Noise Reduction by Projection Direction ... - Semantic Scholar

Noise Reduction by Projection Direction Dependent Diffusion for Low Dose Fan-beam X-ray Computed Tomography Shaojie Tanga,b, Xuanqin Mou a, Yanbo Zhanga, Hengyong Yuc,d Institute of Image processing & Pattern recognition, Xi'an Jiaotong University, Xi'an, Shaanxi, b 710049, China; School of Automation, Xi'an University of Posts and Telecommunications, c Xi'an, Shaanxi, 710121, China; Department of Radiology, Division of Radiologic Sciences, Wake Forest University Health Sciences, Winston-Salem, NC, 27157, USA; dBiomedical Imaging Division, VT-WFU School of Biomedical Engineering and Sciences, Wake Forest University Health Sciences, Winston-Salem, NC, 27157, USA a

ABSTRACT We propose a novel method to reduce the noise in fan-beam computed tomography (CT) imaging. First, the inverse Radon transform is induced for a family of differential expression of projection function. Second, the diffusion partial differential equation (PDE) is generalized from image space to projection space in parallel-beam geometry. Third, the diffusion PDE is further induced from parallel-beam geometry to fan-beam geometry. Finally, the projection direction dependent diffusion is developed to reduce CT noise, which arises from the quantum variation in the low dose exposure of a medical x-ray CT (XCT) system. The proposed noise reduction processes projections iteratively and dependently on x-ray path position, followed by a general CT reconstruction. Numerical simulation studies have demonstrated its feasibility in the noise reduction of low dose fan-beam XCT imaging. Keywords: Noise Reduction, X-ray CT, Low Dose, Fan-beam

1. INTRODUCTION Worldwide there are growing concerns on radiation induced genetic, cancerous and other diseases. Facing the increasing radiation risk, the well-known ALARA (As Low As Reasonably Achievable) principle is widely accepted in the medical community. Therefore, a large number of researches have been carried out to decrease radiation dosage while keep the diagnostic image quality. One of important strategies is the so-called noise reduction technique. First, a patient is irradiated by an x-ray source with a low radiation dose during XCT imaging; Then, a noise reduction scheme is utilized to remove the noise in image or projection space. As for the noise reduction in image space, anisotropic diffusion is a superior processing tool [1,2]. Simultaneously, the research on the noise reduction in projection space has also become a hot topic at present [3-13]. Now, researchers have understood the statistical features of the sinogram data in a medical XCT [3,4], which becomes the solid basis for the developments of the sophisticated algorithms in projection space. Here, we have a simple review for those algorithms. Hsieh proposed an ‘ATM’ algorithm based on the local statistical results in projection space [5], which combined the effects of both mean and median filters together. Kachelrieβ et al. generalized a multidimensional adaptive filtering for the conventional, spiral single-slice, multi-slice, and cone-beam CT. CT image noise is dramatically reduced, while the degradation of resolution can be neglected [6]. La Riviere estimated the line integrals from the noisy projection by maximizing a penalized-likelihood objective function, which was constructed based on the assumption of a compound Poisson distribution of the polychromatic XCT imaging [7]. Based on the experimental study result on the noise properties of XCT sinogram data [4], Wang et al. developed a sequence of noise reduction algorithms mainly in projection space [8-12], which include Maximum a posteriori (MAP), Penalized weighted least-squares (PWLS), and Multi-scale Penalized Weighted Least-Squares (Multi-scale PWLS). Recently, Zhu et al. explored an improved PWLS algorithm in projection space. There is a substantial insight to the essential effect of scatter estimation and correction on the performance of noise reduction and resolution [13]. In this paper, we will generalize the isotropic diffusion PDE to the projection space in fan-beam geometry, and develop a noise reduction scheme by using a projection direction dependent diffusion in projection space. Begin the Medical Imaging 2011: Physics of Medical Imaging, edited by Norbert J. Pelc, Ehsan Samei, Robert M. Nishikawa, Proc. of SPIE Vol. 7961, 79613L · © 2011 SPIE · CCC code: 1605-7422/11/$18 · doi: 10.1117/12.877635

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Introduction two lines below the Keywords. The manuscript should not have headers, footers, or page numbers. It should be in a one-column format. References are often noted in the text1 and cited at the end of the paper.

