- Research
- Open Access

# Image reconstruction of fluorescent molecular tomography based on the tree structured Schur complement decomposition

- Wei Zou
^{1, 2}, - Jiajun Wang
^{1, 2, 3}Email author and - David Dagan Feng
^{2, 3, 4}

**9**:20

https://doi.org/10.1186/1475-925X-9-20

© Zou et al; licensee BioMed Central Ltd. 2010

**Received:**31 December 2009**Accepted:**20 May 2010**Published:**20 May 2010

## Abstract

### Background

The inverse problem of fluorescent molecular tomography (FMT) often involves complex large-scale matrix operations, which may lead to unacceptable computational errors and complexity. In this research, a tree structured Schur complement decomposition strategy is proposed to accelerate the reconstruction process and reduce the computational complexity. Additionally, an adaptive regularization scheme is developed to improve the ill-posedness of the inverse problem.

### Methods

The global system is decomposed level by level with the Schur complement system along two paths in the tree structure. The resultant subsystems are solved in combination with the biconjugate gradient method. The mesh for the inverse problem is generated incorporating the prior information. During the reconstruction, the regularization parameters are adaptive not only to the spatial variations but also to the variations of the objective function to tackle the ill-posed nature of the inverse problem.

### Results

Simulation results demonstrate that the strategy of the tree structured Schur complement decomposition obviously outperforms the previous methods, such as the conventional Conjugate-Gradient (CG) and the Schur CG methods, in both reconstruction accuracy and speed. As compared with the Tikhonov regularization method, the adaptive regularization scheme can significantly improve ill-posedness of the inverse problem.

### Conclusions

The methods proposed in this paper can significantly improve the reconstructed image quality of FMT and accelerate the reconstruction process.

## Keywords

- Inverse Problem
- Mean Square Error
- Singular Value Decomposition
- Regularization Parameter
- Tikhonov Regularization

## Background

Near-infrared (NIR) light can travel several centimeters through biological tissue, and hence has the potential to qualify the molecular information by fluorochromes in tissue [1]. Recently, there has been increasing interest in the molecularly-based medical imaging method, such as fluorescent molecular tomography (FMT) [2–4], in which the injected fluorophore may accumulate in diseased tissue. During the imaging process, the tissue surface is illuminated with excitation light. Then, the fluorophores are excited to emit the light, which is detected as fluorescence [5]. The process of fluorescent light generation and transportation through tissues can be described by a forward model, so that the surface measurements can be predicted on the basis of a guess of the system parameters and the given source positions. To reconstruct an image, it is necessary to calculate the internal optical and fluorescent properties with the given measured data and sources [6].

One of the major challenges in the reconstruction of FMT is its high computational complexity resulted from extremely large-scale matrix manipulations. Generally, the iterative solution approaches, such as CG method [7] and Gauss-Newton (GN) method [8], are more efficient than the direct solution approaches. Additionally, the iterative methods based on the reduced system can be more efficient than those based on the global system. One of such systems is the Schur complement system, which was firstly used by Haynsworth [9]. The condition number of the Schur complement of a matrix is never greater than that of the given matrix, and hence the convergence properties of iterative solving of linear systems can be significantly improved [7, 10]. In this paper, we propose to adapt this idea for the FMT reconstruction. The most important innovation of our method lies in its tree structured level-by-level decomposition strategy, where decompositions in each level are performed in two ways. This strategy is quite different from that in [10] where only one component of the global solution is derived in the Schur complement system. The advantages of our method are obvious because a further improvement in the reconstruction accuracy and speed can be achieved with level-by-level Schur complement decomposition. Another contribution of this paper is that we propose a modified spatially variant regularization method incorporating the objective function to tackle the ill-posed nature of the inverse problem.

## Methods

### Forward Model and Finite Element Formulation

where the subscript *x* denotes the excitation wavelength; ∇ is the gradient operator; *S*
_{
x
}(W/cm^{3}) is the excitation light source; Φ_{
x
}(W/cm^{2}), *D*
_{
x
}(cm), and *k*
_{
x
}(cm^{-1}) represent the photon fluence, the diffusion coefficient, and the decay coefficient, respectively; Ω denotes the bounded domain of reconstruction.

*m*indicates the emission wavelength,

*ω*(

*rad/s*) denotes the modulation frequency of the source.

