Tài liệu 26 Algorithms for Computed Tomography pptx

Herman University of Pennsylvania 26.1 Introduction 26.2 The Reconstruction Problem 26.3 Transform Methods 26.4 Filtered Backprojection FBP 26.5 The Linogram Method 26.6 Series Expansion

Gabor T Herman “Algorithms for Computed Tomography”

2000 CRC Press LLC <http://www.engnetbase.com>.

Algorithms for Computed

Tomography

Gabor T Herman

University of Pennsylvania

26.1 Introduction 26.2 The Reconstruction Problem 26.3 Transform Methods

26.4 Filtered Backprojection (FBP) 26.5 The Linogram Method 26.6 Series Expansion Methods 26.7 Algebraic Reconstruction Techniques (ART) 26.8 Expectation Maximization (EM)

26.9 Comparison of the Performance of Algorithms References

26.1 Introduction

Computed tomography is the process of reconstructing the interiors of objects from data collected based on transmitted or emitted radiation The problem occurs in a wide range of application areas Here we discuss the computer algorithms used for achieving the reconstructions

26.2 The Reconstruction Problem

We want to solve the following general problem There is a three-dimensional structure whose internal composition is unknown to us We subject this structure to some kind of radiation, either by transmitting the radiation through the structure or by introducing the emitter of the radiation into the structure We measure the radiation transmitted through or emitted from the structure at a number

of points Computed tomography (CT) is the process of obtaining from these measurements the

distribution of the physical parameter(s) inside the structure that have an effect on the measurements The problem occurs in a wide range of areas, such as x-ray CT, emission tomography, photon migration imaging, and electron microscopic reconstruction; see, e.g., [1,2] All of these are inverse problems of various sorts; see, e.g., [3]

Where it is not otherwise stated, we will be discussing the special reconstruction problem of estimating a function of two variables from estimates of its line integrals As it is quite reasonable for any application, we will assume that the domain of the function is contained in a finite region of the plane In what follows we will introduce all the needed notation and terminology; in most cases these agree with those used in [1]

Supposef is a function of the two polar variables r and φ Let [Rf ](`, θ) denote the line integral

off along the line that is at distance ` from the origin and makes an angle θ with the vertical axis.

This operatorR is usually referred to as the Radon transform.

The input data to a reconstruction algorithm are estimates (based on physical measurements) of the values of[Rf ](`, θ) for a finite number of pairs (`, θ); its output is an estimate, in some sense,

off More precisely, suppose that the estimates of [Rf ](`, θ) are known for I pairs: (` i , θ i), 1

≤ i ≤ I We use y to denote the I-dimensional column vector (called the measurement vector)

whoseith component, y i, is the available estimate of[Rf ](` i , θ i ) The task of a reconstruction

algorithm is:

given the datay, estimate the function f

Following [1], reconstruction algorithms are characterized either as transform methods or as series expansion methods In the following subsections we discuss the underlying ideas of these two approaches and give detailed descriptions of two algorithms from each category

26.3 Transform Methods

The Radon transform has an inverse,R−1, defined as follows For a functionp of ` and θ,

h

R−1pi(r, φ) = 1

2π2

Z π

0

−∞

1

r cos(θ − φ) − ` p1(`, θ)d`dθ , (26.1)

wherep1(`, θ) denotes the partial derivative of p with respect to its first variable ` [Note that it is

intrinsically assumed in this definition thatp is sufficiently smooth for the existence of the integral

in Eq (26.1)] It is known [1] that for any functionf which satisfies some physically reasonable

conditions (such as continuity and boundedness) we have, for all points (r, φ),

h

Transform methods are numerical procedures that estimate values of the double integral on the right hand side of Eq (26.1) from given values ofp(` i , θ i ), for 1 ≤ i ≤ I We now discuss two such

methods: the widely adopted filtered backprojection (FBP) algorithm (called the convolution method

in [1]) and the more recently developed linogram method.

