backprojection by upsampled fourier series expansion and interpolated fft

IEEE TRANSACTIONS ON IMAGE PROCESSING. VOL. I. NO. I . JANUARY 1992 77

Backprojection by Upsampled Fourier Series Expansion and Interpolated FFT

Makoto Tabei and Mitsuhiro Ueda

Abstract-A fast backprojection method through the use of interpolated fast Fourier transform (FFT) is presented. The computerized tomography (CT) reconstruction by the convolution backprojection (CBP) method has been proved to produce precise images. Specifically, the realization of “de-blur” filtering by linear convolution is crucial for precise reconstruction. However, the backprojection part of the conventional CBP method is not very efficient: it leads to high computational complexity, and arbitrary control of the frequency characteristics of the interpolation function is not possible. In this paper, we propose an alternative approach to interpolating and backprojecting the convolved projections onto the image frame. First, the upsampled Fourier series expansion of the convolved projection is calculated. Then, using a Gaussian function, it is projected by the aliasing-free interpolation of FFT bins onto a rectangular grid in the frequency domain. The total amount of computation in this procedure for a 512 x 512 image is 1 /5 of the conventional backprojection method with linear interpolation. This technique also allows the arbitrary control of the frequency characteristics.

I. INTRODUCTION

COMPUTATIONALLY efficient method of comput- A erized tomography (CT) reconstruction that follows directly from the projection theorem is to fill the two-dimensional (2-D) Fourier space by the one-dimensional (1-D) transforms of the projections and then to take the 2-D inverse Fourier transform. This type of reconstruction technique relies on an interpolation of discrete Four- ier transform (DFT) bins to convert projection data from a polar to a raster grid in the frequency domain. However, the interpolation which is based on an ordinary fill-in con- cept does not work well with DFT. Therefore, it is prone to yielding an inferior reconstructed image [ 11-[3].

Recently, we have proposed an interpolation technique of the DFT bins based on a deconvolution technique using Gaussian functions [4]. This procedure can interpolate DFT bins precisely, i.e., it produces arbitrary frequency 1- and 2-D sinusoids with the DFT. However. when we

Manuscript received May 26, 1990; revised October 23, 1990. M. Tabei was with the Research Laboratory of Electronics, Massachu-

setts Institute of Technology, Cambridge, MA. He is now with the Preci- sion and Intelligence Laboratory, Tokyo Institute of Technology, Midori- ku, Yokohama, 227 Japan.

M. Ueda is with the International Cooperation Center for Science and Technology, Tokyo Institute of Technology, Meguro-ku, Tokyo, 152 Ja- pan.

IEEE Log Number 9104330.

apply this technique to the above CT reconstruction, a strong obstacle still exists to precise reconstruction. The CT reconstruction requires the “de-blur’’ filtering of the projection data. The filter has a response proportional to the absolute value of the frequency. The singularity of such a filter (the derivative of frequency response is not continuous at the origin) leads to a slowly decaying spatial response, and it causes significant aliasing with the DFT.

In contrast, the convolution backprojection (CBP) algorithm ingeniously separates the filtering operation into two steps: the projection data are linearly convolved with the de-blur filter in the spatial domain to realize the singular frequency response, then it is interpolated (low-pass filtered) and backprojected onto the image frame [ 11-[3], [5]. This algorithm has been shown to be accurate and amenable to implementation. However, the backprojection is a very time consuming process, especially when higher order interpolation is employed to realize good low- pass filter characteristics. This computational complexity is due to the fact that the contribution of the convolved projection at each projection angle must be evaluated and summed at each pixel in the image frame.

As a result, we propose an alternative approach to interpolating and backprojecting the convolved projection onto the image frame. The procedure is as follows. The convolved projection is transformed using a 1-D fast Fourier transform (FFT) and is replicated at higher frequencies to perform upsampling [6], [7]. (The interpolation generally requires upsampling before filtering out higher frequency components.) It is then multiplied with the frequency response of the interpolation function. The resultant sequence provides the Fourier series coefficients of the convolved projection to be backprojected. Each of the coefficients is given along a radial line in polar coordinates, and the Gaussian function is used to project them onto a rectangular grid. When the convolved projections for all angles have been projected in the frequency domain, a 2-D inverse fast Fourier transform (IFFT) operation is performed and correction of its result by division with the Gaussian function produces a reconstructed image.

The computation required in this procedure for a 512 x 512 image is 1 / 5 of that required for conventional backprojection with linear interpolation. The use of any higher order interpolation functions does not affect the

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78 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. I. NO. I. JANUARY 1992

computational complexity, since it only requires a re- placement of the tables representing the frequency response of the interpolation function.

