Image features vary in size and thus feature analysis often requires a multi-scale approach. Typically, this is achieved using a bank of filters centred at discrete scales. We introduce a novel filter bank constructed from Fourier series basis functions in the logarithmic frequency domain. The filter bank responses can be used to obtain a continuous approximation of the response to another filter shifted through scale. Using the Riesz transform of the filter bank, we can create a vector-valued monogenic signal scale response. The amplitude of this response is a phase- invariant distribution of the local energy of the image across scale, from which statistics such as mean scale and variance can be calculated. We demonstrate the usefulness of the filter bank by using principal component analysis to design filters, using k-means clustering to classify pixels by scale response and local structure, and creating novel continuous methods of blob detection and phase congruency.