Analysis of Iterative Deblurring: No Explicit Noise

TL;DR

The paper analyzes iterative deblurring without explicit noise using a one-dimensional $1$2-pixel model to study how a transfer matrix $T$ and its SVD govern information retention and null-space behavior. It compares Richardson-Lucy (RL), Landweber (LW), and explicit SVD-based deblurring, showing that nonnegativity in RL can recover null-space content for high-contrast images while regularization is necessary for low-contrast cases; however, over-regularization blurs real structure. The work highlights how initialization, blur width (3-bin vs 5-bin), and pixel count influence restoration, and connects these findings to nuclear/high-energy physics data analyses such as reaction-plane deblurring, suggesting that noise inclusion and multi-dimensional extensions are natural future steps. Overall, the results provide a principled understanding of when deblurring methods succeed or fail in quantitatively demanding physics applications and how to tune regularization and initialization accordingly.

Abstract

Iterative deblurring, notably the Richardson-Lucy algorithm with and without regularization, is analyzed in the context of nuclear and high-energy physics applications. In these applications, probability distributions may be discretized into a few bins, measurement statistics can be high, and instrument performance can be well understood. In such circumstances, it is essential to understand the deblurring first without any explicit noise considerations. We employ singular value decomposition for the blurring matrix in a low-count pixel system. A strong blurring may yield a null space for the blurring matrix. Yet, a nonnegativity constraint for images built into the deblurring may help restore null-space content in a high-contrast image with zero or low intensity for a sufficient number of pixels. For low-contrast images, control over null-space content can be achieved through regularization. When regularization is applied, the blurred image is, in practice, restored to one that is still blurred but less than the starting image.

Figures (10)

• Figure 1: Illustration of the three-bin blurring function. (a) Image from blurring an original with pixel 5 at 1 and other at 0, for $n=12$. (b) Singular values of the blurring function, ordered by their magnitude. Lines join the values to guide the eye.
• Figure 2: Illustration of the five-bin blurring function. (a) Image from blurring an original with pixel 5 at 1 and other at 0, for $n=12$. (b) Singular values of the blurring function, ordered by their magnitude. Lines join the values to guide the eye.
• Figure 3: Singular vectors for the five-bin blurring function. Within degenerate spaces, pairs of vectors are chosen to transform onto the other or itself under reflection and/or translation. Panel (a) shows the two unique vectors that transform onto themselves under the transformations.
• Figure 4: Deblurring test with a single pixel with index 5 at the intensity 1 and all other pixels at the intensity of 0. (a) The original image $F$ (circles), image $g$ (triangles) from blurring $F$ with the 5-pixel blurring function, and the restored images $f$ from different deblurring methods are displayed. As the image restored by the RL method is indistinguishable from the original, the circles represent that image, too. The image restored using the LW method is shown as diamonds. Finally, the image restored with SVD, where the operational space was reduced ($m=6$ in Eqs. \ref{['eq:SVDinverse']} and \ref{['eq:SVD0']}) compared to LW, is shown as circles. The symbols corresponding to each case are connected with lines. (b) Coefficients of decomposition in the basis of singular vectors vs singular value index for the original and blurred images and images restored with the RL and LW methods.
• Figure 5: Deblurring test with a 4-pixel ramp. In the original image, the intensity is 1 for the pixels 5-8 and 0 for the others. The operational space for SVD is here the full row space of $T$.