*S*

_{ m }is the emission light source. The diffusion coefficient

*D*

_{ x,m }(cm), and the decay coefficient

*k*

_{ x,m }(cm

^{-1}) are defined, respectively, as[6]

where *μ*
_{
ax,mi
}(cm^{-1}) denotes the absorption coefficient due to endogenous chromophores; *μ*
_{
ax,mf
}(cm^{-1}) represents the absorption coefficient due to exogenous fluorophores;
is the reduced scattering coefficient; *q* is the quantum efficiency of the fluorophore; *τ*(s) is the lifetime of fluorescence; and finally, *c*(cm/s) is the speed of light in the medium.

where **n** is a vector normal to the boundary ∂Ω, *b*
_{
x,m
}is the Robin boundary coefficient.

*P*elements and joined at

*N*vertex nodes. The solution Φ

_{ x,m }is approximated by the piecewise linear function , with

*φ*

_{ i }(i = 1...

*N*) being basis functions [13]. Hence, equations (1) and (2) can be rewritten as

with Ω_{
h
}and Γ_{
h
}being the bounded domain and its boundary, respectively.

### Inverse Process of FMT

where *G* is the forward operator, || || is L_{2}-norm, **x** and **y** are the calculated optical or fluorescent properties of the tissues and the detector readings, respectively.

*E*attains its extremum at

**x**+

*Δ*

**x**, expanding the gradient of the objective function

*E*' about

**x**in a Taylor series and keeping up to the first-order term leads to

where *T* denotes the transpose, *Δ* **y** = **y** - *G*(**x**) is the residual data between the measurements and the predicted data. The Jacobian matrix **J** is a measure of the rate of change in measurement with respect to the optical parameters. It describes the influence of a voxel on a detector reading. **H** is the Hessian matrix, whose entries are the second-order partial derivatives of the function with respect to all unknown parameters describing the local curvature of the function with respect to many variables [16].

where *λ* is a regularization parameter, which can be determined by the Morozov discrepancy principle [17], **I** is an identity matrix.

### Adaptive Regularization Scheme

The problem of image reconstruction for FMT is ill-posed [18]. The Tikhonov regularization technique, as mentioned above, is one of the major methods to reduce the ill-posedness of the problem [19]. However, there exists one protrudent difficulty for this technique in the determination of the regularization parameter. A general unexpected characteristic of the NIR imaging is that the resolution and contrast of the reconstructed images degrade with the increased distance from the sources and the detectors [20]. Considering the fact that the value of the regularization parameter has important effect on the contrast and resolution of the resultant images, one strategy to solve this problem is to use a spatially variant regularization parameter. Meanwhile, it can be inferred that the objective function is related to the regularization parameters [15]. During the process of minimizing the objective function, decreasing *λ* will speed up the convergence if the value of objective function is decreasing, otherwise increasing *λ* can enlarge the searching area (trust-region). Upon the basis of these considerations, we propose a modified regularization method both adaptive to the spatial variations and the objective function.

*M*and

*N*, respectively. Thus, we have for the matrices in equation (15):

*Δ*

**x**∈

**R**

^{ N × 1},

*Δ*

**y**∈

**R**

^{ M × 1},

**J**∈

**R**

^{ M × N }, and

**I**∈

**R**

^{ N × N }. To construct a spatially variant regularization framework, the inverse term of (

**J**

^{ T }

**J**+

*λ*

**I**)

^{-1}in equation (15) is replaced with (

**J**

^{ T }

**J**+ λ)

^{-1}, which results in the following equation

with *Δ* *x*
_{
i
}(*i* = 1, 2, ..., *N*) being the component of the vector *Δ* **x**. It can be easily seen that each node *p*
_{
i
}(*i* = 1, 2,...,*N*) in the reconstructed domain is regularized by a corresponding regularization parameter *λ*
_{
i
}(*i* = 1, 2,...,*N*) respectively. Obviously, the above mentioned Tikhonov regularization can be regarded as a special case of equation (17) when *λ*
_{1} = *λ*
_{2} = ⋯*λ*
_{N} = *λ*.

*λ*

_{ i }can be defined as follows to compensate the decrease of the resolution and contrast with the increased distance from the sources and detectors:

where **r**
_{
i
}is the position of node *p*
_{
i
}, **r**
_{
s
}and **r**
_{
m
}respectively denote the positions of the source and detector closest to the node *p*
_{
i
}, *c*
_{1} and *c*
_{2} are two positive parameters determined empirically in our paper.

