  Image processing is a fascinating field for exploring quantum algorithms.
However, achieving quantum speedups turns out to be a significant challenge. In
this work, we focus on image filtering to identify a class of images that can
achieve a substantial speedup. We show that for images with efficient encoding
and a lower bound on the signal-to-noise ratio, a quantum filtering algorithm
can be constructed with a polynomial complexity in terms of the qubit number.
Our algorithm combines the quantum Fourier transform with the amplitude
amplification technique. To demonstrate the advantages of our approach, we
apply it to three typical filtering problems. We highlight the importance of
efficient encoding by illustrating that for images that cannot be efficiently
encoded, the quantum advantage will diminish. Our work provides insights into
the types of images that can achieve a substantial quantum speedup.