26.4 Filtered Backprojection (FBP)

In this algorithm, the right hand side of Eq (26.1) is approximated by a two-step process (for derivational details see [1] or, in a more general context, [3]) First, for fixed values ofθ, convolutions

defined by

[p ∗ Y q] (`0, θ) =

Z ∞

−∞p(`, θ)q `0− `, θ
d` (26.3)

are carried out, using a convolving function q (of one variable) whose exact choice will have an

important influence on the appearance of the final image Second, our estimatef∗off is obtained

by backprojection as follows:

f∗(r, φ) =

Z π

0

[p ∗ Y q] (r cos(θ − φ), θ) dθ. (26.4)

To make explicit the implementation of this for a given measurement vector, let us assume that the data functionp is known at points (nd, m1), −N ≤ n ≤ N, 0 ≤ m ≤ M − 1, and M1 = π Let

us further assume that the functionf is to be estimated at points (r j , φ j), 1≤ j ≤ J The computer

algorithm operates as follows

A sequencef0, , f M−1 , f M of estimates is produced; the last of these is the output of the algorithm First we define

for 1≤ j ≤ J Then, for each value of m, 0 ≤ m ≤ M − 1, we produce the (m + 1)st estimate from

themth estimate by a two-step process:

1 For−N ≤ n0≤ N, calculate

p c n0d, m1
= d XN

n=−N

p (nd, m1) q n0− n
d
, (26.6)

using the measured values ofp(nd, m1) and precalculated values (same for all m) of q((n0− n)d) This is a discretization of Eq (26.3) One possible (but by no means only) choice is

2 For 1≤ j ≤ J , we set

f m+1 r j , φ j

= f m r j , φ j

+ 1p c r jcos m1 − φ j

This is a discretization of Eq (26.4) To do it, we need to interpolate in the first variable

ofp cfrom the values calculated in Eq (26.6) to obtain the values needed in Eq (26.7)

In practice, oncef m+1 (r j , φ j ) has been calculated, f m (r j , φ j )is no longer needed and the

computer can reuse the same memory location forf0(r j , φ j ), , f M−1 (r j , φ j ), f M (r j , φ j ).

In a complete execution of the algorithm, the uses of Eq (26.6) requireM(2N +1) multiplications

and additions, while all the uses of Eq (26.7) requireMJ interpolations and additions Since J is

typically of the order ofN2andN itself in typical applications is between 100 and 1000, we see that

the cost of backprojection is likely to be much more computationally demanding than the cost of convolution In any case, reconstruction of a typical 512× 512 cross-section from data collected by a typical x-ray CT device is not a challenge to state-of-the art computational capabilities; it is routinely done in the order of a second or so and can be done, using a pipeline architecture, in a fraction of a second [4]

26.5 The Linogram Method

The basic result that justifies this method is the well-known projection theorem which says that “taking

the two-dimensional Fourier transform is the same as taking the Radon transform and then applying the Fourier transform with respect to the first variable” [1] The method was first proposed in [5] and the reason for the name of the method can be found there The basic reason for proposing this method is its speed of execution and we return to this below In the description that follows, we use the approach of [6] That paper deals with the fully three-dimensional problem; here we simplify it

to the two-dimensional case

For the linogram approach we assume that the data were collected in a special way (that is, at points whose locations will be precisely specified below); if they were collected otherwise, we need

to interpolate prior to reconstruction If the function is to be estimated at an array of points with rectangular coordinates{(id, jd)| − N ≤ i ≤ N, −N ≤ j ≤ N} (this array is assumed to cover the

object to be reconstructed), then the data functionp needs to be known at points

(nd m , θ m ) , −2N − 1 ≤ n ≤ 2N + 1, −2N − 1 ≤ m ≤ 2N + 1 (26.8)

and at points



nd m , π

2 + θ m, −2N − 1 ≤ n ≤ 2N + 1, −2N − 1 ≤ m ≤ 2N + 1, (26.9) where

θ m= tan−1 2m

4N + 3 andd m = d cos θ m . (26.10)

The linogram method produces from such data estimates of the function values at the desired points using a multi-stage procedure We now list these stages, but first point out two facts One is

that the most expensive computation that needs to be used in any of the stages is the taking of discrete

Fourier transforms (DFTs), which can always be implemented (possibly after some padding by zeros)

very efficiently by the use of the fast Fourier transform (FFT) The other is that the output of any stage

produces estimates of function values at exactly those points where they are needed for the discrete computations of the next stage; there is never any need to interpolate between stages It is these two facts which indicate why the linogram method is both computationally efficient and accurate (From the point of view of this handbook, these facts justify the choice of sampling points in Eqs (26.8) through (26.10); a geometrical interpretation is given in [7].)

