^{1, 2, 3}

^{1, 2, 3}

^{1}

^{2, 3, 4}

^{1}

^{2}

^{3}

^{4}

Image reconstruction of fluorescent molecular tomography (FMT) often involves repeatedly solving large-dimensional matrix equations, which are computationally expensive, especially for the case where there are large deviations in the optical properties between the target and the reference medium. In this paper, a wavelet-based multiresolution reconstruction approach is proposed for the FMT reconstruction in combination with a parallel forward computing strategy, in which both the forward and the inverse problems of FMT are solved in the wavelet domain. Simulation results demonstrate that the proposed approach can significantly speed up the reconstruction process and improve the image quality of FMT.

The use of near-infrared (NIR) light in biomedical research has made significant progress over the past few years [

Reconstruction of tomographic data from diffusing sources involves the generation of a forward model that predicts the photon distribution striking the detectors for a given source location and medium [

The generation and propagation of the excitation and fluorescence (emission) light in random highly scattering media can be described by two coupled diffusion equations which are the _{1} approximation to the radiative transport equation (RTE). In the frequency domain, the diffusion equations become elliptic and can be expressed as

The solutions to (

Suppose

The accuracy and speed of solving the forward problem as discussed in Section

For the convenience of the following discussion, a brief introduction of the theory of wavelet transform is presented here. The wavelet transform is a tool that cups up data of functions or operators into different frequency components and then studies each component with a resolution matched to its scale. By a proper design of the basis, the wavelet can project the signal onto a chain of embedded approximations and details at various levels of resolutions, and, as a result, the wavelet transform is usually referred to as the multiresolution analysis. For example, the two-level wavelet-based multiresolution representation of one dimensional discrete signal

Similarly, the two-level wavelet-based multiresolution representation of a 2D image

In order to exploit the multiresolution property of the wavelet and reduce the forward computational time, the forward problem is first represented in the wavelet domain. For such a purpose, multiplying both sides of (

It is well known that the most important feature of the wavelet transforms lies in the fact that most information of the signal is contained in a small number of entries with other entries being very small and therefore can be neglected. As a result, the dimension of the forward problem can be reduced level by level by using only the approximation components of the wavelet coefficients to describe the forward problem, that is,

Using the above multiresolution representation, the forward problem can be solved in a fine-to-coarse-to-fine procedure which can be summarized as in Algorithm

Owing to the fact that some important features are contained in the coarse resolution solution, as a result, it will be very helpful for speeding up the iterative process when solving the forward problem at a higher level resolution with the solution obtained at a coarser resolution as an initial guess. Therefore, we can expect to expedite the process of solving the forward problem by using Algorithm

After the discussion of the wavelet-based algorithm for the forward problem corresponding to the excitation light, the next task for us will be that of solving the forward problem for the emission photons. For the case where there are large deviations between the referenced and target medium, the forward equations must be solved repeatedly during the process of reconstruction following a model-based iterative image reconstruction scheme. Therefore, a rapid and accurate computational implementation of the forward problem is of critical importance for fluorescent molecular tomographic image reconstruction. From (

In summary, the whole forward computation process in our proposed algorithm can be realized with Algorithm

The most important aspect of the parallel computing strategy is decoupling of the two coupled equations. In order to illustrate the improvement of the parallel computing strategy in computational complexity as compared with the sequential one quantitatively, the computing efficiency is analyzed as follows.

Because the maximum computational complexity of solving linear equations defined by a matrix sized

Particularly, if we are interested only in the reconstruction of the absorption coefficient

The forward and inverse problem of FMT can be, respectively, formulated as

Generally,

By introducing two quantities of

Using (

(4) Set

(6) Solve

(7)

From Algorithm

Actually the fluorescent measurements are used as the input to reconstruct the image for FMT according to Section

In the presence of the background fluorescence, the fluorescence concentration can be formulated as follows [

Furthermore, the fluorescence concentration can be described as follows [

According to (

In order to improve the reconstruction quality in the presence of the background fluorescence, the reconstruction results can be corrected as follows [

The algorithm proposed in this paper has been firstly tested in a 2D simulated phantom with two anomalies existing within it as illustrated in Figure

Optical and fluorescent properties.