2. THEOREM 2.1 Radon Transform n

R , assume ( x) = ( x1 , x2 ,L , xn ) be the coordinate of an arbitrary point, and an unity vector is referred to as θ = (θ1 , θ 2 ,L , θ n ) . If f ( x) is a bounded image function with a finite support, then the Radon transform of the

In

image function can be expressed as [14],

Rf (θ , s) =

∫

R

where

f ( x)δ ( s − x ⋅θ )dx,

(2.1)

n

s denotes the projection position, and δ represents the Dirac function.

2.2 Hilbert Transform Hilbert transform of a projection function p(θ , s ) = Rf (θ , s ) can be expressed as, +∞

Hp (θ , s ) = pv ∫ p (θ , z ) −∞

π s−z

dz ,

(2.2)

where “pv” stands for the principal-valued integral. 2.3 Central Slice Theorem Based on the Radon and Fourier transforms, central slice theorem can be written as [14],

Rf ∧ (θ , ρ ) = f ∧ ( ρθ ), where sign

∧

(2.3)

denotes Fourier transform.

2.4 Imaging Geometries

Fig.1. Parallel-beam and equi-distant fan-beam geometries.

o - x1x2 is a Cartesian coordinate system, where o is the origin and rotational center; ξ and u are the view-angle and projection position for equi-distant fan-beam geometry; R0 and D are respectively the distances from x-ray source S to the origin o and to the x-ray linear detector in equi-distant fan-beam geometry; θ and s are the view-angle and projection position in parallel-beam geometry; θ = (θ1 ,θ 2 ) = ( cos θ ,sin θ ) and As shown Fig.1,

θ ⊥ = ( sin θ , − cosθ ) . Hereafter, we will discuss in R 2 .


2.5 Inverse Radon Transform of a Family of Differential Expression In parallel-beam geometry, the inverse Radon transform, i.e., the filtered backprojection (FBP) of a family of differential expression π

∫H 0

∂ 2m Rf (θ , s ) of projection function Rf (θ , s) within an angular interval [0, π ) has ∂s 2 m

∂ ∂ 2m Rf (θ , x ⋅θ )dθ ∂s ∂s 2 m

π

= ∫ 2π 0

π +∞

+∞

sgn( ρ ) 2m 2m ∧ i 2πρ x⋅θ i 2πρ x⋅θ ∧ ∫−∞ i i ρ (i ρ ) Rf (θ , ρ )e d ρdθ = 2π ∫0 −∞∫ sgn( ρ ) ρ (i ρ ) f ( ρθ )e d ρdθ

π +∞

= 2π ∫

i 2πρ x⋅θ 2m ∧ | ρ | d ρ dθ = 2π ∫ (i ρ ) f ( ρθ )e

0 −∞

+∞ +∞

∫ ∫ ( (iv ) 1

2

+ (iv2 ) 2 ) F (v)ei 2π v⋅ x dv1dv2 m

−∞ −∞

= 2πΔ m f ( x).

(2.4)

If m = 1 , there is

∂2 Rf (θ , s ) = 2π RΔf (θ , s ). ∂s 2

(2.5)

2.6 Diffusion Partial Differential Equation A general equation for an anisotropic diffusion PDE is

∂ f ( x) = ∇ ⋅ ( D ( f ) ∇f ( x) ) , ∂t where

(2.6)

D ( f ) is a direction dependent conduction coefficient. As for isotropic diffusion PDE, there is ∂ f ( x) = d Δf ( x), ∂t

(2.7)

where d is a direction independent conduction coefficient. 2.7 Projection Direction Dependent Diffusion in Projection Space 2.7.1. In Parallel-beam Geometry Calculating the radon transforms of the left- and right-hand sides of Eq.(2.7) gives