In equation (19), the arctan function is used to guarantee a relatively small fluctuation range of the regularization parameters and avoid too large values of them. Obviously, regularization parameters determined from equation (19) relate to the objective function in a similar manner to that as pointed out before. In such a way, the regularization parameters are adaptive not only to the spatial variations but also to the variations of the objective function to accelerate the convergence.

### Reconstruction Based on the Schur Complement System

As has been pointed out previously, the iterative methods based on the Schur complement system can be more efficient to solve large-scale problems. Hence, we propose to reconstruct the tomographic image of FMT with level-by-level decomposition in the Schur complement system.

where **k** = **J**^{
T
}
**J** + **λ** and **b** = **J**^{
T
}
*Δ* **y**.

To solve the inverse problem of FMT in the Schur complement system, the solution space **R**^{
n
}is firstly decomposed into two subspaces of U and V with dimensions *m* and *n*-*m*, respectively. Let [**Γ Ψ**] be an orthonormal basis of the solution space **R**^{
n
}. The basis of the *m*-dimensional coarse subspace U is formed by the columns of **Γ** ∈ **R**^{
n × m
}and the columns of **Ψ** ∈ **R**^{
n×(n-m)
}form the basis of the (*n* - *m*) dimensional subspace V.

where *u* and *v* are the projections of *Δ* **x** on the subspaces U and V, respectively.

*u*and

*v*level by level with the Schur complement decomposition along two paths in a tree structure, and then solve the subsystems in the Schur complement systems. Our approach is different from that proposed in [10], where only the projection

*v*is solved in the Schur complement system. The level-by-level Schur complement decomposition can be schematically illustrated as in Figure 1. We derive the iterative system in the following discussions.

**S**

_{(i, j)}is the Schur complement matrix with the subscript (

*i*,

*j*) being the

*j*th (

*j*= 0, 1,..., 2

^{ i }) term at the

*i*th (

*i*= 0, 1,...,

*L*) level in the tree structure as illustrated in Figure 1. Particularly,

**S**

_{(0,0)}is the global matrix

**k**as defined in equation (20). To solve this system in the Schur complement system, equation (22) will be further decomposed at the

*i*+1th level. Thus, the solution

*Δ*

**x**

_{(i,j)}is firstly expressed with the bases of the two subspaces as

where *Δ* **x**
_{(i+1,2j-1)} and *Δ* **x**
_{(i+1,2j)} are the projections of *Δ* **x**
_{(i,j)} on the subspaces formed by the columns of **Γ**
_{(i,j)} and **Ψ**
_{(i,j)}, respectively.

**Γ**

_{(i,j)}

**Ψ**

_{(i,j)}]

^{ T }, we can obtain

where
,
,
, and
, while the two components on the right-hand side (RHS) of equation (26) are
, and
. From equation (26), it can be seen that **S**
_{(i,j)11} and **S**
_{(i,j)22} correspond to the equations for the unknowns of *Δ* **x**
_{(i+1,2j-1)} and *Δ* **x**
_{(i+1,2j)}, respectively, while **S**
_{(i,j)12} and **S**
_{(i,j)21} define the coupling between these two sets, which will be eliminated in the following discussions.

**S**

_{(i,j)11}[7], . From equation (27), we have

It can be found that the condition number of matrix **S**
_{(i+1,2j)} is smaller than that of matrix **S**
_{(i,j)}
[9]. Hence, solving the inverse problem in the Schur complement system at the *i*+1th level will be more efficient than solving it at the *i* th level. We herein solve equation (29) using the biconjugate gradient method [23]. Its advantage is that it does not square the condition number of the original equations [24]. Basically, the biconjugate gradient method can be used to solve the large-scale systems with the fastest speed among all the generalized conjugate gradient methods in many cases [25]. The algorithm for solving equation (29) can be summarized as follows

### Algorithm 1

- 1.
Input an initial guess

*Δ***x**_{(i+1,2j)0}; - 2.
Initialize

*d*_{0}=*f*_{0}=*r*_{0}=*p*_{0}←**b**_{(i+1,2j)}-**S**_{(i+1,2j)}*Δ***x**_{(i+1,2j)0}; - 3.