Excitation light | ||||||
---|---|---|---|---|---|---|

Background | 0.06 | 0.06 | 5.0 | 0.3 | 0.5 | 0.06 |

Anomalies | 0.06 | 0.15, 0.2 | 5.0 | 0.3 | 0.5 | 0.06 |

Fluorescent light | ||||||

Background | 0.02 | 0.006 | 2.0 | 0.3 | 0.5 | 0.02 |

Anomalies | 0.02 | 0.05, 0.1 | 2.0 | 0.3 | 0.5 | 0.02 |

Model of reconstruction.

Uniform mesh.

The data correction strategy is implemented after the reconstruction for improving the reconstruction quality. Figure

Reconstructed image of absorption coefficient due to fluorophore

Figures

Performance comparison of reconstruction algorithms.

Performance | Reconstruction without data correction | Method in [ | Proposed algorithm |
---|---|---|---|

1.114 × 10^{-7} | 6.573 × 10^{-8} | 2.312 × 10^{-8} | |

NRMS | 6.798×10^{-3} | 5.238 × 10^{-3} | 3.471 × 10^{-3} |

Reconstructed image of absorption coefficient due to fluorophore

Recently, there has been a great amount of interest in developing multimodality imaging techniques for oncologic research and clinical studies with the aim of obtaining complementary information and, thus, improving the detection and characterization of tumors [

Model of prior image.

Adaptively refined mesh.

Furthermore, in order to demonstrate the advantage of the proposed algorithm as compared with the traditional single resolution method without wavelet transform, Figures

Performance comparison of reconstruction algorithms.

Performance | Single resolution method | Proposed multiresolution algorithm |
---|---|---|

Computation time (s) | 247 | 185 |

NRMS | 6.623 × 10^{-3} | 2.679 × 10^{-3} |

7.985 × 10^{-8} | 1.572 × 10^{-8} |

Reconstructed image of absorption coefficient due to fluorophore

To further validate the proposed algorithm for 3D reconstruction, we extend the methods previously defined for triangular elements to tetrahedral elements. Therefore, the shape functions in the local coordinate system

In the 3D case, a phantom of radius 10 mm and height 40 mm with a uniform background

Schematic diagram of the phantom of radius 10 mm and height 40 mm with a uniform background of

3D mesh for image reconstruction with 858 nodes and 3208 tetrahedral elements.

Reconstructed images using the proposed algorithm, which are 2D cross sections through the reconstructed 3D volume. The right-hand side corresponds to the top of the cylinder

Reconstructed images using the single resolution method, which are 2D cross sections through the reconstructed 3D volume. The right-hand side corresponds to the top of the cylinder

Table

Performance comparison of reconstruction methods.

Performance | Single resolution method | Proposed multiresolution algorithm |
---|---|---|

Computation time (s) | 3196 | 2279 |

NRMS | 9.114 × 10^{-2} | 2.043 × 10^{-2} |

1.963 × 10^{-5} | 4.566 × 10^{-6} |

Furthermore, the proposed algorithm decouples the two coupled equations for the forward problem of FMT, and thus it is quite suitable for parallel computing of the two independent equations with two processors. Table

Efficiency analysis of parallel computing strategy.

2D case | 3D case | |||
---|---|---|---|---|

Number of processors | 1 | 2 | 1 | 2 |

Computation time (s) | 292 | 185 | 3712 | 2279 |

Speedup | 1.00 | 1.58 | 1.00 | 1.63 |

In summary, a wavelet-based multiresolution reconstruction algorithm is proposed in combination with the parallel forward computation strategy for the purpose of speeding up the reconstruction process with an improved reconstruction accuracy. The most important contribution of this paper is the novel extension of the multiresolution reconstruction approach originally developed for the diffuse optical tomographic reconstruction to the case of fluorescent molecular tomographic reconstruction and for the case where there are large deviations of the optical parameters between the target and the reference medium. Different from the algorithm proposed in [

This work was supported by the National Natural Science Foundation of China (no. 60871086), Natural Science Foundation of Jiangsu Province China (no. BK2008159), ARC, and PolyU Grants.