**Histogram calculation**

To build a histogram, it is necessary to determine a bin width ("bin" parameter). The indices derived from the histogram will depend on this bin width parameter.

This dependence, similar to that found in texture index calculations, is often overlooked in publications.

In LIFEx, with the absolute model the histogram is built a number of bins equal to that entered by the user in the "number of grey level" and "size of bin" fields of the resampling menu.

In LIFEx, with the relative model the histogram is built only with "number of grey level" fields of the resampling menu that entered by the user and min and max are extracted values of each ROI.

**DISCRETIZED_HISTO_Skewness** is the asymmetry of the grey-level distribution in the histogram.

\begin{equation}

DISCRETIZED\_HISTO\_Skewness=\frac{\frac{1}{E}\sum_{i}(I(i)-\overline{I})^{3}}{\left(\sqrt{\frac{1}{E}\sum_{i}(I(i)-\overline{I})^{2}}\right)^{3}}\end{equation}

where \(I(i)\) corresponds to the number of voxels with intensity \(i\), \(E\) the total number of voxels in the Volume of Interest and \(\overline{I}\) the average of grey-levels in the histogram.

**DISCRETIZED_HISTO_Kurtosis** reflects the shape of the grey-level distribution (peaked or flat) relative to a normal distribution.

\begin{equation}

DISCRETIZED\_HISTO\_Kurtosis=\frac{\frac{1}{E}\sum_{i}(I(i)-\overline{I})^{4}}{\left(\frac{1}{E}\sum_{i}(I(i)-\overline{I})^{2}\right)^{2}}\end{equation}

where \(I(i)\) corresponds to the number of voxels with intensity \(i\), \(E\) the total number of voxels in the Volume of Interest and \(\overline{I}\) the average of grey-levels in the histogram.

**DISCRETIZED_HISTO_Entropy_log10** reflects the randomness of the distribution.

\begin{equation}

DISCRETIZED\_HISTO\_Entropy_{log10}=-\sum_{i}p(i)\cdot log_{10}(p(i)+\varepsilon)

\end{equation}

where \(p(i)\) is the probability of occurrence of voxels with intensity \(i\) and \(\varepsilon\) = 2e-16

Be aware of the logarithm used in the formula. We use the logarithm with base 10 in LIFEx but the logarithm base 2 is sometimes used in other software ; see _log2 formula.

**DISCRETIZED_HISTO_Entropy_log2** reflects the randomness of the distribution.

\begin{equation}

DISCRETIZED\_HISTO\_Entropy_{log2}=-\sum_{i}p(i)\cdot log_{2}(p(i)+\varepsilon)

\end{equation}

where \(p(i)\) is the probability of occurrence of voxels with intensity \(i\) and \(\varepsilon\) = 2e-16

**DISCRETIZED_HISTO_Energy** reflects the uniformity of the distribution.

\begin{equation}

DISCRETIZED\_HISTO\_Energy=\sum_{i}p(i)^{2}

\end{equation}