End for

After the derivation of *Δ* **x**
_{(i+1,2j)} from equation (29) with algorithm 1, the next task is to obtain the other component of *Δ* **x**
_{(i+1,2j-1)} for the synthesis of the solution *Δ* **x**
_{(i,j)}. Here, *Δ* **x**
_{(i+1,2j-1)} is also solved in the Schur complement system due to its low condition number.

**S**

_{(i,j)12}in equation (26) using block Gaussian elimination with

**S**

_{(i,j)22}as pivot block, we have

where
and
. Thus **S**
_{(i+1,2j-1)} is the Schur complement with respect to **S**
_{(i,j)22}.

Thus, the solution *Δ* **x**
_{(i+1,2j-1)} can be obtained in a same manner as in Algorithm 1, and the only difference is that *Δ* **x**
_{(i+1,2j)}, **S**
_{(i+1,2j)}, and **b**
_{(i+1,2j)} should be replaced with *Δ* **x**
_{(i+1,2j-1)}, **S**
_{(i+1,2j-1)}, and **b**
_{(i+1,2j-1)}, respectively. Solving equation (31) is computationally efficient because of the reduced condition number in the Schur complement system [7]. Moreover, such a strategy of deriving both *Δ* **x**
_{(i+1,2j-1)} and *Δ* **x**
_{(i+1,2j)} in the Schur complement system can be implemented in a parallel manner, since equations (29) and (31) are decoupled. Therefore the subsystem at the *i* th level as in equation (22) can be decomposed into the two linear subsystems at the *i*+1th level, i.e., Schur complement systems as in equations (29) and (31). After obtaining *Δ* **x**
_{(i+1,2j-1)} and *Δ* **x**
_{(i+1,2j)}, they are then substituted into equation (23) to yield the solution *Δ* **x**
_{(i,j)} at the *i* th level. The whole reconstruction algorithm is summarized as follows

### Algorithm 2

- 1.
Set

**x**_{0}to an initial guess; - 2.
**x**←**x**_{0}, calculate**b**and**k**at**x**in equation (20) with the adaptive regularization scheme as in equation (19); - 3.
The global system of equation (20) is decomposed with the Schur complement system level by level in a same manner as the decomposition of equation (22) into equations (29) and (31) to obtain the subsystem

**S**_{(i,j)}*Δ***x**_{(i,j)}=**b**_{(i,j)}at the*i*th level for*i*=1,...,*L*and*j*=1,..., 2^{ i }, the subspaces at the*i*th level are formed by the columns of**Γ**_{(i,j)}and**Ψ**_{(i,j)}, respectively; - 4.
Set

*i*=*L*;

For *j* = 1,..., 2^{
i
}do

Combining equations (26), (27), and (30), solve **S**
_{(i+1,2j)}
*Δ* **x**
_{(i+1,2j)} = **b**
_{(i+1,2j)} and **S**
_{(i+1,2j-1)}
*Δ* **x**
_{(i+1,2j-1)} = **b**
_{(i+1,2j-1)} with Algorithm 1, where *Δ* **x**
_{(i+1,2j-1)}, **S**
_{(i+1,2j-1)}, and **b**
_{(i+1,2j-1)} are used instead of *Δ* **x**
_{(i+1,2j)}, **S**
_{(i+1,2j)}, and **b**
_{(i+1,2j)} when Algorithm 1 is employed for the latter case;

- 5.
While

*i*≥ 0

{

For *j* = 1,..., 2^{
i
}do

Substitute the solutions *Δ* **x**
_{(i+1,2j)} and *Δ* **x**
_{(i+1,2j-1)} into equation (23) to obtain the solution *Δ* **x**
_{(i,j)} at the *i* th level;

End for

*i* = *i* - 1;

- 6.
**x**_{0}←**x**_{0}+*Δ***x**_{(0,1)}; - 7.
If ||

*Δ***x**_{(0,1)}|| >*ε*

go to 2;

Else

**x** ← **x**
_{0}, output **x**.

As mentioned before, the Schur complement system has a smaller condition number than that of the system from which it is constructed [7]. As a result, iterative methods based on the Schur complement systems can be more efficient than the methods based on the global matrix as in equation (20) due to its reduced scale and the smaller condition number. Therefore, the proposed algorithm can be expected to be more efficient than the conventional ones, as the results demonstrated in the next section.