1 Fourier transforming of the data — For each value of the second variable, we take the DFT

of the data with respect to the first variable in Eq (26.8) and Eq (26.9) By the projection theorem, this provides us with estimates of the two-dimensional Fourier transformF of

the object at points (in a rectangular coordinate system)



k (4N+3)d0, (4N+3)d k tanθ m



, −2N − 1 ≤ k ≤ 2N + 1 ,

and at points (also in a rectangular coordinate system)



k (4N+3)dtan π2 + θ m

, (4N+3)d k , −2N − 1 ≤ k ≤ 2N + 1 ,

2 Windowing — At this point we may suppress those frequencies which we suspect to be noise-dominated by multiplying with a window function (corresponding to the convolving

function in FBP)

3 Separating into two functions — The sampled Fourier transform F of the object to be

reconstructed is written as the sum of two functions,G and H G has the same values

asF at all the points specified in Eq (26.11) except at the origin and is zero-valued at all other points.H has the same values as F at all the points specified in Eq (26.12) except at the origin and is zero-valued at all other points Clearly, except at the origin,F = G+H.

The idea is that by first taking the two-dimensional inverse Fourier transforms ofG and

H separately and then adding the results, we get an estimate (except for a DC term which

has to be estimated separately, see [6]) off We only follow what needs to be done with G; the situation with H is analogous.

4 Chirp z-transforming in the second variable — Note that the way the θ m were selected implies that if we fixk, then the sampling in the second variable of Eq (26.11) is uniform Furthermore, we know that the value ofG is zero outside the sampled region Hence, for

each fixedk, 0 < |k| ≤ 2N+ 1, we can use the chirp z-transform to estimate the inverse

DFT in the second variable at points

(4N + 3)d , jd



, −2N − 1 ≤ k ≤ 2N + 1, −N ≤ j ≤ N. (26.13) The chirp z-transform can be implemented using three FFTs, see [7]

5 Inverse transforming in the first variable — The inverse Fourier transform of G can now

be estimated at the required points by taking, for every fixedj, the inverse DFT in the

first variable of the values at the points of Eq (26.13)

26.6 Series Expansion Methods

This approach assumes that the function,f , to be reconstructed can be approximated by a linear

combination of a finite set of known and fixed basis functions,

f (r, φ) ≈XJ

j=1

and that our task is to estimate the unknowns,x j If we assume that the measurements depend linearly on the object to be reconstructed (certainly true in the special case of line integrals) and that

we know (at least approximately) what the measurements would be if the object to be reconstructed was one of the basis functions (we user i,jto denote the value of theith measurement of the jth basis

function), then we can conclude [1] that theith of our measurements of f is approximately

J

X

j=1

Our problem is then to estimate thex j from the measured approximations (for 1 ≤ i ≤ I) to

Eq (26.15) The estimate can often be selected as one that satisfies some optimization criterion.

To simplify the notation, the image is represented by aJ -dimensional image vector x (with

compo-nentsx j) and the data form anI-dimensional measurement vector y There is an assumed projection matrix R (with entries r i,j) We letr i denote the transpose of theith row of R (1 ≤ i ≤ I) and so

the inner producthr i , xi is the same as the expression in Eq (26.15) Theny is approximately Rx

and there may be further information thatx belongs to a subset C of R J, the space ofJ -dimensional

real-valued vectors In this formulationR, C, and y are known and x is to be estimated Substituting

the estimated values ofx j into Eq (26.14) will then provide us with an estimate of the functionf

The simplest way of selecting the basis functions is by subdividing the plane into pixels (or space into voxels) and choosing basis functions whose value is 1 inside a specific pixel (or voxel) and is

