This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 124.16.148.19

This content was downloaded on 27/03/2015 at 04:55

Please note that terms and conditions apply.

A multi-scale geometric flow method for molecular structure reconstruction

View the table of contents for this issue, or go to the journal homepage for more

2015 Comput. Sci. Disc. 8 014002

(http://iopscience.iop.org/1749-4699/8/1/014002)

Home Search Collections Journals About Contact us My IOPscience

A multi-scale geometric flow method for molecularstructure reconstruction

Guoliang Xu1, Ming Li and Chong ChenState Key Laboratory of Scientific and Engineering Computing, Institute of ComputationalMathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences,Beijing 100190, Peopleʼs Republic of ChinaE-mail: xuguo@lsectioncc.ac.cn, liming@lsectioncc.ac.cn and chench@lsectioncc.ac.cn

Received 13 April 2014, revised 16 July 2014Accepted for publication 11 February 2015Published 26 March 2015

Computational Science & Discovery 8 (2015) 014002

doi:10.1088/1749-4699/8/1/014002

AbstractWe have previously reported an L2-gradient flow (L2GF) method for cryo-electron tomography and single-particle reconstruction, which has a reasonablygood performance. The aim of this paper is to further upgrade both the com-putational efficiency and accuracy of the L2GF method. In a finite-dimensionalspace spanned by the radial basis functions, a minimization problem combininga fourth-order geometric flow with an energy decreasing constraint is solved by abi-gradient method. The bi-gradient method involves a free parameterβ ∈ [0, 1]. As β increases from 0 to 1, the structures of the reconstructedfunction from coarse to fine are captured. The experimental results show that theproposed method yields more desirable results.

Keywords: reconstruction, electron microscopy, tomography, single-particleanalysis, geometric flow

1. Introduction

In recent decades, cryo-electron microscope imaging techniques have been established asindispensable tools for determining the three-dimensional (3D) structures of large macro-molecules and biological machinery. These techniques can be separated into several imagingmodalities, including single-particle analysis (multiple copies of a low- or high-symmetry unit),electron tomography (single copy of a low-symmetry unit) and electron crystallography(multiple copies of a low-symmetry unit symmetrically arranged in a lattice) (see [21]).

1 Author to whom any correspondence should be addressed.

Computational Science & Discovery 8 (2015) 0140021749-4699/15/014002+26$33.00 © 2015 IOP Publishing Ltd

In this paper, we focus on single-particle analysis in which multiple copies of identicalparticles are imaged at different, randomly chosen orientations. We assume that the alignmentproblem would already have been solved using existing methods (see [6, 11, 12, 17, 18, 20]).The next step is to conduct the 3D image reconstruction, which is the problem addressed by thealgorithm introduced in this paper. The reconstruction algorithms are often weighted back-projection (WBP) methods [16], direct Fourier methods [16] or iterative methods, including thealgebraic reconstruction technique [9], the simultaneous iterative reconstructive technique [8]and the simultaneous algebraic reconstruction technique [1]. In order to accelerate convergenceof these iterative algorithms, block iterative techniques have been proposed previously(see [9, 14]).

The problem of producing 3D reconstructions from a series of two-dimensional (2D)projection images is typically an inverse problem. Theoretically, an object can be reconstructeduniquely from its projections when all of its projections are known. In practice, however, only alimited number of projection images can be obtained. Hence, in most cases, the inverse problemis ill-posed. Thus some constrained conditions should be imposed so that the problem is well-posed. One such technique that imposes these constrained conditions is the projection ontoconvex sets, which assumes that the object f belongs to the intersection of some closed convexsets (see [19, 25]). Another technique that can be used to find a well-posed approximationsolution is the regularization technique. It should be noted that classic Tikhonov regularizationhas been employed in tomography reconstruction [15].

An L2-gradient flow method (L2GF) has been presented previously [13, 24] for cryo-electron tomography reconstruction (see [24]) and single-particle analysis (see [13]). Byminimizing an energy functional consisting of a fidelity term and a regularization term, anL2GF is derived. The flow is solved by a finite-element discretization in the B-spline functionspace in the spatial direction [22] and an explicit Euler scheme in the temporal direction. Theexperimental results show that the proposed method is stable, reliable and robust. A completetheoretical analysis of the convergence has been published previously [5]. The convergenceanalysis for solving the L2GF using the semi-implicit finite element method has been given in[4]. Therefore, the L2GF method is effective.

The purpose of this paper is to further enhance both the computational speed and accuracyof the L2GF method, using B-spline radial basis functions rather than B-spline basis functionsas the basis functions, and a fourth-order geometric flow as a regularizer. We present a bi-gradient method to solve the involved variational model in a finite-dimensional space spannedby the B-spline radial basis functions. There are several advantages of using the B-spline radialbasis functions. First, it is possible to obtain a C2-smooth object, and the local support propertyof the B-spline basis can accelerate the reconstruction process. Second, the x-ray transformationof the B-spline basis has a closed-form representation, hence, no integration is required,resulting in a remarkable reduction in computational cost. Third, as an electric potential, f is thesolution of the Poisson and Boltzmann equations. This solution can be approximated by aGaussian map, which is the sum of certain radial basis functions, hence using the radial basis ismore reasonable.

The paper is organized as follows. In section 2, some basic setting is defined, includingimage size, the B-spline radial basis function grid and the volume grid. The geometric flow usedand its level-set formulation are presented in section 3. In section 4, we explain the algorithmsin details, while section 5 gives some illustrative examples and discussions. Finally, we

2

Comput. Sci. Discov. 8 (2015) 014002 G Xu et al

conclude this paper in section 6. Some detailed derivations are presented in appendices Aand B.

2. Problem setting

Let =g u{ ( )} lp

d 1l, ∈ ∈ Sd u,l

2 2, be a set of 2D projections measured from an unknown 3Dfunction (electric potential) f by the x-ray transform Xdl

i.e.

∫= = + ∈

( )g X f f t t Su u e e u d d( ) ( ) : , d , ,l ld d d d(1) (2) 2

l l l l3

⎡⎣ ⎤⎦where ed

(1)land ed

(2)lare two directions in ⊥dl satisfying

∥ ∥ = ∥ ∥ = =e e e e1, , 0, (1)d d d d(1) (2) (1) (2)

l l l l

∥ ∥· stands for the Euclidian normal of a vector in 3. ⊥dl is the space consisting of all the vectorsperpendicular to dl. ed

(1)land ed

(2)lalso determine the in-plane rotation of the projection. Our

problem is to reconstruct f x( ), Ω∈ ⊂ x 3, such that X f u( )dlis as close to g u( )dl

as possiblein the following sense.

=f J f* arg min ( ),where

∫∑= −=

J f

pX f gu u u( )

1( ) ( ) d . (2)

l

p

d d1

2

l l2

⎡⎣ ⎤⎦

We assume that all the measured images have the same size + × +n n( 1) ( 1), and the

pixel values g u( )dlof each image are defined on the integer grid points ∈ −i j( , ) ,T n n

2 2

2⎡⎣ ⎤⎦ (we

assume n is an even number). Since the values of g u( )dlare the projection of f, we therefore

define a sphere Ω as

Ω = ∈ ∥ ∥ ⩽ +{ }nx x:

21 .3

For simplicity, we put the sphere Ω into a cube defined as Ω = − − +1, 1cn n

2 2

3⎡⎣ ⎤⎦ , with theassumption that

Ω Ω= ∈ ⧹

= + >

f

g i j i jn

x x( ) 0, if ,

( , ) 0, if2

.

c

d2 2

l

Then the image values g u( )dlat the grid points are defined as

∫= + + ∈ −−∞

∞( )g i j f i j t t i j

n ne e d( , ) d , ( , )

2,

2,l

Td d d

(1) (2)2

l l l

⎡⎣⎢

⎤⎦⎥

for the unknown function f.Because the measured images g u( )dl

are affected by heavy noise, a regularizationmechanism is absolutely necessary. To avoid the regularization effect destroying the accuracyof the reconstruction result, we reconstruct the electric potential f using the following twostages.

