Polar Separable Transform for Efficient Orthogonal Rotation-Invariant Image Representation

• URL: http://arxiv.org/abs/2510.09125v1
• Date: Fri, 10 Oct 2025 08:26:09 GMT
• Title: Polar Separable Transform for Efficient Orthogonal Rotation-Invariant Image Representation
• Authors: Satya P. Singh, Rashmi Chaudhry, Anand Srivastava, Jagath C. Rajapakse,
• Abstract summary: We introduce textbfPSepT (Polar Separable Transform), a separable orthogonal transform that overcomes the non-separability barrier in polar coordinates.<n>Results show better numerical stability, computational efficiency, and competitive classification performance on structured datasets.<n>The framework enables high-order moment analysis previously infeasible with classical methods, opening new possibilities for robust image analysis applications.
• Score: 10.420952921069759
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Orthogonal moment-based image representations are fundamental in computer vision, but classical methods suffer from high computational complexity and numerical instability at large orders. Zernike and pseudo-Zernike moments, for instance, require coupled radial-angular processing that precludes efficient factorization, resulting in $\mathcal{O}(n^3N^2)$ to $\mathcal{O}(n^6N^2)$ complexity and $\mathcal{O}(N^4)$ condition number scaling for the $n$th-order moments on an $N\times N$ image. We introduce \textbf{PSepT} (Polar Separable Transform), a separable orthogonal transform that overcomes the non-separability barrier in polar coordinates. PSepT achieves complete kernel factorization via tensor-product construction of Discrete Cosine Transform (DCT) radial bases and Fourier harmonic angular bases, enabling independent radial and angular processing. This separable design reduces computational complexity to $\mathcal{O}(N^2 \log N)$, memory requirements to $\mathcal{O}(N^2)$, and condition number scaling to $\mathcal{O}(\sqrt{N})$, representing exponential improvements over polynomial approaches. PSepT exhibits orthogonality, completeness, energy conservation, and rotation-covariance properties. Experimental results demonstrate better numerical stability, computational efficiency, and competitive classification performance on structured datasets, while preserving exact reconstruction. The separable framework enables high-order moment analysis previously infeasible with classical methods, opening new possibilities for robust image analysis applications.

