**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" or "bin width" (\(bw\)) fields of the resampling menu.

\begin{equation}

bw = \frac{max-min}{nbGreyLevel}

\end{equation}

where \(max\) is maximum of intensity (in ROI), \(min\) is minimum of intensity (in ROI) and \(nbGreyLevel\) is the number of grey level.

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_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}

**DISCRETIZED_AUC_CSH** reflects the cumulative intensity of histograms \(p(i)\) produce by a per cent volume of a ROI (derived from ROI semi-automatic ROI delineation methods) with an intensity above a certain threshold is plotted against that threshold value \(j\), which is varied from 1 to number of grey [Van Velden 2011].

The area under of this new histogram (DISCRETIZED_AUC_CSH feature) is a quantitative index of tracer uptake heterogeneity and/or heterogeneous response where lower values correspond with increased heterogeneity. In this case DISCRETIZED_AUC_CSH is independent of value max (here, number of grey).

\begin{equation}

DISCRETIZED\_AUC\_CSH=\sum_{j}{\sum_{i \geq j}p(i) * bw}

\end{equation}

where \(p(i)\) is the probability of occurrence of voxels with intensity \(i\), \(j\) is the number of grey and \(bw\) the width of one bin.

[Van Velden 2011] Floris H. P. van Velden, Patsuree Cheebsumon, Maqsood Yaqub, Egbert F. Smit, Otto S. Hoekstra, Adriaan A. Lammertsma, Ronald Boellaard. Evaluation of a cumulative SUV-volume histogram method for parameterizing heterogeneous intratumoural FDG uptake in non-small cell lung cancer PET studies ; Eur J Nucl Med Mol Imaging (2011) 38:1636?1647. DOI 10.1007/s00259-011-1845-6 ; https://link.springer.com/article/10.1007/s00259-011-1845-6