MRI machines use superconducting magnets to create an image. However, these magnets are very expensive. It is possible to use weaker magnets in a low-eld MRI, but those will result in a lower signal-to-noise ratio, meaning the images will be polluted. An image can be smoothed by viewing it as the initial condition of a partial dierential equation (PDE) and changing it through time integration. The choice for the PDE determines the way the image changes. This paper compares four PDE's: a second-order equation originally proposed by Perona & Malik, a fourth-order equation as proposed by You & Kaveh, and both aforementioned equations with a delity term added to them. Said delity term ensures the result does not deviate too far from the original image. All methods use a diusion coecient specially desiged to preserve edges. These methods are tested on two versions of the Shepp-Logan phantom, one having been corrupted with 'salt-and-pepper' noise, and the other one having been treated with a Gaussian lter, blurring the image. The salt-and-pepper phantom is improved most by applying the Perona- Malik method with a delity term. This method gives a good balance between removing noise and preserving edges and details within the image. For the blurry phantom the best result is seen using Perona-Malik, where some of the edges become more dened. However, a delicate balance has to be kept between rening the edges and blurring out any lower-contrast detail, and the total eect is limited. The methods are also tested on images that were created using a prototype of a low-eld MRI machine. The noise in these images is mostly the 'saltand- pepper' type. Though the preferred result is somewhat subjective, the Perona-Malik method with delity once again gives the clearest image here.