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Computed Tomography: Introduction and Instrumentation Primary Source: Medical Imaging Signals and Systems By Jerry Prince and Jonathan Links

Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

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Page 1: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

Computed Tomography: Introduction and Instrumentation

Primary Source: Medical Imaging Signals and Systems

By Jerry Prince and Jonathan Links

Page 2: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each
Page 3: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each
Page 4: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

CT: Physics

• Same as x-ray radiography: – Image contrast from photoelectric effect

– Image blurring from compton scatter

Particulate ionizing radiation: Electromagnetic ionizing radiation:

Page 6: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each exposure • Shield all of x-ray beam but slice • Obtain1D projections of 2D axial cross section • A Radon transform takes 1D projections of a 2D object over many angles

and it has an inverse

“raw data” (indirect and tomographic modality)

Page 7: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

• The Radon Transform: the integral transform consisting of the integral of a function over straight lines

Projections – general definition/concepts

x

y

θ

l l

l

θ=45o

θ=90o

θ=135o

l θ=0o

θ=179o

l

dxdyyxyxfg )sincos(),(),(

where δ(xcosθ+ysinθ-l) is a line impulse that exists along a line normal to

θ a distance l from the origin.

δ(xcosθ+ysinθ-l)

f(x,y)

g(l,θ), the Radon Transform:

Page 8: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

EW

1

EW

2

EW

3

EW

4

EW

5

EW

6

EW

7

EW

8

EW

9

NS1

NS2

NS3

NS4

NS5

NS6

NS7

NS8

NS9

BATTLESHIP

East - West Data

N

o

r

t

h

/

S

o

u

t

h

D

a

t

a

Battleship:

New Rules

Old way

EW 2/ NS 3 Hit!

EW 5/ NS 6 Miss!

Page 9: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

EW

1

EW

2

EW

3

EW

4

EW

5

EW

6

EW

7

EW

8

EW

9

NS1

NS2

NS3

NS4

NS5

NS6

NS7

NS8

NS9

1 0 4

BATTLESHIP

East - West Data

N

o

r

t

h

/

S

o

u

t

h

D

a

t

a

Battleship:

New Rules

Tell the other

player how

many times you

see a battleship in

a square along

each column.

Page 10: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

EW

1

EW

2

EW

3

EW

4

EW

5

EW

6

EW

7

EW

8

EW

9

NS1

NS2

NS3

NS4

NS5

NS6

NS7

NS8

NS9

1 1 1 0 1 1 1 4 0

BATTLESHIP

East - West Data

N

o

r

t

h

/

S

o

u

t

h

D

a

t

a

Battleship:

New Rules

Here it is finished.

Page 11: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

EW

1

EW

2

EW

3

EW

4

EW

5

EW

6

EW

7

EW

8

EW

9

NS1

NS2

NS3

NS4

NS5

NS6

NS7

NS8

NS9

1 1 1 0 1 1 1 4 0

BATTLESHIP

East - West Data

N

o

r

t

h

/

S

o

u

t

h

D

a

t

a

If this is all your

opponent told you,

could you find

where the

battleships were?

Need more info.

Repeat the

procedure for

rows.

Page 12: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

EW

1

EW

2

EW

3

EW

4

EW

5

EW

6

EW

7

EW

8

EW

9

NS1 1

NS2

1

NS3 4

NS4 1

NS5 0

NS6 0

NS7 3

NS8 0

NS9 0

1 1 1 0 1 1 1 4 0

BATTLESHIP

East - West Data

N

o

r

t

h

/

S

o

u

t

h

D

a

t

a

Here are all the

answers.

Page 13: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

EW

1

EW

2

EW

3

EW

4

EW

5

EW

6

EW

7

EW

8

EW

9

NS1 1 1 1 0 1 1 1 4 0 1

NS2

1 1 1 0 1 1 1 4 0 1

NS3 1 1 1 0 1 1 1 4 0 4

NS4 1 1 1 0 1 1 1 4 0 1

NS5 1 1 1 0 1 1 1 4 0 0

NS6 1 1 1 0 1 1 1 4 0 0

NS7 1 1 1 0 1 1 1 4 0 3

NS8 1 1 1 0 1 1 1 4 0 0

NS9 1 1 1 0 1 1 1 4 0 0

1 1 1 0 1 1 1 4 0

BATTLESHIP

East - West Data

N

o

r

t

h

/

S

o

u

t

h

D

a

t

a

Can you find the

ships now if you just

knew the gray squares?

One idea:

Smear the East West

numbers all the way

up the columns.

This tells us that we

should not spend much

time looking along

EW4 or EW9.

