Abstract—Total variation (TV) minimization algorithm is

widely used for solving incomplete data problems in computer

tomography. Based on the POCS-TV method, we propose a new

scheme of TV minimization algorithm, B-POCS-TV-β, which is

similar to the Bregman iteration in the sense of “adding the

residual back”. Numerical simulations are implemented with

few-view and limited-angle problems. We compare the

reconstruction error and image quality of B-POCS-TV-β and

POCS-TV. B-POCS-TV-β improves the convergence speed a lot.

Fewer iterations are needed to obtain the same accuracy.

Index Term — Computer tomography (CT), TV minimization,

POCS-TV, few-view, limited-angle, B-POCS-TV-β.

I. INTRODUCTION

In computer tomography (CT), there are various situations

where it is impossible to measure the complete data due to

scanning geometry or imaging hardware. These problems can

arise in mammography, dental CT or industrial CT, known as

limited-angle problems. In other applications, a complete

scanning is available, but under-sampling projections from only

a few angles are taken for a tradeoff between image quality and

radiation dose or scanning time, leading to a few-view problem.

Since the incomplete data problem is ill-conditioned,

conventional reconstruction methods, such as filtered back-

projection, usually lead to artifacts in reconstructed images due

to the incompleteness of data. Various methods are proposed to

solve the incomplete data problems. Among all these methods,

Total variation (TV) minimization algorithm is a great success

for piecewise constant or piecewise continuous image

reconstruction and is widely used in recent years. The

breakthrough work in compressive sensing (CS) of Candes et al.

accurately recovers an image from highly sparse sampling

along 22 random selected radial lines in the Fourier domain via

TV regularized optimization [1].

The well-known TV minimization algorithm is stated as

follows：

Authors are all with the Department of Engineering Physics, Tsinghua

University and Key Laboratory of Particle & Radiation Imaging (Tsinghua

University), Ministry of Education, Beijing, P. R. China, 100084 (e-mail:

zhangli@nuctech.com, blueplasmaice@gmail.com).

min . . , 0TV

f s t Mf g f= ≥ (1)

where f is the discrete image vector, g is the measured

projections, and M is the system matrix. Under certain

assumption that the image is piecewise constant, the TV norm,

which is in fact the l1-norm of the gradient of image, is usually

sparse, or approximately sparse, which is usually the case in

medical imaging and other applications. TV minimization

algorithm takes the advantage of sparseness and effectively

seeks the solution with the sparsest gradient, hence suppressing

the artifacts and greatly improving the image quality.

Sidky et al.[2] solve the constrained problem in a hybrid

scheme, alternately performing TV-minimization and

projection onto convex sets (POCS). TV-minimization is

performed by the gradient descent method. This algorithm

appears to be highly accurate and easy for implementation. We

denote this algorithm as POCS-TV through this article.

In this article, we combine the POCS-TV method with a

Bregman framework for solving the constrained problem (1).

Bregman iteration is originally adopted for solving

unconstrained optimization problems and extended by

Goldstein and Osher to solve constrained problems, with a

simplified form. They applied the Bregman framework to solve

not only the general l1-regularized optimization problem but

also TV denoising and CS problems [3]. The Bregman iteration

has many advantages, e.g. it converges very quickly.

Our method is in a Bregman-wise framework, but differs

from the traditional Bregman iteration in two senses: first,

traditional Bregman iteration solves the constrained problem by

solving sequential unconstrained problems that combines the

objective function with the constrained function via a Lagrange

multiplier into a Lagrangian, however, we just take advantage

of the effectiveness of Bregman updating and consequently

solve a sequence of constrained problems directly, using

POCS-TV to obtain an approximate solution at each iteration;

second, experiments prove that a decreasing factor is often

required to ensure convergence while in traditional Bregman

iteration it is unnecessary.

