Computed Tomography Notes, Part 1. The equation that governs the image intensity in projection imaging is:

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1 Noll 3 CT Notes : Page Compute Tomograph Notes Part Challenges with Projection X-ra Sstems The equation that governs the image intensit in projection imaging is: z I I ep µ z Projection -ra sstems are the most inepensive an wiesprea meical imaging evice but there are some major rawbacks: There is no epth z inormation in the images we can t tell where along a particular line where a lesion is locate. Lack o contrast large changes in attenuation coeicient ma results in ver small changes in image intensit: Solution: Compute Tomograph see Macovski pp. 3-4 Deinition: Tomograph is the generation o cross sectional images o anatom or structure. irst we will reuce the imensionalit o the problem through collimation o the -ra source to a gle slice through the object choose a gle z location:

2 Noll 3 CT Notes : Page The intensit along this D row o etectors is now: We now eine a new unction: I I ep µ I g ln µ I where g is the projection through some unknown unction that we wish to etermine. We can also escribe g as the line integral through in the irection: g Another wa o writing the line integral is: g δ where eines a line along with the integration is to occur this is the onl place where the elta unction is non-zero. We can now escribe the line integral at an arbitrar angle :

3 Noll 3 CT Notes : Page 3 with the ollowing epression: g δ cos This collection o projections g is known as the aon transorm o. The Central Section Theorem projection-slice theorem Perhaps the most important theorem in compute tomograph is the central section theorem which sas: The D T o a projection g is the D T o evaluate at angle. Taking the D T o the projection we get: G D { g } δ cos ep i cos ep i cos Observing that the D T o is: ep i u v ep i u v an that uv in polar coorinates is cos we can see that: G u v u cos v

4 Noll 3 CT Notes : Page 4 To make an image then we can acquire projections at man ierent angles over to ill in the uv space an then inverse D T to get the input image : u vep i u v uv G ep i cos Eample Suppose we have an object that has the same projection at all angles: g c We can see that that: an thereore { c } rect / D circ Jr r jinc r r

5 Noll 3 CT Notes : Page 5 Thus i we project through a circularl smmetric jinc unction we will get a c unction: Eample We can also use the central section theorem to etermine projections through a known object. or eample suppose we wante to know the projection through rectrect at an angle o /4. u v c uc v G c cos c c / c / c / g r { c / } tri r D

6 Noll 3 CT Notes : Page 6 Sinograms In general we have ata or g or man ierent angles that can be place into a big matri that we call a ogram. or eample let s take some point object at e.g. δ - then: g δ an letting then: δ cos δ g δ cos δ cos a elta unction locate at cos. That is a point traces out a usoi in the - space an thus the name ogram. This is also known as aon space. or a more comple object.

7 Noll 3 CT Notes : Page 7 In the ogram the maimum eviation escribes an object s istance rom the origin an the point o peak eviation escribes the angular location o object. The three objects above rom smallest to largest are locate at r 3 /4 an - or. inall observe the smmetr: p r p r.

8 Noll 3 CT Notes : Page 8 Methos or Image econstruction Image reconstruction rom a set o projections is a classic inverse problem. This a particularl rich problem in that there are man ierent was to approach this problem an we will present several o these below.. Direct ourier Interpolation Metho This metho makes irect use o the central section theorem. The steps in the image reconstruction are:. D T each o the projections: D { g } G. Interpolate to ˆ u v polar to rectangular coorinates e.g. ou coul use Matlab unctions interp or griata 3. Inverse D T. Backprojection-iltering Backprojection means that we smear the projection ata back across the object space. The backprojection operator or a gle projection looks like this: b g δ cos

9 Noll 3 CT Notes : Page 9 an the total backprojecte image is the integral sum o this over all angles: δ g b b cos Now consier that: { } i g D ep then the backprojecte image can be written as: δ i i b cos ep ep cos This is nearl the inverse T ormulation o in polar coorinates but two changes are neee: a. limits o integration shoul be an an b. we nee or an integration in polar coorinates We can aress the irst issue b recognizing that - an we can aress the secon issue multipling an iviing b. r i D D b ** ** cos ep This sas that the backprojecte image is equal to the esire image convolve with a /r blurring unction.