But there is something

probably up in EW8

Page 14: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

EW

1

EW

2

EW

3

EW

4

EW

5

EW

6

EW

7

EW

8

EW

9

NS1 1+1

=2

4+1

=5 1

NS2

1

NS3 4

NS4 1

NS5 0

NS6 0

NS7 3+1

=4 3

NS8 0

NS9 0

1 1 1 0 1 1 1 4 0

BATTLESHIP

East - West Data

N

o

r

t

h

/

S

o

u

t

h

D

a

t

a

Next, smear the

North/ South

numbers to the left

and add them

to what was in the

grid before.

Where do you

think the ships are?

By the biggest

numbers?

Is this always true?

Page 15: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

EW

1

EW

2

EW

3

EW

4

EW

5

EW

6

EW

7

EW

8

EW

9

NS1 2 2 2 1 2 2 2 5 1 1

NS2

2 2 2 1 2 2 2 5 1 1

NS3 5 5 5 4 5 5 5 8 4 4

NS4 2 2 2 1 2 2 2 5 1 1

NS5 1 1 1 0 1 1 1 4 0 0

NS6 1 1 1 0 1 1 1 4 0 0

NS7 4 4 4 3 4 4 4 7 3 3

NS8 1 1 1 0 1 1 1 4 0 0

NS9 1 1 1 0 1 1 1 4 0 0

1 1 1 0 1 1 1 4 0

BATTLESHIP

East - West Data

N

o

r

t

h

/

S

o

u

t

h

D

a

t

a

Here I finished

smearing the

north/south

numbers to the left

and adding them to

the east/west

numbers.

Where do you

think the ships are?

By the biggest

numbers?

Is this always true?

Page 16: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

EW

1

EW

2

EW

3

EW

4

EW

5

EW

6

EW

7

EW

8

EW

9

NS1 2 2 2 1 2 2 2 5 1 1

NS2

2 2 2 1 2 2 2 5 1 1

NS3 5 5 5 4 5 5 5 8 4 4

NS4 2 2 2 1 2 2 2 5 1 1

NS5 1 1 1 0 1 1 1 4 0 0

NS6 1 1 1 0 1 1 1 4 0 0

NS7 4 4 4 3 4 4 4 7 3 3

NS8 1 1 1 0 1 1 1 4 0 0

NS9 1 1 1 0 1 1 1 4 0 0

1 1 1 0 1 1 1 4 0

BATTLESHIP

East - West Data

N

o

r

t

h

/

S

o

u

t

h

D

a

t

a

Next, smear the

North/ South

numbers to the left

and add them

to what was in the

grid before.

Are the battleships

where the biggest

numbers are?

All of the time?

Some of the time?

Page 17: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

1 2 3 4 5 6 7 8 9

A

B

C

D

E

F

G

H

I

BATTLESHIP

0

0

1

1

1

0

0

0 1 1 2 1 1

1

0 0 0

What if we can measure along the diagonals?

Page 18: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

1 2 3 4 5 6 7 8 9

A 2 2 3 2 3 2 2 6 2 1

B

2 3 3 2 2 2 3 6 2 1

C 6 6 6 4 5 6 6 9 6 4

D 3 3 2 1 3 3 3 7 2 1

E 2 1 1 1 2 2 3 5 1 0

F 1 1 2 1 2 3 2 5 0 0

G 4 5 5 4 6 5 5 7 3 3

H 2 2 2 2 2 2 1 4 0 0

I 2 2 3 1 2 1 1 4 0 0

1 1 1 0 1 1 1 4 0

BATTLESHIP

Now add the

diagonal

information to our

totals.

Are we doing any

better?

Are the battleships

where the biggest

numbers are more

often?

Page 19: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

How is “BMEN 420 Battleship” similar to what we do in CT?

http://www.colorado.edu/physics/2000/index.pl

Page 20: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

• Step 1: Backprojection – form 2D images (in (x,y) Cartesian space) from your 1D projections (collected in (ℓ,θ) polar space)

Examples of backprojecting: “smearing” your 1-D projection data collected at

an angle back into x-y space

Image Reconstruction from Projections Backprojection Summation

),sincos(),( yxgyxb

b0 (x,y)

b90 (x,y)

b35 (x,y) x

y

θ

l

Page 21: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

• Step 1: Backprojection – form 2D images (in (x,y) Cartesian space) from your 1D projections (collected in (ℓ,θ) polar space)

Examples of backprojecting: “smearing” your 1-D projection data collected at

an angle back into x-y space

Image Reconstruction from Projections Backprojection Summation

),sincos(),( yxgyxb

b0 (x,y)

b90 (x,y)

b35 (x,y)

Page 22: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

• Step 1: Backprojection – form 2D images (in (x,y) Cartesian space) from your 1D projections (collected in (ℓ,θ) polar space)

Examples of backprojecting: “smearing” your 1-D projection data collected at

an angle back into x-y space

Image Reconstruction from Projections Backprojection Summation

),sincos(),( yxgyxb

b0 (x,y)

b90 (x,y)

b35 (x,y)

Page 23: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

• Step 2: Summation – sum the backprojections from step 1 to obtain the reconstructed object fbs(x,y).