∂ Rf (θ , s) = dRΔf (θ , s). ∂t

(2.8)

∂ d ∂2 Rf (θ , s ) = Rf (θ , s ). 2π ∂s 2 ∂t

(2.9)

Substituting Eq.(2.5) into Eq.(2.8) has

2.7.2. In Equi-distant Fan-Beam Geometry Between parallel-beam and equi-distant fan-beam geometries, there is a following relationship in their projection variables,


θ = ξ + tan −1 ( u D ) , s = R0u

(

ξ = θ − tan −1 s

u 2 + D2 ,

)

R02 − s 2 , u = sD

R02 − s 2 ,

(2.10) (2.11)

Therefore, we have

Rf (θ , s) = g f (ξ , u ) ,

(2.12)

∂ Rf (θ , s) ∂s DR02 ∂ f ∂u ∂ f ∂ξ ∂ f ∂ f 1 g (ξ , u ) + g (ξ , u ) g (ξ , u ) g (ξ , u ) = = − 3 2 ∂u ∂s ∂ξ ∂s ∂u R0 − s 2 ( R02 − s 2 ) 2 ∂ξ =

1

( R02 − s 2 )

3

2

⎛ ⎞ ∂ f 2 ∂ g f (ξ , u ) − ( R02 − s 2 ) g (ξ , u ) ⎟ , ⎜ DR0 ∂u ∂ξ ⎝ ⎠

(2.13)

and

∂2 Rf (θ , s ) ∂s 2 ⎧ 1 ∂ ⎪ = ⎨ ∂s ⎪ R 2 − s 2 3 2 ) ⎩( 0 = +

⎛ ∂2 f ⎞ 2 DR02 1 ∂2 f ∂2 f ξ ξ g , u g , u g (ξ , u ) ⎟ − + ( ) ( ) ⎜ 2 2 2 2 2 ∂u ( R0 − s ) ⎝ ∂ξ ⎠ ( R 2 − s 2 ) ∂ξ∂u

D 2 R04

(R

2 0

− s2 )

3

0

2 0

3DR s

(R (u =

2 0

− s2 )

⎫ ⎛ ⎞⎪ ∂ f 2 ∂ f 2 2 g (ξ , u ) − ( R0 − s ) g (ξ , u ) ⎟ ⎬ ⎜ DR0 ∂u ∂ξ ⎝ ⎠⎪ ⎭

5

2

s ∂ f ∂ f g (ξ , u ) − g (ξ , u ) 3 2 2 2 ∂ξ ∂u ( R0 − s )

2 (u 2 + D2 ) ∂2 u 2 + D2 ) ⎛ ∂2 f + D2 ) ∂ 2 f ( ⎞ f g (ξ , u ) − g (ξ , u ) + ⎜ 2 g (ξ , u ) ⎟ 4 2 2 3 2 2 2 D R0 ∂u D R0 D R0 ⎝ ∂ξ ∂ξ∂u ⎠

2

3

2

3(u 2 + D2 ) u ∂ f u 2 + D2 ) u ∂ f ( + g (ξ , u ) − g (ξ , u ) . D 4 R02 ∂u D 3 R02 ∂ξ 2

In this situation, Eq.(2.9) can be converted as

∂ f g (ξ , u ) ∂t


(2.14)

=

2 2 d ( D 2 + u 2 ) ⎧⎪ 2 ⎞ ∂2 2 2 ∂ 2 2 2⎛ ∂ f f f u + D g u − D u + D g u + D ξ , 2 ξ , ( ) ( ) ) ( ) ⎨( ⎜ 2 g (ξ , u ) ⎟ 4 2 2 ∂u ∂ξ∂u 2π D R0 ⎩⎪ ⎝ ∂ξ ⎠

⎛ ⎞⎫ ∂ ∂ f +u ⎜ 3 ( u 2 + D 2 ) g f ( ξ , u ) − D g (ξ , u ) ⎟ ⎬ . ∂u ∂ξ ⎝ ⎠⎭

(2.15)

