A new two-dimensional transform is derived and implemented that is able to discriminate with respect to angular and radial frequency. This octave-band directional filter bank (OBDFB) is maximally decimated, has a separable polyphase implmentation, provides perfect reconstruction, and can be implemented in a tree structure allowing for a somewhat arbitrary number of angular and radial divisions. This decomposition is based on the directional filter bank (DFB) and is compared to other transforms with similar properties. Additionally, the OBDFB is used in three applications. Texture segmentation results are provided with comparisons to both decimated and undecimated transforms. With hyperspectral data, the OBDFB is used to increase classification accuracy using texture augmentation and likelihood score combination. Finally, ultrasound despeckling is addressed with respect to real-time implementations, and subjective test results are presented. A non-uniform two-dimensional transform is also designed that is a modified version of the OBDFB. It is rationally sampled and maximally decimated, but it provides both angular and radial frequency passbands from the initial stage instead of making separate divisions like the OBDFB. It also does not create subband boundaries on the principal frequency axes and allows for further decomposition as well.