Quantum morphological Haar wavelet transform for multi-subband image decomposition

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) = 7n+ 12, and the number of auxiliary qubits does not scale with image size. A D-map metric D= Lmean−Lmin 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 independent of image size while the total runtime scales with the number of blocks; the quantum advantage of the proposed circuit lies in the O(n)-depth per-block transform, independent of image size, rather than in end-to-end wall-clock time.