2.7.3. In Equi-angular Fan-Beam Geometry Between parallel-beam and equi-angular fan-beam geometries, there is a following relationship in their projection variables, (2.16)

θ = ξ + γ , s = R0 sin γ ,

)

(

ξ = θ − tan −1 s

(

R02 − s 2 , γ = tan −1 s

)

R02 − s 2 ,

(2.17)

Therefore, we have

Rf (θ , s) = g f (ξ , γ ) ,

(2.18)

∂ Rf (θ , s ) ∂s =

∂ f ∂γ ∂ f ∂ξ ∂ f 1 ∂ f 1 g (ξ , γ ) g (ξ , γ ) g (ξ , γ ) g (ξ , γ ) + = − 2 2 2 ∂γ ∂s ∂ξ ∂s ∂γ ∂ξ R0 − s R0 − s 2 =

⎛ ∂ f ⎞ ∂ f g (ξ , γ ) − g (ξ , γ ) ⎟ , ⎜ ∂ξ R − s ⎝ ∂γ ⎠ 1

2 0

2

and

∂2 Rf (θ , s ) ∂s 2 ⎛ ∂ f ⎞ ⎫⎪ ∂ ⎪⎧ ∂ f 1 − , , g ξ γ g ξ γ ( ) ( ) ⎨ ⎜ ⎟⎬ ∂s ⎪ R02 − s 2 ⎝ ∂γ ∂ξ ⎠ ⎭⎪ ⎩ ⎧∂ f ⎫ ∂ f s = g (ξ , γ ) − g (ξ , γ ) ⎬ + 3 ⎨ ∂ξ ⎭ ( R02 − s 2 ) 2 ⎩ ∂γ

=

1 2 R0 − s 2

⎧ ∂2 f ⎫ ∂ ∂ f ∂2 g (ξ , γ ) + 2 g f (ξ , γ ) ⎬ ⎨ 2 g (ξ , γ ) − 2 ∂ξ ∂γ ∂ξ ⎩ ∂γ ⎭


(2.19)

=

sin γ

⎧∂ f ⎫ ∂ f g (ξ , γ ) ⎬ + ⎨ g (ξ , γ ) − ∂ξ R cos γ ⎩ ∂γ ⎭ 3

2 0

1 2 R0 cos 2 γ

⎧ ∂2 f ⎫ ∂ ∂ f ∂2 g (ξ , γ ) + 2 g f (ξ , γ ) ⎬ . ⎨ 2 g (ξ , γ ) − 2 ∂ξ ∂γ ∂ξ ⎩ ∂γ ⎭

(2.20)

In this situation, Eq.(2.9) can be converted as

∂ f g (ξ , γ ) ∂t ⎧⎪ ⎛ ∂2 f ⎞ ∂ ∂ f ∂2 f = g (ξ , γ ) + 2 g (ξ , γ ) ⎟ cos γ ⎜ 2 g (ξ , γ ) − 2 3 ⎨ 2 ∂ξ ∂γ ∂ξ 2π R0 cos γ ⎪⎩ ⎝ ∂γ ⎠ d

⎛ ∂ f ⎞⎫ ∂ f + sin γ ⎜ g (ξ , γ ) − g (ξ , γ ) ⎟ ⎬ . ∂ξ ⎝ ∂γ ⎠⎭

(2.21)