3

Comput. Sci. Discov. 8 (2015) 014002 G Xu et al

Stage one. Compute an approximate minimizer

≈∈

f J farg min ( ), (3)f V

1

using our bi-gradient iterative method (see section 4.3), where V is a given function space (weuse radial basis function space defined by B-splines; see section 4.1).

Stage two. Compute a regularized function of f1, denoted as f, using a geometric flow withthe following constraint (see section 4.6)

⩽ ( )J f J f( ) . (4)1

Remark 2.1. We should point out that stage 2 is not simply a smoothing or a post-processing ofthe result from the first stage. The second stage requires that the energy functional is notincreasing.

3. Gradient and geometric flow

We want to minimize J(f) by adjusting f iteratively. This goal is achieved by a bi-gradient flow,in a radial basis function space, combining with a geometric flow. In the following, we firstcompute the gradient and then introduce the geometric flow. Suppose f is represented as

∑ ϕ= =f f i j kx x i( ) ( ), [ , , ] .T

ii i

Then it is easy to see that

∫∑ ϕ∂∂

= −=

( )J f

f pX f g Xu u u u

( ) 2( ) ( ) ( ) d .

l

p

id d d i

1l l l2

⎡⎣ ⎤⎦The gradient of J(f) with respect to fi is denoted as J f( ( )).

Owing to the insufficient number of projection directions, the minimization problem (3) isnot well-posed, meaning there may be an infinite number of solutions. To overcome thisdifficulty, we combine the minimization process with the surface diffusion flow. In theparametric form, the surface diffusion flow is represented as (see [23], p 56),

Δ∂∂

= −t

Hx

n2 , (5)s

where x is the surface point, n is the surface normal, H is the mean curvature and Δs denotes theLaplace–Beltrami operator over the surface. For evolving a close surface, the surface diffusionflow is volume-preserving and area-shrinking; hence, this flow has a very desirableregularization effect. In the level-set form, the surface diffusion flow is represented as (see[23], p 75)

Δ∂∂

= −∥ ∥

∥ ∥f

t

f

ffdiv , (6)f

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

where Δ f stands for the Laplace–Beltrami operator on the level-set Γ = ∈ = f cx x{ : ( ) }c3

(see [23], p 28). The weak form of the surface diffusion flow can be written as

4

Comput. Sci. Discov. 8 (2015) 014002 G Xu et al

∫ ∫ϕ Δϕ ϕ ϕ∂∂

= + − ∥ ∥

f

tH H fx n n n xd 2 2 d , (7)T T 2

3 3

⎡⎣ ⎤⎦where ϕ2 is the Hessian matrix of ϕ and

= −∥ ∥

Hf

f

12

div ,⎛⎝⎜

⎞⎠⎟

is the mean curvature of the level-set surface Γc. The derivation of this weak form is given inappendix A. The surface diffusion flow is a fourth-order equation. Using the weak form, onlythe second-order derivatives are required, while the tri-cubic B-spline functions have sufficientsmoothness. This is the main reason for using the weak-form equations.

We denote the right-handed side of (7) by δ ϕf( , ). Take ϕ as ϕi, and arrange δ ϕf( , )i as avector. We regard this vector as the negative gradient − G f( ) of certain unknown functional G(f). In our minimization method, we need only the vector without knowing the energy.

Note that δ ϕf( , ) is linear with respect to ϕ. For a given vector h, let

∑ ϕ= =h h i j kx x i( ) ( ), [ , , ] .T

i

i i

Then if we take ϕ = h, we have δ = −f h G f h( , ) ( )T . Therefore, the directional derivative ofG(f) in the direction h can be computed.

4. Numerical minimizations

In this section, we discretize the minimization problem in a function space. Then we solve thediscretized minimization problem by the bi-gradient method.

4.1. Discretization

Given an even integer ⩾m 4, suppose the domain Ω = − − +1, 1cn n

2 2

3⎡⎣ ⎤⎦ is uniformlypartitioned with grid points

= − ⩽ ⩽ = ( )hm m

i i ix i i i,2 2

, , , ,i 1 2 3

where = +h n

m

2 . The function f is represented as

∑ ϕ Ω= = ∈− + ⩽ ⩽ −

f f h x x xx x x( ) ( , ), [ , , ] , (8)m

Tc

ii i

22

2

1 2 3m2

where

ϕ = ∥ − ∥h N h hx x i( , ) ( ),i

and N(s) is a cubic B-spline basis function. The cubic B-spline basis function, defined on theuniform knots − −2, 1, 0, 1, 2, is

=

− + ⩽ <

− ⩽ <

⩽

N s

ss

s

s s

s

( )

23 2

, 0 1,

16

(2 ) , 1 2,

0, 2 .

(9)

23

3

⎧

⎨⎪⎪

⎩⎪⎪

5

Comput. Sci. Discov. 8 (2015) 014002 G Xu et al

The support of N s h( ) is the interval − h h( 2 , 2 ). Hence, the support of ϕ hx( , )i is∈ ∥ − ∥ < h hx x i{ : 2 }3 . It is not difficult to show that

Theorem 4.1. The functions in the set ϕ − + ⩽ ⩽ −{ }i i2 2m m2 2

are linearly independent.

Therefore, they form a basis functional space. Since N(s) is a C2-continuous function,ϕ hx( , )i is also C2 continuous and

ϕ

ϕ

ϕ

∂∂

=′ ∥ ∥

∥ ∥∥ ∥ ≠

∥ ∥ ==

∂∂ ∂

=″ ∥ ∥

∥ ∥−

′ ∥ ∥

∥ ∥∥ ∥ ≠

∥ ∥ == ≠

∂∂

=

″ ∥ ∥

∥ ∥−

− ∥ ∥ ′ ∥ ∥

∥ ∥∥ ∥ ≠

− ∥ ∥ =

=

( )

h

x

y N h

h

j

h

x x

y y N h

h

y y N h

h

j k j k

h

x

y N h

h

y N h

h

hj

xy

yy

y

xy

y

y

yy

y

xy

y

y y

yy

y

( , )( )

, 0,

0, 0,

1, 2, 3,

( , )( ) ( )

, 0,

0, 0,

, 1, 2, 3, ,

( , )( ) ( )

, 0,

2, 0,

1, 2, 3,

j

j

j k

j k j k

j

j j

i

i

i

2

2 2 3

2

2

2

2 2

2 2

3

2

⎧⎨⎪

⎩⎪

⎧⎨⎪

⎩⎪

⎧

⎨⎪⎪

⎩⎪⎪

where = − =h y y yy x i [ , , ]T1 2 3 .

The volumetric electron density maps of molecules are often approximated by Gaussianmaps in the literature (see [2, 10, 26]). In such approximations, each atom is simulated by a

Figure 1. Left: iso-contours of the bi-cubic B-spline basis. Right: iso-contours of thecubic B-spline radial basis.

6

Comput. Sci. Discov. 8 (2015) 014002 G Xu et al

sphere. Using radial basis functions, the spherical property of atoms can be ideallyapproximated. The left figure of figure 1 shows the iso-contours of a bi-cubic B-spline basisfunction. When the iso-value approaches 0, the iso-contours differ greatly from the circles. Theright figure shows the iso-contours of the cubic B-spline radial basis function. Furthermore, theprojections of the cubic B-spline basis functions have no closed form, while the projections ofthe cubic radial B-spline basis functions can be exactly evaluated from their closed forms. Thisincreases greatly the computational efficiency.

