3.2 Inverse Problem: Reconstruction
3.2.3 Least Squares Estimation
3.2.3.2 Numerical Optimisation
Optimisation is ubiquitous, and it has been used in many areas including engineering, economics, operations research, and others. Essentially, the numerical optimisation is a mathematical problem identifying optimum or optima, which may be expressed in calculus-based formulas. There are plenty of choices of numerical optimisation schemes, which includes line search methods, trust-region based methods, gradient (steepest) descent method, conjugate gradient methods, Levenberg-Marquardt algorithm, Gauss-Newton algorithm, quasi-Newton methods, the simplex method, and other variants of existing methods. A thorough investigation on numerical optimisation algorithms [21] [304] [30] [217] showed that the conjugate gradient method, quasi-Newton method and trust-region types of
line search method, steepest descent method, and Nelder-Mead method are suitable for small-scale or medium-scale reconstruction problems.
We emphasise the conjugate gradient method and quasi-Newton method in this thesis. The per- formance of the linear conjugate gradient method is determined by the distribution of the eigenvalues of the coefficient matrix. By transforming, or preconditioning the linear system, we can make this distribution more favourable and improve the convergence of the method significantly. The linear conjugate gradient method was proposed by Hestenes and Stiefel in the 1950s[110]. It is an iterative
method, which was motivated by the alternative to Gaussian elimination that was well suited for solving large problems. In addition, Fletcher and Reeves[78]introduced the first nonlinear conjugate
gradient method in 1964. It was one of the earliest known techniques for solving large-scale nonlinear optimisation problems. Over the years, many variants of this original scheme were proposed, and some were widely used in practice. The key features of these algorithms were that they required no matrix storage, and they were faster than the steepest descent method [217]. Vogel [304] has shown
that for extremely large and highly structured linear systems, the conjugate gradient method pro- vided a very efficient solution. In addition, the conjugate gradient method could also be adapted to solve non-quadratic optimisation problems, and a general non-linear conjugate gradient algorithm is summarised in AppendixD.
Furthermore, the motivation to develop the quasi-Newton method was the attempt of solving a long optimisation calculation in the mid 1950s. The quasi-Newton method is one of the most creative ideas in nonlinear optimisation[217]. Similar to Newton’s methods and the steepest descent method,
the optimisation is supplied with the gradient information of the objective function as the guidance of the searching direction. The advantages of the quasi-Newton method are: firstly, it possesses rapid superlinear convergence; secondly, it is designed to prevent the calculation of the Hessian matrix or its inverse at each iteration that happens in other Newtons methods; thirdly, due to the efficiency of the quasi-Newton, it makes the development and solution of the large-scale optimisation problems became reality. Thus, the core of the quasi-Newton theory is the approximation of the inverse Hessian matrix. The most popular approximation methods used in quasi-Newton algorithms are DFP and BFGS methods (Appendix D), named using the acronymic combinations of the scientists’ names: Davidon-Fletcher-Powell and Broyden-Fletcher-Goldfarb-Shanno. In addition, the limited-memory quasi-Newton methods use a few vectors instead of manipulating fully dense approximation of the inverse Hessian matrix. There are various limited-memory methods, and the one based on BFGS is named L-BFGS. Based on the description of the BFGS method, the L-BFGS method, which has been used in our application, is outlined as follows.
The objective of BFGS method is to find an appropriate approximation for the inverse Hessian matrix, that is Hk+1= VkTHkVk+ αksksTk, (3.65) in which αk= 1 zT ksk , Vk = I − αzksTk, (3.66) and sk = fk+1− fk, zk = g(fk+1) − g(fk). (3.67)
In addition, the updating at k-th iteration can be formulated using
Hk+1= VkTV T k−1· · · V T 0 HinitialV0· · · Vk−1Vk + VkT· · · VT 1 α0s0sT0V1· · · Vk .. . + VkTVk−1T αk−2sk−2sTk−2Vk−1Vk + VkTαk−1sk−1sTk−1Vk + αksksTk (3.68)
In order to circumvent storing all these vector pairs of {sk, zk}, L-BFGS method only use infor-
mation from last m iterations that is
Hk+1= VkTV T k−1· · · V T k− ˆmHinitialVk− ˆm· · · Vk−1Vk + VkT· · · VT k− ˆm+1αk− ˆmsk− ˆmsTk− ˆmVk− ˆm+1· · · Vk .. . + αksksTk (3.69)
where m is defined by the user according to empirical. In addition, a typical value 3 ≤ m ≤ 20 could produce satisfactory results, and ˆm = min{k, m − 1}. Therefore, when k > m we can discard the vector pair {sk−m, zk−m} from storage that is the main difference between L-BFGS and its parent
3.3
Conclusions and Discussion
In this chapter, we have surveyed various classical and novel reconstruction methods, which can be applied to reconstruct DBT volumes with particular geometric setup. The reconstruction algorithms can be coarsely divided into two categories, namely analytical and iterative methods.
