The Linogram Algorithm And Direct Fourier Method With . - DiVA Portal
Institutionen för medicin och vårdAvdelningen för radiofysikHälsouniversitetetThe Linogram Algorithm and DirectFourier Method with LinogramsPaul R. EdholmDepartment of Medicine and CareRadio PhysicsFaculty of Health Sciences
Series: Report / Institutionen för radiologi, Universitetet i Linköping; 65ISSN: 1102-1799ISRN: LIU-RAD-R-065Publishing year: 1991 The Author(s)
1991-01-22SN 0348-7679The Linogram Algorithm and Direct FourierMethod with LinogramsPaul R. EdholmAvdelningen för diagnostisk radiologiUniversitetet i LinköpingREPORTULI-RAD-R-065
1The Linogram Algorithm and Direct Fourier Method with LinogramsPaul R. EdholmThe conventionai map of projection data will here be called the sinogrammap. Among the algorithms used with this map of data there are two of maininterest for this paper: the Filtered Back Projection (FBP) and the DirectFourier Method (DFM). FBP is the most popular algorithm used in commercialeT machines although it is computationally expensive compared to the DFM.The reason for its popularity is that FBP gives better pictures than theDFM uniess the lat ter is used with very careful interpolations.In this paper another map of projection data will be presented, here calledthe linogram map. The FBP may be implemented with this map in aparticularly simple way, here called the Linogram Algorithm (LA). In thisthe back projection, which is so computationally expensive with thesinogram map, can be reduced to a computationally inexpensive series ofFFT's.The DFM may also be implemented with the linogram map in a particular wayso that it turns out that the two methods FBP and the DFM arecomputationally identical.The LA has been described in [1-5]. In [4] a definition of linograms wasintroduced, which is slightly different from the definition given in [1-3],and leads to a somewhat simplified mathematical description of thealgorithm.This text is an attempt to describe the Linogram Algorithm based on some ofthe ideas in [4] and also such that the mathematical description is moresimilar to the actual digital implementation.The DFM method with linograms will also be presented. The two methods,which are different conceptually, have a very similar structure. In boththe image is reconstructed without any interpolations.Geometry. Let the object to be reconstructed be the function f: R2such thatf(x,y) { O for lxi 1 or lyl 1 O otherwise R,(1)
2In image reconstruction from projections it is necessary to specify lineintegrals through the plane of the object, i.e. projecting rays in allpossible directions. A line through the object may be specified by twocoordinates. Based on the geometry of the detector array, the usual way isto let the first coordinate de fine the length of the normal to the linefrom the origin, and the second coordinate the angle from the x-axis tothis normal in the counterclockwise direction. If these coordinates are(r p ' ep)' then the line through the (x,y)-plane has the equation(2)All possible lines in the (x,y)-plane may be specified in this way. In theplane of the rectangular coordinate system (r, e), each point defines aline through the (x, y)-plane, (Fig 1) which may be writtenx r/(cose) - y tane(3)The point (x p ' yp) in the (x,y)-plane defines the sinusoidr xp cose yp sine(4)in the (r,e)-plane. If we now have chosen (x p ,y3 so that this point is onthe line (2) in the (x,y)-plane, the sinusoid (4) must pass through thepoint (rp,ep)' which represents all points on (2) (Fig 2). The other pointson this sinusoid define all other lines in the (x,y)-plane passing throughthe point (xp,yp)' This is seen by inser ting (4) in (3)orx (xp cosex xp yp sine)/cose - y tane (Yp - y) tana(5)e varies through the range n, the expression (5) describes all linesin the (x,y)-plane passing through (xp,yp)' henA line through the (x,y)-plane may be specified by a number of ways. Is itpossible to find another coordinate system, say (u, v), in which, asbefore, every point defines a line through the (x, y)-plane and also suchthat the set of points defining all lines through a specific point (x p' yp)instead of a sinusoid, forms a straight line in the (u, v)-plane?
