged.• Gray value distance (voxel intensity)• Neighbors and same gray value interval• Homogeneity (Texture analysis)


Segmentation (13)

Merging Strategies of two regions A,B: pA∈A, pB∈B, E is separation edge of A,B

• Single Linkage: At least one voxel pair (pA,pB) on E satisfies distance metric

• Contiguity Constraint Complete Linkage: All voxel pairs (pA,pB) on E satisfy distance metric

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Segmentation (14)

Merging Strategies of two regions A,B: pA∈A, pB∈B, E is separation edge of A,B

• Contiguity Constraint Average Linkage: Average of all voxel pairs on E satisfy Distance metric D , < D

• Centroid Linkage: Average of A,B (not only E)

, < D


Segmentation (15)

Merging Strategies of two Regions : • Complete Linkage: Centroid

Linkage and difference of grayvalue intervals of A,B must bebelow threshold

( , )< D and (( , ),( , )) < D

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Segmentation (16)

Merging Strategies of many regions :• Sequence is important (not commutative).


Example Segmentation (1)

Tübinger Middle Ear Implant:[M. Maassen, HNO Tübingen]

Transformer

Microphone

Main

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Brain

Outer ear

Mastoid



• Region-oriented segmentation

• Step-wise refinement of over-segmented image data

Segmentation Correction

• Object mask• Morphology• Clipping• Connected components

• Local features• 2d/3d region growing

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Segmentation Correction3d region growing

M1

Homogeneous image regionZK

M2

Morphological OpeningZK

2d DilatationM3

Subtraction: M1 – M3ZK

ClippingZK

M4

3d region growing.

2d region growing.M5 Subtraction: M4 – M5



Over-segmentationOriginal

M1

3d region growing.M1

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Anatomical connectionsImage artifacts

3d region growing.M1



Segmentation Correction3d region growing.

M1

Homogeneous Image RegionZK

M2


2d DilatationM3


ClippingZK

M4

3d region growing.


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M2


2d DilatationM3





M2


2d DilatationM3


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M2


2d DilatationM3





M1


M2

Morphological OpeingZK

2d DilationM3


ClippingZK

M4

3d region growing.


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3d region growing. ClippingZK

M4→ Limited grow



3d region growing. ClippingZK

M4

343536

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M1


M2


2d DilatationM3


ClippingZK

M4

3d region growing.



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4849 4950 50

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Result of Segmentation:



Segmentation of bordering Mastoid Bone

Expansion of Mask

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Result of Segmentation:


Image Source

• Department of Neuroradiology, University Hospital Tübingen

• Department of Radiology,University Hospital Tübingen

• Kindlmann, Durkin: VolVis 1998

• Pekar et al. IEEE Visualization 2001.

medical imaging and virtual medicine - ggg

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