Topic 8: Tomographic Imaging

Transcription

1 Topic 8: Tomographic maging Aim This section covers tomographic imaging and some of its applications mainly related to medical imaging. Contents: ntroduction The adon Transform ourier nversion iltered back Projection an eam reconstruction ummary APPL PTC P PATMT of PYC Tomographic maging -1- emester 1

2 ntroduction Technique to allow determination of internal structure of object. z f(x,y,z) x y p(x) f(x,y) mage gives absorption characteristics of slice through object. APPL PTC P PATMT of PYC Tomographic maging -2- emester 1

3 ntroduction ull 3- structure built up from slices. f(x,y,z ) 0 z y f(x,y,z) f(x,y,z ) 1 x f(x,y,z 2) APPL PTC P PATMT of PYC Tomographic maging -3- emester 1

4 Applications Medical x-ray: (CT-can). X-ray analysis of human body, normally of head, but other sections possible. eological/tructural x-ray: X-ray analysis rock samples and fossils. nternal analysis of mechanical components (parts of aero engines) Astronomical mages: an-beam radio astronomy uses same techniques. ltra-sound Tomography: Three-dimensional images from ultra-sound scans. M and PT Medical maging: ses same mathematics of reconstructions from projections. APPL PTC P PATMT of PYC Tomographic maging -4- emester 1

5 Characterisation of ystem ystem assumes ACT, two possible geometries, 1) Collimated eam : lice of object illuminated by collimated beam with information collected by a linear detector array. f(x,y) Collimated x ray eam 1 ensor APPL PTC P PATMT of PYC Tomographic maging -5- emester 1

6 Characterisation of ystem 2) an eam : bject illuminated by an expanding beam with linear detector array giving a fan section of object. f(x,y) lit ource an X ray beam 1 Arc etector Assumptions valid for object with absorption variations that are large compared to wavelength. eg. X-ray of human body. APPL PTC P PATMT of PYC Tomographic maging -6- emester 1

7 Collimated eam eometry Take a thin collimated beam at angle θ of intensity 0, θ f(x,y) y Q (t) θ0 1 t s x 0 2- slide through object of absorption f(x, y), intensity detected at position t from ay s given by, Z ) Q θ (t) = 0 (1 f(x,y)ds so normalised absorption of the object is Z p θ (t) = ay ay f(x,y)ds APPL PTC P PATMT of PYC Tomographic maging -7- emester 1

8 Collimated eam eometry The ray in direction θ that intersects the detector at position t has equation, xcosθ+ysinθ = t o a line can be represented by so we have that Z Z p θ (t) = δ(xcosθ+ysinθ t) f(x,y)δ(xcosθ+ysinθ t)dxdy which can be formed for any θ by, either: 1. otating the bject (problem if object is a person!!). 2. otating the etector/ource system. p θ (t) is known as adon Transform of f(x,y). The problem: Collect adon Transform, we want to form f(x,y). APPL PTC P PATMT of PYC Tomographic maging -8- emester 1

9 ourier nversion Theorem Take case of θ = π 2, so Z Z p π/2 (t) = f(x,y)δ(y t)dxdy so, from shifting property, we have, Z p π/2 (t) = f(x,t)dx ourier transform of f(x,y) is Z Z (u,v) = f(x,y) exp( ı2π(ux+vy)) dxdy so with u = 0, we get Z [ Z ] (0,v) = f(x,y)dx exp( ı2πvy) dy so that Z (0,v) = p π/2 (t) exp( ı2πvt)dt APPL PTC P PATMT of PYC Tomographic maging -9- emester 1

10 ourier nversion Theorem Which says that ourier transform of the projection forms one-line in ourier space. f(x,y) p (t) π/2 u=0 (u,v) u APPL PTC P PATMT of PYC Tomographic maging -10- emester 1

11 ourier nversion Theorem xpress (u, v) in polar coordinates, u = wcosθ and v = wsinθ to give function (w,θ). Also define T of p θ (t) as Z P θ (w) = p θ (t) exp(ı2πwt)dt Projection at angle θ then f(t,s) is rotated version of f(x,y) with t = xcosθ+ysinθ and s = ycosθ xsinθ so that Z p θ (t) = f(t,s)ds giving, Z [ Z ] P θ (w) = f(t,s)ds exp( ı2πwt) dt which in (x, y) coordinates, gives, Z Z P θ (w) = f(x,y) exp( ı2πw(xcosθ+ysinθ)) dxdy APPL PTC P PATMT of PYC Tomographic maging -11- emester 1

