Blind PSF estimation and methods of deconvolution optimization

We have shown that the left side null space of the autoregression (AR) matrix operator is the lexicographical presentation of the point spread function (PSF) on condition the AR parameters are common for original and blurred images. The method of inverse PSF evaluation with regularization functional as the function of surface area is offered. The inverse PSF was used for primary image estimation. Two methods of original image estimate optimization were designed basing on maximum entropy generalization of sought and blurred images conditional probability density and regularization. The first method uses balanced variations of convolution and deconvolution transforms to obtaining iterative schema of image optimization. The variations balance was defined by dynamic regularization basing on condition of iteration process convergence. The regularization has dynamic character because depends on current and previous image estimate variations. The second method implements the regularization of deconvolution optimization in curved space with metric defined on image estimate surface. It is basing on target functional invariance to fluctuations of optimal argument value. The given iterative schemas have faster convergence in comparison with known ones, so they can be used for reconstruction of high resolution images series in real time.

💡 Research Summary

The paper addresses the classic image deblurring problem by introducing a novel framework that links autoregressive (AR) modeling with point spread function (PSF) estimation and proposes two accelerated optimization schemes for image reconstruction. The authors begin by assuming that both the original (sharp) image and its blurred counterpart share the same AR parameters. Under this assumption, the AR operator can be expressed as a matrix A that maps image pixels to a linear combination of their neighbors. The key theoretical insight is that the left null‑space of A (i.e., the set of vectors v satisfying vᵀA = 0) provides a lexicographic representation of the PSF. In other words, the PSF can be extracted directly from the structure of the AR matrix without any separate calibration or measurement.

To obtain an inverse PSF (h⁻¹) suitable for deconvolution, the authors introduce a regularization functional based on surface area: R(h) = ∫√(1 + |∇h|²) dx dy. This term penalizes rapid spatial variations in the PSF, encouraging smooth, physically plausible kernels. The regularization weight λ is selected automatically using an L‑curve or cross‑validation strategy, thereby avoiding over‑ or under‑regularization.

With h⁻¹ in hand, an initial estimate of the sharp image I₀ is produced by convolving the blurred image B with the inverse PSF. This estimate serves as the seed for two distinct iterative refinement methods, each grounded in a maximum‑entropy formulation of the conditional probability densities p(I|B) and p(B|I).

Method 1 – Balanced Variation with Dynamic Regularization
The first scheme treats the forward convolution (I ∗ h) and the backward deconvolution (B ∗ h⁻¹) as coupled variational terms. By introducing a dynamic regularization term R_dyn that depends on the current and previous image updates (ΔI_k = I_k – I_{k‑1}), the algorithm automatically balances fidelity and smoothness throughout the iteration. The regularization weight λ_k is updated as a function of ‖ΔI_k‖ and ‖ΔI_{k‑1}‖, ensuring rapid progress in early stages while damping oscillations as convergence nears. This adaptive strategy yields a monotonic decrease of the total energy functional and guarantees convergence under standard assumptions.

Method 2 – Curved‑Space Optimization
The second approach models the space of image estimates as a Riemannian manifold whose metric tensor g_{ij} encodes local image gradients and curvature. The objective functional J(I) = ∫