Accurate $$$\Delta f_0$$$ measurements are essential for fat-water separation and for removing off-resonance related blurring in spiral images. However, areas of low signal make it difficult to estimate $$$\Delta f_0$$$ values from MRI data. Additionally, extreme off-resonance areas may exhibit signal dropout that further disrupt field map measurements. Robust and simple segmentation (e.g. thresholding) can be used to restrict $$$\Delta f_0$$$ map estimation to areas it is likely to succeed (Figure 1, left), but this leaves holes in the resulting field map. Filling these holes requires some form of data extrapolation or regularized solution [1], which may also be useful near the edges of the imaged object.

Image inpainting (interpolation or extrapolation) is typically used to synthesize data to repair damaged portions of digital images. Several methods for inpainting have been proposed; most propagate known valid image information into the regions to be inpainted while attempting to preserve smoothness and continuity of isophotes (lines of equal gray value) inside the interpolated region. Proposed methods vary in speed and difficulty of implementation [2, 3]. This work describes the application of an image inpainting technique based on the fast marching method (FMM) [4] for efficient and robust extrapolation of $$$\Delta f_0$$$ map data.

The method shown treats the regions to be inpainted as level sets, and uses the FMM to progressively interpolate the data starting from the edges. This method has the benefit of being both efficient and simple to implement, with comparable results to alternative methods. This implementation also offers flexibility with respect to the specific image propagator, such that it may be tuned to a given application. The FMM solves the Eikonal equation $$$|\nabla T| = 1$$$ for the region to be inpainted $$$\Omega$$$, such that the solution $$$T$$$ is the signed distance function (Figure 1, right) representing the distance of a pixel from the boundary of the region to be inpainted ($$$\delta\Omega$$$, where $$$T = 0$$$). Thus, level sets of $$$T$$$ are the successive boundaries for the unknown region as it shrinks due to inpainting. An inpainting propagator is then applied progressively on the level sets of $$$T$$$ to calculate a value for each pixel $$$p$$$ in the unknown region. This work used the propagator proposed by Telea, which calculates $$$p$$$ using a combination of the image intensity and gradient at each pixel $$$q$$$ in its neighborhood $$$\varepsilon$$$:

$$I(p) = \frac{\sum_{q\in\varepsilon}{w(p,q)[I(q) + \nabla I(q)(p-q)]}}{\sum_{q\in\varepsilon}{w(p,q)}}$$

where $$$\omega$$$ is a weighting function with range $$$[0, 1)$$$ that comprises three measures of similarity: a) the dot product of $$$\vec{pq}$$$ and $$$\vec{N}$$$, the normal vector of $$$T$$$ at $$$p$$$; b) the distance $$$|\vec{pq}|$$$; and c) the level-set distance $$$|T(p)-T(q)|$$$.Figure 3 shows results for a representative slice of a 3D dataset, though this implementation operates in a slice-by-slice manner. This technique produces visual consistency, but not necessarily accurate values. Inpainted areas, especially if they are relatively large, may suffer from some amount of blurring. In the case of a $$$\Delta f_0$$$ map, we expect relatively little high-frequency data to begin with, mitigating such issues. Future improvements include using a model-based propagator to incorporate physical properties–such as estimated magnetic susceptibility–during inpainting, and extending the method to operate in 3D.