In the following, we describe only the backprojection part of the reconstruction procedure as the linear convolution part is exactly the same as that described in the lit- erature [ 11, [3]. In Section 11, we explain the interpolation of the convolved projection through the use of its Fourier series expansion. The fast evaluation of the Fourier series sum at each pixel in the image frame, i.e., backprojection, is discussed in Section 111. The results are shown in Section IV. In Section V we compare with other reported work.

11. INTERPOLATION BY FOURIER SERIES In CT reconstruction by the CBP method, the recon-

structed imagef(x, y), where (x, y) are the coordinates in the image plane, is given by the following backprojection of the convolved projection q(0, r ) :

+ y . sin (r 6)) where NO is the number of projections, 7r(iO/No) is the discretized angular coordinate 0, and r is the coordinate which is given by the (signed) distance from the origin to the projection of (x, y) onto the projection line at angle a ( i o / N O ) (see Fig. 1). In the digital system, q ( r ( i o / N o ) , r ) is given only at the sampling points of r , i,. A r ( A r is a sampling interval of r ) . Thus (1) requires the interpolation of q with r from its values at the sampling points.

Here we interpolate through the use of a Fourier series coefficient Q(?r(iO/Ne), kr), which is given by multiplying the frequency response of the interpolation function F by the DFT of q:

exp ( - j 2 r e) N r - 1

q ( R- ;O’ r ) = c Q ( R - :0’ k, ) exp ( j21r- N r L r k r ) kr = - N r

Nr Nr 2 2

A r I r 5 - A r . _ - (3)

(4)

The relation between (2) and (3) is similar to that of the DFT and IDFT; however, ( 3 ) uses 2Nr terms of (2), which is twice as many as the Nr used by the IDFT. This enables

Y 4

t Fig. 1. Coordinate system for backprojection.

filter F to cause higher frequency components to decay smoothly. The result is that the basis for interpolation becomes local rather than becoming the rippled and slowly decaying form of the sin (x)/x function.

If we choose F as a “half-band filter,” which has an- tisymmetric characteristics about the folding frequency, this process using the DFT and IDFT satisfies the interpolation property, i.e., the weightings are 1 at the sampling bin and 0 at the other bins [6]. This is a good characteristic; however, it is not necessarily required in most applications, such as CT reconstruction [7]. On the other hand, it is crucial that the frequency response converge to double or higher order zeros to avoid the Gibbs phenom- enon by the truncation of the Fourier series.

The frequency characteristics of two half-band filters F2 and F3 are given as follows [6]:

The frequency response of these filters and their inverse Fourier transforms (which serve as a basis for interpolation) are shown in Fig. 2. F3 is almost identical to the modified cubic spline function given in [8].

We also give the frequency response of the linear interpolation FL to compare the reconstructed image with that of conventional backprojection by linear interpolation:

The function FL is not a half-band filter and does not satisfy the interpolation property when it is used in a finite sum; however, it has the double zero at the sampling frequency, like F2 and F3.


TABEI AND UEDA: BACKPROJECTION BY UNSAMPLED FOURIER SERIES EXPANSION 79

0.0 0.5 1.0 1.5 2.0

Normarized frequency

(a)

-3 -2 - 1 0 1 2 3

Bin distance (./AT)

(b) Fig. 2 . Fz , F3 and their inverse Fourier transforms. (a) Spectrum. (b) In-

verse Fourier transform of (a).

Putting ( 3 ) into ( 1 ) with x and y discretized as i, A x and i, A y , we obtain the following:

where G: is a 2-D Gaussian function in the wrapped and normalized form which we define as follows:

G?,,(s,, s,, P , M,, M,)

= c c m m

m, = --03 m, = -a

P (12 )

In this equation, only the first two arguments of G i . are variables and P is a positive constant which is proportional to the square of the width of the Gaussian function in the frequency domain. The last two arguments are integer constants which are equal to DFT data lengths.

N O - I N , - I T

No i g = O k , = - N , f(i, A X , i, A y ) = -

N, A r

L L

N, si,<- 2 2

The direct evaluation of (8) over all points of i, and i, requires 2NxN, NrNfl summations. This is a significant increase in comparison with the N,N, No required by the conventional backprojection in the spatial domain. There- fore, the described procedure does not have any signifi- cance without the existence of a fast computational procedure for the sinusoidal functions.

