Wavelets have been very popular in the field of image compression and noise reduction. Another interesting application is adaptive mesh refinement. There are many wavelets with various properties, which will have different effects on different applications. There is no consensus on which wavelet is the best option for adaptive mesh refinement. Most commonly used wavelets for adaptive mesh refinement are Donoho's interpolating wavelet and Sweldens wavelet, the latter a lifted version of Donoho's interpolating wavelet. A detailed comparison of both wavelets is done on different data sets. Moreover, different manners of handling the boundaries are tested. An algorithm to construct the meshes using wavelets is tested and optimised. Donoho's interpolating wavelet with the lower order boundary stencil implementation resulted to be the most accurate, whilst resulting in very high compression compared to the original mesh. Furthermore, changing the adaptivity of the algorithm, which constructs the meshes, turned out to be valuable for fast changing shapes. Lastly, an improvement on the inverse transform during the adaptive mesh refinement had promising results.