II. METHODOLOGY

The key procedure of the Bregman iteration is to solve a

sequence of subproblems instead of the original problem,

calculating the subgradient at each iteration and adding the

An Improved TV Minimization Algorithm for

Incomplete Data Problem in Computer

Tomography

Hui Xue, Li Zhang, Member, IEEE, Zhiqiang Cheng, Member, IEEE,

Yuxiang Xing and Yongshun Xiao

2621978-1-4244-9105-6/10/$26.00 ©2010 IEEE

wresidual back. It yields very accurate solutions even if the

subproblems are not solved as accurately.

In Bregman iteration, solving the unconstrained problem

2

2

1min ( )

2J u u bµ + −

(2)

is equivalent to solving a series of problems

( )1k k kb b b u

+ = + − (3)

21 1

2

1min ( )

2

k ku J u u bµ+ += + −

(4)

with the initial value 0 00b u= = . Let the observed signal b be

composed of the true signal and noise, it is well illustrated in [4]

that by “adding residues back” in the iterations, each time both

the “unrecovered good signal” and the “bad noise” are

strengthened in kb , thus giving rise to a more accurate

decomposition and a better approximation to the true signal.

We take advantage of this “updating” step in Bregman

techniques. Our work is in a similar framework, replacing the

TV-minimization problem with a series of subproblems and

solving each subproblem via POCS-TV.

ALGORITHM 1 B-POCS-TV

1) Initialization:0 0f = , 0g g= , 1k = ;

2) POCS-TV: solving

min . . , 0k k k k

TVf s t Mf g f= ≥ (5)

by POCS-TV of PN iterations;

3) Updating: 1k k k

g g g Mf+ = + − (6)

4) 1k k= + .

The B-POCS-TV algorithm appears to converge faster than

POCS-TV at first, but unfortunately, usually after about a few

dozens of iterations, the error starts to increase slowly, leading

to unexpected divergence, which is illustrated in III.A. In order

to ensure the convergence of B-POCS-TV, we use a technique

which resembles the well-known backtracking method[5]. We

use a tunable updating stepsize β and a decreasing factor c is

introduced to control β , so that as 0β → , B-POCS-TV-β

gradually becomes POCS-TV. We denote the number of outer

iterations as BN and the equivalent number of iterations N as

the number of total interior POCS-TV iterations in both

algorithms.

ALGORITHM 2 B-POCS-TV-β

1) Initialization:0

0f = ,0

g g= , 0k = ,0

1β = ,decreasing

factor 1c < , step-updating period T ;

2 )POCS-TV: solving

min . . , 0k k k k

TVf s t Mf g f= ≥ (7)

by POCS-TV of PN iterations;

3)Modified updating:

( )1k k k kg g g Mfβ+ = + − ;

4 )Stepsize updating:

1, mod( 1, ) 0

,

k

k

k

if k T

c otherwise

ββ

β+

+ ==

; (8)

5) 1k k= + .

III. NUMERICAL SIMULATION

We validate our method on numerical experiments for

few-view and limited-angle problems. For ease of comparison

we use the same geometry configuration as in [2]. The detectors

are composed of 512 bins with a total length of 41.3cm. The

distance from the source to the axis is 40cm while the distance

from the source to the central detector is 80cm The parameters

of the experiments are shown in Table I.

A. Few-view

We used the 256×256 Shepp-Logan phantom to evaluate of

our algorithms for few-view problems. Projection data was

generated at 20 view angles specified by (19) in [2]. Both

POCS-TV and B-POCS-TV-β algorithms were tested for

comparison. B-POCS-TV was also tested to illustrate the

necessity of the decreasing factor. In POCS-TV, the number of

iterations N is 200 while each iteration consists of 20 interior

TV iterations. In B-POCS-TV-β, we chose 0.75c = , 1T = for

simplicity, and 10P

N = for an approximate solution. B

N is

chosen in all our experiments to ensure an equivalent POCS

iteration number in both algorithms, i.e. B P

N N N= . The

relative error at each iteration is defined as * *kf f f− ,

where *

f is the exact solution. The convergence speed of the

algorithms can be observed in Figure .1, where the relative error

vs. iteration number for the test image is plotted, on a

logarithmic scale. It can be seen that B-POCS-TV-β converges

faster than POCS-TV, obtaining lower reconstruction error

with the same number of iterations, while B-POCS-TV

diverges.

