Journal of X-Ray Science and Technology 10 001 59 66 59 IOS Press Linear diohantine euations for discrete tomograhy Yangbo Ye a,gewang b and Jiehua Zhu a a Deartment of Mathematics, The University of Iowa, Iowa City, IA 54, USA b Deartment of Radiology, The University of Iowa, Iowa City, IA 54, USA Abstract. In this reort, we resent a number-theory-based aroach for discrete tomograhy DT,which is based on arallel rojections of rational sloes. Using a well-controlled geometry of X-ray beams, we obtain a system of linear euations with integer coefficients. Assuming that the range of ixel values is ai, j =0, 1,...,M 1, with M being a rime number, we reduce the euations modulo M. To invert the linear system, each algorithmic ste only needs log M bit oerations. In the case of a small M, we have a greatly reduced comutational comlexity, relative to the conventional DT algorithms, which reuire log N bit oerations for a real number solution with a recision of 1/N. We also reort comuter simulation results to suort our analytic conclusions. Keywords: Comuted tomograhy CT, discrete tomograhy DT, number theory, Diohantine euations, comutational comlexity 1. Introduction Comuted tomograhy CT refers to comuterized synthesis of an image from external measurements of a satially varying function. Line integrals are the most common external measures, which are also known as rojections. This underlying function is traditionally real-valued. However, the range of the underlying function is ractically often discrete in industrial imaging and other alications. The roblems of discrete tomograhy DT are to determine the underlying function from weighted sums over subsets of its domain. In the most essential sense, DT allows that given a discrete range of the underlying function we find its values that could not be found without the knowledge on the discrete range. Recently, DT attracts increasing interests, and becomes challenging and exciting. Current studies e.g. [3,6] of DT are focused on algorithms derived from horizontal and vertical rojections, lus sometimes diagonal rojections. If the number of unknowns is more than the number of euations, these rojections cannot determine a uniue solution in the general case. Furthermore, existing algorithms do not directly make use of the discreteness of ixel values to reduce the comutational comlexity see [1,7]. In this aer, we formulate a system of linear euations with integer coefficients, and develo a number-theory-based algorithm for DT. In the next section, we describe a re-secified geometry for the formation of aroriate rojections. In the third section, we roose a number-theorybased algorithm that can be alied to rojection data collected according to the re-secified geometry. Then, we resent reresentative results obtained via comuter simulation. In the last section, relevant issues are discussed for futher research, and conclusions are made. 0895-3996/0/$8.00 001 IOS Press. All rights reserved
60 Y. Ye et al. / Linear diohantine euations for discrete tomograhy. Projection formation Suose an image ax, y is divided into a suare of grid n n. The total number of cells is n. Assume each cell has size of 1 1 and in each cell ax, y takes a constant value. For 1 i, j n, let us denote by ai, j the constant function value in the i, jth cell, which is the cell enclosed by the lines: x = i 1, x = i, y = j 1 and y = j. It is easy to have horizontal and vertical rojections. They are rays of width 1. The horizontal rojections give n euations: b 0/1 j = n ai, j, j =1,...,n, 1 i=1 where b 0/1 j is the density detected by the ith ray. Here the subscrit 0/1 denotes the sloe of the rays. Similarly from the vertical rojections we get b 1/0 i = n ai, j, i =1,...,n, j=1 where 1/0 reresents the vertical sloe. There are two diagonal directions, one with sloe 1/1 and the other with sloe 1/1. We take the width of the rays eual to 1/ and divide neighboring rays by lines y = x k for sloe= 1/1, and by y = x k for sloe= 1/1, for k Z. Note that each ray asses through 1/ area of a cell. Therefore the diagonal rojections with sloe 1/1 are b 1/1 k = n 1 aj, j k1 aj 1,j k, n <k n, 3 j=0 where we set ai, j =0if i, j is outside the range 1 i, j n. Similarly from the other diagonal direction we get euations b 1/1 k = n 1 aj, n j k 11 aj 1,n j k 1, n <k n. 4 j=0 The key idea of our new algorithm is to have rojections in other directions, and to do so in such a way that the resulting euations have integral coefficients. In this sense, our rojections are different from those in [1,8]. Let / be a ositive rational number in its reduced form. Then the greatest common divisor, =1. Assume that >>0. We want to construct arallel rays of sloe /, by dividing our suare grid into stries using arallel lines y =/x j, n <j n. Then each strie and hence each ray have width / which is less than 1 but greater than 1/, because >>0. Such a ray asses through several cells. By comuting the area of each cell i, j it asses through, we can determine the coefficient of ai, j in the euation for this ray. The following theorem summarizes our results. Here we denote by [x] the integral art of a real number x, that is, the largest integer x. Denote by {x the fractional art of x, that is, {x = x [x].