N, --

111. FAST BACKPROJECTION The 2-D complex exponential function of frequencies

f, and fy over the range given by (9) and (10) is synthesized through the use of a 2N, X 2N, point inverse DFT

Equation (1 1) is based on the deconvolution technique; therefore, it works well only over a limited range of values of 0, i.e., if we choose to be excessively small, an aliasing error caused by the periodicity of the DFT increases; and if we choose P to be excessively large, a rounding error increases due to the compensation by the exponential function after the DFT. However, if we choose an appropriate value of 6, this process gives a sta- ble and suprisingly precise value on the interval given by (9) and (10).

In practice, this procedure still has the problem of eval- uating the infinite sum in (12 ) . However, we can truncate


80 IEEE TRANSACTIONS ON IMAGE PROCESSING. VOL I . NO. I. JANUARY 1992

the Gaussian distribution without affecting the result, if the values become sufficiently small. (See Appendix A for the discussion on the choice of 0 and truncation condition.)

Now we put (11) into (8), and obtain

f ( i x Ax, i, AY)

where

2 N , - 1 2 N r - 1 / Nd-I N r - l / . \

N, e AX c, = ~

N, * A r

If we choose C, = Cy = 1 , the size of the reconstruction region becomes equal to the size of the projection region.

IV. RESULTS The convolved projection q of length N , can be calcu-

lated by the linear convolution of the projection data with the filter function using a 2N, point FFT [ 11. Here we use a Shepp-Logan type filter function [ 5 ] .

To calculate the Fourier series coefficients Q , convolved projection q is circularly shifted by N , / 2 (Fig. 3(a)), and its N , point 1-D real FFT is used to give N , / 2 + 1 points of complex data. It is then replicated at the higher frequencies by complex conjugate folding, and multiplied by the frequency response of the interpolation function (Fig. 4(a)).

The resultant sequence is an N, + 1 point complex Fourier series set of coefficients (Fig. 4(b)). It is projected with blurring by the Gaussian function onto the 2N, X (N, + 1) complex grid along the corresponding projection line (Fig. 5(a)). The Fourier series extends to twice the folding frequency; therefore, the higher frequency components exceed the boundary, which should be treated by complex conjugate folding or wrapping.

Then a 2-D real inverse FFT is calculated to produce 2N, x 2N, real data. It is then circularly shifted by N, and

Fig. 3 .

----

(b) Circular shift of the data.

-1

(a) (b)

Fig. 4. Generation of Fourier series coefficient. (a) DFT of q . (b) Q(a(is/W U.

(b) (C)

Fig. 5. Backprojection in frequency domain. (a) Backprojection in frequency domain. (b) Backprojection before compensation. (c) Backprojec- tion after compensation.

Ny according to (9) and (10) (see Fig. 3(b); only its central N, X N, points are shown in Fig. 5(b)). Its division by the Gaussian function produces the backprojection (Fig 5(c)). In actual reconstruction, the convolved projection for all projection angles are projected in the frequency domain before the 2-D FFT.

The Gaussian function in the frequency domain is calculated with 0 = 4.65 on a 7 x 7 point distribution. It satisfies relative precision better than lop3 (see Appen- dexes A and B). In this case, the magnitude of the exponential correction term after the DFT is less than 3 in the reconstruction circle; therefore, the increase of rounding error is negligible. Furthermore, note that while this correction magnifies the arithmetic rounding error, it does not magnify the noise in the projection data. (In general,


TABEI AND UEDA: BACKPROJECTION BY UNSAMPLED FOURIER SERIES EXPANSION

(b) =

Fig . 6. Reconstructed image. (a) -0.1-2.1. (b) 0.99-1.04.

TABLE I R E C O N S T R U C T I O N E R R O R

Proposed Conventional

(linear) interpolation F, F. F ,

Inside Skull 8.421 x IO-' 8.357 x 10 7.983 x I0 ' 7 507 x IO- ' Whole Area 6.249 x IO-' 6.242 x I O - ' 5.859 x 10 ' 5.259 x I O - '

the effective bit length for arithmetic is far larger than that for projection data.)

A 2-D Gaussian function is separable to the product of two 1-D Gaussian functions which can be calculated by second-order recursion [9] (Appendix B) . Therefore, the major part of the computation is the generation of a 2-D Gaussian function as the product of a complex Fourier series coefficient and two 1-D Gaussian functions. It takes around 2 X 7 X 7 X N , 2: 100 N, real multiplications and additions for every projection line. Therefore, the total number of operations is 100 A'," multiplications and additions. This is a significant reduction from the NoN,Ny multiplications and 3NB N,N, additions of conventional backprojection with linear interpolation.