12 ourier nversion Theorem which gives that, so by forming we can form P θ (w) = (w,θ) = (u,v) pθ(t) 0 θ π P θ (w) T of absorption f(x,y) APPL PTC P PATMT of PYC Tomographic maging -12- emester 1

13 n terms of iagrams Projection p θ (t) gives a strip at angle θ in the ourier Plane, θ f(x,y) y p (t) θ t s x θ (u,v) so if we take a whole series of projections at θ = 0 π then we can fill-in the whole ourier plane, (u,v), and then inverse ourier Transform to get f(x,y). APPL PTC P PATMT of PYC Tomographic maging -13- emester 1

14 nterpolation Problem Practical case : sample p θ (t) at t, & at θ. o in ourier space, sample on PLA of (u,v) θ t w = 1 and θ t To form f(x,y) must re-sample to a Cartesian grid in ourier space of u, v spacing. Major problem with this method, usually use 1. Zero rder (earest neighbour) 2. irst rder (Linear interpolation). Also range of high order techniques, which can help in reducing interpolation noise. APPL PTC P PATMT of PYC Tomographic maging -14- emester 1

15 Limited Angle econstruction f limited range of angles available, regions of ourier space will be undefined. imilar to effect of ourier filtering, giving a detected ourier space function, (u,v) = (u,v) (u,v) which results in an effective P of h(x,y) with the reconstruction given by g(x,y) = f(x,y) h(x,y) ifficult to reconstruct f(x, y) from limited angle data since parts of the ourier plane are completely missing. easonable results possible in radio astronomy where f(x,y) know to be isolated objects. CLA technique possible. APPL PTC P PATMT of PYC Tomographic maging -15- emester 1

16 iltered ack Projection ased on filtering the projections p θ (t) followed by a back projection. n polar coordinates the inverse T if (w,θ) is given by: f(x,y) = Z Z 2π 0 0 (w,θ)exp(ı2πw(xcosθ+ysinθ))wdwdθ ow if f(x,y) is real, then due to symmetry condition of the T, we have that, (w,θ+π) = ( w,θ) we can change the limits of the integration and get: f(x,y) = Z Z π Consider a rotated coordinate system at angle θ, so that we get the simpler expression that: 0 f(x,y) = where we not from previous that (w,θ)exp(ı2πw(xcosθ+ysinθ)) w dw dθ Z π 0 t = xcosθ+ysinθ [ Z ] P θ (w) w exp(ı2πwt)dw dθ P θ (w) = (w,θ) = (u,v) where P θ (w) is the 1- ourier Transform of the projection at angle θ. APPL PTC P PATMT of PYC Tomographic maging -16- emester 1

17 ow define a filtered projection of q θ (t) = iltered ack Projection Z P θ (w) w exp(ı2πwt)dw which, from the Convolution Theorem, we have have that: q θ (t) = p θ (t) h(t) where h(t) is the filter function h(t) = { w } 10 9 abs(x) igh Pass filtering the 1- projection. Then final image is given by f(x,y) = Z π 0 q θ (xcosθ+ysinθ)dθ APPL PTC P PATMT of PYC Tomographic maging -17- emester 1

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19 iltered ack Projection 1. rom each angle θ detect p θ (t). 2. or each p θ (t) form q θ (t) by 1- Convolution. 3. ack-project q θ (t) across reconstruction at angle θ. θ f(x,y) y p (t) θ t s x h(t) ^ f(x,y) θ q (t) θ Mathematics of both reconstruction techniques are identical, and get same problem of non-linear sampling in ourier space. Why use: Computationally simpler, no final 2- T needed. ot such as issue as it used to be. APPL PTC P PATMT of PYC Tomographic maging -18- emester 1

20 Typical esults imple CT section of sheeps neck. CT mage ourier Transform CT scan of human head showing injury: APPL PTC P PATMT of PYC Tomographic maging -19- emester 1