3Define a line in the x,y-plane aswhere (u p , v p ) is a specific point in the (u, v)-plane (Fig 3). The pointsin the (u,v)-plane defines all possible lines through the (x,y)-planex u - yv(7)Assume that our previous point, (x p , y p ), is a point somewhere on the line(6), thenu xp ypv(8)is a straight line through the (u, v)-plane (Fig 4). As we have chosen(xp,yp) to be on the line (6) in the (x,y)-plane, the line (8) in the(u,v)-plane must pass through (up' v p )' which represents all points on(6). The other points on this line define all other lines in the (x, y)plane passing through (x p ' yp)'This may be seen by inserting (8) in (7).x or(9)When v varies from - to , the expression (9) describes all lines in the(x,y)-plane passing through (x p ' yp)'There is a kind of symmetry between the (x,y) and the (u,v) systems as isseen by (6) and (8). A point in one system defines a line in the other.Sinograms and Linograms. When projection data, in the form of lineintegrals through a function f(x,y), are mapped in the (r,e)-coordinates,where data for a point in the object form the sinusoid (4), this map isusually called a sinogram [6], and it may be denoted p(r,e) and defined asp(r,e) J f(rcose - tsine, rsine tcose)dt(10)
4If the projection data instead are mapped in the (u,v) coordinates wheredata for a point in the object form the straight line (8), this map will becalled a linogram analogous to the term sinogram [l].Why linograms instead of sinograms? Well, in the most popular algorithm,called filtered back projection, the projection data are first subjected toa filtration of their spatial frequencies and then "back projected". Thisis a somewhat unprecise expression. The mathematical definition is thateach point in the object is reconstructed by making an integration of thefiltered projection data representing all line integrals through the pointin question.In the sinogram case we then have to do an integration along a sinusoid foreach point, but in the linogram case this integration is along a straightline. This may be a simpler operation, and as we will see later, the use oflinograms allows us to replace the computationally expensive backprojection by a series of fast Fourier transforms.The relation between the coordinates for the sinogram (r,a) and thelinogram (u,v) arer {a uOJ(l v Z )arctan v,roru cosav tanaLinograms. A linogram mapping all lines through f(x,y) has to have arangein v from - to , corresponding to a range of n for e in the sinogram.In order to have a finite range for v we are going to use two finitelinograms.The first will be called go (u, v) and contains projection data of f(x, y)for the range - 1 v 1.The second will be called g. (u, v) and is achieved by first rotating theobject c10ckwise the angle and then taking projections of it for therange - 1 v 1, exactly as for go (u, v).From the two linograms two partial images will be reconstructed, f (x, y)oand f.(x, y). The final image is formed by first rotating f.(x,y)counterclockwise the angle n/2 and then adding it to fo(x,y).