{/=0 Y. Ye et al. / Linear diohantine euations for discrete tomograhy 61 Theorem 1. Let and be ositive integers with, =1and >. The euations from rojections of sloe / are b / j= aj /,l 1 aj / 1,l 0<{/</ 1 for n <j n. { { { aj [/],l {/ / { 1 { aj [/],l { aj [/]1,l { aj [/]1,l aj [/],l, We will rove this Theorem in the next section. Here we oint out that the coefficients are all rational numbers, and indeed, if multilied by, they become integers. This means that all our Es 1 through 5, together with Es 6 to 8 below, can be rewritten as linear euations of integral coefficients. If we seek integer solutions ai, j, we are dealing with a system of linear Diohantine euations. If the cells are suosed to take values ai, j =0, 1,..., M 1, we can further reduce these linear Diohantine euations modulo M. By simle reflections or change of coordinates, we can get euations for rojections of sloes /, /, and /. Corallary 1. Let and be ositive integers with, =1and >. The euations from rojections of sloe / are b / j= al, j / 1 al, j / 1 {/=0 0<{/</ 1 { { { al, j [/] { al, j [/]1 5
6 Y. Ye et al. / Linear diohantine euations for discrete tomograhy for n <j n. { al, j [/] {/ / { 1 { al, j [/]1 al, j [/], 6 Corallary. Let and be ositive integers with, =1and >. The euations from rojections of sloe / are b / j= an 1 j /,l 1 an j /,l for n <j n. {/=0 0<{/</ 1 { {/ / { { { an 1 j [/],l { { an j [/],l an j [/] 1,l 1 { an j [/],l an j [/] 1,l, Corallary 3. Let and be ositive integers with, =1and >. The euations from rojections of sloe / are b / j= al, n 1 j / 1 al, n j / {/=0 0<{/</ { { al, n 1 j [/] 7
for n <j n. Y. Ye et al. / Linear diohantine euations for discrete tomograhy 63 1 { { al, n j [/] 8 { al, n j [/] 1 1 { al, n j [/] {/ / { al, n j [/] 1, 3. Algorithm and simulation We select arallel rays whose sloes are rational numbers of the form ±/, and control the width and location of each ray in a secific way. The resulting linear euations then have all coefficients being integers. Because of these integral coefficients, we may reduce this system of linear Diohantine euations modulo M, if the cells are suosed to take values ai, j =0, 1,..., M 1. In other words, we seek solutions in the field Z/M Z, when M is a rime number. This linear system can be solved over Z/M Z in the same way as over R, using the standard Gaussian elimination method, for examle. The main advantage of our aroach is that, over Z/M Z, each oeration uses only log M bit oerations cf. Cohen []. On the other hand, for solution rocedures of a linear system over R, each oeration needs log N bit oerations for a real number solution of recision 1/N. For small M, the time log M reresents a big saving of comutation time over the usual log N time. As an examle, let us look at the case of M =, i.e., the case when every cell can only take value 0 or 1. Now Z/Z is the finite field of two elements 0 and 1. Oerations in Z/Z are fast: 0 0=0, 0 1=0, 1 1=1, 00=0, 01=1, 11=0, etc. Each oeration only need log =1bit oeration. On the other hand, the solution rocedure for the same linear system is much slower if we seek a real number solution to the recision of, let us say, 0.01 = 1/100. With N = 100, one oeration here need log 100 44 bit oerations. In other words, our new algorithm uses only 1/44 of the comutation time of a standard CT algorithm. This huge saving of comutation time was observed in a comutation of a 30 30 DT rocedure using Matlab. For our new modulo algorithm, the CPU time is significantly faster than using a standard algorithm seeking real number solutions. The linear euations are given in Section from Es 1 to 8. In these euations, Es 1 and are derived from horizontal and vertical rojections and are of integral coefficients. Euations 3 and 4 are from diagonal rojections; we need to multily them by a factor to get the euations of integral coefficients we want. Note that if the cells are suosed to take integral values ai, j =0, 1,...,M 1, b 1/1 k and b 1/1 k will be integers or half integers. Conseuently after multilying them by, Es 3 and 4 become linear euation of integral coefficients. Similarly, we can multily Es 5 to 8 by and change them to linear euations of integral coefficients.