0 everywhere else However, there are other choices that may be preferable; for example, [8] uses spherically symmetric basis functions that are not only spatially limited, but also can be chosen to be very smooth The smoothness of the basis functions then results in smoothness of the reconstructions, while the spherical symmetry allows easy calculation of ther i,j It has been demonstrated [9], for the

case of fully three-dimensional positron emission tomography (PET) reconstruction, that such basis

functions indeed lead to statistically significant improvements in the task-oriented performance of series expansion reconstruction methods

In many situations only a small proportion of ther i,j are nonzero (For example, if the basis functions are based on voxels in a 200× 200 × 100 array and the measurements are approximate

line integrals, then the percent of nonzeror i,jis less than 0.01, since a typical line will intersect fewer than 400 voxels.) This makes certain types of iterative methods for estimating thex j surprisingly efficient This is because one can make use of a subroutine which, for anyi, returns a list of those

js for which r i,j is not zero, together with the values of ther i,j [1,10] We now discuss two such

iterative approaches: the so-called algebraic reconstruction techniques (ART) and the use of expectation

maximization (EM).

26.7 Algebraic Reconstruction Techniques (ART)

The basic version of ART operates as follows [1] The method cycles through the measurements repeatedly, considering only one measurement at a time Only thosex j are updated for which the correspondingr i,jfor the currently considered measurementi is nonzero and the change made to x j

is proportional tor i,j The factor of proportionality is adjusted so that if Eq (26.15) is evaluated for the resultingx j, then it will match exactly theith measurement Other variants will use a block of

measurements in one iterative step and will update thex jin different ways to ensure that the iterative process converges according to a chosen estimation criterion

Here we discuss only one specific optimization criterion and the associated algorithm (Others can be found, for example, in [1]) Our task is to find thex in R Jwhich minimizes

(k · k indicates the usual Euclidean norm), for a given constant scalar r (called the regularization

parameter) and a given constant vector µ x

The algorithm makes use of anI-dimensional vector u of additional variables, one for each

mea-surement First we defineu (0)to be theI-dimensional zero vector and x (0)to be theJ -dimensional

zero vector Then, fork ≥ 0, we set

u (k+1) = u (k) + c (k) e i k ,

x (k+1) = x (k) + rc (k) r i k , (26.17) wheree iis anI-dimensional vector whose ith component is 1 with all other components being 0 and

c (k) = λ (k) r y i k − hr i k , x (k)i

− u (k) i k

withi k = [k(modI) + 1].

µ x be any element of R J Then for any real numbers λ (k) satisfying

0< ε1 ≤ λ (k) ≤ ε2< 2 , (26.19)

the sequence x (0) , x (1) , x (2) , determined by the algorithm given above converges to the unique vector

x which minimizes Eq ( 26.16 ).

The implementation of this algorithm is hardly more complicated than that of basic ART which

is described at the beginning of this subsection We need an additional sequence ofI-dimensional

vectorsu (k), but in thekth iterative step only one component of u (k)is needed or altered Since the

i ks are defined in a cyclic order, the components of the vectoru (k) (just as the components of the

measurement vectory) can be sequentially accessed (The exact choice of this — often referred to as

the data access ordering — is very important for fast initial convergence; it is described in [11] The underlying principle is that in any subsequence of steps, we wish to have the individual actions to be

as independent as possible.) We also use, for every integerk ≥ 0, a positive real number λ (k) (These

are the so-called relaxation parameters They are free parameters of the algorithm and in practice

need to be optimized [11].) Ther i are usually not stored at all, but the location and size of their nonzero elements are calculated as and when needed Hence, the algorithm described by Eq (26.17) and Eq (26.18) shares the storage efficient nature of basic ART and its computational requirements are essentially the same Assuming, as is reasonable, that the number of nonzeror i,j is of the same order asJ , we see that the cost of cycling through the data once using ART is of the order IJ , which is

approximately the same as the cost of reconstructing using FBP (That this is indeed so is confirmed

by the timings reported in [12].) An important thing to note about Theorem26.1is that there are