4.2. Fast computation of partial derivatives of J(f)

In our bi-gradient method, we need to compute the partial derivatives of J(f) with respect to thecoefficients of f. Hence, we consider first the computation of these partial derivatives. It is easyto see that

∫ ∫∑ ϕ∂∂

= −= − −

J f

f pX f g X u v

( ( )) 2d d . (10)

l

p

id d d i

1n

n

n

n

l l

2

2

2

2 ⎡⎣ ⎤⎦Now let us explain how each of the terms in (10) is efficiently computed.

Computing ϕXd il.

Let ∈ Sd 2 be a given direction. Then the projection of ϕi in the direction d is a 2Dfunction, defined as

∫∫∫∫∫

ϕ ϕ= + +

= ∥ + + − ∥

= − + − +

= +

= +

−∞

∞

−∞

∞

−∞

∞

−∞

∞

−∞

∞ ( )

( )

( )( )

( ) ( )

( )

( )

X u v u v t h t

N u v t h h t

N u h v h t h t

h N u v t t

h N a t t

e e d

e e d i

i e e i e e d

a d

( , ) , d

d

d

( , ) d

d .

T T

d i i d d

d d

d d d d

i

(1) (2)

(1) (2)

(1) (1) (2) (2)

2 2

For the cubic B-spline radial basis function ϕi, we have

∫ ∫

∫

ϕ

=

+ + + ⩽ ⩽

+ < <

⩽ < ∞

−

−

−

−

( ) ( )( )

( )X u v

h N a t t h N a t t a

h N a t t a

a

( , )

2 d 2 d , 0 1,

2 d , 1 2,

0, 2 ,

a

a

a

a

d i

0

12 2

1

42 2

0

42 2

2

2

2

2

⎧

⎨⎪⎪⎪

⎩⎪⎪⎪

where

= ∥ ∥

= − + −( ) ( )a u v

u v u h v h

a

a i e e i e e

( , ) ,

( , ) ,T T

i

i d d d d(1) (1) (2) (2)

7

Comput. Sci. Discov. 8 (2015) 014002 G Xu et al

ed(1) and ed

(2) are two directions defined by (1), which span the (u,v)-plane in space 3. Theintegrations above can be exactly computed using the expression (9) and the following integralformulas

∫

∫

+ = + + + + +

+ = + + + + +

− + + + +

( ) ( ) ( )

( ) ( )( ) ( )

( )

t a t t t a a t t a C

t a t t a t t a a t t a

a ta

t a t C

d12

log ,

d1

162 5 2 8 log

log2

.

2 2 2 2 2 2 2

2 2 2 2 2 2 4 2 2

4 22

2 2

12

12

12

32

12

12

12

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥

Computing X fdl. Here we propose an efficient approach for computing X fdl

.

∑

∑

ϕ

ϕ

=

=

( )

( )

X f u v X f u v

f X u v

( , ) ( , )

( , ),

d d

ii i

ii d i

l l

l

⎛⎝⎜⎜

⎞⎠⎟⎟

where ϕX u v( )( , )d ilis computed as above. Notice that N(s) is locally supported. The cost for

computing X fdlis O m( )3 . The total cost for the projection is O pm( )3 , where p denotes the total

number of projections. Thus, compared with using fast Fourier transform, the cost of thismethod is one order higher; however, its performance is much better. The computation can beaccelerated by removing small coefficients.

∑ ϕ≈ϵ>

( ) ( ){ }

X f u v f X u v( , ) ( , ),f

d i d il l

i

where ϵ > 0 is a given small number.

4.3. Refinement of the B-spline radial basis functions

Let

∑ ϕ Ω= = ∈− + ⩽ ⩽ −

f f h x x xx x x( ) ( , ), [ , , ] . (11)m

Tc

ii i

22

2

1 2 3m2

In the following, the vector consisting of the coefficient of f x( ) is denoted as f . Now we refinef x( ) by replacing h and m with h

2and m2 , respectively. To obtain a representation in the form

∑ ϕ Ω= = ∈− + ⩽ ⩽ −

f f h x x xx x x( ) ( , 2), [ , , ] ,m m

Tc

ii i

2 2

1 2 3

from (11), we need to refine formulas for N s h( , ). It is easy to derive that

= − + −

+ + + + +

N s h N s hh

N sh h

N sh

N sh h

N s hh

( , )18

,2

12 2

,2

34

,2

12 2

,2

18

,2

.

⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝

⎞⎠

8

Comput. Sci. Discov. 8 (2015) 014002 G Xu et al

Hence

∑ϕ ϕ= ∥ ∥ ≈− ⩽ ⩽

h N h n hx x x( , ) ( , ) ( , 2),0i

i i2 2

where values for ni are computed by interpolation. There are a total of 125 basis functionsinvolved, and the coefficients could be determined by interpolating 125 points− −h h h h[ , 2, 0, 2, ]3.

4.4. L2-gradient flow

One method to minimize the energy J(f) is to use the following L2GF

∫ ϕ∂∂

= − ∂∂

− ⩽ ⩽

f

t

J f

f

m mx id

( ( )),

2 2.i

i3

In matrix form, the flow can be written as

∂∂

= −Mt

J ff

( ),

where ∫ ϕ ϕ=

M xdi j3⎡⎣ ⎤⎦ is a sparse matrix and = −J f R Gf( ) with R and G are defined

by the right-hand side of (10). We call the vector −M J f( )1 as L2-gradient of J(f). To compute−M J f( )1 efficiently, matrix elements of M are approximated by

∫ ∫∏ϕ ϕ ≈ − − = ==

( )( )N

x

hi N

x

hj x i i i j j jx i jd d , , , , , , .

k

kk

kk ki j

1

3

1 2 3 1 2 33⎜ ⎟ ⎜ ⎟⎛⎝

⎞⎠

⎛⎝

⎞⎠

Using this approximation, −M 1 can be approximately computed quickly.

4.5. Stage one: minimize J(f)

We present the main steps of the bi-gradient method. The method depends on a parameterβ ∈ [0, 1].

Algorithm 4.1. Bi-gradient method.

(i) Compute a threshold ε > 0stop of J(f) for stopping the iteration.

(ii) Given an initial value =f 0(0) . Set k = 0.

(iii) Compute = − J fr : ( )kk( ) and then = −Mh r:k k

1 .

(iv) Compute αk and βk such that

α β+ + =( )J f r h min, (12)kk

kk

k( ) ( ) ( )

where r k( ) and h k( ) are spline functions with rk and hk as their coefficient vectors,respectively. The real numbers αk and βk are obtained by solving a 2 × 2 linear systemderived from (12).

(v) Set α β= +d r h( )k k kk k( ) ( ). Then for the given β = ∈α

α+ [0, 1]1

, determine a τk, suchthat

τ β β+ − + =( )( )J f r d(1 ) min,kk

kk

( ) ( )

9

Comput. Sci. Discov. 8 (2015) 014002 G Xu et al

(vi) Compute

τ β β= + − ++ ( )f f r d(1 ) .k kk

kk

( 1) ( ) ( )

(vii) If the following stopping condition

ε⩽ = −+( )J f k Mor 1 (13)k( 1)stop

is satisfied, stop the iteration. Then = +f f: k1

( 1) is the required result. Otherwise, set k as+k 1 and then go back to step 3. The integer >M 1 in (13) is a given bound for the

iteration.

