Computational Framework for Estimating Relative Gaussian Blur Kernels between Image Pairs

TL;DR

The paper addresses estimating spatially varying relative Gaussian blur between image pairs to recover depth information via defocus (DFD). It introduces a zero-training, forward computational framework that analytically links a defocused image to a sharper reference and discretizes this relation to compute per-pixel blur maps $\sigma(x,y)$ efficiently. Key contributions include a matrix-based discretization using a radial Gaussian PSF, a practical per-pixel sigma estimation pipeline with decimation to manage $C_{max}$, and validation on real datasets showing MAEs below $2\%$ in blur estimation and accurate re-estimation of blurred intensities. The work enables real-time, training-free depth-from-defocus with robustness to partial blur while acknowledging limitations such as ignoring lens aberrations and noise, and it demonstrates practical viability on real-world data (Real-MFF, DPDD).

Abstract

Following the earlier verification for Gaussian model in \cite{ASaa2026}, this paper introduces a zero training forward computational framework for the model to realize it in real time applications. The framework is based on discrete calculation of the analytic expression of the defocused image from the sharper one for the application range of the standard deviation of the Gaussian kernels and selecting the best matches. The analytic expression yields multiple solutions at certain image points, but is filtered down to a single solution using similarity measures over neighboring points.The framework is structured to handle cases where two given images are partial blurred versions of each other. Experimental evaluations on real images demonstrate that the proposed framework achieves a mean absolute error (MAE) below $1.7\%$ in estimating synthetic blur values. Furthermore, the discrepancy between actual blurred image intensities and their corresponding estimates remains under $2\%$, obtained by applying the extracted defocus filters to less blurred images.

Figures (7)

• Figure 1: R-Images are the convolved results of L-Images using Gaussian blur kernels with a spatially-varying standard deviation increases linearly from 1 to 2. For the upper pair, this gradient is applied from top to bottom; for the lower pair, it is applied from left to right. The MAE for relative blur extraction or estimating the known standard deviation is $1.4\%$ for the upper image pair and $1.47\%$ for the lower pair.
• Figure 2: Image formation through ray tracing in geometric optics. Both triples $(f,d_i,d_f)$ and $(f,d_i+\Delta d_i,d_f+\Delta d_f)$ are described by the lens law.
• Figure 3: Computational framework for estimating a pixel's intensity in the R-Image. For a pixel centerd in a local $R_s \times R_s$ image patch in the L-Image, the image intensity in the R-Image for $M$ blur levels $\sigma_m$ is computed by approximating the underlying integral with using an $N$-point Riemann sum. Each block represents either a vector quantity or an operation, characterized by the specified dimensions of the original or transformed data as detailed in the main text.
• Figure 4: Performance evaluation of the computational framework can be done by comparing the actual R-Image with its estimation $\hat{R}$-Image at the pixel level. The framework generates all pixel values as $\mathbfcal{M}(\sigma)$. The estimation is produced by convolving the L-Image with the Gaussian kernel corresponding to the estimated $\sigma$ of that pixel.
• Figure 5: Error calculation architecture for image pairs ($I_B$, $I_F$). For the detected sharper subsets (BH and FH), their respective images ($I_B$ and $I_F$) serve as the L-image. The complementary, less sharp subsets (BL and FL) are then estimated as $\widehat{BL}$ and $\widehat{FL}$ using the process in Fig. \ref{['Fig4']}. The final errors $e_B$ and $e_F$ are obtained by calculating the difference between the original and estimated less sharp subsets.