5Let Q define a counterclockwise rotation the angle n/2 of a functionf:R' R around the origin so that1[Q f] (x, y) f (-y, x),[Q0f] (x, y) f (x, y) and(11)I[Qf] (x, y) f (y, -x).For any 2-dimensional function ! we use, respectively, F[I]!, F[']! and F,!to denote the Fourier transform of ! with respect to its first variable,the Fourier transform of ! with respect to its second variable, and thetwo-dimensional Fourier trans form of !. Variables in the spatial domain arerepresented by small letters and in the Fourier domain by capital letters.In the following expressions, k is an index assuming the two values Oand 1. The range of v is limited to -1 v 1. The two linograms g and gloare then defined bya gk (u,v)J[Q-kf] (u-yv,y) dyk O,l;-'"-1 v 1.(12).The RHS represents line integrals through f(x,y), where x has been replacedby (u-yv) according to (7).The filter. The linograms defined by (12) cannot be used directly forreconstruction, they have first to be filtered.We will now try to find how to make a proper filtration of the spatialfrequencies of the projection data in the linogram. As a preliminary wewill first make an l-dimensional Fourier trans form of gk in the firstvariable[F[1]gk 1 (U, v) r.gk (u, v) e- 12n Uu du-'"(13)Insert (12) in (13)ri r[F[I]g ] (U, v) k Q -k f](u-yv, y) dy } e -i2n Uu du. (14)
6Change variabel from u to xX {uu - yv xdxdu yv(15) 1The LHS of (16) is a l-D Fourier trans form of gk in the first variable andthe RHS of (16) represents a 2-D Fourier trans form of f(x,y). The exponentfor contains the expression U(x yv) which is the inner product of the twovectors (U, Uv) and (x,y). For a constant v, the RHS of (15) is thus anexpression for the lineX U{ Y Uv,(17)in the 2-dimensional Fourier trans form of !(x,y).This means that(18)This is the "projection slice theorem" stated in linogram form.For the moment we leave (18) and start at another end, namely theselfevident fact thatf(x,y) '"JJ[F f]-'"2i2Jl(x,y) (X, Y)(X,Y) edX dY,(19)i.e. if we make an inverse Fourier trans form of the Fourier transform off(x,y), of course we regain f(x,y).Change from coordinates (X,Y) to polar coordinates (R,e) in the Fourierdomain. The usual integration in e from O to 2Jl is replaced by theequivalent range from -Jl/4 to 7Jl/4.f(x,y) 7/4Jl'"J-Jl/4 J o[F 2 f] (R cose, R sine)ei2Jl(x,y) (Rcose,Rsine)IRI d R de.(20)
7If R in (20) with the integration between O and 00 is permitted to benegative and is integrated from 00 to 00 it is sufficient with anintegration of 0 from O to n, that is in (20)is replaced by4nJr3/- 4Divide the angular range for 0 in the two ranges.and(21)For each range we will make an inverse transformation of F,i, eachincomplete and resulting in a partial image fk(x,y). For the second rangein (21) we will rotate f(x,y) clockwise using the operator O defined in(11). The two partial images will be denoted f (x,y) and f,(x,y) and areodefined by[F,O -k fl (Rcos0; Rsin0)ei2n(x,y)(Rcos0, Rsin0)IRI d R de,(22)When the two partial images are added together the result will be acomplete image of f(x,y).1 k n/4f(x,y) E O-n/4k.OJ J00i2n(x,y)(Rcose,Rsine)k[F,O- fl (Rcose, Rsine)eIRld R de.(23)00or,(24)Note that the two fk(x,y); k.O,l are not subject to the same limitations asdefined for f(x,y) in (1).We now change the variables on the RHS of (22) from polar coordinates tothe Cartesian coordinates used in (18), so that U R cose and v tane.From this follows that
8Uv 1Vl(l v') cosa and l(l v') sina.R sina,Ve than have dU l(l v')so that dR and(25)dvl v' de,jUl dU dv IRI dR de.Vhen these changes are inserted in (23) we getfk(x,y) t-1J'"ki2Jt(x,y)(U,Uv)[F,Q- fl (U, Uv)eIUldU dy.(26)'"Compare (26) and (18). The difference between the RHS in 18 and theintegrand on the RHS in (26) is that the latter is multiplied by the factorIUI. This then constitutes the necessary filtration of the spatialfrequencies in the projection data. Define(27)so that the two linograms g'k(u,v) consist of properly filtrated data.V(!U!) is a band-limiting function.The linogram algorithm. Ve are now ready to derive the algorithm. Insert(27) in (26)(U,v)ei2nxUdU dY,(28)and perform the inverse transform in U. Ve then have(29)Change from variabel u to x according to (15).fk(x,y) tg'k (x yv,v) dv(30)-1For a constant y-value the LHS is a line through fk(x,y) paralIeI to the xaxis and the RHS is a set of paralIeI line integrals through the filtered
9linogram (Fig 5). This may be regarded as aparallel projection of thefiltered linogram. Ve can then use the projection theorem as follows.A Fourier transform of fk(x,y) in the first variable may be expressed as-i2n Xxdx.Now inser t (30) on the RHS.[F ['I fkl (X,y) J'"t-i2n Xxg'k (x yv,v)dv edx(31)-'" -1As gkis zero for lvi 1 so is g'k' Ve can then extend the limi ts ofintegration for v to infinity. Change variabel from x to u according to(15). J'" J'" g'k (u,v) e- i2nX (u-yv)dv duCD(32)00The RHS represents a 2-dimensional Fourier transform of g'k (u,v). Theexponent for cofitains X(u-yv), which may be regarded as the inner productof the two vectors(X, -yX) and (u,v).(33)Ve then have that(34)Note the similarity between (34) and (18).Ve said that (18) was an expression for the celebrated projection slicetheorem, which states that the Fourier transform of a projection of afunction is to be found as a central line in the 2-dimensional Fouriertransform of the function.In (34) we have that the Fourier trans form of a line in the function,paraliei to the x-axis, is to be found as a central line in the 2dimensional Fourier trans form of the filtrated linogram to the function.