21 Typical esults ections of head formed into full Three-imensional model of skull. mages from APPL PTC P PATMT of PYC Tomographic maging -20- emester 1

22 Practical ystems To implement Collimated eam tomography system we need to produce a thin collimated beam. This in not possible with x-rays (no lenses available). Collimated eam ystem: se single detector, and collect one line projection at a time. X ray source ingle etector bject otate ystem APPL PTC P PATMT of PYC Tomographic maging -21- emester 1

23 Practical ystems riginal system, but 1. low to collect data. 2. Most x-rays not used (high does to patient) 3. Mechanically complex (linear and rotation scans) asis of system used for rock samples (rotate sample), but not practical for humans. APPL PTC P PATMT of PYC Tomographic maging -22- emester 1

24 an eam econstruction Practical system, illuminate with linepoint source and collect data from a linear array detector, hence an eam. quiangular ays quispaced etectors lit ource f(x,y) f(x,y) an x ray eam 1 Arc etector an x ray eam 1 Linear etector Two eometries: quiangular ays etectors equally spaced along arc of circle. ays at equal angles. quispaced etectors etectors spaced along linear array. ays T equally spaced. n practice, all modern machine are of the quiangluar ay geometry, which is the only one actually used. APPL PTC P PATMT of PYC Tomographic maging -23- emester 1

25 Practical ystems ange of commercial systems, all based on the equiangular rays geometry, schematically piral system by PCK, which has a 360 ring of detectors and three moving x-ray sources, ody move through three spiral scans. APPL PTC P PATMT of PYC Tomographic maging -24- emester 1

26 an beam econstruction eomtery of fan system is y f(x,y) γ M γ β r ( γ ) β x or a range of β in range 0 2π we collect r β (γ) where γ in the range γ M γ M. APPL PTC P PATMT of PYC Tomographic maging -25- emester 1

27 Angular ackprojection The reconstrcuion can be formulated as a back-projection where the colelcted one-dimensional scan is: 1. caled by a cos() of the fan-beam angle. 2. ne-dimensional convolution with a filter function. 3. ack-projected along the rays of the equiangulat fan. f(x,y) ^ y β x s ( γ ) β o allowing the image to be formed by a series of one-dimensinal ourier transforms. APPL PTC P PATMT of PYC Tomographic maging -26- emester 1

28 ther Techniques ay orting: Alternative technique to collect r β (γ) and then sort the rays by finding the corresponding (t,θ) and thus forming p θ (t) (collimated beam), projection. f(x,y) can then be formed by the conventional filtered back projection. AT: Algebraic econstruction Technique that is based on iterative exact inversion of the adon Transform without using ourier techniques. Computational very slow, but gives good smooth images. All detailed in Computer Tomography, econstruction from Projections by erman, Academic Press Also computer simulation package called AK, which simulated both projections and reconstructions. APPL PTC P PATMT of PYC Tomographic maging -27- emester 1

29 ther Tomographic ystems Magnetic esonance maging: (M) ses nuclear magnetic resonance of the proton at points in the body. The imaged point being selected by varying magnetic and field gradients. ffectively measures water content. (a) orizontal can (b) Vertical scan Clinically very save (no ionising radiation), and very good for soft tissue (brain or abdominal), but will not work with bone. APPL PTC P PATMT of PYC Tomographic maging -28- emester 1

30 ther Tomographic ystems Positron mission Tomography: (PT) sed injected radio isotope (usually 15 ), which decays by positron emission (which annihilates with an electron to give 2 γ-rays). etect with a ring detector + e + e annihlation γ etector ing γ A p ecay (a) PT system (b) Typical image ives line on which decay occurs. econstruction similar to tomography. Can be made to work in real-time to give live image. APPL PTC P PATMT of PYC Tomographic maging -29- emester 1

31 ummary asic concepts of tomographic imaging. Collimated beam projection. ourier inversion theorem nterpolation problems and issues. iltered back projection for collimated beam reconstruction. Pratcical systems and problems. an beam reconstrcution. ther tomographic imaging systems. APPL PTC P PATMT of PYC Tomographic maging -30- emester 1