The bitonic filter was recently developed to embody the novel concept of signal bitonicity (one local extremum within a set range) to differentiate from noise, by use of data ranking and linear operators. For processing images, the spatial extent was locally constrained to a fixed circular mask. Since structure in natural images varies, a novel structurally varying bitonic filter is presented, which locally adapts the mask, without following patterns in the noise. This new filter includes novel robust structurally varying morphological operations, with efficient implementations, and a novel formulation of non-iterative directional Gaussian filtering. Data thresholds are also integrated with the morphological operations, increasing noise reduction for low noise, and enabling a multi-resolution framework for high noise levels. The structurally varying bitonic filter is presented without presuming prior knowledge of morphological filtering, and compared to high-performance linear noise-reduction filters, to set this novel concept in context. These are tested over a wide range of noise levels, on a fairly broad set of images. The new filter is a considerable improvement on the fixed-mask bitonic, outperforms anisotropic diffusion and image-guided filtering in all but extremely low noise, non-local means at all noise levels, but not the block-matching 3D filter, though results are promising for very high noise. The structurally varying bitonic tends to have less characteristic residual noise in regions of smooth signal, and very good preservation of signal edges, though with some loss of small scale detail when compared to the block-matching 3D filter. The efficient implementation means that processing time, though slower than the fixed-mask bitonic filter, remains competitive.