10Note also that in the 2-dimensional Fourier transform[F 2 g'k l (U,V) we have that,{U XV -yX.The image fk(x,y) stands in a similar relation to gk (u,v), as gk(u,v) tofk(x,y).From (34) we get(X, -yX)ei2n xXdX,(35)which says that a line in fk(x,y) paralIeI to the x-axis, i.e. with a fixedy-value, is reconstructed by performing an inverse Fourier transform of thevalues along the lineV -yX(36)in the 2-dimensional Fourier transform of the filtered linogram g'k(u,v).The final reconstructed image of the object f(x,y) is acquired byrotating f,(x,y) and adding it to fo(x,y) according to (24). It is repeatedhere.(37)
11Some comments on implementationA description of how to implement the linogram method is given in [2] and acomprehensive description of how to implement linograms with the slightlydifferent definition treated here and in [4] is given in [4]. In [4] theresult from different kinds of band limiting filters, W(U), were comparedwith each other and with the results from filtered back projection. Forthis purpose several phantoms were used. All experiments were simulatedwith the program package SNARK77. In [4] is also given an excellentdescription of this program package by which phantoms and a number ofdifferent projections and reconstruction methods may be simulated.An important feature in the implementation is the two-dimensional Fouriertransform of the filtered linogram. The first trans form in the u-directioncan easily be done by FFT of the Linogram. The filtration is then performedby multiplying with IU/. If now the trans form in the v-direction also isdone by FFT we would get a rectangular grid of points as in Fig 6a. From(36) and (1) we see that only points lying in the sectors defined byv - yXlyl 1,(38)are relevant and from these we would have to interpolate in order to getthe points shown in fig 6b. For small values of X there are then very fewpoints to interpolate from. The points in fig 6b, however, can becalculated exactly and without interpolation by doing a DFT and this inturn can be calculated by the so called chirp-z-transform (CZT) [7], whichperforms the DFT at the cost of 3 FFT's.By the use of CZT it is thus possible to implement the linogram methodwithout any kind of interpolation.
12Direct Fourier methodsThese are methods in which the image is reconstructed from its twodimensional Fourier transform by two inverse FFT's.The two-dimensional Fourier trans form for the image is derived from theone-dimensional Fourier trans forms of the projections of the object,utilizing the "projection slice theorem". This theorem says that when aparalIeI projection is Fourier transformed in its first variable, it isequal to a line through the origin of the two-dimensional Fourier transformof the image. When projection data are in the sinogram form defined in (10)they may be denoted p(r,e). Here E is the coordinate along the axisperpendicular to the rays. For sinograms the projection slice theorem isexpressed as[F[ 'lpl (R,e) [F 2 fl (R cose, R sine)(39)For linograms this theorem has already been stated as (18), it is repeatedhereAs X U, we can rewrite this as(40)When data for the LHS in the two expressions (39) and (40) are known forpoints in a rectangular array, the RHS represents these points on radiallines through the origin.In the sinogram case the points representing the RHS of (39) lie on a polargrid with equal increments in e between the radial lines and equalincrements in R between the points on the radial lines.This polar grid represents points in the two-dimensional Fourier transformof the image but the image cannot be calculated directly from this grid.First we have to change it into a rectangular grid of points and this isdone by interpolation from the polar grid. The image can then bereconstructed by a sequence of two one-dimensional inverse FFT's, oneparalIeI to the X-axis and the other to the Y-axis.
