Quantum median filtering is an important step for many quantum signal processing algorithms. Current quantum median filtering designs show limitations in either computational complexity or incomplete noise detection. We propose a design of quantum median filtering, which uses approximate median filtering with noise tolerance threshold to remove salt-and-pepper noise. Instead of calculating the median, we search an approximate median by sorting four times, which reduces the computational complexity from \(O\left( {21{q^2} + 63q} \right) \) to \(O\left( {12{q^2} + 36q} \right) \). Here, q is the qubit used to represent the gray value. Furthermore, we adopt a two-level threshold to detect the noise points as much as possible. Finally, we design a complete quantum circuit to implement the approximate median filtering. The computational complexity of our proposed circuit is \(O\left( {10{n^2} + 14{q^2}} \right) \) for a NEQR quantum image with a size of \({2^n} \times {2^n}\). The complexity analysis shows that our proposed method significantly speeds up the filtering process compared with the classical filtering methods and the existing quantum filtering methods. In addition, the simulation results prove the proposed approximate median filtering is feasible.

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具有噪声容限阈值的近似中值滤波的量子电路设计

量子中值滤波是许多量子信号处理算法的重要步骤。当前的量子中值滤波设计在计算复杂性或不完整的噪声检测方面显示出局限性。我们提出了一种量子中值滤波设计，该设计使用具有噪声容限阈值的近似中值滤波来去除椒盐噪声。而不是计算中位数，而是通过四次排序搜索近似中位数，这将计算复杂度从\（O \ left（{21 {q ^ 2} + 63q} \ right）\）降低到\（O \ left（ {12 {q ^ 2} + 36q} \ right）\）。在此，q是用于表示灰度值的量子位。此外，我们采用两级阈值来尽可能多地检测噪声点。最后，我们设计了一个完整的量子电路来实现近似中值滤波。对于大小为\（{2 ^ n}的NEQR量子图像，我们提出的电路的计算复杂度为\（O \ left（{10 {n ^ 2} + 14 {q ^ 2}} \ right）\）\ times {2 ^ n} \）。复杂度分析表明，与经典滤波方法和现有的量子滤波方法相比，我们提出的方法大大加快了滤波过程。此外，仿真结果证明了所提出的近似中值滤波是可行的。