Schematic Representation o f the Scanning Geometry

of a CT System

What are inside the gantry?

Scanner without covers

Scanner with covers

DisadvantagesGenerations sourceSource

collimation detectorDetector

collimation

Source- Detector movement

Advantages

single

single

single

single

single

single

multiple

Pencil beam

Fan- beamlet

Fan- beam

Fan- beam

Fan- beam

Fan- beam

Narrow

cone- beam

single

multiple

many

Stationary ring

many

Stationary ring

Multiple arrays

1st Gen.

2nd Gen.

3rd Gen.

4th Gen.

5th Gen.

6th Gen.

7th Gen.

no

yes

no

no

no

yes

yes

Trans.+Rotates Trans.+Rotates Rotates together

Source Rotates only

No movement

3rdGen.+

bed trans.3rdGen.+

bed trans.

No scatter

Faster than 1G

Faster than 2G

Higher efficiency than 3G

Ultrafast for cardiac

faster 3D imaging

faster 3D imaging

slow

Low efficiency

High cost and Low

efficiency

high scatter

high cost

higher cost

higher cost

8th Gen. singlewide

cone- beamFPD no 3rd Gen. Large 3D Relatively

slow

What is displayed in CT images?

HU1000CT# T

water

water

Water: 0HU Air: -1000HU

Typical medical scanner display:

[-1024HU,+3071HU],

Range: 1224096 12 bit per pixel is required in display.

Hounsfield scales for typical tissues

For most of the display device, we can only display 8 bit gray scale. This can only cover a range of 2^8=256 CT number range. Therefore, for a target organ, we need to map the CT numbers into [0,255] gray scale range for observation purpose. A window level and window width are utilized to specify a display.

WL

-1024

+3071

0

255

2/#

2/#2/

2/#

)2/(#

0

scaleGray Displayed

max

max

WLCT

WLCTWL

WLCT

I

IW

WLCT

),( ZE Mass density

Mass attenuation coefficient: attenuation per electron or per gram

Reminder:

4~3

3

Z

E

Windowing in CT image display

Multi-row CT detector (I) GE Light Speed

Multi-row CT detector (II) Siemens Sensation

Future 256-slice cone-beam CT detector

Krestel-Imaging Systems for Medical Diagnosis

SNR is dependent on dose, as in X-ray.Notice how images become grainier and our ability to see small objects decreases as dose decreases. Next slides discuss analysis of SNR in CT. We will see some similarities with X-ray. But we also see some important differences.

In CT, the recon algorithm calculates the of each pixel.x-ray = No e -∫ dz

recorder intensity

For each point along a projection g(R), the detector calculates a line integral.

n0=Incoming photon density

(x,y)

Detector

ith line integral

Ni

X-ray Sourceof area A

Ni = n0 A exp ∫i - dl = N0 exp ∫i - dl where A is area of detector and N0 = n0 A

The calculated line integral is ∫i dl = ln (N0/Ni)

Mean = ≈ ln (N0/Ni)2(measured variance) ≈ 1/Ni

Now we use these line integrals to form the projections g(R). These projections are processed with convolution back projection to make the image.

SNR = C /

M

(x,y) = ∑ g(R) * c(R) * (R-R’) ∆ i = 1

add projections convolution back projection where R’ = r cos ( - )

π

Since ∆ = π/M = M/π ∫ g(R) * c(R)* (R-R’) d 0

We can view this as:

û = h(r, ) ** (r, )estimate Entire system input image or

and recon process desired image

Discrete Backprojection over M projections

Recall = M/π ∫ g(R) * c(R)* (R-R') dH(p) = (M/π) (C() / ||) is system impulse response of CT system

C() is the convolution filter that compensates for the 1/|| weighting from the back projection operation

Let’s get a gain (DC) of 1. Find a C() to do this.We can consider C() = || a rect(/ 20). Find constant a

H(0) = (M/π) a a = π/M

If we set H(0) = 1, DC gain is 1.

Therefore, C(p) = (π/M) || rect(/ 2).

C(p)

p0

This makes sense – if we increase the number of angles M, we should attenuate the filter gain to get the same gain.

At this point, we have selected a filter for the convolution-back projection algorithm. It will not change the mean value of the CT image. So we just have to study the noise now.

The noise in each line integral is due to differing numbers of photons. The processes creating the difference are independent.

- different section of the tube, body paths, detector

What does this imply about the noise properties along the projection? The set of projections? What does this say for a plan of attack?