13A crucial step is the interpolation from polar to rectangular coordinates.A simple bilinear interpolation is not sufficient to achieve a good result.More correct interpolations [B], [9] give results comparable to thoseperformed by filtered back projection.In the linogram case, for each of the two linograms, the pointsrepresenting the RHS ,of (40), form apattern similar to the grid in fig 6b.The radial lines do not have equal increment in e but in v, whichrepresents tana. On the radial lines the points lie on equal increment inthe X-coordinate so that the points lie on straight lines pa,allel to theY-axis.The two grids [F 2 f ] (X,Xv) can be combined in one grid by rotating thekgrid [F 2 f,] (X,Xv) the angle n/2.We would then have a grid as in fig 7. This grid represent points in thetwo-dimensional Fourier transform of the complete image. As in the sinogramcase a rectangular grid may be interpolated. The image may then bereconstructed by two FFT's as in the sinogram case.A better way, however, would be to reconstruct each partial image f fromkits own partiaI grid by doing a two-dimensional inverse Fourier transformof it.If the discrete points representing [F 2 f k ] (X,Xv) were a perfectrectangular grid in (X,Y)-space, we could have used an inverse FFTalgorithm two times, first in the direction of one of the axes than in thedirection of the other in order to calculate fk(x,y). Now this is not thecase because even though the X-coordinates for the points conform to thegrid lines in a rectangular grid, the Y-coordinates do not. In order to dothe inverse transform in the Y-direction we have to do a DFT. But this canbe achieved by the chirp-z-transform mentioned on p. 10 [7] at the cost of3 FFT's.By this transform it is possible to calculate the correct values for
14[F[l]f ] (X,y), without interpolation. We then have to start with thekfiltered linograms defined in (27) and here restated. As the Fouriervariable U is equal to X, U is replaced with X.The filtered linogram (the LHS of 41) is then remapped as grid points inthe two-dimensional Fourier transform of f k ,[F,f ] (X,vX)k [F [ l ] g'k] (X,v)(42)The next step is to use the CZT to do an inverse DFT in the secondvariable.[F[']f k ] (X,y) [(F['])lF,f k ] (X,vX)(43)We can then calculate fk(x,y) by doing an inverse FFT in the firstvariable. The complete image is then calculated by (24).Comparison between the Linogram method (LM) and the Direct Fourier Methodwith Linograms (DFM). In both methods the steps are the same up to thestage where we have filtered linograms which are Fourier transformed in thefirst variable, i.e. [F[']g'k] (X, v).From this point they take different routes (Fig 8) and arrive at the sameresult, namelya partial image Fourier transformed in its first variable,Le. [F[ l ]f ] (X, y).kIn both methods all operations are only in the second variable, which istransformed from v to y. The first variable is all the time X.In the LM a Fourier transform of the linogram in the v-coordinate iscarried out by CZT, which changes this coordinate into V, expressed as-yX. A multiplication of this coordinate with (-l/X) is then performed,which gives the desired result.In the DFM the linogram is first remapped in to the 2-dimensional Fouriertrans form of the partial image. The v-coordinate is multiplied with X which