Computing stopping threshold. Let f* be the exact function to be reconstructed, which isunknown. Then the measured image gdl

can be represented as

= +( )g X f nu u u( ) * ( ) ( ),d d dl l l

where n u( )dlare additive noise images. Therefore

∫ ∫∑ ∑= − == =

( ) ( )J f

pX f g

pnu u u u u* 1 * ( ) ( ) d

1( ) d . (14)

l

p

l

p

d d d

1

2

1

2l l l2 2

⎡⎣⎢

⎤⎦⎥ ⎡⎣ ⎤⎦

Hence we can take

∫∑ε ==

pn u u

1( ) d .

l

p

dstop

1

2l2

⎡⎣ ⎤⎦To compute εstop, we need to estimate n u( )dl

. Though n u( )dlare unknown in general, some

parts of n u( )dlare presented in the image g u( )dl

. We can use these known parts to estimate εstop.

In this paper, we regard the part − ⧹ ∥ ∥ ⩽R R Ru u[ , ] { : }2 in g u( )dlas noise. We therefore

compute εstop as follows

∫∑

∑ ∑

επ

π

≈−

≈−

=− ⧹ ∥ ∥⩽

= + ⩾

pg

pg i j

u u4

(4 )( ) d

4(4 )

( , ) ,

l

p

R R R

l

p

i j

u ud

d

stop

1[ , ] { : }

2

1

2

l

nl

2

2 22

⎡⎣ ⎤⎦⎡⎣ ⎤⎦

where − R R[ , ]2 is the domain of the measured images. The integrations above areapproximated by summations of the image values over the integer grid points.

4.6. Stage two: combining the geometric flow

The second stage for reconstructing f is as the following steps.

Algorithm 4.2. Combining geometric flow.

(i) Given an initial value =f f(0)1, where f1 is the output of algorithm 4.1 Set k = 0.

(ii) Compute J f( )k( ) and G f( )k( ) , where J is defined by (2) and G is defined by thegeometric flow in section 3.

10

Comput. Sci. Discov. 8 (2015) 014002 G Xu et al

(iii) Compute combined direction α β= − −J f G fh ( ) ( )k kk

kk( ) ( ) (λ = β

αkk

k, see section 4.6.1

for detail).

(iv) Compute τk (see (21) and section 4.6.1 for detail) and then compute

τ= ++f f h , (15)k kk

k( 1) ( ) ( )

where h k( ) is a linear combination of the B-spline radial basis function with hk as itscoefficient vector.

(v) If the following stopping condition

ϵ

∥ ∥

∥ ∥< = −

( )( )

G f

G fk Nor 1 (16)

k( )

(0)

is satisfied, stop the iteration. Then = +f f: k( 1) is the final result. Otherwise, set = +k k 1and then go back to step 2. In (16), ϵ is a small number: we take it as 10−5. The integer

>N 1 in (16) is a given bound for the iteration.

4.6.1. Combined direction. Let

= −

∥ ∥= −

∥ ∥

( )( )

( )( )

J f

J f

G f

G fr g: , : .k

k

k k

k

k

( )

( )

( )

( )

If =r g 1kT

k ( =g rk k), then the combine direction is taken as rk. If = −g r 1kT

k ( = −g rk k), thenthe spanned space by gk and rk is one-dimensional (1D). Then we replace gk by −hk 1 and restartthe determination of the combined direction. In the following, we assume ∣ ∣ <r g 1k

Tk . Let

α

α

α

α=

−

−=

−

−r

g rg

r g˜

1, ˜

1,k

k k k

k

kk k k

k2 2

where α = r gk kT

k. Then it is easy to see that

α∥ ∥ = ∥ ∥ = = = = = − ⩾r g g g r r g r r g˜ ˜ 1, ˜ 0, ˜ 0, ˜ ˜ 1 0.k k kT

k kT

k kT

k kT

k k2

Let

θ α= − − ( )r gsign( ) cos ˜ .k kT

k01

Then

θ π< 2.0

In the plane spanned by the vectors gk and rk, we first introduce an angle θ measured fromvector rk to vector gk. Then we define the two step-size curves

τ θθ

θ θ θ θθ π π=

+ +∈ −

a

b c d( )

2 cos ( )

cos ( ) 2 cos ( ) sin ( ) sin ( ), ( 2, 2), (17)r

r

r r r2 2

τ θϑ

ϑ ϑ ϑ ϑϑ θ θ π=

+ += − ∈

a

b c d( )

sin ( )

sin ( ) 2 cos ( ) sin ( ) cos ( ), (0, ), (18)g

g

g g g2 2 0

11

Comput. Sci. Discov. 8 (2015) 014002 G Xu et al

for reducing the energies J(f) and G(f) in the directions

θ θ θ θ π πθ ϑ ϑ ϑ θ θ π

= + ∈ −= + = − ∈

r r rg g g

( ) cos ( ) sin ( ) ˜ , ( 2, 2),

( ) sin ( ) cos ( ) ˜ , (0, ),k k

k k 0

respectively. The numbers ar, br, cr and dr are given by (24)–(27). Constants ag, bg, cg and dgare given by (29)–(31). Note that

θ θ θ θ π= ∈ [ ]r g( ) ( ), , 2 .0

Let

τ θ τ θ τ θ θ θ π= ∈{ } [ ]( ) min ( ), ( ) , , 2 .r g 0

Then the curve τ θ( ) is the right step-size for the direction θr( ), which makes J(f) not increaseand G(f) decrease. We want to find a θ θ π∈* [ , 2]0 , that leads to the maximal reduction of theenergy G(f). Since

τ θ θ τ θ θ τ θ

θ θ τ θ τ θ

θ θ τ θ τ θ

+ = + +

= + − +

= − − ∥ ∥ +

( )( )( )

( )

( )

G f G f G f O

G f G f O

G f G f O

g g

g

( ( ) ( )) ( ) ( ) ( ) ( ) ( )

( ) sin ( ) ( ) ( )

( ) sin ( ) ( ) ( ) .

T

Tk

2

02

02

Omitting the higher order term τ θO ( ( ))2 , we determine our best θ* as follows

θ θ θ τ θ= −θ θ π∈

( )* arg max sin ( ).[ , 2]

00

Then the combined direction is given as

θ θ θ θ θ θ= − + − = +( ) ( ) ( ) ( )h g g r rsin * cos * ˜ cos * sin * ˜ . (19)k k k k k0 0

Remark 4.2. Since =r r˜ 0kT

k , we know from (19) that

θ= >( )h r cos * 0. (20)kT

k

Hence hk is a descent direction of J(f).

