The sieve is a morphological scale-space operator that filters an input signal by removing intensity extrema at a specific scale. In images, this processing can be carried out along a path – the 1D sieve – or over a connected graph – the 2D sieve. The 2D version of the sieve generally performs better; it is however much more complex to implement. In this paper we present the 1.5D sieve, a Hamiltonian path-based version of the sieve algorithm that behaves “in between” the 1D or 2D sieve algorithms, depending on the number of paths used. Experiments show that its robustness to the presence of noise and its performance in texture classification are similar to the original 2D sieve formulation, while being much faster and simpler to implement.