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arXiv:2608.00053

Fast Trainable Multilinear Bases for Image Compression

Shiwen An, Zhongyi Ni, Huanhai Zhou, Jin-Guo Liu

eess.IVcs.CVcs.LGmath.OCquant-phSubmitted Jul 26, 2026• Announced Jul 28, 2026• Revised Aug 17, 2026 (v2)
v1 · Jul 26, 2026v2 · Aug 17, 2026v3 · Aug 29, 2026v4 · Oct 11, 2026
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Abstract

The Discrete Fourier Transform, the Discrete Cosine Transform, and their block-wise variants underpin most deployed image and video codecs. Their effectiveness rests on three properties: they run in near-linear time (linear up to a polylogarithmic factor), they are exactly invertible, and they carry few to no parameters. In this work, we generalize these bases to isometric multilinear bases, allowing a small number of extra parameters, polylogarithmic in the image size, while preserving all three properties. Given an image dataset, we develop a systematic framework that searches this family for the basis compressing the dataset most effectively: the basis is parameterized as an isometric tensor network, inspired by quantum many-body theory, and trained with Riemannian optimization on the manifold of unitary matrices. Across natural photographs and line drawings, the trained bases consistently improve on their fixed, non-parametric counterparts. On Quick Draw line-drawing compression, they store images in roughly $20\%$ fewer bytes than JPEG's $8 \times 8$ block cosine transform at the same reconstruction quality.

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The Discrete Cosine Transform (DCT) is an integral part of classical image compression, which becomes the basis of the JPEG standard. With advances in quantum computing, there is growing interest in exploring quantum analogues of classical algorithms for such transforms. This paper presents a simulation of the Quantum Discrete Cosine Transform (QDCT) using the Qiskit Python library, applied to standard grayscale images segmented into 8x8 and 4x4 blocks. The QDCT algorithm is formulated as a unitary operator and evaluated using state-vector simulation. Due to the current constraints of quantum hardware and challenges in circuit synthesis, this study focused on simulation results, as direct quantum circuit measurement leads to amplitude collapse and unusable output. The performance of QDCT is quantitative compared to classical DCT using Peak Signal-to-Noise Ratio (PSNR), Structural Similarity Index (SSIM), and compression ratio metrics. The result of this study reveals the feasibility and current limitations of QDCT-based image compression while providing a reproducible benchmark for future quantum image processing research. This study offers a realistic assessment of the potential and technical limitations of QDCT in the context of emerging quantum technologies.
This work introduces quantum-inspired tensor-network circuits as trainable transforms for image inpainting. Among the proposed architectures, the diagonal quantum Fourier transform (QFT) relaxation is invertible with O(N2log⁡N)O(N^2 \log N)O(N2logN) computational cost for N×NN\times NN×N images, inherently preserving minimum coherence throughout training via its circuit structure and eliminating the need for explicit coherence penalties. Unconstrained gradient-based phase optimization (Riemannian-optimization free) enables efficient learning from randomly sampled training data, allowing the learned transform to generalize to test images observed through fixed sampling masks. Numerical tests show that the learned models outperform fixed transforms and per-image optimization while matching the performance of much larger unitary architectures, yet with far fewer parameters.
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The Discrete Fourier Transform (DFT) is central to the analysis of uniformly sampled signals, yet many practical applications involve non-uniform sampling, requiring the Non-Uniform Discrete Fourier Transform (NUDFT). While quantum algorithms for the standard DFT are well established, a corresponding framework for the non-uniform case remains underdeveloped. This work introduces a quantum algorithm for the Non-Uniform Quantum Fourier Transform (NUQFT) based on a low-rank factorization of the NUDFT matrix. The factorization is translated into an explicit quantum construction using block encodings, Quantum Signal Processing, and the Linear Combination of Unitaries framework, yielding an εεε-accurate block encoding of the NUDFT matrix with controlled approximation error from both classical truncation and quantum implementation. Under standard oracle access assumptions for non-uniform sampling points, we derive explicit, non-asymptotic gate-level resource estimates. The resulting complexity scales polylogarithmically with target precision, quadratically with the number of qubits through the quantum Fourier transform, and logarithmically with a geometry-dependent conditioning parameter induced by the non-uniform grid. This establishes a concrete and resource-efficient quantum analogue of the NUDFT and provides a foundation for quantum algorithms on irregularly sampled data.
Quantum computing has emerged as a revolutionary paradigm in numerous fields owing to its basis in quantum mechanical concepts such as quantum entanglement and parallelism. This has potential in various applications including computer vision, particularly, image processing and compression. Applications that depend on largescale image data for storing, processing, and transmission have to rely on compression methods for real-time processing. While the classical methods based on discrete cosine transformations (DCT used in JPEG) and discrete wavelet transformations (DWT) perform image compression efficiently, they suffer from trade-off between loss-less compression and compression ratios. In the current work, we would like to explore the usefulness of implementing quantum wavelet transforms for image compression for better PSNR values and execution times. We compare the application of quantum Haar wavelet transform (QHWT) with different quantum encoding methods such as NASS and FRQI and find that NASS-encoded images on application of QHWT gives better PSNR values and shorter runtime with visibly closer details to the original images, showcasing potential for efficient and faster image compression methods.