How Ct Works

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MIPR Lecture 7Copyright Oleh Tretiak, 2004

1

Medical Imaging and Pattern

Recognition

Lecture 7

Computed TomographyOleh Tretiak


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2

Computer Tomography:

How It Works

Only one plane is illuminated. Source-subject motion provides

added information.


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3

How it Works

Original CT Scanner

Head only

One minute scanning time

Two sections

Ten minutes to compute images

Extremely successful!


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4

First CT Scanner


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5

Before and After CT


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6

Contemporary CT


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7

Fan-Beam Computer

Tomography


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8

Contemporary Spiral Scanner

Configuration:

40 slices per rotation maximum

Other options are 32 slices or 16 slices

40 mm axial distance scanned in one

rotation

0.4 sec per rotation 60 kW generator


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9

Example: Head

Bleeding due to injury

Can cause brain injury

if not treated

Blood between brain

and dura, easy to

treat


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10

Example: Head

FRONTAL CONTUSION

WITH SUBARACHNOID

HEMORRHAGE


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11

Chest Study

Pneumothorax (airbetween lung and chest)

Also note the bilateral

lower lobe consolidationof lungs, right beinggreater than left. Thereis a chest tube within theright hemithorax.


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12

Abdomen

Appendicitis

(arrow)

Contrast agents instomach and in

blood


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13

Mathematics of Computed

Tomography Model for measurements

Direct problem

Inverse problem

Algorithm for computed tomography


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14

Direct Problem

Beam with

intensity I0 enters

body with varyingattenuation

Each layer has

thickness t

I I

I II I I

Ia e1tI0, Ib e

2tIa,

Ic e3tIb, I1 e

4 tIc

I1 e4 te

3te2te

1tI0 e(1 2 3 4 )tI0

ln(I0 /I1) (1 2 3 4 )t


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15

Integral Equation

x

I (y)I

I1(y) exp( (x,y)dx)I0

ln(I0 /I1(y)) (x,y)dx

x

I (t, )

I

y

q

t

ln(I0 /I1(t,)) (tcos lsin,tsin lcos)dl


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16

Radon Transform

x

I (t, )

I

y

q

t

f(x,y)

g(t, )

g(t,) f(tcos lsin,tsin lcos)dl


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17

Inverse Radon Transform

Given: X-ray transmission

measurements I1(t, ). Find (x, y)

Given: g(t, ). Find f(x, y)

Method:

(a) convolution

(b) backprojection

f(x,y) g1(x cos y sin,)d

0

g1(t,) h(t)g(,)d


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Example

-1

0

1

-1

0

1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

f(x, y)

Lines: g(t, ), same for all

Dots: g1(t, ), after convolution


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Backprojection

-1

0

1

-1

0

1

-0.5

0

0.5

1

1.5

One, two, and four angles of backprojection

-1

0

1

-1

0

1

-0.5

0

0.5

1

1.5

-10

1

-1

0

1

-0.5

0

0.5

1


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More Backprojection

-1

0

1

-1

0

1

-0.5

0

0.5

1

-10

1

-1

0

1

-0.5

0

0.5

1

-1

0

1

-1

0

1

-0.5

0

0.5

1

8, 15, and 30 angle backprojection


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21

Pictures

f(x, y) g(t,

Theta horizontal


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22

Backprojection at 4, 16, and

100 angles


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23

History

1917, Joachim Radon

Solved formal inverse problem. Interest in

theory of integration and geometry 1958, Simeon Tetelbaum of KPI

publishes a paper about X-raytomography. Publishes valid inverse problem solution.


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24

More History

1963 John Cormack publishestheoretical and experimental results.

Experiment with cylindrical objects 1972 Godfrey Hounsfield develops CT

scanner

1979 Hounsfield and Cormack receiveNobel prize in Medicine


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MIPR Lecture 7Copyright Oleh Tretiak, 2004 25

Example of Contrast

QuickTime and aTIFF (Uncompressed) decompressor

are needed to s ee this picture.

12 bit image, full

contrast range.

Window for low

densities


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More Contrast Operations

QuickTim e and a

TIFF (Uncompressed) decom pressorare needed to see this picture.

Window for highdensities


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3-D Images

Spiral scan procedures produce sets of

sectional images suitable for 3-D imaging

Resectioning: Compute new section plane

Projection: Compute sums along rays

Rendering: Segment image and show

surface.


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Bronchoscopy

Path View


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Colonoscopy

Path View


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Medical Practice

In the fall of 2003 Siemens became the firstCT supplier ever to receive clearance fromthe FDA for a computer-aided technique of

identifying nodules, that is, possible tumors,in the lung. CT is also used for the diagnosisof colon cancer: A virtual flight through thehuman colon can detect even the smallest

polyps. If these are removed in time, anoutbreak of colon cancer can very probablybe prevented.


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Comparison

Left: A polyp seen with optical endoscopy. Right: View in virtual endoscopy.


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MIPR Lecture 7C i ht Ol h T ti k 2004 32

Summary

Computer tomography became successfulbecause it showed soft tissue differences thatcould not be seen on X-rays.

Evolution of high-speed (spiral scan)machines came about through improvementsin X-ray detectors

This has led to 3-D imaging methods Surgery planning

Virtual endoscopy

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