no restrictions of consistency in its statement Hence, the algorithm of Eqs (26.17) and (26.18) will converge to the minimizer of Eq (26.16) — the so-called regularized least squares solution — using

the real data collected in any application

26.8 Expectation Maximization (EM)

We may wish to find thex that maximizes the likelihood of observing the actual measurements,

based on the assumption that ith measurement comes from a Poisson distribution whose mean is given by Eq (26.15) An iterative method to do exactly that, based on the so-called EM (expectation

maximization) approach, was proposed in [13] Here we discuss a variant of this approach that was designed for a somewhat more complicated optimization criterion [14], which enforces smoothness

of the results where the original maximum likelihood criterion may result in noisy images

Let RJ

+denote those elements of RJin which all components are non-negative Our task is to find

thex in R J

+which minimizes

I

X

i=1

[hr i , xi − y ilnhr i , xi] + γ

where theJ × J matrix S (with entries denoted by s j,u ) is a modified smoothing matrix [1] which has the following property (This definition is only applicable if we use pixels to define the basis functions.) LetN denote the set of indexes corresponding to pixels that are not on the border of the

digitization Each such pixel has eight neighbors, letN j denote the indexes of the pixels associated with the neighbors of the pixel indexed byj Then

x T Sx =X

j∈N



x j−1 8

X

k∈N j

x k





2

Consider the following rules for obtainingx (k+1)fromx (k).

p j (k) =

I

X

i=1

r i,j

9γ s j,j − x j (k)+ 1

9s j,j

J

X

u=1

s j,u x (k)

q j (k) = x

(k) j

9γ s j,j

I

X

i=1

r i,j y i

x j (k+1) = 1

2 −p (k) j +r

p j (k)2+ 4q j (k)

!

Since the first term of Eq (26.22) can be precalculated, the execution of Eq (26.22) requires essentially no more effort than multiplyingx (k)with the modified smoothing matrix As explained

in [1], there is a very efficient way of doing this The execution of Eq (26.23) requires approximately the same effort as cycling once through the data set using ART; see Eq (26.18) Algorithmic details of efficient computations of Eq (26.23) appeared in [15] Clearly, the execution of Eq (26.24) requires

a trivial amount of computing Thus, we see that one iterative step of the EM algorithm of Eq (26.22)

to Eq (26.24) requires, in total, approximately the same computing effort as cycling through the data set once with ART, which costs about the same as a complete reconstruction by FBP A basic difference between the ART method and the EM method is that the former updates its estimate based on one measurement at a time, while the latter deals with all measurements simultaneously

x (0) , x (1) , x (2) , generated by the algorithm of Eqs ( 26.22 ) to ( 26.24 ) converges to the minimizer

+.

26.9 Comparison of the Performance of Algorithms

We have discussed four very different-looking algorithms and the literature is full of many others, only some of which are surveyed in books such as [1] Many of the algorithms are available in general purpose image reconstruction software packages, such as SNARK [10] The novice faced with a prob-lem of reconstruction is justified in being puzzled as to which algorithm to use Unfortunately, there

is not a generally valid answer: the right choice may very well be dependent on the area of application and on the instrument used for gathering the data Here we make only some general comments regarding the four approaches discussed above, followed by some discussion of the methodologies that are available to comparatively evaluate reconstruction algorithms for a particular application Concerning the two transform methods we have discussed, the linogram method is faster than FBP (essentially anN2logN method, rather than an N3method as is the FBP) and, when the data are collected according to the geometry expressed by Eqs (26.8) and (26.9), then the linogram method

is likely to be more accurate because it requires no interpolations However, data are not normally collected this way and the need for an initial interpolation together with the more complicated-looking expressions that need to be implemented for the linogram method may indeed steer some users towards FBP, in spite of its extra computational requirements