15ehanges it into Y. An inverse Fourier transform by CZT in this eoordinatethen gives the desired result.Although the two methods are eoneeptually different the ealeulations arepraetieally identical. Both methods arrive at exaetly the same result andno interpolations are performed.This is elear from the following. Assume that we only eompute one eolumn in[F[llfkl (X,y), (i.e. in the following X 1 and is a constant). Call thissequenee of values I(y). It is eomputed from a corresponding column ofvalues in [F[llg'k 1 (X,v) (with the same eons tant X). Call this sequenceL(v).In the Linogram method we make a forward Fourier trans form into a newsequence [F,g'k 1 (X,V). We do not, however, compute this sequence forinteger values of V but at the fractional values for which V -yXH -1[F2gi](X, -yX) N-i2(-yXjYLL(v)e-12Jr-N-nN ·1n - N -IzL12Jr yXv nL(v)e 1IN -I0 -2-The "remapping" of the LHS into I (y) consists in letting the secondvariable assumeinteger values. As the LHS in fact only is a sequenceof values, it can directly be accept ed as I (y).The "remapping step" is therefore only a coneeptual step. It is no step inthe caleulations. So we have that
16In the Direct Fourier method we start with a similar conceptual "remappingstep" by considering the sequence L(v) to be values at the fractionalcoordinate points Y vX, in [F,fkl (X,vX).We then do an inverse Fourier transform of this sequence resulting in I (y)I(y) N-iThus the calculations are exactly the same for both methods.
Id17sFig l. A line specified bythe point (s ,e ).p pFig 2. A sinusoid (onlyapartof it is shown), specified bythe point (x p ' yp)'Id1\X u.p - VP!:l':fp\IXpXvvp,-----,UpUlA.Fig 3. A line specified byFig 4. A line specified bythe point (up' v p )'the point (X p ' yp)'
18 xFig 5. The row of points in fk(x,y) paralIeI to the x-axis arereconstructed by a paralIeI projection of g'(u,v).
19vIv--.Ii IIIII,IrxxIhaFig 6. a) The resulting grid if the Pourier transform in the v-direction isperformed by PPT. b) The grid of points needed for the reconstruction. Thisgrid can be achieved without interpolation by CZT.yxPig 7. The grid resulting from a remapping and combination of the two[p[llg;'l (X,v).
20v, .c xlx(f; 9:J ex) -!IX)remapremap!IXFig 8. Pictorial comparison of the different routes taken by the LinogramMethod and the Direct Fourier Method with linograms. Both start from afiltered and Fourier transformed linogram (upper left) and end at a partialimage Fourier transformed in its first variable (lower right). The LinogramMethod goes via the 2-D Fourier transformed linogram (upper right), theDirect Fourier Method via the 2-D Fourier transformed image (lower left).
21REFERENCES[1]P.R. Edholm and G.T. Herman, "Linograms in image reconstruction fromprojections", IEEE Trans. Med. Imaging, vol MI-6, pp 301-307, 1987.[2]P.R. Edholm, G.T. Herman and D. Roberts, "Image reconstruction fromlinograms: Implementation and evaluation", IEEE Trans. Med. Imaging,vol MI-7, pp 239-246, 1988.[3]P.R. Edholm, "Linograms", Report ULi-RAD-R-058, University ofLinköping.[4]M. Magnusson, "Implementation of the linogram method for CTreconstruction", in preparation.[5]L. Axel, G.T. Herman, J. Listerud and D. Roberts, "Magnetic resonanceimaging based on linograms" , in preparation.[6]This term was introduced in a poster presentation at the 1975 meetingon Image Processing for 2-D and 3-D Reconstructions from Projectionsat Stanford, CA. The material appeared in a collection of postdeadlinepapers for that meeting. PD5 - Tomogram Construction by PhotographicTechniques. Paul Edholm and Bertil Jacobsson.[7]L.R. Rabiner, R.II. Schafer and C.M. Rader, "The chirp-z-transformalgorithm", IEEE Trans. Audio Electroacoust., Vol. AU-17 , pp. 86-92,1969.[8]H. Stark, J.II. lIoods, I. Paul and R. Hingorani, "An investigation ofcomputerized tomography by direct Fourier inversion and optimuminterpolation, IEEE Trans. Biorned. Eng., Vol. BME-28, pp. 496-505,1981.19]F. Natterer, "Fourier reconstruction in tomography", Numer. Math. Vol.47, pp. 343-353, 1985.[10] G.T. Herman and S.II. Rowland, "SNARK77": A programming system forimage reconstruction from projections", Dep. Comput. Sci., State Univ.New York at Buffalo, Tech. Rep. 130, 1978.