10 Noll 3 CT Notes : Page Up until this point we ve onl one the backprojection. In orer to get the inal image we nee to uno this blurring unction. Thus the steps in backprojection-ilter metho are. Backproject all projections g to get b.. orwar D T to get

11 Noll 3 CT Notes : Page 3. ilter with or to get. This is a cone -like weighting H u v v u applie to uv 4. Inverse D T to get. One isavantage to this metho is that it is oten necessar to backproject across an etene matri because the blurre image etens beon the original object ue to the long etent o the /r unction. In aition the eblurring iltering one in the ourier omain will lea to artiact i the object isn t pae out. 3. Direct ourier Superposition an iltering Metho This metho makes use o the act that backprojection is mathematicall equivalent to aing a line to the ourier ata. We show this b eamining looking at the backprojection operator in a rotate coorinate sstem where: r cos cos r The backprojection operator is:

12 Noll 3 CT Notes : Page b b r g r δ cos g g δ r r The D T in the rotate rame is: B ur vr G ur δ vr an substituting back to the stanar coorinate sstem we get: B u v G u cos v δ u vcos This is the T o the projection G positione along a elta line at angle. The steps in this metho are then:. D T each projection to get G.. Place ata irectl into ourier matri a or superimpose each G. 3. ilter with or to get. 4. Inverse D T to get. 4. iltere Backprojection Metho In this metho we reverse the orer o backprojection an iltering. The steps are:. ilter the backprojection with a ilter. This is sometimes calle a ramp ilter. a. ourier metho: b. Convolution metho g' D g'. Backproject or all angles to get. g { { g } D where c * c D { } Putting it all together we get:

13 Noll 3 CT Notes : Page 3 { } { } { } δ δ δ i i g D D D cos ep cos ep cos cos ˆ an changing the limits o integration to an we get: { } cos ep ˆ i D

14 Noll 3 CT Notes : Page 4 One eample: An then backprojecting Now let s look at the convolution unction c. c { }

15 Noll 3 CT Notes : Page 5 oes not eist. However we can in the T o a variet o unctions that approach in the limit. or eample: lim c ε { ep ε } lim ε 4 ε ε 4 Thus - or small c will approach /ε - or large c will approach / Above the unction ep ε clips o the high spatial requenc parts o with an approimate cuto requenc o. There are numerous other unctions that can o that. or eample we coul clip o the high spatial requencies ug a rect unction.

16 Noll 3 CT Notes : Page 6 tr rect rect i C c c c This ilter has substantial ringing artiact. One can also appl a Gaussian or Hanning ilter:

17 Noll 3 CT Notes : Page 7 rect ep C C The ilter resulting rom a Hanning apoize ramp ilter is: 5. Algebraic econstruction Technique AT This is like regular backprojection but this metho uses iterative corrections. There are numerous variants but I will iscuss the simplest aitive AT. Here we let g i be the measure projections an ij q be the image at iteration q. Then: N g j q ij i q ij q ij

18 Noll 3 CT Notes : Page 8 Here s an eample or a noise ree object. Consier the object with 4 piel values an 6 pieces o projection inormation. We initialize the ata to all zeros e.g.. Looking at the projections rom top to bottom we can in the piel values at q : 5.5; 5.5; Now looking at the let-right projections: ; ; Now we look at the iagonal projections: ; ;

19 Noll 3 CT Notes : Page 9 which we can see is consistent with all o the projections. While this proceure ma look ine in the presence o noise that is the projections aren t eactl consistent this metho converges ver slowl an in some cases not at all. Usuall one goes thought the entire set o projections multiple times.