What makes an image feature stable?
Without being an expert in the image processing field there is something that comes to mind that maybe could point you in the right direction. Usually we express a feature as a number, to which we assign an estimator, which is a random variable, with some probability distribution (for which we normally care about mean value and variance). If we take the noise example which i think is the easier. You could try to express this resistance in term of the variation of the probability distribution of the feature estimator given the perturbing noise source. Let the output of your estimator for the feature be the random variable $\mathbf{X}$, and let the noise be characterized by $\mathbf{Y}$. Some measure of the resistance could be $\frac{P(\mathbf{X})}{P(\mathbf{X}|\mathbf{Y})}$. If this value is close to one, then the feature would be resistent to perturbations by noise, if it is not then it won't be. Maybe you could define a function of this to adjust the values to be more significant. For the other transformations such as rotation, scaling etc, you could use the same but do $\frac{P(\mathbf{X})}{P(\mathbf{f(X)})}$, where $f(\mathbf{X})$ denotes the transformation of the random variable that characterizes the estimator when you apply the rotation to the image. Of course, I think this $f()$ would be very hard to find in general but perhaps if the feature is not hard to estimate you can do some math and some change of variable and arrive at closed expressions for it.

Hope this helps.