1Utgivna rapporter vid Radiofysiska Institutionen, Universitetet i 7.18.19.20.21.22.23.24.25.Leif Kusoffsky: HTF-begreppet och dess applikation. (1973-05-23)Bengt Nielsen: Undersökning av uranraster. (1973-06-15)Per Spanne: High dose RPL-dosimetry. (1973-09-30)har utgåtti Är ersatt av rapport 041.Carl Carlsson: Spridd strålning i röntgendiagnostik. (1973-09-10)Leif Kusoffsky och Carl Carlsson: Hodulationsöverföringsfunktionen,HTF. (1973-09-12)har utgåtti Är ersatt av rapport 052.Carl Carlsson: Grundläggande fysik inom röntgendiagnostik.(1973-09-14)Paul Edholm: Bildbehandling. (1973-09-20)har utgåttl Är ersatt av rapport 026.Bengt Nielsen: Investigation of Roentgen Focal Spot. (1973-11-12)Gudrun Alm Carlsson: Kärnfysikaliska grunder för radioaktiva nuklider.(1974-11-11)Carl Carlsson: Strålningsdosimetri med radioaktiva nuklider i människa.(1974-11-13)Carl Carlsson: Växelverkan mellan materia och joniserande strålningfrån radioaktiva nuklider. (1974-11-29)Per Spanne: Strålningsdetektorer. (1974-11-29)Gudrun Alm Carlsson: Statistisk precision vid radioaktivitetsmätning.(1974-12-05)Carl Carlsson: Aktivitetsbestämning ur uppmätt räknehastighet.(1974-12-05)Gudrun Alm Carlsson: Pulshöjdsanalys. (1974-12-12)Gudrun Alm Carlsson: Kvantelektrodynamik för elektroner - Feyman-diagram och strålningskorrektion för tvärsnitt. (1975-01-07)Gudrun Alm Carlsson: Klassisk elektrodynamik. Växelverkan mellan laddade partiklar och elektromagnetiska fält. (1975-01-07)Sten Carlsson: Vätskescintillatorn. (1975-01-09)Per Spanne och Gudrun Alm Carlsson: Problem vid radioaktivitetsmätningar vid höga räknehastigheter. (1975-01-21)Carl Carlsson: Signal och bakgrund vid mätning av låga radioaktiviteter. (1975-02-24)Bertil Persson: Val av radionuklider och radioaktiva markörer för användning in vivo. (1975-03-17)Carl Carlsson: Användning av logaritmer och exponetiaifunktioner inomröntgendiagnostik. (1975-04-03)
226. Ulf Boström: Röntgenbildförstärkare och Röntgen-TV. (1975-04-07)(Ersätter rapport nr 010).27. Gudrun Alm Carlsson: Riskuppskattningar vid små stråldoser och strålskyddsrekomendationer. (1975-04-10)28. Gudrun Alm Carlsson: Analys av Honte Carlo metoder för simulering avfotontransporter. (1975-09-12)29. Leif Kusofsky: Rutinbeskrivningar. Honte Carlo program för fotontransportsimuleringar. (1975-09-05)30; Leif Kusoffsky: Jämförelse mellan två olika växelverkansmodeller för15-200 keY fotoner använda i Honte Carlo beräkningar av spridd strålning. (1975-09-12)31. Gudrun Alm Carlsson: A critical analysis of concepts of ionizing radiation and absorbed dose. (1977-01-21)32. Gudrun Alm Carlsson: A different formulation of the definition of energy imparted. (1977-01-21)33. Carl Carlsson: Vectorial and plane energy fluences - useful concepts inradiation physics. (1977-06-01)34. Gudrun Alm Carlsson och Carl Carlsson: Strålningsdosimetri i röntgendiagnostiken. (1979-10-01)35. Gudrun Alm Carlsson: Absorbed dose equations. The general solution ofthe