## Results and Discussion

*μ*

_{ axf }. Two phantoms as illustrated in Figure 2 are used to evaluate the proposed algorithm. Figure 2(a) contains one object, and Figure 2(b) contains two objects of different shapes. Table 1 and Table 2 outline the optical and fluorescent parameters in different regions of the simulated phantoms corresponding to Figures 2(a) and 2(b), respectively. Four sources and thirty detectors are equally distributed around the circumference of the simulated phantom. The simulated forward data are obtained from equations (1) and (2), in which Gaussian noise with a signal-to-noise ratio of 10dB is added to evaluate the noise robustness of the algorithms. The parameters

*c*

_{1}and

*c*

_{2}in equation (19) are, respectively, set to 0.2 and 2. The initial guesses for solutions

*Δ*

**x**

_{(i+1,2j)}and

*Δ*

**x**

_{(i+1,2j-1)}of equations (29) and (31) are set to 0. The initial value of

**x**

_{0}is set to 5

*mm*

^{-1}. The subspace created from the right singular vectors of the singular value decomposition (SVD) is optimal. Since SVD is computationally expensive, it is expected that a subspace close to SVD subspace will do almost as good. Thus, the choice of an oscillatory basis can be a basis created by sine or cosine functions with increasing frequency [26]. Here discrete cosine basis is employed in the simulations. To reliably evaluate the performance of different methods for the inverse problem, the best way is to use an independent forward model, which is different from the one employed in the inverse problem, to generate the synthetic data [27]. Therefore, in our case, a finer mesh as shown in Figure 3 with 169 nodes and 294 triangular elements is used to generate the forward simulated data.

Optical and fluorescent properties of one-object phantom

Optical and fluorescent properties of two-object phantom

*μ*

_{ axf }for one object phantom using the adaptive regularization scheme and Tikhonov regularization method. Figures 7(a) and 7(b) depict the results for two objects phantom from the above two different algorithms. Here, both of the results from Figures 6 and 7 are based on the CG method. As seen from Figures 6 and 7, better reconstructed results can be achieved from the adaptive regularization scheme. To quantitatively assess the accuracy of the different algorithms, the mean square error (MSE) is introduced

*N*is the total number of nodes in the domain. The superscript

*calc*denotes the values obtained using reconstruction algorithms; and

*actual*denotes the actual distribution of

*μ*

_{ axf }which is used to generate the synthetic image data set. Table 3 lists the performance of the reconstruction algorithms in terms of MSE. It can be seen that the adaptive regularization scheme can significantly improve the quality of the reconstructed images and achieve a smaller MSE in either case.

Comparison of performance of methods

Methods | One object | Two objects |
---|---|---|

Adaptive regularization scheme | 2.973 × 10 | 2.860 × 10 |

Tikhonov regularization method | 5.352 × 10 | 4.892 × 10 |

*μ*

_{ axf }for one object phantom using the different algorithms after 1, 15, and 30 iterations, respectively. After 30 iterations, the reconstructed image from the proposed algorithm has a relatively higher contrast than those obtained from the other two algorithms. Figure 9 depicts the reconstructed images of

*μ*

_{ axf }for two objects phantom using the different algorithms, from which it can be seen that the proposed method can reconstruct the images more accurately than the other two methods even after the first iteration. According to the third column of Figure 9, the reconstructed image quality based on our algorithm is significantly improved as compared with that based on the other two methods.

Computation time of FMT image reconstruction for 30 Iterations

Methods | One object | Two objects |
---|---|---|

CG | 62s | 86s |

Schur CG | 203s | 281s |

Our algorithm | 141s | 179s |

Performance comparison of reconstruction methods for 3D case

Methods | Schur CG | Our algorithm |
---|---|---|

Computation time (s) | 3527 | 2215 |

MSE | 3.629 × 10 | 1.241 × 10 |

## Conclusion

In this paper, we developed a novel image reconstruction method of FMT, based on the tree structured Schur complement decomposition in combination with the adaptive regularization scheme. The proposed approach decomposes the global inverse problem level by level with the Schur complement decomposition, and the resultant subsystems are solved with the biconjugate gradient method. The spatially variant regularization parameter is determined adaptively according to the objective function. Simulation results demonstrate that the proposed method outperforms the previous methods, such as the CG and the Schur CG methods, in both reconstruction accuracy and speed.

## Declarations

### Acknowledgements

This research is supported by the National Natural Science Foundation of China, No.60871086, the Natural Science Foundation of Jiangsu Province China No.BK2008159, the CSC Scholarship, PolyU, and ARC grants.

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.