Unsharp Measurement with Adaptive Gaussian POVMs for Quantum-Inspired Image Processing

We propose a quantum measurement-based framework for probabilistic transformation of grayscale images using adaptive positive operator-valued measures (POVMs). In contrast, to existing approaches that are largely centered around segmentation or thresholding, the transformation is formulated here as a measurement-induced process acting directly on pixel intensities. The intensity values are embedded in a finite-dimensional Hilbert space, which allows the construction of data-adaptive measurement operators derived from Gaussian models of the image histogram. These operators naturally define an unsharp measurement of the intensity observable, with the reconstructed image obtained through expectation values of the measurement outcomes. To control the degree of measurement localization, we introduce a nonlinear sharpening transformation with a sharpening parameter, $γ$, that induces a continuous transition from unsharp measurements to projective measurements. This transition reflects an inherent trade-off between probabilistic smoothing and localization of intensity structures. In addition to the nonlinear sharpening parameter, we introduce another parameter $k$ (number of gaussian centers) which controls the resolution of the image during the transformation. Experimental results on standard benchmark images show that the proposed method gives effective data-adaptive transformations while preserving structural information.