Fig. 6 shows a 128 x 128 reconstructed image of the Shepp-Logan phantom with interpolation function FL. In graphic representation, it looks exactly the same as the image reconstructed by conventional backprojections with linear interpolation. Table I shows a comparison of reconstruction errors. (The error is evaluated as the rms distance from the phantom.) F3 gives the best accuracy. We used highly aliased projection data, which is produced by direct sampling of the phantom without any prefiltering

TABLE I1 COMPUTATIONAL T I M E (SECONDS BY SUN SPARKSTATION 1)

(A ) ( B ) Conventional Proposed

N r X Ns Nr x N , B-P B-P ( A ) I ( B )

64 x 64 64 X 64 0.9 1.6 0.6 128 X 128 128 X 128 7.3 6 . I 1.20 256 x 256 256 x 256 58.4 25.0 2.34 512 X 512 512 X 512 465.5 fQ1.7 4.58

before sampling. For bandlimited projection data, the advantage of F3 will become more apparent.

Table I1 shows the computational times. From the above discussion, the proposed method should work 10 times faster than conventional backprojection for a 512 X 512 image. The difference is due to the computation of 1- and 2-D FFT and other miscellaneous computations.

V . DISCUSSION

Many reports exist on CT reconstruction with FFT [lo]-[13]. In fact, some methods are even more computationally efficient than the procedure presented in this pa-


82 IEEE TRANSACTIONS ON IMAGE PROCESSING. VOL I , NO I . JANUARY 1 Y Y 1

per [ 131. However, these methods are based on the con- cept related to the sampling theorem that states that the bandlimited signal can be reconstructed from the frequency components up to folding frequency. Although they generally work well, in practice, the assumption con- cerning the bandlimiting conditions makes these proce- dures somewhat vulnerable to aliased data and/or reduces the resolving power in comparison with the CBP method.

In contrast, the proposed method is intended to mimic the backprojection operation of the CBP method in the frequency domain. The frequency response of the linear interpolation function of conventional backprojection has a first zero at the sampling frequency, i.e., twice the folding frequency. To achieve these characteristics, the DFT of the convolved projection is extended up to the sampling frequency by replication using complex folding. As a result, the proposed method becomes less susceptible to aliased projection data while maintaining high resolving power.

Fig. 7(a) and (b) shows the overall frequency characteristics and spatial response of the method. The singular characteristic of the deblur filter is realized precisely by the linear convolution, and its repetitive images beyond the sampling frequency are filtered out by low-pass filter characteristics of interpolation. Note that the interpolation does not have any response beyond sampling frequency in constrast to the conventional CBP method. Ac- tually, the results for the proposed method converge with the results of the CBP method if we extend the Fourier series expansion further with frequency response FL. However, the reconstruction error does not improve, because the frequency components beyond the sampling frequency do not contribute to the reconstruction at all.

On the other hand, better reconstruction precision is obtained by adjusting the frequency Characteristics below the sampling frequency; for instance, the use of a half- band filter F2 or F3 gives better precision. This is the reason why the higher order interpolation function has been introduced in the conventional CBP method. With the proposed method, arbitrary frequency characteristics can be obtained without increasing computational complexity.

In the proposed procedure, prior to backprojection, the projection data are processed by a 2Nr-point DFT and IDFT to realize the linear convolution, and then its N,-point DFT is calculated. This back and forth operation between spatial and frequency domains may seem redun- dant. However, in the strict sense, the deblur filter is only realizable as a linear convolution because of its singular frequency characteristics at the origin; therefore, it is crucial to go back to the spatial domain once and abandon the wrapped portion of the convolved projection and then calculate the DFT.

For the latter half of the method, a similar procedure for the generation of the arbitrary frequency sinusoidal function has been reported with the prolate spheroidal function or Kaiser-Bessel function [ 121. This previously

-e: ' -2 I ' -1 I ' ; ' ; ' ; ' ; Bin distance ( r / A r )

(b) Fig. 7. Overall frequency characteristics. (a) Overall frequency response.

(b) Overall spatial response.

reported work is optimal in the sense that a minimum number of distributions is required to attain the required precision. The method proposed in this paper is not optimal in the same sense. It requires slightly more points of distribution for the same precision because the Gaus- sian distribution never reaches zero, thus we need to calculate the value until it becomes negligible. For instance, for the relative precision of the Kaiser-Bessel function requires a 6 x 6 distribution and the Gaussian function requires a 7 x 7 distribution; for lop6, 12 x 12 and 14 x 14 distributions are required, respectively. The difference corresponds to a 36% increase in the grid points in the two dimensions.