absorbed dos e equation and solutions under different kinds ofradiation equilibrium. (1978-01-27)36. har utgåttI Är ersatt av 057.37. Paul Edholm: Konturen. En radiologisk studie. (1978-05-10)38. Gudrun Alm Carlsson: Burlins kavitetsteori. (1979-08-15)39. Bengt Nielsen: Upplösningförmåga, oskärpa och HTF. (1980-01-23)40. Gudrun Alm Carlsson, Karl-Fredrik Berggren, Carl Carlsson och RolandRibberfors: Beräkning av spridningstvärsnitt för ökad noggrannhet idiagnostisk radiologi. I Energibreddning vid Comptonspridning.(1980-01-25)41. Paul Edholm: Röntegenprojektionens geometri. (1980-09-05)(Ersätter rapport 004)42. Per Spanne och Carl Carlsson: Kontroll av kärnkraftsindustrins TLDsystem för persondosimetri. (1980-10-30)43. Gudrun Alm Carlsson: Kavitetsteori - allmänna grunder. (1981-01-20)44. Carl Carlsson och Bengt Nielsen: Kvalitetsvärdering av raster förbekämpning av spridd strålning vid röntgenundersökningar. Del I Teori(1981-08-21)
345. Carl Carlsson och Bengt Nielsen: Kvalitetsvärdering av raster förbekämpning av spridd strålning vid röntgenundersökningar. Del IIExperimentella resultat. (1981-08-21)46. Bengt Nielsen: Mätmetoder för att bestämma modulationsöverföringsfunktionen för radiologiska system. (1981-08-21)47. Gudrun Alm Carlsson: Skalära och vektoriella fysikaliska storheter.Deras betydelse för ,förståelsen av röntgendetektorernas uppträdande iett strålningsfält. (1981-09-23)48. Gudrun Alm Carlsson: Fotonspridningsprocessen vid röntgendiagnostiskastrålkvaliteter. (1981-09-23)49. Gudrun Alm Carlsson: Effective use of Monte Carlo methods for simulating photontransport with special reference to slab penetrationproblems in X-ray diagnostics. (1981-10-19)50. Anders Björk och Bengt-Olof Dahl: Konstruktion av experimentell datortomograf. Utarbetande av datorproigram för styrning av rörelseenheter,insamlande av mätdata och presentation av bilder. (1982-06-23)51. Georg Matscheko: Utnyttjande av Comptonspridning vid bestämning avprimärspektrum av röntgenstrålning från diagnostiska röntgenrör.(1982-11-12)52. Paul Edholm: Praktisk tomografi. (1982-12-08)53. Sune Eriksson, Carl Carlsson, Olof Eckerdahl och JUri Kurol: Riktlinjerför klinisk och röntgenologisk övervakning av överkäkshörntändernaseruption hos barn och ungdomar mellan 8 och 15 år. Analys av indikationer och metoder för röntgenundersökning med hänsyn tagen till stråldoser och diagnostisk utfall (december 1984)54. Paul Edholm: Diagnostisk radiologi för propedeutkursen. (1985-01-31)55. Börje Forsberg och Per Spanne: Stråldoser till personal vid klinikerför gynekolog
to denote the Fourier transform of ! with respect to its first variable, the Fourier transform of ! with respect to its second variable, and the two-dimensional Fourier transform of !. Variables in the spatial domain are represented by small letters and in the Fourier domain by capital letters. expressions, k is an index assuming the two values O
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