Image reconstruction through multiple 1D approximations
OA Version
Citation
Abstract
Function approximation is a fundamental aspect of computational models and machine learning, often relying on neural networks due to their ability to effectively model complex functions and relationships. However, neural networks can be computationally intensive and lack interpretability. In this thesis, we explore an alternative approach to approximating two-dimensional (2D) functions by decomposing them into multiple one-dimensional (1D) approximations. Our method aims to enhance computational efficiency and interpretability while maintaining high approximation quality. We propose a framework that projects to approximate 2D functions through a series of 1D interpolations and also uses greedy sampling. By generating uniformly distributed projections and projecting pixel coordinates onto these projections, we form 1D curves and use interpolation to predict the values of the original function. Linear interpolation is employed for its simplicity and speed in estimating values between sampled points. A greedy algorithm is used to select sampling points that significantly reduce approximation error, optimizing the sampling strategy. We conducted extensive experiments on some images to evaluate the performance of our method. Metrics such as Mean Squared Error (MSE) and Peak Signal-to-Noise Ratio (PSNR) were used to assess reconstruction quality. Additionally, we ran neural network model and some other traditional models for comparison. Our results demonstrate that the proposed method provides a different focus compared to other methods, especially excelling in the restoration of high-contrast details in images. The findings suggest that multiple 1D approximations can reconstruct 2D functions with efficiency. Contrary to our initial intuition, the results reveal that increasing the number of sample points has a more significant impact on reconstruction quality than increasing the number of projections. Specifically, we observed that under the same parameter count, using as many sample points as possible led to better reconstruction results. Increasing the number of projections, while beneficial for reducing artifacts, has a less pronounced effect compared to increasing sample points. However, adding more projections can improve edge clarity and enhance the accuracy of each step in the greedy selection process, which helps in achieving better sample point locations during reconstruction. Additionally, we tested various sampling methods, such as uniform sampling and greedy MSE selection, and found that greedy selection of sample points based on MSE yielded significantly improved clarity, particularly around key features of the image. The experiments also showed that incorporating spatial diversity and edge information into the selection process did not always yield better results, highlighting the importance of selecting sample points that balance both edge and surrounding details. This work contributes to the field by providing an alternative method for function approximation that addresses some limitations of neural networks, particularly in terms of computational efficiency. Future work includes extending the approach to higher-dimensional data, exploring advanced interpolation techniques, and integrating the method with machine learning models to balance performance and transparency. Additionally, further research is needed to optimize the balance between projections and sample points to achieve the best reconstruction quality under different parameter constraints.
Description
2025
License
Attribution-NoDerivatives 4.0 International