The sampled values of the distribution must be generated each time depending on the central position of the distribution. The Gaussian function has an advantage in the practical sense, because it can be generated by a re- cursive procedure (see Appendix B), whereas the Kaiser- Bessel function requires interpolation of precomputed tables. Our implementation of the procedure requires only two multiplications per grid point whereas the reported work requires seven multiplications. Their procedure does not utilize the fact that the distribution is separable in two directions; however, even with an improvement in that respect, their procedure requires more computations on the whole. The published research has not reported the actual computation time; however, we can infer it might be even slower than the conventional CBP method because of the significant increase in the amount of data during the filtering operation. With the Gaussian function, the optimization procedure is explicit, and we can evalu-


TABEI AND UEDA: BACKPROJECTION BY UNSAMPLED FOURIER SERIES EXPANSION 83

ate the necessary condition immediately from the required precision.

The use of the Gaussian function has a further advantage. One can modulate the amplitude and frequency of the synthesized signal because the procedure maintains the natural characteristics of the Gaussian distribution [4]. This is quite useful in the analysis of many physical prob- lems, and will be discussed in detail in a future publica- tion.

VI. CONCLUSION

In this paper, we described the use of upsampled Four- ier series to simulate the CBP method in the frequency domain, and we also discussed its fast evaluation with an interpolated FFT. Although many reports exist on suc- cessful CT reconstruction with the FFT technique, in the strict sense, the accuracy of reconstructed images has not been equivalent to that of the CBP method. The proposed method improves the computational efficiency by an order of magnitude without loss of precision.

With recent advances in hardware, the computational consideration has become less important. However, there has remained a long-standing question: why is the CT reconstruction by FFT not equivalent to that of the CBP method? This paper gives one answer to that question.

Besides its computational efficiency, this procedure gives versatility to the CT reconstruction. One aspect of this versatility is the arbitrary choice of the frequency characteristics of the interpolation. This provides an adaptive filtering capability, depending on the signal-to- noise ratio (SNR) of the projection data. Furthermore, this procedure may be directly applicable to the fast computation of back-propagation algorithms.

APPENDIX A

In the following, we will discuss the derivative of (1 1). For the sake of simplicity, we will start the discussion with a 1-D case, and extend it to the 2-D case afterwards.

The Fourier transform of a Gaussian distribution is also a Gaussian distribution [ 141. By shifting the center of the distribution to p , in the frequency domain, the next relation holds:

where x and U are the variables in the spatial and frequency domain, respectively. is a positive constant that is proportional to the square of the width of the Gaussian distribution in the frequency domain.

By replacing the infinite integral on the right-hand side of (16) with the infinite sum using Poisson’s sum formula, we obtain the following [ 141 :

exp ( j 2 a g) . Here we express the discrete coordinates of x and U as i x / 2 N x and k,, respectively, where 2N, is length of the 1-D (inverse) DFT. This equation states that by replacing the integral of the Fourier transform with the summation, we obtain the wrapped form of the transform as a result.

Substituting t, = p , / 2 N , and dividing both sides of (17) by exp ( - rP( i , /2N,)2) , we obtain the following:

m

exp ( j 2 a i , t X ) + c exp (-rp 6, i + 1;)) l x = -m, # 0

* exp ( j 2 a ( i x + 2N,l,)t,)

exp ( j 2 r 5) where G$ is defined as

(19)

For the convenience of the following discussion, we limit the range of i, as follows:

-N, I i, 5 N, - 1. (20)

Then the absolute value of the aliasing term, which ap- pears as the summation on the left-hand side of (1 8), can be evaluated as follows:

We neglect the terms other than the two major ones because they have smaller values by several orders of magnitude.


84 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. I . NO. I . JANUARY 1992

Now, we denote the desired precision for the result as E . Then EA must satisfy the following in the interval given by ( 9 ) (see Fig. 8).

E! takes the largest value at the both ends of the interval. Therefore, E! should satisfy

E ! ( - + ) I E

and we obtain

(24) 2 p 2 - - log ( E ) . a

In practice, it is not possible to evaluate the infinite Gaussian distribution of ( 1 9 ) . Typically, we wish to ap- proximate it by calculating only NI points around its center and by truncating the others. Fig. 8 illustrates the truncation. The error resulting from this truncation is given by

* (sum at points outside of N I ) . (25)

This is due to the fact that the DFT is bounded by the absolute sum of the input data. The total sum at the outside is dominated by the values of the first two points just outside the NI points. With a typical value of 0, the next adjacent values are less than lo-* of the inner neighbors and the ratio becomes even larger as it gets further from the center. The distance d from the center of the Gaussian distribution to the first truncation is given by

Nr NI - 1 d ~ - + 1 . 2 2

The largest error occurs when the truncation is most as- symetrical. When it is symetrical, i.e., when d is bal- anced at both sides, the error sum is much smaller due to the convergent nature of the Gaussian distribution. Thus the largest value of E: is evaluated as follows:

E: also has to satisfy the following in the interval given by (9):

Fig. 8 . Truncation of Gaussian distribution.