64 Y. Ye et al. / Linear diohantine euations for discrete tomograhy In order to get enough euations from Es 5 through 8, we may fix a ositive integer L and select all and with 1 < L and, =1. These fractions /, together with 0 and 1, form a Ferry seuence of level L. There are about 3/π L such airs of and, which roduce about 1/π L directions: ±/ and ±/. In each of these directions we can get more than n but less than n rays. Since we need n linearly indeendent euations, we may set 1/π L n = n. This means that L may be set to eual to π/ 1 n. For examle, if n = 56, we may set L =14. 4. Discussions and conclusion Much efforts are needed to otimize imaging rotocols and reconstruction algorithms. Clearly, image uality would deend on domain knowledge, imaging geometry, data comleteness, noise level, and algorithmic details. Extensive simulation and hysical exeriments must be erformed in selected cases to gain some insight into interlays among all the major factors. It is ackowledged that the most oular imaging geometry is fan-beam-oriented, instead of arallelbeam-oriented. However, our findings are still of significance, because we can always rebin fan-beam rojection data to arallel-beam rojection data in the format we refer. This rebinning rocess is always much faster than the reconstruction rocess. Furthermore, the rebinning is even a necessary intermediate ste in siral/helical CT, in which an X-ray source rotation and an object translation are erformed simultaneously. We believe that fan-beam DT should be a most ractical roblem to solve in near future. Although we have not addressed data inconsistence in this roof-of-concet reort, we oint out that noise in data can be suressed in the same sirit of the conventional linear system solution algorithms. On the other hand, given a secific knowledge on the discrete range of the underlying function, more efficient algorithms may be designed for various DT alications. We are articularly interested in ursuing research along a number-theory direction. Relevant results will reorted in future ublications. In conclusion, we have roosed a number-theory-based aroach for discrete tomograhy DT, and demonstrated that our method can greatly reduce the comutational comlexity. Further work is underway to generalize the findings, refine the algorithm and aly it to various alications. Aendix. Proof of Theorem 1. Suose the sloe of a ray is /, where, =1, >>0, and the ray is bounded by y =/x and y =/x 1. Then this ray asses through cells in three different cases. The first case is that there exists an integer m Z such that l 1/ < m < /. This condition is euivalent to 0 < {/ </. In this case, the ray runs through three cells on level l with unknowns a[/],l, a[/]1,l, and a[/],l. The areas assed by the ray for the three cells are w [/],l = 1 [ ] [ ] l 1 l 1 = { w [/]1,l =1 {, { 1 [ ] l l l [ ] l
=1 Y. Ye et al. / Linear diohantine euations for discrete tomograhy 65 { {, w [/],l = 1 1 = {. [ ] l 1 l [ ] Thus the rojection over these three cells roduces three terms for the linear euation { 1 { { a[/],l { a[/]1,l { a[/],l. A1 The second case is that / Z, i.e., {/ =0. In this case the ray asses through two cells on level l. The coefficients of the unknowns a/,l and a/ 1,l are w /,l = / and w /1,l =1 /, resectively. This case therefore contributes two terms to the linear euation a/,l 1 a/ 1,l. The third case is when {/ /. In this case, the ray runs through two cells on level l and the corresonding coefficients are: w l,[/]1 = 1 [ ] 1 = { 1 [ ] 1 l 1 A w l,[/] =1 1 { = {. Hence the rojection over these two cells has the following two terms 1 { a[/]1,l { a[/],l. A3 Collecting Es A1, A, and A3, we rove Theorem 1.
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