# Image Reconstruction with Maclaurin Series Expansion

## TL;DR

This paper explores a theoretical method for image reconstruction using a Maclaurin series expansion in the Fourier domain under idealized conditions.

## Contribution

The novel approach uses a Maclaurin series to reconstruct images from minimal angular scanning data without prior knowledge or training.

## Key findings

- The Maclaurin series expansion converges across the entire Fourier space under ideal conditions.
- Computer simulations demonstrate successful 2D image reconstruction using a truncated Fourier-domain series.
- The method requires minimal data and avoids the need for training or prior knowledge.

## Abstract

This is a forward-thinking theoretical investigation and may not have practical values for current imaging systems. This investigation assumes that there is no noise in the measurements, the signals are continuous (not sampled), the computer has perfect precession, and there are no round-off errors. Under these unrealistic conditions, we can form a Maclaurin series expansion in the Fourier domain with measurements in a small scanning angular range. We show that this Maclaurin series expansion converges in the entire Fourier space. As a result, a complete data set is available for image reconstruction. The Fourier domain is complex; the expansion coefficients are most likely complex with real parts and imaginary parts. Computer simulations are performed to illustrate a 2D spatial-domain image can be obtained if a Fourier-domain truncated Maclaurin series expansion is available. Our goal is to use minimum data for trust-worthy reconstruction without any prior knowledge and training data.