-112 -114 0 114 112

i x l ( 2 N x ) Fig. 9 . Aliasing and truncation errors.

and takes its largest values at the extremes of this interval (see Fig. 9 ) . Therefore, it follows that:

E:(- : ) I E . (29)

For simplicity, we neglect p'" in the denominator of (27 ) , and substitute ( 2 4 ) and ( 2 7 ) into ( 2 9 ) , to obtain

In practical implementation, it is advantageous to choose a value for /3 which is as large as possible but which does not increase N, to the next integer value, because such an increase will allow some margin for error without increasing the amount of necessary computation.

The roundoff error of the arithmetic should also be taken into consideration. The roundoff error E: is given by

where E , is a representation error of floating point numbers because

E , exp ( j217 - ;:)I = E , .


TABEI AND UEDA: BACKPROJECTION BY UNSAMPLED FOURIER SERIES EXPANSION

~

85

The typical values of E , is 2 - 2 4 for the IEEE single precision floating point number and the magnitude of E : is usually much smaller than Ef, or Ef . However, E: im- poses a certain limitation on the achievable precision E .

The roundoff also takes the largest value at both sides of (9). Substituting (24) into (31) gives

El (- i) 2 E - ( ’ / ’ ) * E , . (33)

Therefore:

E , . (34) E 2 E - ( l / 8 ) .

Then. hence:

E 2 €;I9. (35) As a result, the achievable precision is limited to 8/9 of the arithmetic bit length.

Now we can discuss the 2-D case, using the results of the above discussion. The 2-D sinusoidal function with frequencies t, and ty is calculated by

exp ( j27r(ixt, + i,t,))

= exp (j27rix t,) exp ( j 2ai , ty)

2Nr - I

exP (TO ( $ ) 2 ) k , c = 0 Gk(k, - 2Nxtx, P , 2N,)

* G,!& - 2Nyty, P , 2NJ * exp (j27r $) (36)

(37)

The aliasing error E: and truncation error E : in the inter- vals in (9) and (10) are given by the following:

* (1 + EA(&)) - 1

As a result, the values of /3 and NI should be chosen with the same limit as for the 1-D case.

The roundoff error in the 2-D case is given in a somewhat different way as follows:

This is because, due to the random nature of the roundoff error, the errors given by the implementation of (36) and (37) are different. (Note that (36) requires fewer numbers of operations than (37); however, (36) cannot be used for the calculation if there is more than one Gaussian distribution.) However, in practice, the roundoff will not yield significantly different results from the simple sum of (31) unless we choose an excessively large 0. (The reason for the near equivalency of results is that since we evaluate E ; as the absolute sum of the input to the 2-D DFT in analysis, the practical error becomes much smaller by the statistical effect of different signs of many roundoff components. of the 2-D distribution.) We found by numerical experiment that the condition given by (35) is also suffi- cient in a 2-D case.

APPENDIX B

Fig. 10 shows a C subroutine which generates a Gaus- sian distribution by second-order recursion. The subroutine starts the recursion from the center of the distribution and works out to either side. (Otherwise, although it may not be a serious problem for CT reconstruction, the second-order recursion produces a fairly large numerical error, which is proportional to the square of the number of steps [9].) The 2-D distribution is generated as the product of two 1-D distributions in orthogonal directions. Fig. 1 l(a) and (b) show the error distribution of the generated complex exponential function using this subroutine (the folding part was omitted from the subroutine) followed by a 64 x 64 2-D FFT. (We generate a complex exponential, rather than a real sinusoid, because the error from real sinusoids has many zeros and is difficult to show in graph- ical representation. The error from real sinusoids is bounded by the error of complex exponentials.) The amplitude of the synthesized complex exponential is 1, and the error is evaluated as the absolute value of the difference with the complex exponential function calculated by double precision arithmetic. The frequencies are t, = 14.O1/2Nx and ty = 2 1 . 5 / 2 N y for both cases. and NI are 4.65, 7 , and 9.3 , 14, respectively. The error distribution is not affected significantly by the choice of frequencies.