Having θ* and the combined direction hk, we compute the step-size in the direction hk asfollows

τ τ θ= ( )* . (21)k

4.6.2. Step-size curves. We first consider the step-size curve τ θ( )r . Given a descent directionh, from minimizing τ+J f h( )k( ) , we have

τ′ + =( )J f h 0,k( )

12

Comput. Sci. Discov. 8 (2015) 014002 G Xu et al

we can determine τ as

τ = − N

D

1,

1

with

∫∑= −=

N X f g X hu u( ) d , (22)

l

pk

d d d1

1

( )l l l2

⎡⎣ ⎤⎦

∫∑==

D X h ud , (23)

l

p

d1

1

2l2

⎡⎣ ⎤⎦To make J(f) not increase, we redefine

τ = −N

D

2.1

1

Taking

θ θ= +h r rcos ( ) sin ( ) ˜ ,k k

then we obtain (17) with

∫∑= − −=

a X f g X ru u( ) d , (24)r

l

pk

kd d d

1

( )l l l2

⎡⎣ ⎤⎦

∫∑==

b X r ud , (25)r

l

p

kd

1

2l2

⎡⎣ ⎤⎦

∫∑==

c X r X r u˜ d , (26)r

l

p

k kd d

1l l2

∫∑==

d X r u˜ d . (27)r

l

p

kd

1

2l2

⎡⎣ ⎤⎦

Now we consider step-size curve τ θ( )g . Since

τ θ τ θ

τ θ θ τ

+ = +

+ + ( )G f g G f G f

G f O

g

g g

( ( ( ))) ( ) ( ) ( )12

( ) ( ) ( ) ,

T

T2 2 3

then from τ θ′ + =G f g( ( ( ))) 0 and omitting the higher order term τO ( )2 , we obtain

τ θθ θ

= − G f

G f

g

g g

( ) ( )

( ) ( ) ( ). (28)

T

T 2

Substituting

θ ϑ ϑ= +g g g( ) sin ( ) cos ( ) ˜ ,k k

into (28), we obtain (18), with

= − = −( ) ( )a G f G fg , (29)gk

T

k gk( ) ( )

k

⎡⎣ ⎤⎦

13

Comput. Sci. Discov. 8 (2015) 014002 G Xu et al

= =( ) ( )b G f G fg g , (30)g kT k

k g gk2 ( ) ( )

k k

= =( ) ( )c G f G fg g̃ , (31)g kT k

k g gk2 ( )

˜( )

k k

= =( ) ( )d G f G fg g˜ ˜ , (32)g kT k

k g gk2 ( )

˜ ˜( )

k k

where G f( )gk( )

kand G f( )g g

k( )k k

are the first- and second-order variations of G with respect to gk.Appendix B gives the computational details.

4.6.3. Discussions. Now we show that the f produced by algorithm 4.2 satisfies the condition(4). To illustrate this, let us consider a general form functional

∫ℰ =

f E f x x( ) ( ( )) d .3

Then the partial derivative with the coefficient of the spline function f is

∫ ϕℰ∂

= = − ′

f

fr r E f x

( )with ( ) d .

ii i i3

Let

∑ ϕ=r r .i

i i

Then the changing rate of ℰ f( ) in the direction r is

∫

∫∫∑

∑

ϕ

ℰ + = +

= ′

= ′

= −

= =

tf tr

tE f tr

E f r

r E f

r

x

x

x

dd

( )dd

( ) d

( ) d

( ) d

.

t t

i

i i

ii

0 0

2

3

3

3

Hence, ℰ f( ) is non-increasing in the direction r. Let ϕ= ∑h hi i i be another function, if

∑ ⩾h r 0, (33)i

i i

then it is easy to derive that

∑ℰ + = − ⩽=t

f th h rdd

( ) 0.t i

i i0

Hence, ℰ f( ) is non-increasing in the direction h.From the discussion above, we know that J(f) is not increasing in the direction hk at f k( ), as

(20) is satisfied. Furthermore, the step-size τk given by (21) makes ⩽+J f J f( ) ( )k k( 1) ( ) .Therefore, the function sequence f{ }k( ) produced by algorithm 4.2 causes the sequenceJ f{ ( )}k( ) to decrease monotonically.

14

Comput. Sci. Discov. 8 (2015) 014002 G Xu et al

5. Numerical experiments and discussions

To evaluate the performance of our reconstruction algorithms, we present several examples inthis section. We first look at the performance of our reconstruction algorithms for clean data insection 5.1. Then, we look at the performance of algorithms for noisy data in sections 5.2 and5.3. We compare our results with that of WBP.

5.1. Numerical experiments for clean data

Given a volume data =F f{ }ijk with size × ×131 131 131, we first generate 5000 projection

images =I{ }i i 15000 with size 131 × 131 at randomly chosen projection directions =d{ }i i 1

5000 on theunit sphere (figure 2 shows the first five projection images). Then we reconstruct F usingdifferent reconstruction algorithms. Let = =R r{ }l

ijkl

ijk( ) ( )

1131 , = …l 1, , 5, be the reconstructed

volumes using WBP, and our bi-gradient method with β = 0, β = 1

3, β = 1

2and β = 1,

respectively. Table 1 lists the L2-errors E l( ), where E l( ) is defined as

∑∑∑= − = … == = =

Em

f r l m1

, 1, , 5, 131.l

i

m

j

m

k

m

ijk ijkl( )

31 1 1

( ) 2⎡⎣ ⎤⎦

Since the data is clean, we do not use the regularization term in the reconstruction equation. TheL2-errors are monotonically decreasing as the iteration number increases. It can be clearlyobserved that as β increases from 0 to 1, the errors decrease. Choosing β = 1 gives the mostaccurate result. Following β = 1, the next best result is given by WBP with L2-error 0.08 131.Following WBP are β = 1

2, β = 1

3, and β = 0. More iterations will make the case β = 1

2better

than WBP. For instance, the L2-error is 0.07 855 after 40 iterations. In the next subsection, we

Figure 2. Five projection images of the volume data F.

Table 1. L2-errors …E E, ,(2) (5) for different iteration numbers for the clean data.=E 0.08 131(1) .

Iteration number β = 0 β = 1

3β = 1

2 β = 1

5 0.15 519 0.15 368 0.14 793 0.07 80310 0.13 726 0.13 419 0.12 313 0.05 79015 0.12 838 0.12 416 0.10 977 0.05 68620 0.12 167 0.11 635 0.09 931 0.05 67425 0.11 733 0.11 115 0.09 246 0.05 67230 0.11 354 0.10 652 0.08 653 0.05 671

15

Comput. Sci. Discov. 8 (2015) 014002 G Xu et al

will illustrate that the most accurate method may not be the best method for data with highnoise.

Figure 3 shows the central slices of the reconstructed volume data, where figure (a) is fromthe initial data. Figures (b–f) are produced from the reconstruction results of the WBP, bi-gradient method with β = 0, β = 1

3, β = 1

2, and β = 1, respectively. These figures show the

same conclusions as the numerical results.

Remark 5.3. The increase in the accuracy of the bi-gradient method as β increases from 0 to 1 isdue to the fact that when β = 0, we search the minimal point in the 1D space rspan [ ]k , butwhen β = 1, we search the minimal point in the 2D space r hspan [ , ]k k . When β ∈ (0, 1), themethod is an average of the cases β = 0 and β = 1.

5.2. Numerical experiments for noised data with SNR = 1.0

At this point, we generate a new set of data by adding additive white Gaussian noise to each ofthe images Ii, produced in the previous subsection with SNR = 1.0. Figure 4 shows five noisyimages of the ones shown in figure 2. We then reconstruct f using different reconstructionalgorithms from the noisy images. Because the data is noisy, a regularization term is used. As

Figure 3. Central slices of the reconstructed volume data from the clean projections. (a)is from the initial data. (b) is from WBP. (c)–(f) are from the reconstruction results ofour bi-gradient method with β = 0, β = 1

3, β = 1

2and β = 1, respectively, after 30

iterations.

Figure 4. Five noised projection images of the volume data f for SNR = 1.0.

Table 2. Energies of different methods after 30 iterations.

SNR β = 0 β = 1

3β = 1

2 β = 1

1.0 101 747 81.537 409 101 552 48.877 989 100 916 05.869 403 990 8452.694 144

16

Comput. Sci. Discov. 8 (2015) 014002 G Xu et al

before, we use R r( ) to represent the reconstructed volume. For our bi-gradient method, we iterate30 steps. In table 2, we list the energies, defined by (2), for the reconstruction results after 30iterations. The energies show that the bi-gradient method with β = 1 yields minimal energy.Hence, it is indeed the most accurate method. However, the most accurate method may not leadto the most meaningful results.

