# Second derivative analysis and alternative data filters for multi-dimensional spectroscopies: a Fourier-space perspective

## TL;DR

This paper analyzes the second derivative image method in Fourier space for multi-dimensional spectroscopies, showing how to improve image quality by filtering and extending the approach to higher dimensions.

## Contribution

It introduces a Fourier-space representation of SDI, discusses noise reduction strategies, and extends the method to higher-dimensional data with spectral feature knowledge.

## Key findings

- Fourier-space SDI acts as a multi-band pass filter.
- Eliminating higher Fourier harmonics improves image quality.
- Extensions to higher dimensions enable more effective filtering.

## Abstract

The second derivative image (SDI) method is widely applied to sharpen dispersive data features in multi-dimensional spectroscopies such as angle resolved photoemission spectroscopy (ARPES). Here, the SDI function is represented in Fourier space, where it has the form of a multi-band pass filter. The interplay of the SDI procedure with undesirable noise and background features in ARPES data sets is reviewed, and it is shown that final image quality can be improved by eliminating higher Fourier harmonics of the SDI filter. We then discuss extensions of SDI-like band pass filters to higher dimensional data sets, and how one can create even more effective filters with some a priori knowledge of the spectral features.