86 IEEE TRANSACTIONS ON IMAGE PROCESSING. VOL. I . NO. I . JANUARY 1992

#include <stdio.h> #include <math.h> #define real float #define PI 3.141592653589793238

typedef struct cmplx

void gaus2d( gw, mx, my, beta, sx, sy. q )

I* This subroutine adds one Gaussian distribution to the two dimensioal array "gw[]" on each call. The array should be cleared externally before the first call.

To form the image, we process the resulting array "gw" with two dimensional real inverse FFT, rearrange the area (swap first half and latter half in each dimension), and divide by Gaussian function (See text for detail).

gw[] : Two dimensional complex array which holds the

mx, my : Size of "gw[]". Physical size of "gw(]" will be

( real re; real im; ) complex;

spectrum of the image.

'lmx+(my/2+1)" when spectrum is folded for real valued image.

beta : Squared width of Gaussian function in spectral domain. sx, sy : Coordinate of the center of the Gaussian function. q : Complex amplitude of the Gaussian function.

"mx" , "my" and "beta" are supposed to be identical on each call.

*I

complex g w [ l . q; real beta, sx, sy; int mx, my;

i real vx[20], vy[20], vy-re, vy-im; real dx, dy. wd, el-m, el-p, e2; int nt, jx, jy, kxO, kyO, kx, ky, kx-f;

nt = ceil( 1.5 * beta ) ; I* "nt" : number of distributions. * I wd = - PI 1 beta; / * These three variables are identical * I e2 = exp( 2.0 * wd ) ; I* on each call. * I

kxO = ceil( sx - 0.5 nt ) ; I * (kx0,kyO) designates the lower left * I kyO = ceil( sy - 0.5 nt ) ; I * indices of mapping area. * I

dx = ( kxO + nt 1 2 ) - sx; / * -0.5 <= dx. dy < 0.5, "nt" : odd. *I dy = ( kyO + nt 1 2 ) - sy; I * 0.0 <= dx, dy < 1.0, "nt" . ' even. * I

if( ( kxo = kxo % mx ) < 0 ) kxO += mx; if( ( kyO = kyO % my ) < 0 ) kyO += my;

/ * 0 <= kxO <= mx-1 * / I * 0 <= kyO <= my-1 * I

I * Generate two 1-D Gaussian distributions * I

vx[ nt 1 2 ] = exp( wd * ( dx * dx + dy * dy ) ) 1 beta; vy[ nt I 2 J = 1.0; el-m = exp( 2.0 * wd * ( 0.5 - dx ) ) ; el-p = e2 I el-m;

for( jx = nt 1 2 - 1; jx >= 0; jx -- ) ( / * downward recursion * I vx[ jx ] = vx[ jx + 1 ] * el-m; el-m = el-m * e2;

1

for( jx = nt 1 2 + 1; jx < nt; jx ++ ) ( / * upward recursion * I

> el-m = exp( 2.0 * wd * ( 0.5 - dy ) ) ; el-p = e2 1 el-m;

for( jy = nt I 2 - 1; jy >= 0; jy -- ) ( / + downward recursion * /

vx[ jx ] = vx[ jx - 1 ] el-p; el-p = el-p * e2;

vy[ jy 1 = vy[ jy + 1 1 * e1-m; el-m = el-m e2;

> for( jy = nt 1 2 + 1; jy < nt; jy ++ ) ( / * upward recursion * I

vy[ jy 1 = vy[ ju - 1 1 * e1-p; el-p = el-p * e2;

}

/ * Map 2-D Gaussian distribution on "gw[l" * I

ky = kyO; for( jy = 0; jy < nt; jy ++ ) (

v-re = q.re * vy[ jy I; vy-im = q.im * vy( jy 1; if( ky <= my 1 2 ) ( / * check folding * I

kx = kxO; for( jx = 0 ; jx < nt: jx ++ ) {

qw[ mx ky + kx ].re += vy re * vx[ jx I; gw[ mx * ky + kx ].im += vyIim * vx[ jx 1 ; if( ( ++ kx ) == mx ) kx = 0;

)

} else ( / * delete following if folding is not required * I

if( kxo == 0 ) kx-f = 0; else kx-f = mx - kxO;

for( jx = 0 ; jx < nt: jx ++ ) ( gw[ mx 4 ( my - ky ) + kx f ].re += vy re * vx[ jx I; gw[ mx * ( my - ky ) + kxIf ].im -= vyIim * vx[ jx I ; if( ( -- kx-f ) < 0 ) kx-f = mx - 1;

)

I

if( ( ++ ky ) == my ) ky = 0; 1

)

Fig. 10. A C subroutine to generate Gaussian distribution.