Figure 5 shows the central slices of the reconstructed volume data. The figures, from thefirst to the fifth, are the central slices of the reconstruction volumes of WBP, our bi-gradientmethod with β = 0, β = 1

3, β = 1

2, and β = 1, respectively. It can be clearly observed that the

bi-gradient method with β = 1 gives the noisiest result with detailed structures. After that, thebi-gradient method with β = 1

2, β = 1

3and β = 0 follow successively.

In table 3, we list the L2-errors between the reconstructed volumes and the exact initialvolume data for iterations 5, 10, 15, 20, 25, and 30. The errors are not monotonically decreasingas the iteration number increases. It is easy to see that the most accurate method (β = 1) for theclean data leads to the maximal L2 error, while the other three cases lead to similar L2 errorbounds. All these three cases are better than WBP for iterations 10, 15, 20, 25, and 30. The iso-surfaces of the reconstructed volume data, as shown in figure 6, demonstrate that:

(i) The bi-gradient method with β = 0 gives the most coarse level structure, then followed byβ = 1

3, β = 1

2and β = 1.0.

(ii) The bi-gradient method with β = 1 does not yield valuable iso-surfaces.

Figure 5. Central slices of the reconstructed volume data for the noised data withSNR = 1.0. The figures from the first to the fifth are from the reconstruction results ofWBP, our bi-gradient method after 30 iterations with β = 0, β = 1

3, β = 1

2and β = 1,

respectively.

Table 3. L2-errors …E E, ,(2) (5) for different iteration numbers. The data are noised withSNR = 1.0. =E 0.15 005(1) .

Iteration number E (2) E (3) E (4) E (5)

5 0.15 521 0.15 379 0.15 071 0.36 86310 0.13 770 0.13 567 0.14 074 0.45 72215 0.13 374 0.12 946 0.13 406 0.45 47620 0.13 394 0.12 671 0.12 680 0.45 22125 0.13 419 0.12 618 0.12378 0.45 17030 0.13 428 0.12 571 0.12 203 0.45 145

17

Comput. Sci. Discov. 8 (2015) 014002 G Xu et al

Remark 5.4. The noise is added to the 2D images Ii, not to the volume data F. If the 2D imageis highly polluted, such that the noise level is much higher than the signal, the reconstructedvolume data from these polluted images differs greatly from the initial volume data F. Hence, itis not reasonable to require the reconstruction volumes to be close to the initial volume.Therefore, the L2-error between the initial data and the reconstructed data may not be areasonable measure to evaluate the reconstruction method. In the next subsection, we explainhow we used Fourier shell correlation (FSC) to evaluate the performance of the reconstructionmethods.

Remark 5.5. In the previous section, we showed that, for the clean data set, as β increases from0 to 1, the accuracy of the bi-gradient method also increases. In this subsection, we observe thatβ = 0 leads to the smoothest result. Now we explain the reason. From our previous work [5],we know that if we choose =f 0(0) and search the minimal point in the direction rk, f k( )

converges to the minimal solution f* of the equation =R Gf in the space

= = ∀ ∈⊥N R N Rx x y y( ) { : 0, ( )}T in the sense of the Euclidean norm ∥ ∥ =f f fT , whereN(R) is the null space of the matrix R. While searching the minimal point in the direction hk, f k( )

converges to the minimal solution f* of the equation =R Gf in the space= = ∀ ∈⊥N R M N Rx x y y( ) { : 0, ( )}TM

in the sense of the norm ∥ ∥ = Mf f fMT . The

minimal property of the Euclidean norm makes the case that β = 0 yields the most smoothresult. Table 4 lists some Euclidean norms for the cases β = 0, , , 11

3

1

2. It can be seen that as β

increases, the Euclidean norm also increases.

Figure 6. Iso-surfaces of the reconstructed volume data at the middle iso-values. Figure(a) is from WBP. Figures (b)–(e) are from the reconstruction results of the bi-gradientmethod after 30 iterations with β = 0, ,1

3

1

2and 1, respectively.

Table 4. Euclidean norms of the reconstruction results. For WBP, it is 4.067 898e+02.

Iteration number β = 0 β = 1

3β = 1

2 β = 1

5 3.046 299e+02 3.048 806e+02 3.061 394e+02 3.759 363e+0210 3.225 103e+02 3.231 880e+02 3.270 741e+02 4.281 541e+0215 3.321 209e+02 3.335 966e+02 3.378 084e+02 4.239 439e+0220 3.340 637e+02 3.356 739e+02 3.410 555e+02 4.227 851e+0225 3.355 638e+02 3.377 481e+02 3.444 411e+02 4.227 860e+0230 3.360 086e+02 3.384 513e+02 3.451 244e+02 4.227 148e+02

18

Comput. Sci. Discov. 8 (2015) 014002 G Xu et al

5.3. Numerical experiments for noised data with SNR=0.1

Given a volume data =F f{ }ijk , of size × ×151 151 151, we first generate 200 00 projection

images =I{ }i i 1200 00 with size 151 × 151 at randomly chosen projection directions =d{ }i i 1

200 00 on theunit sphere. These images are polluted by adding additive white Gaussian noise with SNR=0.1(figure 7 shows five of these ‘noised’ images). These noised images are split randomly intodatasets A and B, with each dataset containing 100 00 images. Then, we reconstruct F using ourbi-gradient method with β = 0, , ,1

3

1

2

2

3for each of the datasets (in this section, we do not take

β = 1, as the examples in the previous subsection have shown that taking β = 1 does not lead todesirable results). Because the data are extremely noisy, a regularization term is used. The aimof splitting the dataset into two parts is to allow us to examine the correlation of thereconstructed results using FSC.

As before, we use R r( ) to represent the reconstructed volume. For each value of β in ourmethod, the algorithm iterates 30 steps. In table 5, we list the energies for the reconstructionresults after 30 iterations. These energies are decreasing as the value of β increases and β = 2

3yields minimal energies for the two sets of data.

Figure 8 shows the central slices of the volume data. Figures on the first to the fifthcolumns are from the reconstruction results of WBP, our bi-gradient method withβ = 0, , ,1

3

1

2

2

3. It can be clearly observed that the case β = 2

3gives the noisiest result with

detailed structures. After that, the cases β = , , 01

2

1

3follow successively.

In tables 6 and 7, we list the L2-errors between the reconstructed volumes and the exactinitial volume data for iterations 5, 10, 15, 20, 25, and 30. It is easy to see that the bi-gradientmethod with β = 0 leads to the minimal L2 error, followed by the cases β = 1

3, β = 1

2, and

β = 2

3, and all of these are better than WBP after 30 iterations. The iso-surfaces of the

reconstructed volume data, as shown in figure 9, demonstrate that the bi-gradient method withβ = 0 gives the coarsest level structure, followed by β = 1

3, β = 1

2, and β = 2

3.

Figure 7. Five noised projection images of the volume data f for SNR = 0.1.

Table 5. Energies of different methods after 30 iterations.

Data β = 0 β = 1

3β = 1

2β = 2

3

Dataset A

135 429 907.779 135 138 283.715 134 896 618.185 134 531 987.307

Dataset B

135 409 723.391 135 117 692.553 134 875 769.350 134 510 933.838

19

Comput. Sci. Discov. 8 (2015) 014002 G Xu et al

Figure 8. Central slices of the volume data. The first and the second rows are for thenoised data sets A and B, respectively. The first, second, …, fifth columns are from thereconstruction results of WBP, the bi-gradient method after 30 iterations withβ = 0, ,1

3

1

2and 2

3, respectively.

Table 6. L2-errors …E E, ,(2) (5) for different iteration numbers and for data set A.=E 0.29 193(1) .

