The Radon Transform. Definiion For a given funcion f defined in he plane, which may represen, for insance, he aenuaion-coefficien funcion in a cross secion of a sample, he fundamenal quesion of image reconsrucion calls on us o consider he value of he inegral of f along a ypical line l,θ. For each pair of values of and θ, we will inegrae f along a differen line. Thus, we really have a new funcion on our hands, where he inpus are he values of and θ and he oupu is he value of he inegral of f along he corresponding line l,θ. Bu even more is going on han ha because we also wish o apply his process o a whole variey of funcions f. So really we sar by selecing a funcion f. Then, once f has been seleced, we ge a corresponding funcion of and θ. Schemaically, inpu f oupu (, θ) l,θ fds. This muli-sep process is called he Radon ransform, named for he Ausrian mahemaician Johann Karl Augus Radon (887 956) who sudied is properies. For he inpu f,wedenoebyr( f ) he corresponding funcion of and θ showninhe schemaic. Tha is, we make he following definiion. Definiion.. For a given funcion f, whose domain is he plane, he Radon ransform of f is defined, for each pair of real numbers (, θ),by R f (, θ) := l,θ fds= f ( cos(θ) ssin(θ), sin(θ)+scos(θ))ds. (.) T.G. Feeman, The Mahemaics of Medical Imaging, Springer Undergraduae Texs in Mahemaics and Technology, DOI.7/978--387-97-, c Springer Science+Business Media, LLC
The Radon Transform A few immediae observaions are ha (i) boh f and R f are funcions; (ii) f is a funcion of he Caresian coordinaes x and y while R f is a funcion of he polar coordinaes and θ; (iii) R f (, θ) is a number (he value of an inegral); (iv) in he inegral on he righ, he variable of inegraion is s, while he values of and θ are preseleced and so should be reaed as consans when evaluaing he inegral.. Examples Example.. As an example, le f be he funcion defined by { x f (x, y) := + y if x + y, if x + y >. (.) The graph of f is a cone, shown in Figure.. We have already observed ha, on he line l,θ,wehave x + y =( cos(θ) ssin(θ)) +( sin(θ)+scos(θ)) = + s. I follows ha, on he line l,θ, he funcion f is given by { f ( cos(θ) ssin(θ), sin(θ)+scos(θ)) := + s if + s, if + s >. (.3) From his, we see ha he value of R f (, θ) depends only on and no on θ and ha R f (, θ) = whenever >. For a fixed value of such ha, he condiion + s will be saisfied provided ha s. Thus, for any value of θ and for such ha, we have f ( cos(θ) ssin(θ), sin(θ)+scos(θ)) { := + s if s, (.4) oherwise; whence ( ) fds= l,θ s= + s ds. (.5) This inegral requires a rigonomeric subsiuion for is evaluaion. Sparing he deails for now, we have ( ( ) s= + s ds = + ) ln. (.6)
. Examples.8.6.4..5.5 3 Fig... The figure shows he cone defined in (.) and he graph of is Radon ransform for any fixed value of θ. In conclusion, we have shown ha he Radon ransform of his funcion f is given by ln + if, R f (, θ ) := (.7) if >. In his case, where R f is independen of θ, he value of R f (, θ ) corresponds o he area under he verical cross-secion of he cone defined by z = f (, y). Several of hese cross-secions are visible in Figure.. Example.3. Consider a crescen-shaped region inside he disc x + y = /4 and ouside he disc (x /8) + y = 9/64. Assign densiy o poins in he crescen, densiy / o poins inside he smaller disc, and densiy o poins ouside he larger disc. In oher words, he aenuaion funcion is if x + y /4 and (x /8) + y > 9/64; (.8) A(, θ ) :=.5 if (x /8) + y 9/64; if x + y > /4..6.4. x..4.6.6.5 3.4.5..5 y.5..4.5.6 θ Fig... The figure shows he graph of he aenuaion funcion A defined in (.8), alongside a sinogram of is Radon ransform RA(, θ ).
4 The Radon Transform Figure. shows he graph of his aenuaion funcion alongside a graph of is Radon ransform in he (, θ) plane. Such a graph is called a sinogram and essenially depics all of he daa generaed by he X-ray emission/deecion machine for he given slice of he sample. The funcion values are shown in greyscale, wih whie corresponding o, neural grey o.5, and black o. Figure.3 shows graphs of he Radon ransform for he angles θ = and θ = π/3..8.6.4.6.4...4...4.4...4 Fig..3. For he funcion A defined in (.8), he figure shows graphs of is Radon ransform RA(, ) (lef) and RA(, π/3) (righ), for / /..3 Lineariy Suppose ha wo funcions f and g are boh defined in he plane. Then so is he funcion f + g. Since he inegral of a sum of wo funcions is equal o he sum of he inegrals of he funcions separaely, i follows ha we ge, for every choice of and θ, R( f + g)(, θ) = = = + ( f + g)( cos(θ) ssin(θ), sin(θ)+scos(θ))ds { f ( cos(θ) ssin(θ), sin(θ)+scos(θ)) + g( cos(θ) ssin(θ), sin(θ)+scos(θ))}ds f ( cos(θ) ssin(θ), sin(θ)+scos(θ))ds = R f (, θ)+rg(, θ). In oher words, R( f + g)=r f + Rg as funcions. g( cos(θ) ssin(θ), sin(θ)+scos(θ))ds
.4 Phanoms 5 Similarly, when a funcion is muliplied by a consan and hen inegraed, he resul is he same as if he funcion were inegraed firs and hen ha value muliplied by he consan; i.e., α f = α f. In he conex of he Radon ransform, his means ha R(α f )=αr f. We now have proven he following proposiion. Proposiion.4. For wo funcions f and g and any consans α and β, R(α f + βg)=αr f + βrg. (.9) In he language of linear algebra, we say ha he Radon ransform is a linear ransformaion; ha is, he Radon ransform R maps a linear combinaion of funcions o he same linear combinaion of he Radon ransforms of he funcions separaely. We also express his propery by saying ha R preserves linear combinaions. Example.5. For a fixed value of R,define { if x F R (x, y) := + y R, oherwise. From he exercises, we know ha R(F R )(, θ)= { R if R, if > R. Now consider he funcion.5 if x + y.5, f (x, y) :=. if.5 < x + y., oherwise. Tha is, f = F (.5)F.5. By (.9), i follows ha R f (, θ) =R(F )(, θ) (.5)R(F.5 )(, θ) (.5) if.5, = if (.5) <, if >..4 Phanoms The fundamenal quesion of image reconsrucion asks wheher a picure of an aenuaion-coefficien funcion can be generaed from he values of he Radon ransform of