Figure. 1. Few-view reconstruction, relative error vs. iteration number for

POCS-TV(dash dot line) , and B-POCS-TV-β(solid line):, with the result of

B-POCS-TV(dashed line).

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B. Limited-angle

In limited-angle problem, various experiments were taken for

different scanning angular ranges. Projections uniformly

distributed over an angular range of 180°, 120°and 90°were respectively obtained for reconstruction, with

0.8,0.9, 0.95c = in B-POCS-TV-β and 500, 1000, 2000

equivalent iterations.

The relative error vs. iteration number for the test image is

shown in Figure .2. Results of 500 iterations from projections

over 90°are shown in Figure .3, it is obvious from (a) and (b)

that B-POCS-TV-β provides a smoother and more accurate

image.

IV. CONCLUSION

We have proposed a new method to improve the convergence

speed of the TV minimization algorithm for few-view and

limited-angle problems, lower reconstruction error and better

(a) (b)

(c)

(d)

Figure. 3. Results of 500 iterations from projections over 90°: (a) B-POCS-TV-β,

(b) POCS-TV, (c) Vertical profile, (d) Horizontal profile. The exact value is in

dashed line, while the result is in solid line for B-POCS-TV-β and in dash dot line

for POCS-TV

(a)

(b)

(c)

Figure. 2. Relative error vs. iteration number for POCS-TV(dash dot line) , and

B-POCS-TV-β(solid line), limited angel case: (a) 180°, (b) 120°, (c) 90°.

TABLE I

PARAMETERS OF THE EXPERIMENTS

Angular Range c T B

N P

N N

Few-view

0.75 1 20 10 200

Limited-angle

90° 0.95 1 50 10 500

120° 0.90 1 50 20 1000

180° 0.80 1 50 40 2000

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quality can be obtained with the same number of iterations.

APPENDIX

A. POCS-TV

Sidky et al. developed an accurate reconstruction algorithm

for few-view and limited-angle problems [2]. The POCS-TV

algorithm is employed as a CS approach solving the

optimization problem (1). In this algorithm, the projection data

constraint and the minimization of the image TV are handled

by the gradient descent method and POCS separately. In the

POCS step, both data consistency and positivity are enforced.

The POCS-TV steps are as follows:

1) Initialization: 1n = ,0 0f = ;

2) POCS-step, data consistency: 1

,1 1

2

nn n Ti i

i

i

g M ff f M

M

−− −

= + ;

3)POCS-step, data positivity: ,1 ,1

,2

,1

, 0

0, 0

n n

i in

i n

i

f ff

f

>=

≤;

4)TV-step, ,2 1

2

n n

id f f

−= − ,

TV

i

i

fv

f

∂=

∂,

1,2

2

n n

i i

vf f ad

v

−= − ;

5) 1n n= + and return to 2)

ACKNOWLEDGMENT

This work is supported by the National Natural Science

Foundation of China under the project No. 10875066.

REFERENCES

[1] E. Candes, J. Romberg and T. Tao, “Robust Uncertainty Principles: Exact

Signal Reconstruction From Highly Incomplete Frequency Information,”

IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 489–509, 2006.

[2] E.Y. Sidky, C.-M. Kao, and X. Pan, “Accurate image reconstruction from

few-views and limited-angle data in divergent-beam CT,” Journal of X-ray

Science and Technology, vol. 14, pp. 119–139, 2006.

[3] T. Goldstein, S Osher, The split Bregman method for l1-regularized

problems, SIAM J. Imaging Sci., vol. 1, no.1, pp. 323–343, 2009. [4] W. Yin, S. Osher, D. Goldfarb, and J. Darbon, “Bregman Iterative

Algorithms for l1-Minimization with Applications to Compressed Sensing”,

SIAM J. Imaging Sci., vol. 1, no.1, pp. 143–168, 2008.

[5] D.P. Bertsekas,”Nonlinear Programming, 2nd Edition”, Athena Scientific,

1999.

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