In a near future, the Surface Water Ocean Topography (SWOT) mission will provide images of altimetric data at kilometric resolution. This unprecedented 2-dimensional data structure will allow the estimation of geostrophy-related quantities that are essential for studying the ocean surface dynamics and for data assimilation uses. To estimate these quantities, i.e. compute spatial derivatives of the Sea Surface Height (SSH) measurements, the small-scale noise expected to affect the SWOT data must be smoothed out while minimizing the loss of relevant, physical SSH information. This paper introduces a new technique for de-noising the future SWOT SSH images. The de-noising model is formulated as a regularized least-square problem with a Tikhonov regularization based on the first, second, and third-order derivatives of SSH. The method is implemented and compared to other, convolution-based filtering methods with boxcar and Gaussian kernels. This is performed using a large set of pseudo-SWOT data generated in the Western Mediterranean Sea, from a 1/60 • simulation and the SWOT simulator. Based on Root Mean Square Error and spectral diagnostics, our de-noising method shows a better performance than the convolution-based methods. We find the optimal parametrization to be when only the second-order SSH derivative is penalized. This de-noising reduces the spatial scale resolved by SWOT by a factor of 2, and at 10 km wavelengths the noise level is reduced by 10 4 and 10 3 for Summer and Winter respectively. This is encouraging for the processing of the future SWOT data.