Iteration number E (2) E (3) E (4) E (5)

5 0.17 066 0.16 985 0.17 464 0.19 61510 0.14 953 0.16 200 0.19 536 0.28 07415 0.14 286 0.15 583 0.19 542 0.29 51620 0.14 033 0.14 994 0.18 813 0.28 67125 0.13 990 0.14 878 0.18 605 0.28 40430 0.13 981 0.15 085 0.18 804 0.28 388

Table 7. L2-errors …E E, ,(2) (5) for different iteration numbers and for data set B.=E 0.29 208(1) .

Iteration number E (2) E (3) E (4) E (5)

5 0.17 064, 0.16 984 0.17 466 0.19 62110 0.14 956 0.16 210 0.19 553 0.28 10515 0.14 290 0.15 600 0.19 576 0.29 57220 0.14 035 0.15 018 0.18 841 0.28 76825 0.13 992 0.14 895 0.18 635 0.28 45230 0.13 984 0.15 097 0.18 836 0.28 435

20

Comput. Sci. Discov. 8 (2015) 014002 G Xu et al

Table 8 lists the resolutions for each of the reconstructed volume pairs after 10, 20 and 30iterations. The resolution is computed at the place where the value of FSC is 0.5. It can be seenthat for most of the cases, the resolution of each reconstruction result is higher than thatproduced by WBP. Figure 10 shows the FSC curves for the five cases and different numbers ofiterations.

Figure 9. Iso-surfaces of the reconstructed volume data at the middle iso-values. Firstrow is for the noised data set A. Second row is for the noised data set B. Column (a) isfrom WBP. Columns (b)–(e) are respectively from the reconstruction results of the bi-gradient method with β = 0, ,1

3

1

2and 2

3after 30 iterations.

1

0.9

0.8

0.7

0.6

0.5

0.4

1

0.9

0.8

0.7

0.6

0.5

0.4

1

0.9

0.8

0.7

0.6

0.5

0.40 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

1/Angstrom 1/Angstrom 1/Angstrom

Four

ier S

hell

Cor

rela

tion

Four

ier S

hell

Cor

rela

tion

Four

ier S

hell

Cor

rela

tion

WBPbeta=0

beta=1/3beta=1/2beta=2/3

WBPbeta=0

beta=1/3beta=1/2beta=2/3

WBPbeta=0

beta=1/3beta=1/2beta=2/3

Figure 10. The FSC curves of WBP and our iterative method after 10 (left), 20 (middle)and 30 (right) iterations, respectively.

Table 8. Resolution of different methods. The resolution of WBP is 12.362 563Å.

Iteration number β = 0 β = 1

3β = 1

2β = 2

3

10 12.055 628 Å 12.107 415 Å 12.152 935 Å 12.227 004 Å20 12.079 924 Å 12.199 175 Å 12.280 208 Å 12.391 474 Å30 12.108 918 Å 12.334 633 Å 12.476 160 Å 12.527 178 Å

21

Comput. Sci. Discov. 8 (2015) 014002 G Xu et al

Remark 5.6. Both the numerical and illustrative results in this subsection reveal that thereconstruction results from datasets A and B are very similar, giving further evidence that thebi-gradient method is robust.

5.4. Multi-scale reconstruction

The reconstruction in the previous subsection is conducted for each β. The computationconducted in this way allows examination of the performance of our method. In a realconstruction, we do not need to compute the reconstruction result for each β. We have noticedthat as β increases from 0 to 1, structures from coarse to fine are captured. We now combinethese computations into one loop, aiming to obtain a sequence of multi-scale reconstructionresults. For a given >K 1, we assume we are going to reconstruct a sequence of volume data

…F F F, , , K0 1 from coarse to fine. Then we carry out the following:

(i) Construct an initial volume data F0 using the bi-gradient method with β = 0. The iterationnumbersM and N in the the bi-gradient method are taken as 12 for both the first and secondstages.

(ii) For = …k K1, 2, , we do the following: set β = αα+1with α = + −2k K1 and reconstruct

volume data Fk using the bi-gradient method with −Fk 1 as the initial value. The iterationnumbers M and N in the bi-gradient method are taken as 5 for both the first and secondstages.

For instance, if we take K = 3, and reconstruct the volume data F F F F, , ,0 1 2 3 using the data

in the previous section, we obtain the results corresponding to β = 0, , ,1

3

1

2

2

3. The obtained

results were very close to the results obtained in the previous subsection; hence, they are notpresented again.

6. Conclusions

We have presented a multi-scale bi-gradient flow method for single-particle reconstruction,which enhances both the computational speed and accuracy of the earlier L2GF method wereported. Using the space spanned by the B-spline radial basis functions, a bi-gradient method iscombined with geometric flow with decreasing energy constraints. The experimental resultshave shown that the proposed method yields very desirable results. The multi-scale property ofthe method provides the user with the degree of freedom to reconstruct the volumetric data withthe desired level of detail.

Acknowledgments

This research was supported in part by the NSFC under the grants (11101401, 81173663,11301520) and the NSFC Fund for Creative Research Groups of China under the grants(11021101, 11321061).

22

Comput. Sci. Discov. 8 (2015) 014002 G Xu et al

Appendix A. Derivations of weak form

Now we derive the weak form of the surface diffusion flow (6). Multiplying both sides of (6) bya trial function ϕ ∈ C ( )2 2 with compacted support and then using co-area formula (see [7])and Green formula (see [23] p 24), we have

∫ ∫

∫ ∫

∫ ∫

∫

ϕ Δ ϕ

Δ ϕ σ

Δ ϕ σ

Δ ϕ

∂∂

= −∥ ∥

∥ ∥

= −∥ ∥

= −∥ ∥

= ∥ ∥

Γ

Γ

f

t

f

ff

f

fc

f

fc

H f

x xd div d

div d d

div d d

2 dx, (A.1)

f

f

f

f

c

c

3 3

3

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

⎛⎝⎜

⎞⎠⎟

where (see [3])

Δ ϕ ϕϕ

ϕ ϕ

==

= −

( )P

P n P n

div

div ( )

div ( ) ( ) ,

f f f

f

T

and = −IP nnT . Hence

Δ ϕ Δϕ ϕ ϕ ϕ= − − +( ) ( )nn n n n nn ndiv .fT T T T2

Since

ϕ ϕ ϕ

ϕ ϕ ϕ

= +

= − +∥ ∥

+

( ) ( )H

f

f

nn n n n n

n nP

n

div div ( )

2 ,

T T T T

T T2

2⎡⎣⎢

⎤⎦⎥

ϕ ϕ ϕ

ϕ ϕ ϕ

= +

= +∥ ∥

+

( ) ( )f

f

nn n n n n

n nP n

nn

( )

( ) ,

T T T T

TT

T2

2

and

=∥ ∥

−∥ ∥

=∥ ∥

f

f

f f f

f

f

fn

P( ),

T2 2

3

2

we have

ϕ ϕ ϕ=∥ ∥

+( ) f

fn nn n

n Pn n.T

TT

22

Therefore

Δ ϕ Δϕ ϕ ϕ= + −Hn n n2 . (A.2)fT T 2

Substituting (A.2) into (A.1), we obtain the weak form of the surface diffusion flow as (7).

23

Comput. Sci. Discov. 8 (2015) 014002 G Xu et al

Appendix B. Second-order variation

The weak form of the surface diffusion flow has been written as (see (7))

∫ ∫ϕ Δϕ ϕ ϕ∂∂

= + − ∥ ∥

f

tH H fx n n n xd 2 2 d . (B.1)T T 2

3 3

⎡⎣ ⎤⎦We have known

=∥ ∥

= −∥ ∥

f

fH

f

fn ,

12

div .⎛⎝⎜

⎞⎠⎟

By expansion, we have

Δ

Δ Δ

=∥ ∥

−∥ ∥

=∥ ∥

−∥ ∥

+∥ ∥

( )

Hf f f

f

f

f

Hf

ff

f f f

f

f f f

f

12

12

,

212

12

.

