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Journal of Optics

From retrodiction to Bayesian quantum imaging

Fiona C. Speirits, Matthias Sonnleitner, Stephen Mark Barnett

Published Jan 31, 2017• 6 citations
Publisher

Abstract

We employ quantum retrodiction to develop a robust Bayesian algorithm for reconstructing the intensity values of an image from sparse photocount data, while also accounting for detector noise in the form of dark counts. This method yields not only a reconstructed image but also provides the full probability distribution function for the intensity at each pixel. We use simulated as well as real data to illustrate both the applications of the algorithm and the analysis options that are only available when the full probability distribution functions are known. These include calculating Bayesian credible regions for each pixel intensity, allowing an objective assessment of the reliability of the reconstructed image intensity values.

DOI
10.1088/2040-8986/aa5ccf

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Abstract Color images (i.e. RGB) contain rich visual information essential for object recognition, spatial context analysis, and decision-making across fields such as medical imaging, remote sensing, and computer vision. Processing such data efficiently requires significant computational resources, which can be addressed by the parallelism and speed of quantum computing. Representing RGB image data within quantum states enables efficient and scalable processing compared to classical methods. In recent years, various quantum image representation techniques have been developed based on probabilistic and basis state encoding. Probabilistic encoding methods are efficient for representing quantum binary images but face challenges in accurately reconstructing RGB images. This is due to the randomness introduced by measurement. To overcome this limitation, several basis state encoding approaches have been proposed. These methods map classical pixel values directly to quantum basis states. Thus enabling accurate grayscale and RGB image representation. However, they often require a larger number of qubits and complex gate operations, leading to higher circuit preparation costs. So, there exists substantial potential for quantum probabilistic RGB image representation using randomized measurements. Hence, our objective is to develop a probabilistic quantum image representation, capable of representing RGB images using fewer qubits. This method aims to reconstruct the classical image with high accuracy using randomized measurements, thereby addressing the major limitations of probabilistic methods. To balance image quality with resource efficiency, we propose an Improved Probabilistic Quantum Image Representation (IPQIR) method based on equal superposition states. This approach utilizes a quantum circuit with Hadamard Gate to encode classical pixel values into equal superposition states. The original image is then reconstructed by performing multiple quantum measurements and applying expectation value estimation to average the outcomes. To align with existing research in quantum image representation, quantum state tomography and different applications are utilized as standard evaluation methods to assess the effectiveness of the proposed algorithm. The tomography achieved a fidelity of 97.51%. In addition, various parametric and non-parametric statistical tests were performed to verify the accuracy of the reconstructed classical image. The high Pearson (0.9983) and Spearman (0.9959) correlation coefficients indicate a nearly perfect linear and monotonic relationship between the original and reconstructed images.
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We present a novel quantum tomographic reconstruction method based on Bayesian inference via the Kalman filter update equations. The method not only yields the maximum likelihood/optimal Bayesian reconstruction but also a covariance matrix expressing the measurement uncertainties in a complete way. From this covariance matrix the error bars on any derived quantity can be easily calculated. This is a first step towards the broader goal of devising an omnibus reconstruction method that could be adapted to any tomographic setup with little effort and that treats measurement uncertainties in a statistically well-founded way. In this first part, we restrict ourselves to the important subclass of tomography based on measurements with discrete outcomes (as opposed to continuous ones), and we also ignore any measurement imperfections (dark counts, less than unit detector efficiency, etc), which will be treated in a follow-up paper. We illustrate our general theory on real tomography experiments of quantum optical information processing elements.
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