The quantum denoising technology efficiently removes noise from images; however, the existing algorithms are only effective for additive noise and cannot remove multiplicative noise, such as speckle noise in synthetic aperture radar (SAR) images. In this paper, based on the grayscale morphology method, a quantum SAR image denoising algorithm is proposed, which performs morphological operations on all pixels simultaneously to remove the noise in the SAR image. In addition, we design a feasible quantum adder to perform cyclic shift operations. Then, quantum circuits for dilation and erosion are designed, and the complete quantum circuit is then constructed. For a 2 n × 2 n quantum SAR image with q grayscale levels, the complexity of our algorithm is O ( n + q ) . Compared with classical algorithms, it achieves exponential improvement and also has polynomial-level improvements than existing quantum algorithms. Finally, the feasibility of our algorithm is validated on IBM Q.
Abstract This paper presents a quantum morphological Haar wavelet transform (QMHWT) that integrates the morphological minimum operator into the low-frequency subband of the Haar wavelet, unifying nonlinear morphological processing with multi-directional wavelet decomposition in a quantum arithmetic circuit framework. For a 2 × 2 pixel block [ a , b , c , d ] , the low-frequency subband is defined as L = min ( a , b , c , d ) , while the three directional detail subbands adopt the linear combination form of the classical Haar transform with floor division for integer-domain compatibility. We design a four-stage quantum circuit: a cascade of forward compare-swap (FCS) units extracts the four-way minimum, a reverse compare-swap recovery (FCS − 1 ) stage recycles auxiliary qubits, a modular addition/subtraction stage computes three detail subbands through separate arithmetic chains, and an arithmetic shift right stage performs integer division by two. The resulting circuit has width W ( n ) = 7 n + 12 , and the number of auxiliary qubits does not scale with image size. A D -map metric D = L mean − L min is derived from the morphological low-frequency subband, and its non-negativity, zero-value condition, and upper-bound relationship with the classical morphological gradient are established. Experiments on three standard 512 × 512 test images at 6-bit quantization confirm exact agreement between the quantum circuit output and classical reference results for all 3 × 65536 pixel blocks. The D -map achieves consistently high Pearson correlation with the morphological gradient ( r > 0.95 , mean r = 0.973 ), demonstrating the validity of the proposed transform and its potential use in quantum image edge analysis. The current large-image validation employs a hybrid quantum–classical scheme in which the per-block unitary cost is
The classical image segmentation algorithm based on grayscale morphology can effectively segment images with uneven illumination, but with the increase of the image data, the real-time problem will emerge. In order to solve this problem, a quantum image segmentation algorithm is proposed in this paper, which can use quantum mechanism to simultaneously perform morphological operations on all pixels in a grayscale image, and then quickly segment the image into a binary image. In addition, several quantum circuit units, including dilation, erosion, bottom hat transformation, top hat transformation, etc., are designed in detail, and then they are combined together to construct the complete quantum circuits for segmenting the NEQR images. For a
2n×2nimage with
qgrayscale levels, the complexity of our algorithm can be reduced to O
(n2+q), which is an exponential speedup than the classic counterparts. Finally, the experiment is conducted on IBM Q to show the feasibility of our algorithm in the noisy intermediate-scale quantum (NISQ) era.