T

T T

2

3

22 2

4

2 2

6

⎛⎝⎜

⎞⎠⎟

Then the surface diffusion flow is represented as

∫ ∫ϕ∂∂

= −Ω Ω

ϕf

tQ f R fx xd ( ) ( )d ,

where

ΔΔϕ

= ∥ ∥ = − += − + − +ϕ ϕ ϕ ϕ

Q f H f f Q f

R f Q f Q f Q f

( ) 2 ( ),

( ) ( ) ( ) ( ),

(0)

(1) (2) (3)

and

Δ ϕ

ϕ ϕ

=∥ ∥

=∥ ∥

=∥ ∥

=∥ ∥

ϕ

ϕ ϕ

Q ff f f

fQ f

f f

f

Q fQ f f

fQ f

f f

f

( ) , ( ) ,

( )( )

, ( ) .

T T

T T

(0)2

2(1)

2

(2)(0)

2(3)

2

2

We regard the right-hand side of the surface diffusion flow with a minus sign as the first-order variation ϕG f( ) with respect to ϕ for an unknown energy G(f). Then we compute thesecond-order variation of G(f) with respect to ψ as

∫= +ϕψ ψ ϕ ϕψ

G f Q f R f Q f R f x( ) ( ) ( ) ( ) ( ) d ,3

⎡⎣ ⎤⎦where

Δψ= −

= − +ψ ψ

ϕψ ϕψ ϕψ ϕψ

Q f Q f

R f Q f Q f Q f

( ) ( ) ,

( ) ( ) ( ) ( ),

(0)

(1) (2) (3)

Now let us compute ψQ f( )(0) , ϕψQ f( )(1) , ϕψQ f( )(2) and ϕψQ f( )(3) . It is easy to see that

ψ ψ ψ Δ Δψ= ∥ ∥ =∥ ∥

= =ψ ψ ψ ψ( )f ff

ff f( ) ,

( ), , ( ) .

T2 2

24

Comput. Sci. Discov. 8 (2015) 014002 G Xu et al

Using these equations, we can obtain that

ψ ψ

Δψ ϕ Δ ψ ϕ

ϕ ψ ϕ

ψ ϕ

= +∥ ∥

−∥ ∥

∥ ∥

= +∥ ∥

−∥ ∥

∥ ∥

=+

∥ ∥−

∥ ∥∥ ∥

=∥ ∥

−∥ ∥

∥ ∥

ψψ

ϕψϕ ψ

ϕψψ ϕ ψ

ϕψϕ ψ

( )

( )

Q ff f f f

f

Q f f

f

Q ff f

f

Q f f

f

Q fQ f f Q f

f

Q f

f

Q ff

f

Q f f

f

( )2( ) ( ) 2

,

2 ( ),

( )( ) ( ) 2 ( )

,

( )2 2 ( )

.

T T

T T

T T

T

(0)2 2

2

(0)

(1)2

(1)

(2)(0) (0)

2

(2)

(3)2

2

(3)

References

[1] Andersen A H and Kak A C 1984 Simultaneous algebraic reconstruction technique: a superiorimplementation of the art algorithm Ultrason. Imaging 6 81–94

[2] Blinn J 1982 A generalization of algebraic surface drawing ACM Trans. Graph. 1 235–56[3] Chen C and Xu G 2010 Construction geometric partial differential equations for level-set surfaces J. Comput.

Math. 28 105–21[4] Chen C and Xu G 2012 Gradient-flow-based semi-implicit finite-element method and its convergence

analysis for image reconstruction Inverse Problems 28 035006[5] Chen C and Xu G 2013 Computational inversion of electron micrographs using L2-gradient flow—

convergence analysis Math. Methods Appl. Sci. 36 2492–506[6] Crowther R A 1971 Procedures for three-dimensional reconstruction of spherical viruses by Fourier synthesis

from electron micrographs Phil. Trans. R. Soc. B 221–30[7] Evans L C and Gariepy R F 1992 Measure Theory and Fine Properties of Functions (Boca Raton, FL: CRC

Press)[8] Gilbert P 1972 Iterative methods for the three-dimensional reconstruction of an object from projections

J. Theor. Biol. 36 105–17[9] Gordon R, Bender R and Herman G T 1970 Algebraic reconstruction techniques for three-dimensional

electron microscopy and x-ray photography J. Theor. Biol. 29 471–81[10] Grant J and Pickup B 1995 A Gaussian description of molecular shape J. Phys. Chem. 99 3503–10[11] Heel M 1987 Angular reconstruction: a posteriori assignment of projection directions for 3D reconstruction

Ultramicroscopy 21 111–24[12] Jonic S, Sorzano C O S, Thévenaz P, de Carlo El-Bez C S and Unser M 2005 Spline-based image-to-volume

registration for three-dimensional electron microscopy Ultramicroscopy 103/104 303–17[13] Li M, Xu G, Sorzano C O S, Sun F and Bajaj C L 2011 Single-particle reconstruction using L2-gradient flow

J. Struct. Biol. 176 259–67[14] Marabini R, Herman G T and Carazo J M 1998 3D reconstruction in electron microscopy using art with

smooth spherically symmetric volume elements (blobs) Ultramicroscopy 72 53–65[15] Natterer F 1986 The Mathematics of Computerized Tomography (New York: Wiley)[16] Natterer F and Wübbeling F 2001 Mathematical Methods in Image Reconstruction (Philadelphia, PA: SIAM)

25

Comput. Sci. Discov. 8 (2015) 014002 G Xu et al

[17] Penczek P A, Grasucci R A and Frank J 1994 The ribosome at improved resolution: new techniques formerging and orientation refinement in 3D cryo-electron microscopy of biological particles Ultramicro-scopy 53 251–70

[18] Radermacher M 1994 Three-dimensional reconstruction from random projections-orientational alignment viaradon transforms Ultramicroscopy 53 121–36

[19] Sezan M I and Stark H 1982 Image restroation by the method of convex projections: II. Applications andnumerical results IEEE Trans. Med. Imaging MI 1 95–100

[20] Sorzano C O S, Jonic S, El-Bez C, Carazo J M, de Carlo S, Thévennaz P and Unser M 2004 Amultiresolution approach to pose assignment in 3D electron microscopy of single particles J. Struct. Biol.146 381–92

[21] Sorzano C O S, Jonic S, Cottevieille M, Larquet E, Boisset N and Marco S 2007 3D electron microscopy ofbiological nanomachines: principles and applications Eur. Biophys. J. 36 995–1013

[22] Unser M 1999 Splines a perfect fit for signal and image processing IEEE Signal Process. Mag. 16 22–38[23] Xu G 2008 Geometric Partial Differential Equation Methods in Computational Geometry (Beijing: Science

Press)[24] Xu G, Li M, Gopinath A and Bajaj C L 2011 Computational inversion of electron tomography images using

L2-gradient flows—computational methods J. Comput. Math. 29 501–25[25] Youla D C and Webb H 1982 Image restoration by the method of convex projections: II. Theory IEEE Trans.

Med. Imaging MI 1 81–94[26] Zhang Y, Xu G and Bajaj C 2006 Quality meshing of implicit solvation models of biomolecular structures

Comput.-Aided Geom. Des. 23 510–30

26

Comput. Sci. Discov. 8 (2015) 014002 G Xu et al