El contexto Distrital

Inverse crimes arise when the same model is used to generate the test data and to compute the reconstruction. They are intrinsically related to the process of convert- ing an infinite-dimensional quantity to a finite-dimensional approximation necessi- tated by numerical processing. This has been highlighted in the inverse problem community [27, 91] and more recently in compressed sensing [92].

Conclusion

where the same discrete model is used both for simulation and reconstruction. Sur- prisingly, researchers in the MR community have only started recently to work out this problem. A notable approach has been proposed by Guerquin-Kern et al. [93] in 2012 who have developed an analytical phantom that can be defined with B´ezier curve instead of rasterised image. This allows to take into consideration the contin- uous nature of data, providing more realistic reconstruction simulations.

Note that in this dissertation, we will also use the same discrete model for both simulation and reconstruction in numerical experiments. It should be acknowledged that it will result in artificially better reconstructions because the continuous nature of the data will not be taken into account.

3.6

Conclusion

In this chapter, we have reviewed linear discrete inverse problems. We have discussed important concepts such as ill-posedness, optimisation and regularisation in inverse problems. This chapter has mainly focused on signal recovery methods from partial data. These inverse problems can be solved using low-dimensional signal models by promoting low-complexity regularisation prior because most of the time signals lie in much lower dimensional spaces than their original domain. We have provided an overview of state of art reconstruction methods for sub-Nyquist dynamic MRI and briefly discussed inverse crimes.

Chapter 4

Optimisation framework:

proximal splitting methods

Contents

4.1 Introduction . . . 71 4.2 Proximal operators . . . 73 4.2.1 Definition . . . 73 4.2.2 Absolute value and `1 norm . . . 74

4.2.3 Nuclear norm . . . 76 4.3 Proximal splitting algorithms . . . 76 4.3.1 Proximal gradient methods . . . 76 4.3.2 Alternating direction method of multipliers . . . 79 4.4 Compressed sensing MRI example . . . 82 4.5 On nonconvex optimisation and greedy approaches . . . 84 4.6 Conclusion . . . 87

4.1

Introduction

In this chapter, we introduce proximal splitting methods, a general framework to solve various convex optimisation problems.

Convex optimisation1 [94–96] has increasingly gained in importance in recent years. One of the reason that explains this popularity is that even when the prob- lem dimensions get large (roughly the number of variables and constraints), convex optimisation problems are still relatively easy to solve in contrast to nonconvex prob- lems. In fact, although a common view is to generally interpret linear problems as easy and nonlinear ones as difficult, Rockafellar in his book Lagrange multipliers and

Optimisation framework: proximal splitting methods

optimality (1993) noted ”In fact the great watershed in optimization isn’t between linearity and nonlinearity, but convexity and non-convexity”. The popularity of con- vex optimisation methods can also be explained thanks to the availability of efficient minimisation algorithms to compute globally optimal solutions, the increase in com- puter power in the past decades (Moore’s law), and the significant impact on the resolution of difficult problems when formulated with the help of convex relaxation (i.e. `1 and nuclear norm problems).

For these reasons, optimisation problems expressed as the minimisation of con- vex functionals are now ubiquitous in various areas such as in inverse problems [97], signal and image processing [98] or machine learning [99], and thus there is a huge interest in developing robust, fast and efficient algorithms for convex optimisation. However, difficulties arise when optimisation problems include nonsmooth and large- scale properties. Proximal splitting methods [100, 101] are first-order iterative al- gorithms for solving such convex optimisation problems. They operate by splitting the convex objective function to minimise which generates individual convex sub- problems. These sub-problems are evaluated easily via proximal operators, a gener- alisation of the projection operator. Proximal splitting methods offer a number of interesting properties that are particularly adapted for the work presented in this thesis:

• Convergence. Most of the proximal algorithms that we will describe in sec- tion 4.3 have convergence guarantees and/or potentially quantified competitive convergence rates.

• Computational speed. First-order methods are well suited for large-scale prob- lems, mainly because iterations of typical first-order methods in the large-scale case remains cheap to evaluate (compared to interior-point methods for exam- ple). In the context of `1 and nuclear norms, first-order methods also possess

nearly dimension-independent convergence rates as discussed by Nesterov and Nemirovski [102].

• Simplicity. These algorithms are in general short (a few lines) and easy to implement with minimal storage requirement.

• Flexibility. These algorithms can handle various general convex problems (po- tentially nonsmooth) and as such are flexible as long as the proximal operators can be evaluated easily. Due to the splitting approach, they also naturally fit the distributed and parallel computation framework [103].

This chapter is organised as follows. In section 4.2, we define the notion of proximal operator and give the analytical solutions of proximal operators relevant to this thesis. We review some proximal algorithms in section 4.3, and apply some of these algorithms in the context of compressed sensing MRI in section 4.4. We

Proximal operators

also comment on nonconvex and greedy approaches in section 4.5, before concluding this chapter in section 4.6.