Sub-pixel confusion-uncertainty matrix for assessing soft classifications

The prevailing concerns on ecological and environmental issues, occurring especially at regional to global scales, have prompted significant advances on the use of remote sensing data for the estimation of land cover information at sub-pixel level. However, the quality of such classification products, as well as the performance of the classification protocol employed, are difficult to quantify. This paper had the objectives of 1) reviewing the existing alternatives, while identifying major drawbacks and desirable properties, for sub-pixel accuracy assessment based on cross-comparison matrices, and 2) developing theoretical grounds, for a more general accuracy assessment of soft classifications, that account for the sub-pixel class distribution uncertainty. It was found that, for a sub-pixel confusion matrix to exhibit a diagonalization characteristic that allows identifying a perfect matching case, the agreement measure must be constrained at pixel level, whereas a disagreement measure can take into account the sub-pixel distribution uncertainty, leading to an underspecified problem termed the sub-pixel area allocation problem. It was demonstrated that the sub-pixel area allocation problem admits a unique solution if, and only if, no more than one class is either over- or underestimated at each pixel. In this case, the sub-pixel confusion can be uniquely determined. When no unique solution exists, the space of feasible solutions can be represented by confusion intervals. A new cross-comparison matrix that reports the confusion intervals in the form of a center value plus-minus maximum error was proposed to account for the sub-pixel distribution uncertainty. The new matrix is referred to as sub-pixel confusion-uncertainty matrix (SCM). Sub-pixel accuracy measures were also derived from this matrix. The practical use of the SCM and derived indices was demonstrated in assessing an invasive species detection method and a fuzzy classification of urban land use/land cover through remote sensing procedures.