Related papers

• Recursive Sketched Interpolation: Efficient Hadamard Products of Tensor Trains [3.1007879499686957]
  The Hadamard product of two tensors in the tensor-train (TT) format is a fundamental operation across various applications.<n>We introduce Recursive Sketched Interpolation (RSI), a scale product'' algorithm that computes the Hadamard product of TTs at a computational cost.
  arXiv  Detail & Related papers  (2026-02-20T04:07:37Z)
• CDFlow: Building Invertible Layers with Circulant and Diagonal Matrices [2.6154833242452864]
  We introduce a novel invertible linear layer based on the product of circulant and diagonal matrices.<n>This decomposition reduces parameter complexity from $mathcalO(n2)$ to $mathcalO(mn)$.<n>We build upon this layer to develop Circulant-Diagonal Flow (CDFlow), which achieves strong density estimation on natural image datasets.
  arXiv  Detail & Related papers  (2025-10-29T09:38:50Z)
• FFT-Accelerated Auxiliary Variable MCMC for Fermionic Lattice Models: A Determinant-Free Approach with $O(N\log N)$ Complexity [52.3171766248012]
  We introduce a Markov Chain Monte Carlo (MCMC) algorithm that dramatically accelerates the simulation of quantum many-body systems.<n>We validate our algorithm on benchmark quantum physics problems, accurately reproducing known theoretical results.<n>Our work provides a powerful tool for large-scale probabilistic inference and opens avenues for physics-inspired generative models.
  arXiv  Detail & Related papers  (2025-10-13T07:57:21Z)
• Generalized tensor transforms and their applications in classical and quantum computing [0.0]
  We introduce a novel framework for Generalized Transforms (GTTs), constructed through an $n$-fold tensor product of an arbitrary $b times b$ unitary matrix $W$.<n>For quantum applications, our GTT-based algorithm achieves both gate complexity and circuit depth of $O(log_b N)$, where $N = bn$ denotes the length of the vector input.<n>We explore diverse applications of GTTs in quantum computing, including quantum state compression and transmission, function encoding and quantum digital signal processing.
  arXiv  Detail & Related papers  (2025-07-03T08:28:00Z)
• E2Former: A Linear-time Efficient and Equivariant Transformer for Scalable Molecular Modeling [44.75336958712181]
  We introduce E2Former, an equivariant and efficient transformer architecture that incorporates the Wigner $6j$ convolution (Wigner $6j$ Conv)<n>By shifting the computational burden from edges to nodes, the Wigner $6j$ Conv reduces the complexity from $O(|mathcalE|)$ to $ O(| mathcalV|)$ while preserving both the model's expressive power and rotational equivariance.<n>This development could suggest a promising direction for scalable and efficient molecular modeling.
  arXiv  Detail & Related papers  (2025-01-31T15:22:58Z)
• Obtaining Lower Query Complexities through Lightweight Zeroth-Order Proximal Gradient Algorithms [65.42376001308064]
  We propose two variance reduced ZO estimators for complex gradient problems. We improve the state-of-the-art function complexities from $mathcalOleft(minfracdn1/2epsilon2, fracdepsilon3right)$ to $tildecalOleft(fracdepsilon2right)$.
  arXiv  Detail & Related papers  (2024-10-03T15:04:01Z)
• Chain of Thought Empowers Transformers to Solve Inherently Serial Problems [57.58801785642868]
  Chain of thought (CoT) is a highly effective method to improve the accuracy of large language models (LLMs) on arithmetics and symbolic reasoning tasks. This work provides a theoretical understanding of the power of CoT for decoder-only transformers through the lens of expressiveness.
  arXiv  Detail & Related papers  (2024-02-20T10:11:03Z)
• Average-Case Complexity of Tensor Decomposition for Low-Degree Polynomials [93.59919600451487]
  "Statistical-computational gaps" occur in many statistical inference tasks. We consider a model for random order-3 decomposition where one component is slightly larger in norm than the rest. We show that tensor entries can accurately estimate the largest component when $ll n3/2$ but fail to do so when $rgg n3/2$.
  arXiv  Detail & Related papers  (2022-11-10T00:40:37Z)
• Scalable First-Order Bayesian Optimization via Structured Automatic Differentiation [4.061135251278187]
  We show that a wide range of kernels gives rise to structured matrices, enabling an exact $mathcalO(n2d)$ matrix-vector multiply for gradient observations and $mathcalO(n2d2)$ for Hessian observations. Our methods apply to virtually all canonical kernels and automatically extend to complex kernels, like the neural network, radial basis function network, and spectral mixture kernels.
  arXiv  Detail & Related papers  (2022-06-16T17:59:48Z)
• Hybrid Model-based / Data-driven Graph Transform for Image Coding [54.31406300524195]
  We present a hybrid model-based / data-driven approach to encode an intra-prediction residual block. The first $K$ eigenvectors of a transform matrix are derived from a statistical model, e.g., the asymmetric discrete sine transform (ADST) for stability. Using WebP as a baseline image, experimental results show that our hybrid graph transform achieved better energy compaction than default discrete cosine transform (DCT) and better stability than KLT.
  arXiv  Detail & Related papers  (2022-03-02T15:36:44Z)
• Linear Time Sinkhorn Divergences using Positive Features [51.50788603386766]
  Solving optimal transport with an entropic regularization requires computing a $ntimes n$ kernel matrix that is repeatedly applied to a vector. We propose to use instead ground costs of the form $c(x,y)=-logdotpvarphi(x)varphi(y)$ where $varphi$ is a map from the ground space onto the positive orthant $RRr_+$, with $rll n$.
  arXiv  Detail & Related papers  (2020-06-12T10:21:40Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.