What effect does the convolution have on the noise?

Recall π

= M/π ∫ g(R) * c(R)* (R-R’) d 0

Then the variance at any pixel π

2 = M/π ∫ g

2(R)(R) * [c2(R)] d 0

variance of any one detector measurementTheorem described further at end of these notes.

Assume with n = average number of transmittedphotons per unit beam widthand h = width of beam

hng

12

convolution

π

2 = (M/π) (1/ (nh)) ∫ d ∫ c2 (R) dR = M/nh ∫ c2 (R) dR

0

Easier to evaluate in frequency domain. Using Parseval’s Rule ∞

2 = M/(nh) ∫ |C()|2 d

-∞

C(p)

p0

2/30

30

22

2/3

3

2

phMnC

CSNR

p

hMn

/M

dpM

phn

Mp

p2

22

0

0

The cutoff for our filter C() will be matched to the detector width w.

Let’s let p0 = K/w where K is a constant

Combine all the constants

n was defined over a continuous projection

Let N = nA = nwh = average number of photons per detector element.

2/3whMnCKSNR

In X-ray, SNR √N

For CT, there is an additional penalty. To see this, cut w in ½. What happens to SNR?

Why Due to convolution operationAnother way of looking at it, there is a penalty for oversampling the center or the Fourier space.

wMNCKSNR

First Order Statistics ( What we have studied) m = E[X] = x2 = E[(x - x)2]

Second Order statistic ( Important if we can’t assume independence)

RN (x1 , x2) = E [N(x1) N(x2)]

Supplementary Random Process Material

The following slides may be interesting to someone who has had some background in random processes. It will show how power spectral density analysis is useful in understanding imaging systems. No exams in the class will cover this material. This material is the foundation for the CT noise derivation.

Given an example random process whereN = cos (2π fx + )f is constant, and is uniformly distributed

0 2π

RN (x1 ,x2) = ∫ cos (2π fx1 + ) cos (2π fx2 + ) p() d

Use cos (a) cos (b) = 1/2 (cos (a - b) + cos (a + b))p() = 1/2π

RN (x1 ,x2) = 1/2π∫1/2[( cos (2π (fx1 + fx2) + 2 )+ (cos(2π(fx1 – fx2)] d

First term integrates to 0 across all = ½ cos(2π(fx1 – fx2))

If mN (x) = m for all x ( i.e. mean stays constant) and the random process is said to be wide sense stationary, then the autocorrelation statistic, RN(), depends only on the relative distance between two points ( time points, voxels, etc). RN ( ) is a measure of the information one can deduce about a random process if we know the value of the random process at another location.

RN (x1 ,x2) = RN ( ) RN ( ) = E [ N(x) N(x + )]

The value of the autocorrelation function at 0 represents average power of the random process. This is helpful in measuring noise power. RN (0) = E [ N2 (x)] Measure of average power of random process

Autocorrelation Statistic: RN ( )

Power spectral density of a Random Process NWe can’t take a meaningful Fourier transform of a random process. But a Fourier transform of RN() gives us its power spectrum. This is an indication of where the random processes power resides as a function of frequency.

SN (f) = ∫ RN ( ) e -i 2π f d

RN ( ) = F-1{SN (f)}= ∫ SN (f) e i 2π xf df

∞

E [N2 (x)] = Rx (0) = ∫ SN (f) df -∞

How do statistics change after random process is operated by a linear system?

HN Y

RY,N ( ) = E [Y(x + ) N(x)] = E [N(x) ∫ N(x + - ) h() d] ∞

= ∫ E [N(x) N(x + - )] h() d -∞

∞

= ∫ RN ( - ) h() d -∞

= RN ( ) * h ( )Cross-CorrelationWhat about the autocorrelation of the output Y? That is RY ( ) .

∞ ∞

E [Y(x) Y(x + )] = E [ ∫ h() N(x - ) d • ∫ h() N(x + - ) d ] -∞ -∞

But h(), h() are deterministic. ∞

= ∫ h() h() E[N(x - ) N(x + - )] d d -∞

∞

∫ h() h() RN ( + - ) d d-∞

∞

∫ h()• [h() * RN ( + )] d-∞

RN ( ) = h(-) * h( ) * RN ( )

h(-) H (-f) if real h( )H(-f) = H*(f)

Sy(f) = H*(f)• H(f) • Sx(f)

Sy(f) = |H(f)|2 • Sx(f)Average power

Ry(0) = ∫ |H(f)|2 • Sx(f) df