The prolonged time required to form an MR image continues to impose different challenges at both theoretical and clinical levels. With this motivation in mind, this work addresses a central topic in MRI, which is how to correct the motion problem, through a new multitask optimisation framework. The significance is that by tackling the reconstruction and registration tasks − simultaneously and jointly − one can exploit their strong correlation reducing error propagations and resulting in a significant motion correction. The clinical potentials of our approach are reflected in having higher image quality with fewer artefacts whilst keeping fine details. We evaluate our approach through a set of quantitative and qualitative experimental results.
In a MRI setting, a target image u∈RN representing a part of the patient body is acquired in spatial-frequency space. The measured samples can be represented in a matrix form as x=Au+ε where x∈CM(M≪N) refers to the k,t−space measurements, A:RN→CM is the Fourier operator (neglecting the phase), and ε models some noise. For a multiple receiver coil, A encodes coils sensitivities and the Fourier transform.In this work, we seek to extract simultaneously from a set of multiple MR acquisitions xi, corrupted by motion, a mean static and clean reconstructed image u as well as the deformation maps φi aligning each image of the set to the mean image. Combining these two tasks in a unified variational framework, our optimisation problem is the following:
min
This minimisation problem for motion correction is composed of three terms: (i) a nonlinear-elasticity-based regulariser that describes the nature of the deformations - we model the organs as homogeneous, isotropic, and hyperelastic materials (more precisely, as Saint Venant-Kirchhoff materials) as shown in ^{5,6}; (ii) a discrepancy term that enforces the deformed mean to match the acquisitions; (iii) a total variation (TV) type regulariser for edge preservation of the reconstructed image. We obtain an approximate solution by an alternating optimisation scheme. Our approach is summarised in Figure 1.
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