In image analysis, the shape of an object in a 2-dimensional image is usually summarised by some suitable feature descriptors. One method is to use the image moments for which we introduce area corrections since images are only observed on a discrete lattice. In particular, moment invariants (Hu, 1962; Barton & David, 1962) are constructed which are invariant under translation, rotation and dilation of spatial variables and rescaling of response variables. We critically examine the low order moments. In particular, it is known that any function of the invariants could be used and we propose invariants which stabilise the variance. Variance stabilising transformations are derived and their behaviour is examined for pure and segmented objects. It is shown that unbiased estimates of the population size are obtained under naive thresholding when the object and background are of equal size. Our results indicate how an improved feature vector can be obtained by utilising a suitable combination of variance stabilised moment invariants and the moment invariants themselves. The synthetic examples given here are motivated by real infra-red images containing objects of interest and back-ground. We also examine other moments and approaches including the use of the shape space itself.