87 TABEl A N D U E D A : BACKPROJECTION BY U N S A M P L E D FOURIER SERIES EXPANSION

[6] R. E. Crochiere and L. R. Rabiner, Multirare Digital Signal Pro- cessing. Englewood Cliffs, NJ: Prentice-Hall, 1983.

[7] H. W. Schussler and P. Steffen, “A hybrid system for reconstruction of a smooth function from its samples,” Circuits, Syst. Signal Pro- cessing, vol. 3, pp. 295-314, 1984.

[8] G . T. Herman, S. W. Rowland, and M . Yau, “A comparative study of the use of linear and modified cubic spline interpolation for image reconstruction,” IEEE Trans. Nucl. Sci., vol. NS-26, pp. 2879-2894, 1979.

[9] L. R. Rabiner, R. W. Schafer, and C. M. Rader, “The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust, vol.

[IO] R. M. Mersereau and A. V. Oppenheim, “Digital reconstructions of multidimensional signals from their projections,” Proc. IEEE, vol.

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interpolation,” IEEE Trans. Biomed. Eng. , vol. BME-28, pp. 496-

[ 121 J. P. O’Sullivan, “A fast Sinc function gridding algorithm for Fourier inversion in computer tomography,” IEEE Trans. Med. Imaging, vol.

[I31 P. Edholm, G . T. Herman, and D. A. Roberts, “Image reconstruction from linograms: Implementation and evaluation,” IEEE Trans. Med. Imaging, vol. 7, pp. 239-246, 1988.

[I41 S. Papoulis, Systems and Transforms with Applications in Optics. New York: McGraw-hill, 1968.

AU-17, pp. 86-92, 1969.

62, pp. 1319-1338. 1974.

2 N z 2

2, 505, 1981.

MI-4, pp. 200-207, 1985. Yy

- 4 2

i, 0 M a k o t o Tabei was born in Osaka, Japan, on Feb- ruary 12, 1956. He received the B.E. degree in information engineering from the University of Electro-Communications, Tokyo, Japan, in 1979, and the M.E. degree in information processing from the Tokyo Institute of Technology, Tokyo, Japan, in 1981.

He worked at the Japan Atomic Energy Re- 2 N z search Institute in Ibaraki, Japan, in 1981. Since

1982, he has been a Research Associate at the Pre- cision and Intelligence Laboratory, Tokyo Insti-

tute of Technology. During 1989-1990 he was a Visiting Scientist at the Research Laboratory of Electronics, Massachusetts Institute of Technol- ogy, MA. His research interests include numerical analysis of wave phe- nomena and signal processing in acoustical applications.

_ - N Y 2

- NY N* ,, - Ns 2

-N, --

(b) Fig. 11. Error contour map of synthesized complex exponential. (a) 0 =

4.65, N , = 7. ( b ) p = 9.3, N , = 14.

ACKNOWLEDGMENT The authors would like to thank the members of DSPG, Mitsuhiro U e d a was born in Tokyo, Japan, on

January 13. 1942 He received the B.E. degree in instrument engineering from Keio University, Yokohama, Japan, in 1964, the M.E. degree i n

REFERENCES electrical engineering and the Dr.Eng degree in control engineering from Tokyo Institute of Tech-

[ I ] A C Kak, “Computerized tomography with x-ray, emission, and nology, Tokyo, Japan in 1966 and 1969, respec- ultrasound sources,” Proc. IEEE, vol 67, pp 1245-1272, 1979 tively

[21 R M. Lewitt, “Reconstruction algorithms Transform methods,” From 1969 to 1972 he was a Research Associ- Proc. IEEE, vol 71, pp. 390-408, 1983 ate and in 1972 he became a Associate Professor

131 A. K. Jain, Fundamentals of Digital Image Processing Englewood of the Precision and Intelligence Laboratory at the Cliffs, NJ: Prentice-Hall, 1989 Tokyo Institute of Technology He is currently a Professor at the Interna-

141 M Tabei and M Ueda, “FFT multi-frequency synthesizer,” in Proc. tional Cooperation Center for Science and Technology, Tokyo Institute of ICASSP-88, Apr 1988, pp 1431-1434 Technology His current research interests are in the areas of theoretical

[5] L A Shepp and B F. Logan, “The Fourier reconstruction of a head analysis of wave propagation. the inversion problem of scattering, and sig- section,” IEEE Trans Nucl. Sci , vol NS-21, pp 21-42, 1974 nal processing.

MIT, for their assistance and encouragement.


backprojection by upsampled fourier series expansion and interpolated fft

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