3. NUMERICAL IMPLEMENTATION In numerical implementation, we will substitute the partial derivative of t with a difference method. In equi-distant fan-beam geometry, the discrete form of Eq.(2.15) is written as,

g

f

(ξ , u , t ) = g (ξ , u , t ) + i

j

k +1

f

i

j

k

Δ t d ( D 2 + u 2j ) 2π D 3 R02

⎞ ∂2 ⎪⎧ ⎛ 2 4 ∂ 2 f 2 ∂ ∂ g f (ξ i , u j , t k ) + 2 g f (ξ i , u j , tk ) ⎟ ⎨ D ⎜ D R0 2 g (ξi , u j , tk ) − 2 DR0 ∂u ∂ξ ∂u ∂ξ ⎠ ⎩⎪ ⎝ ⎛ ⎞⎫ ∂ f ∂ f +u j ⎜ DR02 g (ξi , u j , tk ) − g (ξi , u j , tk ) ⎟ ⎬ , g f (ξi , u j , t0 ) = g f (ξi , u j ) , k = 0,L , K , ∂u ∂ξ ⎝ ⎠⎭ where

g f (ξi +1 , u j , tk ) − g f (ξi −1 , u j , tk ) ∂ f , g (ξi , u j , tk ) ≈ ∂ξ 2Δξ

g f (ξi , u j +1 , tk ) − g f (ξi , u j −1 , tk ) ∂ f , g (ξ i , u j , tk ) ≈ ∂u 2Δ u

g f (ξi +1 , u j , tk ) − 2 g f (ξi , u j , tk ) + g f (ξi − 2 , u j , tk ) ∂2 f g (ξi , u j , tk ) ≈ , ∂ξ 2 Δξ2

g f (ξi +1 , u j +1 , tk ) − g f (ξi −1 , u j +1 , tk ) − g f (ξi +1 , u j −1 , tk ) + g f (ξi −1 , u j −1 , tk ) ∂ ∂ f g (ξi , u j , tk ) ≈ , ∂ξ ∂u 4Δξ Δ u


(3.1)

g f (ξi , u j +1 , tk ) − 2 g f (ξi , u j , tk ) + g f (ξi , u j −1 , tk ) ∂2 f g ξ , u , t ≈ , ( i j k) ∂u 2 Δ u2

(3.2)

In equi-angular fan-beam geometry, the discrete form of Eq.(2.21) is written as,

g f (ξi , γ j , tk +1 ) = g f (ξi , γ j , tk ) +

Δt d 2π R02 cos γ j

3

⎪⎧ ⎨ cos γ j ⎩⎪

⎛ ∂2 f ⎞ ∂ ∂ f ∂2 f − + ξ , γ , 2 ξ , γ , g t g t g (ξi , γ j , tk ) ⎟ ( ) ( ) ⎜ 2 i j k i j k 2 ∂ξ ∂γ ∂ξ ⎝ ∂γ ⎠ ⎛ ∂ f ⎞⎫ ∂ f + sin γ j ⎜ g (ξi , γ j , tk ) − g (ξi , γ j , tk ) ⎟ ⎬ , g f (ξi , γ j , t0 ) = g f (ξi , u j ) , k = 0,L , K , ∂ξ ⎝ ∂γ ⎠⎭

(3.3)

where

∂ f ∂ f ∂2 ∂ ∂ f ∂2 g (ξi , γ j , tk ) , g ( ξ i , γ j , tk ) , 2 g f ( ξ i , γ j , tk ) , g ( ξ i , γ j , tk ) , 2 g f ( ξ i , γ j , tk ) , ∂ξ ∂γ ∂ξ ∂ξ ∂γ ∂γ

(3.4)

are in a similar form as in

∂ f ∂ ∂2 ∂ ∂ f ∂2 g (ξi , u j , tk ) , g f (ξi , u j , tk ) , 2 g f (ξ i , u j , tk ) , g (ξ i , u j , tk ) , 2 g f (ξ i , u j , tk ) , ∂ξ ∂u ∂ξ ∂ξ ∂u ∂u

(3.5)

but with different variables.