6 The Radon Transform ha funcion. We will see evenually ha he answer is Yes if all values of he Radon ransform are available, bu only Approximaely yes in pracice, where only a finie se of values of he Radon ransform are measured by a scanning machine. Consequenly, he nice soluion ha works in he presence of full informaion will spliner ino a variey of approximaion mehods ha can be implemened when only parial informaion is a hand. One mehod for esing he accuracy of a paricular image reconsrucion algorihm, or for comparing algorihms, is simply o apply each algorihm o daa aken from an acual human subjec. The drawback of his approach is ha usually we don know exacly wha we ough o see in he reconsruced image. Tha is wha we are rying o find ou by creaing an image in he firs place. Bu wihou knowing wha he real daa are, here is no way o deermine he accuracy of any paricular image. To ge around his, we can apply algorihms o daa aken from a physical objec whose inernal srucure is known. Tha way, we know wha he reconsruced image ough o look like and we can recognize inaccuracies in a given algorihm or idenify dispariies beween differen algorihms. Noneheless, his approach can be misleading. Alhough he inernal srucure of he objec is known, here may be errors in he daa ha were colleced o represen he objec. In urn, hese errors may lead o errors in he reconsruced image. We will no be able o disinguish hese flaws from errors caused by he algorihm iself. To resolve his dilemma, Shepp and Logan (see [4]) inroduced he concep of a mahemaical phanom. This is a simulaed objec whose srucure is compleely defined by mahemaical formulas. Thus, no errors occur in collecing he daa from he objec. When an algorihm is applied o produce a reconsruced image of he phanom, all inaccuracies are due o he algorihm. This makes i possible o compare differen algorihms in a meaningful way. Since measuremen of he Radon ransform of an objec forms he basis for creaing a CT image of he objec, i makes sense o use phanoms for which he Radon ransform is known exacly. We can hen es a proposed algorihm by seeing how well i handles he daa from such a phanom. For example, we have compued he Radon ransform of a circular disc of consan densiy cenered a he origin. Using he lineariy of R, we can compue he Radon ransform of any collecion of nesed discs, each having consan densiy, cenered a he origin. Such a phanom is oo simplisic, hough, o serve as a useful model for any serious applicaion. An acual slice of brain issue is likely o include ineresing feaures in a variey of shapes and sizes locaed in all regions wihin he slice, no only near he cener. To address hese concerns, Shepp and Logan developed he phanom shown in Figure.4. The Shepp Logan phanom is composed of eleven ellipses of various sizes, eccenriciies, and locaions. (The MATLAB R version shown here does no include an ellipse ha models a blood clo in he lower righ near he boundary.) The densiies are assigned so ha hey fall ino he ranges ypically encounered in a clinical seing. Since
.5 The domain of R 7 5 5 5 5 5 5 θ Fig..4. The Shepp Logan phanom is used as a mahemaical facsimile of a brain for esing image reconsrucion algorihms. This version of he phanom is sored in MATLAB R. On he righ is a sinogram of he phanom s Radon ransform, wih θ in incremens of π/8 ( ). we can compue exacly he Radon ransform of any ellipse, he Shepp Logan phanom has proven o be a reliable model on which o es reconsrucion algorihms..5 The domain of R As we can see from he definiion (.), he Radon ransform R f of a funcion f is defined provided ha he inegral of f along l,θ exiss for every pair of values of and θ. Each of hese inegrals is osensibly an improper inegral evaluaed on an infinie inerval. Thus, in general, he funcion f mus be inegrable along every such line, as discussed in greaer deail in Appendix A. In he conex of medical imaging, he funcion f represens he densiy or aenuaion-coefficien funcion of a slice of whaever maerial is being imaged. Thus, he funcion has compac suppor, meaning ha here is some finie disc ouside of which he funcion has he value. In his case, he improper inegrals l,θ fdsbecome regular inegrals over finie inervals. The only requiremen, hen, for he exisence of R f is ha f be inegrable over he finie disc on which i is suppored. This will be he case, for insance, if f is piecewise coninuous on he disc. For a wealh of informaion abou he Radon ransform and is generalizaions, as well as an exensive lis of references on his opic, see he monograph [].
8 The Radon Transform.6 Exercises ( ).. Evaluae he inegral s= + s ds from (.5)... Use a compuer algebra sysem o generae picures such as hose in Figures. and.. { if x and y,.3. Consider he funcion f (x, y) := (Tha is, f has he oherwise. value inside he square where x and y, and he value ouside his square.) (a) Skech he graph of he funcion R f (,), he Radon ransform of f corresponding o he angle θ =. (b) Skech he graph of he funcion R f (, π/4), he Radon ransform of f corresponding o he angle θ = π/4.
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