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

Debashis Saikia, Bikash K. Behera, Mayukha Pal, Prasanta K. Panigrahi

Abstract

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.

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

Abstract

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, , 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 (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.

Paper Structure

This paper contains 35 sections, 2 theorems, 41 equations, 17 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works and Proposed Approach
  3. Research Gap and Motivation
  4. Novelty and Contributions
  5. Organization
  6. Methodology
  7. Problem Formulation
  8. Image Representation in Hilbert Space
  9. Adaptive Gaussian Construction of POVM
  10. Gaussian Response Functions
  11. Construction of Measurement Operators
  12. Normalization and POVM Structure
  13. Sharpening of Measurement Operators
  14. Measurement Interpretation
  15. Image Reconstruction
  16. ...and 20 more sections

Key Result

Theorem 1

Let $I(x,y) \in \{0,1,\dots,255\}$ be a grayscale image and let $\{\mu_k\}_{k=1}^K$ be a set of representative intensities obtained from a statistical model (e.g., a Gaussian mixture model) such that for each intensity level $i$, there exists $\mu_k$ satisfying where $\epsilon_K \to 0$ as $K \to \infty$. Then, for fixed $\gamma$, the reconstruction satisfies for all $(x,y)$. $\blacktriangleleft$

Figures (17)

  • Figure 1: Proposed framework: intensity statistics are used to construct POVMs, followed by sharpening and probabilistic reconstruction.
  • Figure 2: Original Images
  • Figure 3: Comparison of reconstructed images of Lena from Proposed KMeans, Proposed GMM, Unsharp Measurement, Multi-Otsu, and Fast Statistical Recursive methods.
  • Figure 4: Comparison of reconstructed images of Peppers from Proposed KMeans, Proposed GMM, Unsharp Measurement, Multi-Otsu, and Fast Statistical Recursive methods.
  • Figure 5: Comparison of reconstructed images of Barbara from Proposed KMeans, Proposed GMM, Unsharp Measurement, Multi-Otsu, and Fast Statistical Recursive methods.
  • ...and 12 more figures

Theorems & Definitions (4)

  • Theorem 1: Consistency of Adaptive POVM Reconstruction
  • Theorem 2: Sharpness Theorem
  • proof
  • proof