# Partial Order Rank Features in Colour Space
## 5 Conclusions and Future Work  
The lack of a natural order among colours represents an intrinsic impediment to the definition of rank features in colour space. In this paper we have introduced a novel and general approach based on partial orders. Partial orders overcome the problems inherent to ordering multivariate data at the expense of admitting that not all pairs of colours can be compared to each other. We showed that this scheme fits in well with existing grey-scale local image descriptors, that are amenable to extension to the colour domain with little effort. Taking the Texture Spectrum as a model, we showed that its partial-order version in colour space (POTS) can outperform the grey-scale classic descriptor while maintaining the same number of features and with comparable computational complexity. Previous studies have also demonstrated that the use of colour can improve texture discrimination, but at the expense of employing a higher number of features [44; 45; 46]. Notably, our approach improves on published results that use descriptors based specifically on (total) colour space ordering (see Table 3). To the best of our knowledge this is the first time that partial orders have been used to define rank features for pattern recognition. The method is conceptually simple, fairly general and shows potential for application in a wide number of computer vision tasks. Future studies will be focussed on extending the approach to the broader class of descriptors known as Histograms of Equivalent Patterns [9]. The effect of the colour space on the performance of rank features based on partial orders is also an important topic for further investigation. Finally, the insertion of partial order based algorithms in more involved image processing pipelines (e.g., convolutional neural networks) also represents an interesting opportunity for future research; integration at the level of matching [47] has so far been successful. Conceptualization, F.S. and F.B. ; Formal analysis, F.S., F.B. and A.F. ; Methodology, F.S., F.B., A.F. and E.G. ; Software, F.S. and F.B. ; Validation, F.S., F.B., A.F. and E.G. ; Visualization, F.B. and A.F. ; Writing--original draft, F.S. and F.B. ; Writing--review & editing, A.F. and E.G. All authors have read and agreed to the published version of the F.B. This work was partially supported by the Spanish Government under projects AGL2014-56017-R and TN2014-56919-C3-2-R, and by the Department of Engineering at the Universita degli Studi di Perugia (UniPG Eng). Italy, under project _Machine learning algorithms for the control of autonomous mobile systems and the automatic classification of industrial products and biomedical images_ (Fundamental research grants 2017). F.S. performed part of this work as a \begin{table}
\begin{tabular}{l c c c c} \hline \hline \multirow{2}{*}{**Dataset**} & \multicolumn{3}{c}{**Best Result (Literature)**} & \multirow{2}{*}{**Best Result (This Paper)**} \\ \cline{2-2} \cline{4-5}  & **LBP\({}_{\text{cvn}}\)** & **LBP\({}_{\text{lex}}\)** & **LBP\({}_{\text{rcl}}\)** \\ \hline KTH-TIPS & 94.3 [23] & 94.3 [23] & 94.0 [23] & 94.1–94.6 (POTS\({}_{\text{RGB}}^{<\times}\)/\(C_{n}\)) \\ KTH-TIPS2b & 92.3 [23] & 92.3 [23] & 92.1 [23] & 95.3–95.5 (POTS\({}_{\text{ICL}}^{<\times}\)/\(C_{n}\)) \\ Kylberg-Sintorn & N/A & 99.1 [43] & N/A & 95.3–96.5 (POTS\({}_{\text{RGB}}^{<\times}\)/\(C_{n}\)) \\ Outex-13 & 85.3 [1] & 86.3 [1] & 85.9 [1] & 82.3–82.5 (POTS\({}_{\text{PP}}^{>\times}\)/\(D_{n}\)) \\ Outex-14 & 74.3 [1] & 73.4 [1] & 72.3 [1] & 97.7–98.1 (POTS\({}_{\text{OH}}^{<\times}\)/\(D_{n}\)) \\ PapSmear & N/A & N/A & 66.2 [43] & 83.6–84.5 (POTS\({}_{\text{RGB}}^{>\times}\)/\(D_{n}\)) \\ PlantLeaves & 69.9 [23] & 65.2 [23] & 71.9 [23] & 75.9–76.4 (POTS\({}_{\text{RGB}}^{>\times}\)/\(D_{n}\)) \\ RawFoot & N/A & N/A & 80.5 [43] & 96.4–96.5 (POTS\({}_{\text{RGB}}^{>\times}\)/\(C_{n}\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison with the results obtained by other ordering methods as reported in the references indicated.