Holographic displays have always shown great promise, a legitimate solution to current two dimensional stereoscopic displays. Holography is capable of demonstrating fully three-dimensional images without a loss in resolution, however, its current drawbacks such as comparatively limited field of view, limitations in hardware, with spatial light modulators mostly capable of modulating either the amplitude or phase of light, or its subpar image quality, have delayed its introduction and success in the mass market. This work focuses on the latter problem, finding novel algorithmic mechanisms for improving the image quality in replay fields. Given the nature of this issue within this field, there are numerous approaches for resolving this issue, more recently through machine learning, adopting supervised training models. This work follows a different approach. It investigates other industries, such as microscopy, astronomy, and computed tomography to draw parallels with the advances made there and adapted these implementations to computer generated holography. As a result, the Kaczmarz and later the Cimmino iterative algorithms were adapted to this problem and analysed extensively against Gerchberg-Saxton, a common algorithmic benchmark used in holographic projections. The results, in terms of image quality, are positive since in all cases these algorithms converge to a suitably low minima and display the intended replay field, whilst simultaneously outperforming Gerchberg-Saxton in several instances. Much like any other research proposal, there are always drawbacks, and trade-offs. Whilst these algorithms were shown to outperform the reference Gerchberg-Saxton in many scenarios, they are also comparatively computationally intensive and run significantly slower than the benchmark. Much of the work that followed under Cimmino was finding ways of combining these two attributes, i.e., improve the image quality as well as its speed. The Cimmino FFT algorithm was derived and equates to the more formally known Fienup algorithm. This work found a link between the Cimmino algorithm developed in the 1930s solving linear systems, to the Fienup algorithm developed in the 1970s, a well-known phase retrieval algorithm. Nevertheless, it was apparent when analysing the computational intensity of these algorithms the iterative use of discrete Fourier transforms (DFTs) and fast Fourier transforms (FFTs) requires significant processing power and is the bulk of the overall algorithmic run time. The final analysis of this thesis considers reducing this run time further by interpolating the replay field using a Hilbert curve. There is, however, another trade-off between the final image quality and the use of more DFT and FFTs against the use of interpolation, as expected. In conclusion, it can be verified that the larger the resolution of the image, the more benefit will be seen in terms of computational processing and algorithm run time, when adopting Hilbert interpolation.