SAA and BP are straightforward reconstruction algorithms, which mathematically line up the in-focus structures along the X-ray tube motion direction. SAA and BP can reconstruct mass clearly but with serious blurry artefacts, especially for the edges of the mass. FBP is based on the traditional SAA algorithm, which obtains breast structures with deblurring techniques. However, the FBP method is always restricted by the imaging geometry, and the reconstruction efficacy depends on the filter design for limited angle DBT reconstruction problem with highly incomplete sampled spatial frequency domain. FBP contains an inverse filtering procedure for the forward projected images including sampling density information in the frequency domain, and back projects the filtered images to the spatial domain to create the reconstructed slice. Compared to SAA and BP algorithms, FBP suppresses out-of-plane blurring with a comparatively fast reconstruction. FBP is superior for reconstruction of low frequency contents. In addition, it possesses a fast reconstruction, which is feasible for clinical usage, and currently most industrial vendors use FBP to reconstruct DBT volumes.
Iterative methods based on ART technique show their promise in reconstructing DBT volumes, and they are not limited by the imaging geometry. In many studies, SART is favoured for DBT reconstruction because this block-action strategy has a moderate trade-off between ART and SIRT. In the traditional ART method, the measurement noise is considerably amplified due to the contin- ual updating and the intrinsic ill-posed property of this kind of inverse problem. In contrast, the convergence speed is relatively slow using SIRT reconstruction, which averages over all the rays in all projections, and the reconstructed results are always over smoothed. Least squares estimation with proper numerical optimisation provides an alternative way of performing iterative reconstruction. ART based methods are suitable to reconstruct complete and noisy data sets like fully sampled CT projection data with noise; however, DBT systems produce highly incomplete projections with low noise. Therefore, least squares estimation based algorithms outperform ART based methods for DBT reconstruction in theory. However, in real applications, regularisation is necessary when the data are not ideal. Many regularisation schemes are well studied in CT, PET and optical tomography reconstructions, and various Total Variation like (TV-like) regularisation methods[143] [260] [261] are proposed for DBT reconstruction. Furthermore, there is a trade-off between the data fidelity and
the regularisation terms, which are represented by data error and image regularity metric respec- tively (as seen in Figure 3.12). When the regularisation is small, the reconstruction is close to the available data but may contain evident artefacts because of the noisy or inconsistent data, whereas large regularisation term outputs smooth images at the expense of fidelity to the data.
Image Regularity Metric
Da
ta
E
rr
or
Figure 3.12: Diagram of the trade-off between the data fidelity and the regularisation terms represented by data error and image regularity metric respectively. The plot shows the fidelity and regularity plane for an undersampled (blue area) versus a completely sampled tomographic system (red area). For completely sampled systems, a unique image minimises the data error because only one value of the regularity is possible, whereas many possible candidate volumes correspond to the situation of minimum data error for undersampled systems. The two curves represent a generic behaviour of standard iterative algorithms for the case of no regularisation (solid curve) and with regularisation (dashed curve).
In summary, compared to the analytical methods, the iterative methods are conceptually simple with milder assumptions for the missing data, but always converge slowly and may be trapped by local minima. On the contrary, FBP method presents fast reconstruction, but the filter design, parameters tuning are complicated. In our DBT registration and reconstruction framework, we develop a least squares based approach, which is simple to combine with existing numerical optimisation techniques, and it could also incorporate different image constraints and various regularisation schemes.
Chapter
4
Image Registration and Its Applications in DBT
Registration is a major field of research in medical image processing. It searches for the best geomet- rical transformation mapping points from one image to the corresponding points in another image
[19]. Intra-subject registration maximises correspondences of two or more images, which are acquired
using various imaging modalities or at different time points, from the same subject. Whereas, inter- subject registration aim to align images of multiple subjects in order to perform population-based studies on the anatomical variability.
Medical image registration has been used in many clinical and research applications [33] [70] [277]
[185] [169] [18] [116] [348] [223] [292], e.g., image registration can eliminate the effect of patient position
and motion artefacts. Thus, it is often used throughout the imaging, screening and treatment process to capture, visualise and quantitatively analyse the development of disease.
There are several significant concepts, which must be considered when developing image registra- tion algorithms. These include image interpolation, the transformation model used, similarity metric or distance measurement, optimisation, and validation. Firstly, the image processing algorithms of- ten utilise image interpolation. Particularly, in image registration, interpolation contributes in data preparation, image resampling, and transformations. Secondly, the transformation model, including rigid, affine and deformable transforms, parameterises the geometrical deformation field. Distance measurement provides the metric of similarity between the images, and is the registration basis. Many criteria are proposed as the basis of aligning two images, and a common classification is as landmark-based, segmentation-based or intensity-based registration[311] [281]. Next, the registration process can be viewed as an optimisation problem, and the goal is to minimise an associated objec- tive function modelled by the distance measurement. Finally, the validation evaluates the accuracy, robustness, and clinical utility of the registration results. However, the evaluation is difficult because the correct transformation, i.e., the ground truth, is usually unknown in a real application.