Advantages of series expansion methods over transform methods are their flexibility (no special relationship needs to be assumed between the object to be reconstructed and the measurements taken, such as that the latter are samples of the Radon transform of the former) and the ability to control the type of solution we want by specifying the exact sense in which the image vector is to

be estimated from the measurement vector; see Eqs (26.16) and (26.20) The major disadvantage is that it is computationally much more intensive to find these precise estimators than to numerically evaluate Eq (26.1) Also, if the model (the basis functions, the projection matrix, and the estimation criterion) is not well chosen, then the resulting estimate may be inferior to that provided by a transform method The recent literature has demonstrated that usually there are models that make the efficacy

of a reconstruction provided by a series expansion method at least as good as that provided by a transform method To avoid the problem of computational expense, one usually stops the iterative process involved in the optimization long before the method has converged to the mathematically specified estimator Practical experience indicates that this can be done very efficaciously For

example, as reported in [12], in the area of fully three-dimensional PET, the reconstruction times for FBP are slightly longer than for cycling through the data just once with a version of ART using spherically symmetric basis functions and the accuracy of FBP is significantly worse than what is obtained by this very early iterate produced by ART

Since the iterative process is, in practice, stopped early, in evaluating the efficacy of the result of a series expansion method one should look at the actual outputs rather than the ideal mathematical optimizer Reported experiences comparing an optimized version of ART with an optimized version

of EM [9,11] indicate that the former can obtain as good or better reconstructions as the latter, but at

a fraction of the computational cost This computational advantage appears to be due to not trying

to make use of all the measurements in each iterative step

The proliferation of image reconstruction algorithms imposes a need to evaluate the relative per-formance of these algorithms and to understand the relationship between their attributes (free pa-rameters) and their performance In a specific application of an algorithm, choices have to be made regarding its parameters (such as the basis functions, the optimization criterion, constraints, relax-ation, etc.) Such choices affect the performance of the algorithm and there is a need for an efficient and objective evaluation procedure which enables us to select the best variant of an algorithm for a particular task and to compare the efficacy of different algorithms for that task

An approach to evaluating an algorithm is to first start with a specification of the task for which

the image is to be used and then define a figure of merit (FOM) which determines quantitatively

how helpful the image is, and hence the reconstruction algorithm, for performing that task In the numerical observer approach [11,16,17], a task-specific FOM is computed for each image Based

on the FOMs for all the images produced by two different techniques, we can calculate the statistical significance at which we can reject the null hypothesis that the methods are equally helpful for solving

a particular task in favor of the alternative hypothesis that the method with the higher average FOM

is more helpful for solving that task Different imaging techniques can then be rank-ordered on the basis of their average FOMs It is strongly advised that a reconstruction algorithm should not be selected based on the appearance of a few sample reconstructions, but rather that a study along the lines indicated above be carried out

In addition to the efficacy of images produced by the various algorithms, one should also be aware

of the computational possibilities that exist for executing them A survey from this point of view can

be found in [2]

References

[1] Herman, G.T.,Image Reconstruction from Projections: The Fundamentals of Computerized Tomography, Academic Press, New York, 1980.

[2] Herman, G.T., Image reconstruction from projections,J Real-Time Imag., 1, 3–18, 1995.

[3] Herman, G.T., Tuy, H.K., Langenberg, K.J., and Sabatier, P.C.,Basic Methods of Tomography and Inverse Problems, Adam Hilger, Bristol, England, 1987.

[4] Sanz, J.L.C., Hinkle, E.B., and Jain, A.K.,Radon and Projection Transform-Based Computer Vision, Springer-Verlag, Berlin, 1988.

[5] Edholm, P and Herman, G.T., Linograms in image reconstruction from projections,IEEE Trans Med Imag., 6, 301–307, 1987.

[6] Herman, G.T., Roberts, D., and Axel, L., Fully three-dimensional reconstruction from data collected on concentric cubes in Fourier space: implementation and a sample application to MRI,Phys Med Biol., 37, 673–687, 1992.

[7] Edholm, P., Herman, G.T., and Roberts, D.A., Image reconstruction from linograms: imple-mentation and evaluation,IEEE Trans Med Imag., 7, 239–246, 1988.

[8] Lewitt, R.M., Alternatives to voxels for image representation in iterative reconstruction algo-rithms,Phys Med Biol., 37, 705–716, 1992.