Sometimes our spectra are not well resolved. In such cases, we may perform some post processing to enhance the spectral features. You will need astropy module to be installed:

pip install astropy

:::caution

Following modules are currently experimental and under development. Please review the codes and use with caution. You may instead use the Igor Macro implemented by Prof. P. Zhang.

:::

Here is our original data:

```python showLineNumbers url = 'https://pranabdas.github.io/drive/datasets/arpes/sample_spectrum.txt' data, energy, angle = arp.load_ses_spectra(url)

plt.imshow(data, origin = 'lower', aspect = 'auto', \ extent = (angle[0], angle[-1], energy[0], energy[-1])) plt.xlabel("$\theta$ (deg)") plt.ylabel("$E_{kin}$ (eV)") plt.set_cmap('magma_r') plt.show()

<picture>
  <source type="image/webp" srcSet={require("/img/plot-sample-spectra.webp").default} />
  <img src={require("/img/plot-sample-spectra.png").default} alt="plot-sample-spectra" />
</picture>

## Laplacian
We can take the double derivative of the spectra in order to enhance the edges:

$$
I' = \frac{\partial^2 I}{\partial x^2} + w^2 \frac{\partial^2 I}{\partial y^2}
$$

Since the $x$ and $y$ scales represent different quantities (units), we also
have a weight factor $w$.

```python showLineNumbers
# diff2 = arp.laplacian(data, x, y, bw=5, w=1)
diff2 = arp.laplacian(data, energy, angle)

plt.imshow(diff2, vmax=0, cmap='terrain_r')
plt.axis('off')
plt.show()

2D curvature

The curvature method could be more appropriate to enhance spectral features. Two dimensional curvature is given by:

$$ I' = \frac{\left[1 + C_x \left(\frac{\partial I}{\partial x}\right)^2\right] C_y \frac{\partial^2 I}{\partial y^2} - 2 C_x C_y \frac{\partial I}{\partial x} \frac{\partial I}{\partial y} \frac{\partial^2 I}{\partial x \partial y} + \left[1+ C_y\left(\frac{\partial I}{\partial y}\right)^2\right] C_x \frac{\partial^2 I}{\partial x^2}}{\left[1 + C_x \left(\frac{\partial I} {\partial x}\right)^2 + C_y \left(\frac{\partial I}{\partial y}\right)^2 \right]^{3/2}} $$

The details about the curvature method can be found here: P. Zhang et. al., A precise method for visualizing dispersive features in image plots, Review of Scientific Instruments 82, 043712 (2011).

```python showLineNumbers

cv2d = arp.cv2d(data, x, y, bw=5, c1=0.001, c2=0.001, w=1)

cv2d = arp.cv2d(data, energy, angle)

plt.imshow(cv2d, vmax=0, cmap='terrain_r') plt.axis('off') plt.show() ```

You may vary various free parameters ($C_x$, $C_y$, $w$) in order to get the best result specific to your dataset.