λ = Δ t d . Since we have induced the isotropic diffusion PDE from image space to projection space, we may further modify the functionality of parameter λ . One direct method is as follows,

For the convenience of discussion, we let

b − max ( g f ( ξ i , u j ) − g f ( ξ i′ , u j ′ ) ) ⎛ ⎞ f f ( i′ , j′)∈NB( i , j ) λ (ξi , u j ) = a ⎜ g (ξi , u j ) max ( g (ξ m , un ) ) ⎟ e ( m,n) ⎝ ⎠

2

c2

,

(3.6)

where a, b, and c are adjusted empirically; pixel ( m, n ) is in the whole image; and pixel ( i′, j ′ ) is in the fournearest neighbor NB ( i, j ) of pixel ( i, j ) . Note that we have forced

λ as a function of variables ξ and u . In this

way, the diffusion intensity will be modulated according to the projection direction. The diffusion strategy is obviously different from both the isotropic diffusion in Eq.(2.7) and the anisotropic diffusion in Eq.(2.6).

4. COMPUTER SIMULATION In this paper, we only give a computer simulation in equi-distant fan-beam geometry to illustrate the feasibility of the proposed method. The noise-free projection is generated by using an in-house software [15]. Gaussian noise is added into the noise-free projection as follows, f f g noise (ξi , u j ) = g noise − free ( ξ i , u j ) +

1 ⎛ ⎛ k ⎞⎞ f Gaussian ⎜ 0, 4.5 × exp ⎜ 4 g noise − free ( ξ i , u j ) ⎟ ⎟ , k ⎝ 10 ⎠⎠ ⎝

f k = 1.4 × 105 max ( g noise − free ( ξ m , u n ) ) .

( m,n)


(4.1) (4.2)

where Gaussian

( μ ,σ ) 2

indicates a Gaussian quasi-random number with a mean

μ and variance σ 2 . The

projection modeling in Eq.(4.1) is designed based on the reference [4]. Although the projection modeling is not completely consistent with that in a real situation, it is believed that the simulated results are sufficient to illustrate the feasibility of the proposed method. In computer simulation, the projection g will be substituted by the projection g

f noise

f

(ξ , u ) in numerical implementation

(ξ , u ) defined in Eqs.(4.1) and (4.2). i

i

j

j

In Fig.1(b), a black cross is to illustrate the positions, where the images values are used to calculate the spatial resolutions along vertical and horizontal directions. Meanwhile, beside the left bone region, a blue circle is to illustrate a disk region, whose image values are all used to calculate the noise standard deviation (NSD). NSD-resolution tradeoff curves are plotted in Fig.5.

(a)

(b)

(c)

(d)

(e) [-0.05 0.05]

(f) [-0.001 0.001]

Fig.1. (a) Noise-free projection and (b) noise-free reconstruction; (c) noisy projection and (d) noisy reconstruction; (e) the difference image between (a) and (c), and (f) the difference image between (b) and (d).


Fig.2. (top) Noise-reduced reconstructions, and (bottom) the difference images between the corresponding reconstructions on the top row and noisy reconstruction. Parameters are a = 4.0x10-12 and b = 0, 1, 2, 4, and 8 from the left to the right columns.

Fig.3. Same as in Fig.2 but with a = 1.0x10-11.

Fig.4. Same as in Fig.2 but with a = 2.5x10-11.


34 32 30

Resolution [Pixel]

28

Vertical@a=4.0x10-12

26

Horizontal@a=4.0x10-12

24

Horizontal@a=1.0x10-11

Vertical@a=1.0x10-11 Vertical@a=2.5x10-11

22

Horizontal@a=2.5x10-11 Vertical@Noise Horizontal@Noise

20 18 16

4

5

6

7

8

9

10

11

12

13

14

Noise Standard Deviation [10-4/mm]

Fig.5. NSD-resolution tradeoff curves with different parameters a and b.

5. CONCLUSIONS In this paper, we have proposed a novel method to reduce the quantum noise in low dose fan-beam XCT imaging, based on the projection direction dependent diffusion in projection space. Computer simulation has successfully demonstrated the feasibility of the proposed method. Validations by using projection data acquired in a real XCT imaging system will be our future’s works.

ACKNOWLEDGEMENT This work was partially supported by the National Science Fund of China (No.60551003), the program of NCET (No.NCET-05-0828), and the fund of the Ministry of Education of China (No.20060698040).

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