What are inside the gantry? Schematic Representation o f the Scanning Geometry of a CT System Scanner without covers Scanner with covers SourceSource Detector Generations source detector Advantages Disadvantages collimation collimation Detector movement 1st Gen.

single Pencil beam Fanbeamlet single no multiple yes many no 2nd Gen.

single 3rd Gen. single 4th Gen. single Fanbeam Stationary ring no multiple

Fanbeam Stationary ring no 5th Gen. 6th Gen. single 7th Gen. single 8th Gen. single Fanbeam

Fanbeam Narrow many Multiple cone- beam arrays wide FPD cone- beam yes yes No Trans. scatter +Rotate

sTrans. Faster +Rotate than 1G s Faster Rotates than 2G together Source Higher Rotates efficiency only than 3G No Ultrafast movement for cardiac 3rdGen.+

bed trans. 3rdGen.+ bed trans. no 3rd Gen. faster 3D imaging slow Low efficiency High cost and Low efficiency high scatter

high cost higher cost faster 3D imaging higher cost Large 3D Relatively slow What is displayed in CT images? T water

CT# 1000HU water Water: 0HU Air: -1000HU Typical medical scanner display: [-1024HU,+3071HU], Range: 12 4096 2 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. +3071 255 W 0 CT # ( L W / 2) Displayed Gray scale I max W I max

L 0 -1024 CT # L W / 2 L W / 2 CT # L W / 2 CT # L W / 2 ( E , Z ) Mass density Mass attenuation coefficient: attenuation per electron or per gram Reminder:

3 E 3~ 4 Z 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 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. KrestelImaging Systems for Medical Diagnosis 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-ray Source of area A (x,y) ith line integral Ni

Detector 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 / Discrete Backprojection over M projections M (x,y) = g(R) * c(R) * (R-R)

i=1 add projections convolution back projection where R = r cos ( - ) Since = /M = M/ g(R) * c(R)* (R-R) d 0 We can view this as: = estimate h(r, ) Entire system

and recon process ** (r, ) input image or desired image Recall = M/ g(R) * c(R)* (R-R') d) d H(p) = (M/) (C() / ||) is system impulse response of CT system C() is the convolution filter that compensates for the 1/|| weighting from the back projection operation Lets 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/) a a = /M If we set H(0) = 1, DC gain is 1. Therefore, C(p) = (/M) || rect(/ 2).

p0 This makes sense if we increase the number of angles M, we should attenuate the filter gain to get the same gain. C(p) 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

= M/ g(R) * c(R)* (R-R) d 0 convolution Then the variance at any pixel 2 = M/ g2(R)(R) * [c2(R)] d 0 variance of any one detector measurement Theorem described further at end of these notes. 1 Assume n h with n = average number of transmitted 2

g photons per unit beam width and h = width of beam 2 = (M/) (1/ (nh)) d c2 (R) dR = M/nh c2 (R) dR 0 Easier to evaluate in frequency domain. Using Parsevals Rule p 2 M 2 p

dp 2 2 2 = M/(nh) |C()| d nh p M 0 0 - /M C(p) p0 2 2 p03

n hM 3 C 3 / 2n hM 3 / 2 SNR C p0 2 The cutoff for our filter C() will be matched to the detector width w. Lets let p0 = K/w where K is a constant 3/ 2 SNR K C n hM w Combine all the constants n was defined over a continuous projection Let N = nA = nwh = average number of photons per detector element.

SNR K C N M w In X-ray, SNR NN For CT, there is an additional penalty. To see this, cut w in . What happens to SNR? Why Due to convolution operation Another way of looking at it, there is a penalty for oversampling the center or the Fourier space. 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. First Order Statistics ( What we have studied) m = E[X] = x

2 = E[(x - x)2] Second Order statistic ( Important if we cant assume independence) RN (x1 , x2) = E [N(x1) N(x2)] Given an example random process where N = cos (2 fx + ) f is constant, and is uniformly distributed 0 2 RN (x1 ,x2) = cos (2 fx1 + ) cos (2 fx2 + ) p() d Use cos (a) cos (b) = 1/2 (cos (a - b) + cos (a + b)) p() = 1/2 RN (x1 ,x2) = 1/21/2[( cos (2 (fx1 + fx2) + 2 ) + (cos(2(fx1 fx2)] d First term integrates to 0 across all = cos(2(fx1 fx2)) Autocorrelation Statistic: RN ( )

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 Power spectral density of a Random Process N We cant 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 2 f d

RN ( ) = F-1{SN (f)} = SN (f) e i 2 xf df E [N2 (x)] = Rx (0) = SN (f) df - How do statistics change after random process is operated by a linear system? N Y H 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-Correlation What 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() RN ( + - ) d d - h() [h() * RN ( + )] d - RN ( ) = h(-) * h( ) * RN ( ) h(-) H (-f) H(-f) = H*(f) Sy(f) = H*(f) H(f) Sx(f) if real h( ) power

2 SAverage (f) = |H(f)| Sx(f) y Ry(0) = |H(f)|2 Sx(f) df