Image Reconstruction from Projections Backprojection Summation

tionbackprojec theis ),sincos(),( yxgyxbwhere

0

sincos0

0

),(),sincos(),(),( dlgdyxgdyxbyxf yxlbs

*how many projections you “need” depends on what you want to see

Page 24: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

• Step 2: Summation – sum the backprojections from step 1 to obtain the reconstructed object fbs(x,y).

Image Reconstruction from Projections Backprojection Summation

tionbackprojec theis ),sincos(),( yxgyxbwhere

0

sincos0

0

),(),sincos(),(),( dlgdyxgdyxbyxf yxlbs

*how many projections you “need” depends on what you want to see

Page 25: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

CT & General Radiography: Broad Comparisons

Advantages Disadvantages

General Radiography - Inexpensive -“fast” (motion not as big an issue) - no computational power necessary -Broad coverage -Very good resolution

-overlaying structures (projection) -lower contrast

CT -tomography (no overlaying structures) -higher contrast

-more radiation -motion artifacts -expensive -computationally challenging -narrow coverage -big, cumbersome -no long-axis images

Disappearing with computation power, 3D data blocks (allows for reformatting) from spiral, helical, and multislice

Page 26: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

CT: Instrumentation – 1st Generation

From Webb, Physics of Medical Imaging

0-D projection of 1-D object (pencil beam) Translate then rotate

Page 27: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

CT: Instrumentation – 2nd Generation

From Webb, Physics of Medical Imaging

Eliminate some of translation steps… Still have to translate + rotate

Page 28: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

1G and 2G – how large is the time savings?

Consider a 1G or 2G scanner whose source detector apparatus can move linearly at a speed of 1.0 m/sec and the field-of-view has a diameter of 0.5m. Suppose that 360 projections over 180o are required and that it takes 0.5 sec for the source-detector apparatus to rotate one angular increment, regardless of the angle.

-What is the scan time for a 1G scanner?

-What is the scan time for a 2G scanner having 9 detectors

spaced 0.5o apart?

Page 29: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

1G and 2G – how large is the time savings?

Consider a 1G or 2G scanner whose source detector apparatus can move linearly at a speed of 1.0 m/sec and the field-of-view has a diameter of 0.5m. Suppose that 360 projections over 180o are required and that it takes 0.5 sec for the source-detector apparatus to rotate one angular increment, regardless of the angle.

-What is the scan time for a 1G scanner?

-What is the scan time for a 2G scanner having 9 detectors

spaced 0.5o apart?

Example 6.1 in MISS - ans: 6 mins versus 40 secs

Page 30: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

CT: Instrumentation – 3rd Generation

Eliminate translation Just rotate

Page 31: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

CT: Instrumentation – 4th Generation

Eliminate rotation of detectors. Now just source rotates. NO DETECTOR COLLIMATION => More efficient, but more blurring

Page 32: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

• Generations 1-4 – Single x-ray tube (bulky, expensive, difficult to calibrate)

– Moving the source limits scan speed

• Generation 5: electron beam computed tomography (EBCT) – Source: Flying electron beam, steered electromagnetically,

to hit one of four tungsten anode strips. X-rays generated are collimated into a fan beam and detection is as with 4G

– Extremely fast (stop action cardiac without gating)

CT: Instrumentation – 5th ,6th , and 7th Generations

Page 33: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

• Generation 6: helical CT – 3G/4G continuously rotating with moving patient

for 3D volumetric acquisition • 60cm torso scan, 30 secs

• 24cm lung scan, 12 secs

• 15cm angio scan, 30 secs

• Generation 7: multislice CT – “thick” fan beam, parallel rows of detectors

– Collects multiple (up to 64, 256) 1-D projections at one time

• Multislicehelical = larger pitch on helix:

CT: Instrumentation – 5th ,6th , and 7th Generations

Page 34: Computed Tomography: Introduction and Instrumentation · CT: Introduction • Computed Tomography: slice picture • Take a series of conventional projection x-rays, rotate b/t each

CT: Instrumentation

• Instrumentation developments (engineering) have paved the way for modern CT: – Fan beam collimation: 2 pieces of lead with a slit 1-10mm between them,

as close as possible to patient. Controls slice thickness. Done at console. – Copper followed by aluminum filtration to “harden” the beam (make it

more monoenergetic) – Solid state detectors, xenon gas detectors (3G), multiple solid state

detector array – Gantry: holds the x-ray tube, detectors, so can rotate around patient

rapidly and repeatedly : movie of Philips Brilliance 16 slice http://www.youtube.com/watch?v=YAqK-huXQoI http://www.youtube.com/watch?v=2CWpZKuy-NE&feature=related – Slip ring: brush system to deliver power to x-ray tube – Patient table: must account for patient loading and continuous controlled

movement (if helical)